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'~ -… №s xeº. ș3 « *??!！）;-·… …--• ,-•· ·------· · ·******§§}}§§§Wae-·-|- · ·~~:，,， ，,ºaſtſA-ſ-|- - - -- • ， ſ.s.， wyſº 3 *********）（\ſ*\ſ*}}2. （…“ ſſſſſſſſſſ|}}}¿* * … ' • • • • • • • • ¿¿.*¿¿.* º * * * * * * * * · * ** ae. I.B.-0, (3) in which the pseudopotential components Tag are much smaller than the original set 7 (g—g'). Thus, the whole problem is equivalent to the perturbation of free elec- tron waves by a weak pseudopotential and can be solved by elementary computation. For a perfect Bravais lattice the value of 7 (g—g') or of Tag, is a func- tion only of the potential associated with a single atom or ion–in the language of x-ray diffraction, it is just the “atomic form factor” in the formula for diffraction by an assembly of such objects at the appropriate Brag angle. The band structures of most ordinary metals and many semiconductors, can be read at a glance. No only does this provide us with an admirabl parametrization of Fermi surfaces, optical spectra, etc., in perfect crystals, but can be extended to include almost all the properties of thermally excited, impure or disordered materials—electron-phonon interactions, electrical conductivity of solid and liquid metals, lattice dynamics, phase stability of alloys, etc. In moments of enthusiasm [3,4,5] we may perhaps be forgiven for pre- tending that all the problems of the theory of metals are cured by a strong dose of “pseudism”. It is a wonderful model for zeroth order calculations, and the ideal do-it- yourself kit for the enthusiastic amateur. It had the ef. fect of turning band structure theory from a rule of thumb technology into an elegant science. - Nevertheless, the pseudopotential method is not the ultimate solution to the band structure problem. In the first place, the program of replacing the true atomic potential by a localized pseudopotential, independent of energy and momentum, cannot be fulfilled exactly. If, like Herman and his colleagues [6] one is trying to make very accurate first principles calculations, nothing is gained by rewriting the OPW equations in this form. Indeed, there is a danger that the apparent simplicity and rapid convergence of the pseudopoten- tial equations may seduce us into further approxima- tions which hide important effects; once having lost touch with the exact equations, we slide easily into a sloppy mess where qualitative and quantitative, first principles and parametrized, features are inextricably confused. FIGURE 1. The true wave function jºr) in the true potential 601) is replaced by the pseudo wave function (b(r) in the pseudopoten- tial w(r). This type of confusion is compounded by the non- uniqueness of pseudopotentials. The original algebraic proof of this arbitrariness came as something of a sur- prise, but it is really quite obvious. We are asked, in ef- ect to construct a weak potential that will reproduce he effect of a strong potential on an electron wave of iven energy impinging on the atom. The boundary con- ition on the pseudo wave function—that it should atch the true wave function on the outside—is very weak, and amounts to little more than fixing the value of a few integrals over the pseudopotential. We know, for example, that the s-wave scattering phase shift of the true potential will be reproduced at low energies if we choose the spatial average of the pseudopotential correctly—and so on. Almost any function containing a few adjustable parameters can be made to fit these conditions. Of course the problem of finding a fixed local pseudopotential that will imitate the effects of the true potential over a wide range of energy is much more difficult, and has not been solved, but that is not what we are asked to do. This arbitrariness was exploited to the full by Heine and Abarenkov [7] who chose the most elementary pseudopotential functions so as to simplify the rest of the algebra. It was natural to reproduce the core poten- tial of a metallic ion with a square well of depth A1(6), which could be continued outwards as a simple Coulomb potential; or as a screened Coulomb potential, according as one is thinking of an isolated free atom or of a “pseudo atom” in a condensed phase (fig. 2). In fact, the value of A1(3) for a given angular momentum can then be estimated from the optical term values, in the tradition of the quantum defect method of Kuhn and Van Vleck. Such a “model potential” is obviously good physics, and can be more or less justified mathematically. It copes very elegantly with one of the most difficult aspects of the whole theory—the self-consistency problem for the valence electrons — about which, for reasons of brevity, I shall say very little here. According to Shaw [8], the screening corrections can be calcu- lated accurately, although it pays to eliminate the discontinuity at the surface of the square well by treat- VM Á R {O} MI —ºr " -A = z ºf FIGURE 2a. Heine-Abarenkov pseudopotential; before screening. Ves (r) A }*- e-T-s =}+- r — Ra — RM O RM Ra FIGURE 2b. After screening (from [5]). ing the radius of this internal flat region as another ad- justable parameter, depending also on energy and mo- mentum (fig. 3). 8. Rs. Ru F - - - - - - - - | l \, I T." N AL W’ \ / \ / \ | FIGURE 3. Shaw pseudopotential. Notice, however, the dangers of overelaboration. An arbitrarily defined model potential in real space is valu- able only in proportion to its algebraic or geometrical simplicity, and will not bear much “improvement” in the name of numerical precision or in order to get better agreement with experiment. In the event the electronic structure depends on the “form factor”— the Fourier transform of the pseudopotential—which might then just as well be derived directly from the true potential by some more powerful method, or which we could also represent by some simple empirical function [9]. From a formal point of view, the arbitrariness of the pseudopotential is certainly quite worrying. How can the electronic band structure depend uniquely on the periodic lattice potential if this arbitrary function can be interposed in the calculation? Well now, suppose we had tried to solve the equations (2) for the Bloch func- tions expanded in simple plane waves. Since these are an infinite set we should have had to proceed by suc- cessive approximations, just as if we are trying to sum a series term by term. But these equations really have many solutions of much lower energy than the one we are looking for, corresponding to all the narrow tight- bound bands and an expansion in powers of 7 (g-g") simply does not converge for energies in the valence band. We are trying to sum the Born series for scatter- ing by one of the atomic potentials, ignoring the fact that it has numerous deep bound states. The pseu- dopotential trick removes all the effects of these bound states, and gives us a convergent series. It is rather like wanting to evaluate 1/(1+x) when X is about 10: a power series in X will not converge, but we can easily con- struct a new series in somé new variable y = (X–a), say, which can be made to converge in the region of in- terest. The actual terms in the series will depend on the value of a, which may be any arbitrary number larger than about 5–but the final answer will be independent of this choice. Thus the final value of the energy as a function of wave vector comes out the same, whatever form of pseudopotential we introduce into the equa- tions. This suggests a possible criterion for a “best” pseu- dopotential: choose the form of Tag, that causes the se- ries expansion for the Bloch functions to converge most rapidly. There is a rather elaborate mathematical theory of the Born series, due to Weinberg, which can be applied to this problem [10] and which does dis- criminate in principle between various formulae. These investigations are not, perhaps, of very great practical value to the horny-handed programmer of computers, but they are healthy in establishing the basic mathe- matical foundations of the whole technique. 3. The Problem of Bound Bands The most serious limitation of the pseudopotential concept is that it applies only to the so-called “simple” metals—those without d-states in the valence band. There is, of course, a long tradition of representing such states by the tight binding method, as a linear combination of atomic orbitals. The coefficients aſ in such combinations then have to satisfy a set of linear equations of the form (3,-4) al-H > 1, WLL (k) aly - 0, (4) where the index L stands for different angular momen- tum and magnetic quantum numbers; for example, the five values of the component of angular momentum in a band of d-states. The original bound state at 8, is broadened into a band by the various overlap integrals Žulj (k), which can in principle be evaluated, although in practice this is so complicated and inaccurate that one treats them as adjustable parameters. It used to be thought that all the states in metals could be described in this way, by bringing in enough different atomic orbitals. The picture of states over- lapping and broadening to make nice valence and con- duction bands illustrates one of the nursery rhymes o our subject (fig. 4). Unfortunately, this is quite mislead- ing. What happens is that as the atomic potential overlap, and the barriers fall between atomic cells, most of these atomic bound-state orbitals disappear. The ordinary s and p valence levels of the atoms vanis into a nearly free-electron band which can only b described if one includes propagating wave function from above the spectrum of bound states of th separate ions or atoms. p 8. S l l 1 1 I I r FIGURE 4. Conventional picture of energy bands from overlap of atomic orbitals. We thus arrive at an impasse: we can describe ordi- nary S – p bands in pseudopotential language, and d-bands in tight binding language, but there seems no common tongue, even when these bands overlap and hybridize as in the transition metals. This difficulty never seems to have worried the ac- tive calculators of band structures: they used two techniques that gave good numerical results in all cases—the augmented plane wave method and the Green function method. One of the main developments in band structure theory in the past 5 years has been to show the mathematical connections between these use- ful techniques and the concepts of pseudopotential and tight-binding. The idea of an augmented plane wave is quite simple. At some given energy & , one solves the Schrödinger equation inside a spherical potential well, of radius Rs, say. The solution is a linear combination of products or radial functions and spherical harmonics of different values of angular momentum. Now determine these coefficients so that this solution matches on to a plane wave of wave vector k outside the sphere. This function is still not an exact solution of the Schrödinger equa- tion, and has a discontinuity of slope at Rs; but we can build up our Bloch function by combining a set of these with wave vectors k, k + g, etc. just as in (1) and then using the variational principle for the energy. The coefficients satisfy a set of equations exactly like the pseudopotential equations (3) so that we can find –ZººZººZºZº. Zºzº, Y WAYYAve MAY 2, VAVAVA. Y. WX Zºº YNAT& WXV 4p N 2— — — 4s -mſ 5d — — — — — —xz2, (O) 4p -S ---- 4s gº º ºs Bound bond (Ab) FIGURE 5: (a) Conventional LCAO description of formation of metallic conduction band; (b) Description in terms of muffin-tin potentials. FIGURE 6. An augmented plane wave. the energy & as a function of k by finding the roots of the determinant in the usual way. The actual formula for Tag” is rather elaborate, so I will not write it down; it depends upon k, and also upon & through the first derivatives of the radial solu- tions of the Schrödinger equation at Rs. At first sight one might have thought that this could be interpreted as an elaborate energy, and momentum-dependent form factor, derivable from a pseudopotential; but this is not the case. The difficulty is that Tº" does not vanish in the elementary case of an empty lattice — whereas we should certainly expect a pseudopotential to be zero when we remove the true potential to which it is sup- posed to be equivalent. The connection with the tight binding formalism appears even more obscure, even though one can compute perfectly good d-bands by this method. In desperation, we turn to the KKR method of Korrin- ga and of Kohn and Rostoker. This is called the Green function method because it was originally derived in that somewhat abstract language, but it really depends upon a self-consistency argument; as the Bloch wave proceeds through the crystal lattice, and encounters the various atomic spheres, it suffers scattering or dif- fraction – but this diffraction must be exactly what is needed to reproduce the wave and keep it on the move without loss. Again, I will spare you from the algebra, and merely report that, as in the APW method, one uses the radial solutions of the Schrödinger equation in 4TN , j ( [k-g|Rs) ji () k-g"|Rs) N ºxº º º º & C º º /"Yº º º sº R § | º º º º º º º Vº \ (Y& W Qº º \ * ~ &º º N / S. / \\ } , - - -\º () | Q S / / / W º / /S % - 4 -\º V A z / - * z FIGURE 7. Scattered waves recombining as plane waves in KKR method. º º / w each atomic sphere and plane waves outside. The result is yet another set of linear equations—this time for the coefficients of the mixture of solutions of various angular momentum in the sphere: k{cot mi (k)-i} bl–H XL, BLL (k, k)bly = 0. (5) In this formula, the energy & is k”, and mi (k) is the phase shift that would have been produced by the atomic sphere in scattering a plane wave of this energy. The “structure constants” BLL (K, k) depend on the energy and momentum of the state being studied, but otherwise can be laboriously computed from the geo- metrical structure of the lattice. This does not look very much like either of our previ- ous formulae. Indeed, from the pseudopotential point of view it looks quite wrong, for when we apply the empty lattice test we make mi tend to zero, which causes cot mi to blow up. In fact these equations need to be turned upside down if we are to understand them physically [11]. The algebra is again a bit heavy, and depends essentially on some of the analytic properties of the structure constants, each of which is in fact a sum over reciprocal lattice vectors of products of spher- ical harmonics and Bessel functions etc. The result is a set of algebraic equations of the form of (3), with the following expression for the “matrix elements of the pseudopotential”: (6) KKR = — — T; 2 (2l + 1) tan mi where ni (KRs). ji (KRs) cot m} = cot mi- P 6...., |j (KRs)|* I (cos 0.2 ) (7 In this formula, ji and mi are spherical Bessel functions, and P. (cos(\mu) is the ordinary Legendre polynomial for the angle between vectors k—g and k—g'. - This formula is highly instructive, for a number o TeaSOIlS. (i) Consider an empty lattice, for which m = 0. Then m'i will also vanish, and with it tan m'i. Thus Tag is a genuine pseudopotential, which goes to zero with the true potential. (ii) When mi is small, the difference between, say ing the ratios of spherical Bessel functions, Tug. looks just like a scattering amplitude for the ef- fect of our given potential on a single plane wave. This is good physics: the crystal is made up of an assembly of objects, each of which scat- ters the Bloch wave into itself. (iii) A strong potential, with many deep bound states may, nevertheless, have quite small phase shifts, so may behave like a weak pseudopoten- tial. Thus, the principle of subtracting away the divergences due to the bound states amounts to simply representing each phase shift as the smallest possible angle, modulo (TT). This is a well-known property of phase shifts. (iv) As shown by Lloyd [12], this form of matrix element can be derived from a simple model potential. We merely put a delta function singu- larity of potential over the surface of the sphere of radius Rs, of strength to match the phase shift mu outside, for each value of l. (v) The connection with the APW formula was discovered by Morgan [13]. Suppose we write T4” (0) for the values of the APW matrix ele- ments in an empty lattice. Then TAPW F TKKR + TAPW(0). (8) The APW matrix elements have these extra parts to them, which do not really contribute to the band struc- ture, and which do not vanish for any value of l, even for empty space. One can even derive T4” from a model potential [12], but this is much more com- plicated in form than the one for T^* and does not vanish in empty space. FIGURE 8. Pseudopotential for T^*. tan m'i and sin mi exp (im) is negligible. Ignor- These properties of this new form of pseudopotential suggest that it should be much easier than the APW method to use in practice for simple metals, where we need only introduce small phase shifts for a few values of angular momenta. We may also use the computa- tional device of “folding” the determinant for large values of g—g', as if we were treating the diffraction from distant zone boundaries as a small perturbation [14]. This form is also said to be the best for conver- gence of the Born series in the Weinberg sense [10], whatever that may imply. But the whole question of the relative computational efficiency of these methods and their minor variants is quite complicated; all I would say here is that the effort of comparing them is made much more fruitful when we understand the basic algebraic connections. One further mystery needs clarification. Let us recall that the basic algebraic equations (3) are for the pur- pose of discovering the coefficients (3, in some expan- sion of the wave function in the appropriate plane waves. Thus, if we had been using Tº" in these equa- tions, we should have been writing ill, - X, 6,64"(k+g) where bºº" (k+g) is augmented plane wave having the form exp {i(k+g) r} outside of the atomic sphere. Now it turns out [13] that the KKR equations also sup- pose that the wave function has been expanded in aug. mented plane waves—but since the matrix elements (8) are different in these equations the coefficients 3, will be different. In other words, the Bloch function lik, which is supposed to be a unique solution to our band structure problem, has two entirely different represen- tations in terms of the same set of basic functions. This is permissible, because in fact we are only com- bining APW’s to satisfy the Schrödinger equation outside the spheres; the part within each sphere is au- tomatically determined by its adjustment to the boun- dary condition [15]. It is well known that a periodic function defined over only part of the unit cell can be ... * e ~ i ... / ^ z | A A i / Z | 2' / / | e” I / | 2 | - - - - - - - - -- - - - - - - - - - J F --~~<— — — — t .* I 2° 2’ L - .* .* Muffin tin Interstitici] well region %22% FIGURE 9. Function defined as Bloch wave in interstitial region may have arbitrary form in muffin-tin well. f º represented by many Fourier expansions, depending on what properties it is allowed to have in the excluded re- gion. The APW and KKR expansfons both represent lik correctly—yet they are not made up of exactly the same combinations of simple plane waves in the interstitial regions. This point is perhaps worth emphasizing because in either case we have a very explicit represen- tation of the wave function of the Bloch state, in a form that is quite convenient for calculations of electron- electron interactions, self-consistency of potentials, and optical, x-ray, photoemission, and positron- annihilation matrix elements, etc. It has sometimes been held against the APW & KKR methods that they cań only be used for a “muffin-tin potential”—i.e. for a periodic lattice of spherically symmetric wells with “empty space” in between. But this is not an absolute restriction. Suppose there really is a significant nonconstant potential 7" in the intersti- tial region. Then we can take this into account by ad- ding to Tag, the corresponding Fourier component % (g-g') of this potential—made explicit by being given a constant value across the mouths of the muffin- tin wells [16]. Thus, the level which I call the “muffin- tin zero” [17] cuts across the equipotential surfaces, producing muffin-tin wells with bound states, which are eliminated by a pseudopotential device, and ranges of weak potential hills through which the valence elec- trons easily tunnel, and which can be represented adequately by their Fourier transforms. If we go further, and suppose that this interstitial potential had been produced by the superposition of screened Cou- lomb potentials, or charge clouds, carried by the in- dividual atoms, then we can imagine 7 analysed into these spherically symmetrical constituents arranged in a lattice, and reassign these to the corresponding muf- fin-tin wells, whose deep potentials have by now been replaced by a model potential or pseudopotential. In other words, we arrive back precisely at the sort of analysis implied by figure 2 or figure 3: the effect of the atoms on the electrons is equivalent to diffraction by an (b) FIGURE 10. Lattice potential (a) dissected into an interstitial potential and muffin-tin wells. (d) º / º ſº —/TN— Lº ° (àſº Dº —--|-- (e) ºxº~|~ || FIGURE 11. Overlapping potentials (a), summed to make lattice potential (b), dissected into an interstitial potential and muffin- tin wells (c), redefined as pseudopotentials and overlapping ex- ternal parts (d), and recombined as pseudo-atom potentials (e). assembly of screened model potentials, whose outer fields may, within reason, be superposed without hin- drance. Thus we could use T^* +%; as the form factor in any calculation where model potentials are em- ployed, e.g. resistivity of liquid metals, lattice dy- namics, etc. This final demonstration of the equivalence of all three methods of band structure – OPW, APW and KKR – in the case of simple metals and semiconductors is very satisfactory, but I am now worried about one general point. Suppose we have a very anisotropic lat- tice—for example, the chain structure of Te, or the layer structure of graphite. The separation of the poten- tial into muffin-tin wells and an interstitial potentia must be done at a level below the lowest barrier between the atoms — for example, at the level of th potential half way between neighbors along a chain. Bu this may leave very high hills in the interstitial potentia between the chains or layers—and the unwillingness o the electron to tunnel through such hills may not b well expressed by an expansion in plane waves in thi region. Perhaps this is not a serious point after all; bu I mention it to show that we are now gaining confidenc to attack the electronic structure of more comple molecular crystals, a field which has up to now bee dominated by an army of theoretical chemists wieldin innumerable linear combinations of atomic orbitals— º ZºZº. YZ. A^ ^\ § § º high potential hills. o doubt. 4. Resonance Bands FIGURE 12. Potentials in a crystal of long chain molecules: electrons occupy the valleys containing muffin-tin wells, separated by eapon whose fundamental efficacy I now take leave What about d-bands, which can be computed numer- ically by the APW and KKR method, but whose empiri- cal description has usually been handled by the tight binding formula? The answer to this question is per- haps one of the most elegant results of the recent theory. Let us proceed from, say (5), the original KKR equations, which are not unlike the tight-binding equa- tions (4), in that the index L., labelling the unknown coefficients, refers to various spherical harmonics, or components of angular momentum. We might ask, for example, what would happen to the phase shift mi (k) if the energy happened to coincide exactly with a bound state & L of the atomic potential. To answer this ques- tion in general, we should need to study the theory of scattering in the unphysical regions where à lies below the muffin-tin zero, making k pure imaginary; but it turns out that a factor like & L-3 then appears in cot mi(k) just as we might expect. Now look at our formula (6) for the KKR pseudopotential in the recip- rocal lattice representation: if cot m'i were to vanish, at any energy, then tan m'i would become infinite, and everything would go wrong. Thus, if m't should ever go through T/2 the band structure would be seriously affected. Now this is a familiar situation in the general theory of scattering by atoms, molecules or nuclei: the phase shift mi goes through Tl2 in the positive energy region whenever there is a “resonance” of angular momen- tum. Thus, if the atomic or ionic potential has such a resonance, this will give rise to significant band effects in this neighborhood. There is a standard theory of such phenomena, which tells us that we may write tan mi - W_ (9) 3 – ?) for the phase shift of a resonance of width W centered on the energy & I. It is easy to show, using (6), that this has the effect of introducing a band of states of about this width, at about this energy, in the nearly-free- electron spectrum [11]. This argument can be carried further. Starting from the KKR formulae and making systematic transforma- tions and approximations, Heine [18] showed how one could separate out a particular resonance term, and keep this in the angular momentum representation, with indices m, m' for the different components of l, while reproducing a typical pseudopotential expression in the reciprocal lattice representation g.g'. The matrix of these equations can thus be written in the form 3 — k? Tog T go a- (k+g)* am ºn mº mº sº me = >k Ying ygm 3 — à Vmm. Vmºm 3 — à e (10) FIGURE 13. Resonance band crossing nearly free band. Without the submatrices yum etc., this would factorize into a nearly-free-electron, pseudopotential matrix, such as we might expect to find in a simple metal with an s-p band, together with an ordinary tight binding matrix, corresponding to the overlapping and mixing of the 5 degenerate d-levels of the free atom. The coeffi- cients yum etc. then describe the hybridization of these two systems of states, which must necessarily occur when these bands cross one another. As it happens (but not accidentally!) an empirical “model Hamiltonian” of just this form had already been proposed for transition and noble metals [19] before it was deduced directly from the KKR equations. We can now, therefore, justify this type of expression in princi- ple, and even calculate the various coefficients directly from the atomic potential. In fact there are now several different versions of these equations, of varying compu- tability, convergence and analytical simplicity [20] but all essentially equivalent of Heine's formula [5,18]. This reinterpretation of the tight-binding formalism, and its unification with the other band structure methods is very pleasing, but to my mind there is a greater gain. Let us ask how resonances actually arise? For an ordinary one-electron potential, we need to think of the effects of the centrifugal barrier term lºl-H1)/rº in the radial Schrödinger equation, which becomes impor- tant for l-2. A bound d-state is really constrained to avoid the nucleus by this “potential”. Now lower the or- dinary potential at the outer edges of the atom: the ef. fect may be to leave a potential dip within the core, where a “virtual”, long-lived level could still exist, even though, eventually, it would have to decay as the elec- tron tunnelled out into free space. Thus, the original bound d-state has become a d-resonance; if the poten- tial barrier is sufficiently thick, the resonance will be sharp; it is not surprising that the language of over- lapping bound states applies to the bands produced in such cases. From this picture we can learn a lot about the gross features of the density of states of the metal. We see, / ( / 4 |)/r? ~7(/-) *>===-- *HTz 2 * 4. A FIGURE 14. How a bound state of the atom becomes a resonance level of the muffin-tin well (See [17]). for example, that although the little peaks and dips of the d-band can be derived from general tight-binding theory, especially when aided by group theory, the width of this complex of bands will depend chiefly on the width of the resonance, which is governed in turn by the potential barrier produced by the centrifugal force in the outer part of each muffin-tin well. Again, the actual position of this band will be determined mainly by the energy of the original d-state from which it derives—and this is fixed on a scale relative to, say, some deep state of the core. On this scale, however, the position of the ordinary conduction band does not de- pend on any atomic orbitals, but is determined mainly by the muffin-tin zero, which can only be calculated correctly by taking very careful account of screening, correlation energy, overlaps of potential, etc. We thus discover the reason for a well-known difficulty in band structure calculations — that the width of the d-band, and its position relative to the Fermi level is very sensi- tive to the model, and cannot apparently be calculated with the precision we would like. ‘&FL > FIGURE 15. How the position of the d-band within the conduction band depends on the muffin-tin zero (See [17]). 5. Some Thoughts in Disorder Now that we understand the electronic structure of crystalline solids so very well, we are tempted to attack disordered materials — liquids, alloys, amorphous and glassy substances. This campaign has been actively waged now for about a decade, but I am not sure that it has yielded many great prizes. The major difficulty, of course, is that we must abandon Bloch's theorem, which reduces the complexity of the problem in the per- fect lattice by a divisor of the order of 10*. Without crystal momentum as a good quantum number, we flounder about in a mixture of approximate algebra and incomplete intuition, hoping to find some clearcut con- cepts that will guide the interpretation of complicated experiments on messy materials. It is true that the spectrum of the disordered linear array is now well understood [21] – and turns out to be much more spiky than one would have guessed from simple statistical considerations. Some of these fea- tures may persist in three-dimensional systems, but un- fortunately the mathematical methods used in the one- dimensional case seem ill-adapted to generalization. In particular, real solid systems have two properties that cannot be simulated at all by a linear chain. In three dimensions, a localized defect or impurity can be avoided by a detour, so that it does not present an ab- solute barrier to an incident particle or excitation. In three dimensions, also we may have “structural dis- order”, which is no longer topologically equivalent to any regular lattice, whereas in a linear chain the mere succession of atoms prescribes an ordering, however wildly we vary the properties of the individual potential wells. Let me give two examples of simple cases where our present theory is inadequate. It is obvious enough that a disordered transition metal – e.g. liquid iron — should have a d-band arising from the d-resonance, just as in any crystalline phase of about the same atomic volume [22]. The mathematical theory of such a band is still rather uncertain [23], but there is no doubt about the physics. Suppose, however, that we make an alloy– e.g. of Ag and Au – whose constituent atoms have their resonance at different energies; how far apart would these energies need to be to give us two distinct d- bands, and how would this depend on the relative con- centrations and relative ordering of the constituents? he model can be made extremely elementary—equal umbers of A and B type atoms, with a single bound s- tate on each, substituted at random on a regular lattice ith a constant overlap integral V between nearest eighbors. Some highly respected statistical theories hich rely upon defining an average propagator in such a medium, seem to insist that the bands will be drawn out into a continuous broad spectrum as the two levels move apart; others would allow a split to occur when the spacing is rather larger than the width of either band [24]. I feel sure, myself, that the latter pre- diction is correct, but we have still a great deal to do be- fore we can calculate the width of each band the shape of the tails into the gap, and the nature of any levels in these regions. How far, for example, do these bands de- pend upon the possibilities of “percolation”, from one atom to another of the same type, through large distances—a property that depends peculiarly on the dimensionality of the lattice and the relative concentra- tions of the components? Another contradiction between mathematical theo- ries and physical intuition occurs in the case of Localized Localized FIGURE 17. Regions of localized and non-localized states for an “equiconcentration alloy”. 22999 O OOOO O OOOO FIGURE 18. A percolation chain in an equiconcentration alloy. - amorphous semiconductors. Let it be granted, for the sake of argument, that amorphous Ge and Si are “tetrahedral glasses”; each atom has four neighbors, arranged more or less in the regular tetrahedral orienta- tion, just as in the regular diamond lattice, but the con- nectivity of the structure has been altered in a random way, so that there is no long-range order. From the point of view of a chemist, this system is a single covalently bonded molecule: the saturation of all the bonds implies that some energy of excitation is required to create a carrier, so we should expect the material to be a semiconductor. The substantial gap in the optical spectrum of amorphous Ge supports this reasonable interpretation. But suppose we were to treat this by the conventional pseudopotential procedure, as- signing a model potential to each atom and then calcu- lating the diffraction effect on a free electron gas. In the absence of long-range order, there would be no strong Bragg reflections from well-defined lattice planes, and thus no proper band gaps at the zone boundaries, etc.; from the point of view of solid-state theory, this materi- al ought to be a metal. This antinomy needs to be resolved if we are to understand the theory of disor- dered systems — or even the theory of the chemical bond. There is some evidence—as yet merely qualita- tive [27] — that the diffraction approach can be made to give a band gap if one takes into account the higher- order particle correlations. Thus, a glass differs from a liquid in that three neighboring atoms may have a strong tendency to be oriented so as to make a good bond angle; this is a form of short-range order, implying a strong constraint on the three-and four-body statisti- cal distributions of atoms. At the same time, the rela- tionship between the localized molecular orbitals of the chemical bonds and the delocalized “Bloch states” of the crystal or amorphous solid needs to be clarified [28]. But these are only two of the numerous unsolved problems in this field. The above account of the band structure problem is obviously very sketchy and incomplete — especially in the total neglect of all electron-electron effects. We shall obviously learn much more about it as this con- ference proceeds. But I think it is good to look back and see what progress has been achieved—and even better to look forward to whole Alps of ignorance still to be surmounted. 6. References [1] Electrons and Phonons: (Oxford, Clarendon Press 1960). [2] Methods in Computational Physics: Vol. 8: Energy Bands of Solids: Edited by B. Alder, S. Fernbach, M. Roten- berg (New York: Academic Press 1968). [3] Ziman, J. M., Advances in Physics, 13,89 (1964). [4] Harrison, W. A., Pseudopotentials in Metals (New York: Benjamin 1966). ~ [5] Heine, W., The Physics of Metals (ed. J. M. Ziman: Cam- bridge, University Press, 1969) Chap. 1. [6] Herman, F., R. L. Kortum, C. D. Kuglin, J. P. Van Dyke and S. Skillman in Ref. 1, p. 193. [7] Heine, V. and I. V. Abarenkov, Phil. Mag.,9,451, (1964). [8] Shaw, R. W. and W. A. Harrison, Phys. Rev. 163, 604 (1967). R. W. Shaw, Phys. Rev. 174,769 (1968). [9] Ashcroft, N.W., J. Phys. C. 1, 232 (1968). [10] Rubio, J. and F. Garcia-Moliner, Proc. Phys. Soc. 91, 739; 92, 206 (1967). J. B. Pendry, J. Phys. C. 1, 1065 (1968). [11] Ziman, J. M., Proc. Phys. Soc. 86,337 (1965). [12] Lloyd, P., Proc. Phys. Soc. 86,825 (1965). [13] Morgan, G. J., Proc. Phys. Soc. 89, 365 (1966). K. H. Johnson, Phys. Rev. 150, 429 (1966). [14] Lawrence, M. J., Thesis, Bristol University (1969). [15] Slater, J. C., Phys. Rev. 145, 599 (1966). [16] Scholsser, H. C. and P. M. Marcus, Phys. Rev. 131, 2529 (1963). F. Beleznay and M. J. Lawrence, J. Phys. C. 1, 1288 (1968). [17] Ziman, J. M., Proc. Phys. Soc. 91,701 (1967). [18] Heine, V., Phys. Rev. 153,673 (1967). [19] Hodges, L. and H. Ehrenreich, Phys. Letters 16, 203 (1965). F. M. Mueller, Phys. Rev. 153,659 (1967). [20] Hubbard, J., Proc. Phys. Soc. 92, 921 (1967). R. L. Jacobs, J. Phys. C. 1,492 (1968). [21] See, e.g. J. Hori, Spectral Properties of Disordered Chains and Lattices (Oxford: Pergamon Press 1968). [22] Anderson, P. W. and W. L. McMillan, Teoria del mag- netisino nei metalli School XXXVII (1967). [23] Lloyd, P., Proc. Phys. Soc. 90, 207,217 (1967). G. J. Morgan, J. Phys. C. 2, 1446, 1454 (1969). [24] Ziman, J. M., J. Phys. C. 2, 1230 (1969) (II). [25] Ziman, J. M., J. Phys. C. 2, 1532 (1968) (I). D. F. Holcomb and J. J. Rehr (to be published). [26] Tauc, J., R. Grigorivici and A. Vancu, Phys. Stat. Solid 15, 627 (1966). [27] Fletcher, N. H., Proc. Phys. Soc. 91, 724; 92,265 (1967). J. M. Ziman, J. Phys. C. (1969) (III). [28] Anderson, P. W., Phys. Rev. Letters 21, 13, (1968). di transizione: Verenna Summer 12 F. Herman (IBM Res. Center, San Jose): I very much enjoyed Prof. Ziman's remarks. I think that Prof. Ziman’s approach to the problem is in the spirit of someone who tries to give a unified picture. I think, though, it is unfair to neglect appropriate mention of all the work that has been done numerically which in fact had led and has been a stimulus for the very elegant mathematical models that you describe. As long as the subject of band structure remained in the hands of text book writers the subject did not progress very far. But as soon as computers became available, people rolled up their sleeves and began to do actual calculations. Enough empirical progress was made so that the theoretically inclined could make their contributions also. I think it is important to see both sides of the pic- ture. J. W. Gadzuk (NBS): Due to the role of inelastic elec- tron processes at energies far ( ~ 20-100 eV) above the Fermi energy, (for instance as emphasized in current LEED theories) what do you feel is the situation with re- gards to a marriage of band structure and inelastic many-body effects for the excited states of a solid. J. M. Ziman (Univ. of Bristol): I don’t know the answer to this question. P. M. Marcus (IBM Res. Center, New York): In reply to the question as to the validity of band structure con- cepts at energies above the Fermi energy, I can com- ment that observation of LEED spectra indicates that some remnant of a band structure persists to high ener- gies. Reflection maxima in LEED spectra correspond to energy ranges with lowered densities of propagating states, as occur in band gaps, and such maxima are ob- served hundreds of volts above the Fermi energy. The effect of inelastic scattering of electrons is to wipe out any sharp band edges so that the density of the cor- responding propagating states does not drop sharply to zero as it would at the Fermi energy (in fact, all states now attenuate, but do so more strongly in the ranges of the energy gaps) and the diffraction peaks become ower, smoother and, eventually, more spread out in nergy. . J. Freeman (Northwestern Univ.); The complemen- ary question, “How good are band calculations well Discussion on “The Band Structure Problem” by J. M. Ziman (University of Bristol) above the Fermi energy?” is one we can answer only after we get more experiments such as those now being performed in photoemission at high energies and we have more optical experiments of that type. A. R. Williams (IBM, New York): One of the great vir- tues of the pseudopotential method is the ease with which it permits one to go beyond Hartree-Fock theory by means of the RPA dielectric function. Is there any way of implementing the same or a similar screening approximation in KKR and APW calculations? J. M. Ziman (Univ. of Bristol): We can in fact carry through a complete self consistent calculation. We have the wave function explicitly and we can go through the hard work and really slog it all out. I think you have to do this anyway. The RPA dielectric func- tion is only an approximation and there is no theorem saying under what circumstances you may legitimately divide the pseudopotential by the dielectric function and say that is the screened core potential. P. M. Marcus (IBM Res. Center, New York): How im- portant is the interstitial potential between the spheres in these methods? J. M. Ziman (Univ. of Bristol): KKR and APW are equivalent in being able to treat an interstitial potential. All you need to do in either case is to add in the Fourier components of the interstitial potential to the matrix elements in the determinants. In metals the difference between a muffin tin potential with a flat interstitial re- gion and a real potential is probably very small because they are relatively close packed. But in the case of the semi-conductors you get very large interstitial poten- tials indeed, with valleys along which electrons are es- sentially free and hills in other directions in which the electrons are bound. I am even prepared to conjecture that these are characteristic features of semiconduc- tors. The concept of chemical bonding implies a system with certain directions in which the electrons can travel freely without barriers, and other directions where they have to go over or tunnel through the hills. I think we must face this seriously and one of the problems that I hinted at in my talk was how to deal with that case. A. J. Freeman (Northwestern Univ.); D. D. Kelling, F. M. Mueller and I have included the “warped muffin tin” into our Symmeterized Relativistic APW Calcula- tions (cf. Phys. Rev., Feb. 15, 1970). The formalism is identical to the muffin tin case and is readily included into the programs. From actual calculations on Pd, Pt and bec U we find that the effect of the warping is in- deed small. We are carrying out calculations for inter- metallic compounds where the warped muffin tin is necessary in view of the inadequacies of using a muffin tin potential. J. M. Ziman (Univ. of Bristol): I have a student work- ing on long chain hydrocarbon structures which you can pack formally into a crystal. There this is the domi- nant feature. The question is not whether you can do the calculations by this method, which is exactly the same as yours, but whether it is a reasonably conver- gent method which does not seem to have been proved. But we have to test it out and see. 14 BAND STRUCTURE II CHAIRMEN. F. Herman R. C. Caselld RAPPORTEUR: R. E. Watson Electronic In this paper we consider recent calculations of the electronic density of states of nickel [1], palladium [2], platinum [3], scandium [4], iron [5], gold [6]. and plutonium [7], as produced by the QUAD [8] scheme from histogram representations of width 0.001 Ry filled by sampling the appropriate Brillouin zone at more than 1,000,000 random points. Comparisons with the experimental data will be made where appropriate. [1] Zornberg, E. I., and Mueller, F. M., Bull. Am. Phys. Soc. 13,441 (1968), and Zornberg, E. I., (to be published). *Work performed under the auspices of the U.S. Atomic Energy Commission. 417–156 O - 71 – 3 Density of States of Transition, Noble, and Actinide Metals” F. M. Mueller Argonne National Laboratory, Argonne, Illinois 60439 Key words: Electronic density of states: histogram representations; QUAD scheme. [2] Mueller, F. M., Freeman, A. J., Dimmock, J. O., and Furdyna, J. K., (to be published). [3] Mueller, F. M., Ketterson, J. B., Windmiller, L. R., and Horn- feldt, S., (to be published). [4] Koelling, D. D., Freeman, A. J., Mueller, F. M., and Goroff, I., Bull. Am. Phys. Soc. 14, 360 (1969). [5] Preston, R. S., Goroff, I., and Mueller, F. M., Bull. Am. Phys. Soc. 14, 386 (1969). [6] Sommers, C. B., Goroff, I., and Mueller, F. M., (to be published). [7] Mueller, F. M., and Goroff, I., Bull. Am. Phys. Soc. 13, 364 (1968). [8] Mueller, F. M., Garland, J. W., Cohen, M. H., and Bennemann, K. H., The QUAD Scheme (ANL-7556) and (to be published). Discussion on “Electronic Density of States of Transition, Noble, and Actinide Metals” by F. M. Mueller (Argonne National Laboratory) D. J. Fabian (Univ. of Strathclyde); Dr. Mueller, I noticed that in your calculation for platinum you calcu- lated for s-states only; I believe you subtracted all the d-states leaving just a plane wave. You found a very sharp peak at the bottom of the band. Do you really at- tribute that totally to the s-states? F. M. Mueller (Argonne National Lab.): The term s- states is a misnomer. That was why I was trying to be very careful in the beginning of the talk. This is the lowest part of the plane wave structure. If you want an S-like structure you have to include an additional pro- jection operator to project out the jo part of the plane wave. Here we have included all of the lowest part of the plane wave and this, in terms of the interpolation scheme we have used here, represents an orthogonal- ized plane wave. So we have included a lot of structure in there. That is the lowest basis function. I call it an S- state because that is what it is called in the literature. People understand that to be a plane-wave-like struc- ture. F. J. Blatt (Michigan State Univ.); In your calculatio of alpha iron, do you use the magnetization as a parameter? F. M. Mueller (Argonne National Lab.): The mag netization has been put in as a parameter. We have in cluded a sufficient amount of exchange to split th bands up and down so that the resulting number dif ference in the up and down structure conforms to th observed moment for iron. 18 1. Introduction Theoretical energy-band calculations are not estricted, in principle, to pure monatomic crystals. In ractice, however, comparatively few applications have een made to compounds, except for ionic and emiconducting cases. Aside from their intrinsic in- rest, an understanding of the band structures and ermi surfaces of intermetallic compounds is of con- derable importance to the theories of alloy phase sta- ility. Traditional research on alloy formation has been imarily of two types: (1) studies which attempt to cor- late a large amount of data on interatomic spacings, agnetic moments, etc., and (2) the determination of ystal structures and the correlation of certain recur- ng structures with electron concentration, i.e., the ume-Rothery [1] rules for electron compounds. The eory of alloy phase stability formulated by Jones [2] d Konobejewski [3] is based on a nearly-free elec- n approximation and on the thermodynamic princi- e of minimum free energy. Implicit in the theory is e rigid-band approximation in which the density of tes remains fixed as the solute concentration is in- ermanent Address: Center for Materials Science and Engineering, Massachusetts In- te of Technology. Electronic Densities of States and Optical Properties of CsCl Type Intermetallic Compounds J. W. D. Connolly and K. H. Johnson* Advanced Materials Research and Development Laboratory, Pratt and Whitney Aircraft Corp. Middletown, Connecticut 06457 The electronic band structures and densities of states have been calculated from first principles for two intermetallic compounds having the CsCl structure. The nonrelativistic augmented plane wave method has been used in conjunction with an LCAO interpolation technique to determine the band structure and density of states of 8'NiAl to a high degree of accuracy. These theoretical results are in excellent agreement with the measured optical properties if k-conserving (direct) interband transitions are assumed to be dominant. A similar study has been carried out for 8 Aužn, using as a basis the ener- gy bands determined by the relativistic Korringa-Kohn-Rostoker method. The band profiles and density of states of 8 Aužn are qualitatively similar to those of 8'NiAl, except for the appearance of relativistic effects in the former alloy and differences in the relative positions and widths of the respective Au and Nid-bands. The 3'Auzn results have also been compared with the measured optical properties and are again consistent with these measurements if direct interband transitions are assumed. Key words: Auzn; CsCl-type intermetallic compounds; direct interband transitions; electronic den- sity of states; NiAl; optical properties. creased. The newer theories, [4] while challenging the rigid band model, are themselves still only semi-quan- titative in nature. While conventional energy-band techniques rigorously permit us to determine the elec- tronic structure of ordered alloys only at exact stoichiometric proportions, it is nevertheless true that this information is quite important as a starting point for understanding the properties of neighboring con- centrations of disordered solid solution alloys. As an illustration of these ideas, we may cite earlier work on the IB-IIB ordered beta-phase compounds. The results of detailed DHVA [5] and HFMR [6] mea- surements together with fundamental KKR [7] and APW [8] band calculations on this system have in- dicated that the Fermi surface contacts the second Bril- louin zone boundary. This is in accord with Jones” [2] interpretation in which the beta phase occurs at an electron-to-atom ratio of 1.5, when the Fermi sphere touches the second zone boundary. Furthermore, the use of the band profiles in conjunction with the com- position dependence of the optical properties [9] has given some support for rigid-band behavior within the narrow composition limits of the ordered beta phase. 19 The problem of ultimately predicting the particular structure an assembly of atoms will assume as a func- tion of composition, pressure, volume, and temperature is a very complex one. It has not been solved satisfac. torily even for the simplest pure metals, although recently there has been some quantitative research directed to that end [10,11]. We feel that the accurate determination of the Fermi surfaces and band struc- tures of a number of specific intermetallic compounds is an important step toward the goal of being able to predict or explain features of the many alloy phase dia- grams which have been established. 2. Results for Auzn Like the other IB-IIB beta-phase alloys mentioned above, the ordered beta phase of Auzn is stable over a TABLE 1. Constants used in the first principles calculations relatively narrow range of atomic composition bracket- ing stoichiometric B'Auso/n50, which has the ideal CsCl-type crystal structure. At room temperature, the atomic composition limits are 8' Aus/n52 and 8 Aug2.5Zn47.5, the phase boundaries widening in the usual fashion with increasing temperature [12]. Unlike the alloys, 8' CuZn, 6'Ag/n, and 8' AgCd, the present system does not disorder appreciably below the melting temperature. For our band studies of 3' Auso/n50, we have generated a crystal potential in the familiar “muffin- tin” representation from a superposition of neutra atomic Au and Zn charge densities determine originally by Liberman et al. [13]. The key physica parameters, e.g., atomic configurations, lattice con stant, etc., adopted for this work are given in table 1. Auzn Atomic configuration (Au): (5d)” (6s)" Atomic configuration (Zn): (3d)” (4s)* Lattice constant, a = 6.028 a.u." Muffin-tin radius, R(Au)=2.586 a.u. Muffin-tin radius, R(Zn)=2.586 a.u. Zero of potential, Vo-- 0.866 Ry. Maximum angular momentum used in KKR wave function expansions, lmax=3 NiAl Atomic configuration (Ni): (3d)9(4s) Atomic configuration (Al): (3s)*(3p)" Lattice constant, a = 5.442 a.u." Muffin-tin radius, R(Ni) = 2.262 a.u. Muffin-tin radius, R(Al)=2.451 a.u. Zero of potential, Vo = - 1.6000 Ry. Maximum angular momentum used in APW wave function e pansion, limax = 6 Maximum reciprocal vector magnitude used in APW wave fun 5T 2a. tion expansion, |k+ k|max= “R. W. G. Wyckoff, Crystal Structures, (Interscience Publishers, New York, 1948). There are two principal errors in a model crystal potential of this type (relative to the exact Hartree-Fock solution), namely the lack of self-consistency and the over-estimation of the exchange effects through the use of the Slater approximation [14,15]. Recent self-con- sistent band calculations on transition [16] and noble [17] metals have shown that these two errors tend to cancel each other. In any case, for these metals and their alloys, the errors have the primary effect of mere- ly shifting in opposite energy directions the positions and widths of the d-bands with respect to the conduc- tion bands. Thus we can be reasonably confident in adopting the usual Slater-type exchange approximation in non-self-consistent band calculations on intermetal- lics, provided that we allow ourselves the option of using the d-band position and width as empirically ad- justable parameters. We have used the symmetrized, relativistic KKR method to determine the bands of 8 Au/n along five principal symmetry directions of the simple cubic Bril- louin zone. Five k-points were determined between the end points of each of the symmetry directions, for total of 29 nonequivalent points in 1/48 of the zone. Th corresponds to 227 points in the full zone, weighti each nonequivalent point properly. The results are i lustrated in figure 1. Double group notation has be used to label the bands. The broad conduction bands are intersected by relatively flat and narrow set of d-bands arising pri cipally from the 5al electrons of Au, but with consider ble conduction s- and p-state admixture. The Au bands are approximately twice as wide as the Cu bands are in 8' CuZn [7,8]. Immediately below the bº tom of the conduction bands is an extremely narrow of d-bands originating from the Zn 3d electrons. The bands are so flat and narrow on the chosen energy sc that we have merely indicated their boundaries by t shaded profile in figure 1. This same relativistic KKR program has been us to determine the energy band structure of Au. T width of the d-bands and position below the Fermi lev are the same as for 3'Auzn. Also, the width is appro I -0.2 lif = < 5 - 5 s 5 O.O – º 7+ 8 + — > M T R FIGURE 1. mately twice that of the d-bands in pure Cu. A previous 3alculation for Au by Amar and Sommers [18] had the -bands approximately 1 eV wider and 2 eV higher with espect to the Fermi level. This appears to be due to the se of a smaller exchange [2/3 of the Slater value] in a on-self-consistent calculation, which would tend to put he d-bands too high (for the reasons mentioned above). The dominant feature of the measured optical pro- yerties of 8 Aužn is a rapid rise in optical absorption in e spectral range of 2 eV to 3 eV [9,19]. This absorp- ion is responsible for the color of the alloy at room tem- erature. The calculated Au d-bands lie between 2.5 eV nd 5.5 eV below the Fermi level. However, we must re- ard the d-band position and width as uncertain by as much as +0.5 eV, because of the aforementioned un- ertainties in the crystal potential. The present band ructure is therefore only semiquantitatively con- stent with the optical data, in that we would expect ectronic transitions between the top of the Au d- ands and the Fermi level to contribute to the initial se in absorption. Another possible source of interband ansitions in this spectral range are the occupied con- uction states at and immediately below the Fermi vel in the vicinity of the symmetry point M. Possible -conserving transitions from these states to unoccu- ed levels just above the Fermi energy, along with the mputed energy gaps are M7_(EF) → M61 = 2.0+ 0.1 , and T. (EF) → T = 1..6+ 0.1 eV. Similar interband S X A T A R The electronic energy bands for 8 Au/n along the major symmetry directions, calculated by the relativistic KKR method. Double group notation has been used to label the bands. transitions seem to be responsible for the colors of the other beta-brass-type alloys, B' CuZn, 6' Ag/n, and 3'AgCd [7,9,19]. Additional optical absorption between 4 and 5 eV and between 7.3 and 7.8 eV is very likely due to transitions from the lower parts of the Au d-bands to the Fermi energy and to unoccupied conduc- tion states, respectively. Finally, the calculated position of the center of the Zn d-band is approximately 9.0 eV below the Fermi energy. This agrees closely with a measured peak in optical absorption at 8.6 eV. 3. Results for NiAl Ordered beta-phase NiAl, like 3'Auzn, is a Hume- Rothery [l] electron compound of the “3/2” type. In comparison to 8 Au/n, however, the composition range of phase stability is much wider for 8'NiAl. At room temperature the composition limits are 8' Nias Al35 and (3"Nigo Alio [12]. We have used the symmetrized, nonrelativistic APW method to determine the bands of stoichiometric 8'Niso Also. The crystal potential has been generated in an identical fashion to that described above for (3 Au/n, except for the use, in this case, of non- relativistic Hartree-Fock-Slater atomic charge densities calculated by Herman and Skillman [20]. The various physical parameters adopted for this calculation are given in table 1. The bands have been determined at 35 points lying on a cubic grid of spacing T/4a in 1/48 of the Brillouin zone. This grid is equivalent to 512 points in the full zone. The band profiles are shown along six symmetry directions in figures 2 and 3. Single-group notation is used. The nonrelativistic KKR method has also been G. 9: Q >- }*- CD 0.8 Dr. Lll 2. Lld 0.6 H. 0.4 H. 0.2 H. T – T X. FIGURE 2, applied to 8'NiAl. The disagreement is never more than a few thousandths of a Rydberg, supporting previ- ous evidence that the KKR and APW techniques give essentially identical results when applied to the same material for identical crystal potentials. The bands of 8'NiAl are qualitatively similar to those described for 8'Auzn (in the nonrelativistic limit). They are also qualitatively similar to the bands obtained earlier for 8' CuZn [7,8]. The primary difference is the closer proximity of the Nid-bands to the Fermi energy (indicated in figs. 2 and 3 by the solid line), relative to the locations of the d-bands in 8'Auzn and 3' CuZn, respectively. There is no narrow d-band below the con- duction bands of 8'NiAl which is analogous to the Zn d- band in 3 Aužn and 3' CuZn. A density-of-states profile has also been generated for 8'NiAl and is illustrated in figure 4. To obtain a suf- ficiently reliable density of states, it has been necessary M The electronic energy bands for 8'NiAl along the X., T and M directions, calculated by the nonrelativistic APW method. T R S X to interpolate the energy bands to a much finer wav vector mesh in the Brillouin zone than the 512-poin mesh calculated directly with the APW method. Thi has been accomplished by setting up an 18 × 18 LCA type Hamiltonian matrix, the elements of which hav been obtained by a least squares fitting to the AP energies at symmetry points in the zone [21]. The i teraction integrals between atomic orbitals which occ in this fitting scheme have been used as parameters i the manner suggested originally by Slater and Kost [22]. With this procedure, it has been possible to dete mine the bands of 8'NiAl on a mesh of 32,768 points i the full zone (969 nonequivalent points). The resulti histogram for the density of states has been smooth to eliminate statistical scatter. The sharp peaks in the energy range between 0.5 a 0.7 Ry are due to the flat d-bands in this region. T shoulder between .90 and .95 Ry is not due to d-ba 22 | M Z X A T A R FIGURE 3. The electronic energy bands for 8'NiAl along the Z, A, and A directions, calculated by the nonrelativistic APW method, 140 120 H. 8'Ni Al DENSITY OF STATES (ELECTRONS/ATOM-RY.) 100 H. 40+ 20 - O I | | ! O 0.2 0.4 0.6 0.8 1.0 ENERGY (Ry.) FIGURE 4. The electronic density of states for 8' NiAl. E, indicates the Fermi level, E, and E, are the Fermi levels for CoAl and FeAl respectively. 23 structure, but is part of the unoccupied conduction band associated with the critical points M1 and M3 (see fig. 2). In addition to the ordinary density of states, in order to compare with optical experiments [23-26] two joint density of states curves have been generated, assuming direct and nondirect transitions. The results are shown in figure 5. Under the assumptions of (1) a long relaxa- tion time and (2) a constant matrix element, the imagi- nary part of the dielectric function e2 is proportional to A' Ni A Nj (E)/E” NON DIRECT - — — — DIRECT A' Ni A JS - - - - RKB 7 H 6 M- 5 | | | I | | | O l 2 3 4 5 6 7 8 ENERGY (ev.) FIGURE 5. (a) The joint density of states for 3'NiAl divided by the energy E squared, as calculated assuming indirect and direct transitions. (b) The imaginary part of the dielectric function for 6'NiAl. JS indicates the data of Jacobi and Stahl [24] and RKB indicates the data of Rechtien, Kannewurf, and Brittain [23]. the joint density divided by the square of the energy. The es curves have been taken from the optical data of Rechtien et al. [23] and Jacobi and Stahl [24], and are also shown in figure 5. Because of the flatness of the d-bands, the direct and nondirect joint density of states curves are qualitatively similar. However, the former has slightly more struc- ture than the latter, since the nondirect transitions ef- fectively average out any sharp peaks. There are three main peaks in the direct curve, i.e., at 2.1, 4.0 and 5.9 eV, which compares well with the experimental struc- ture in the dielectric function [23] at 2.5, 4.0 and 5.3 eV. The nondirect curve has structure at 1.9 and 4.2 eV, with no pronounced peak at a higher energy, although there is a weak shoulder near 5.4 eV. In both curves there is a shoulder near 3.0 eV. If we denote the structure in the conduction bands between .90 and .95 Ry by C, the major peak in the d-bands at 63 Ry by D, and the two subsidiary peaks at .72 and .50-.54 Ry by D2 and D3, then the following assignments can be made: (1) the structure in the joint density of states at 2.1 eV is due to transitions from the Fermi level to C (Ef-> C), (2) at 3.0 eV is D2 -> C, (3) at 4.0 eV is D1 -> C, and (4) at 5.9 eV is D3 -> C. Thus, the structure in the eg curve is mostly a reflection of the structure in the d-band densi- ty of states. It is also informative to study the variation of the structure in the optical data as a function of composi- tion, as shown in figure 6. According to the x-ray analy- sis [27], on the Al rich side of stoichiometry, £3'NiAl 4.0 E3 3.2H- g J. >- O Cº. Lil Z Lll E 2 2.4 H. El 1.6 40 50 56 AT. 9% A LUMINUM FIGURE 6. The variation of the structure in the e9 function for 8' Ni as a function of composition. E. E., and Ea are defined in figure 5. This figure is taken from ref. [24]. forms a defect structure in which there are vacancie at the Ni sites, whereas on the other side stoichiometry, the Ni atoms occur substitutionall Using this information, it is a simple matter for th 24 energy band structure to explain the variation seen in figure 6. A charge analysis of the LCAO eigenvectors shows that the density of states at the Fermi level is al- most entirely due to Al s- and p-functions so that the number of Al atoms determines the position of the Fermi level. Therefore, on the Ni rich side of stoichiometry, the number of Al atoms decreases and the Fermi level is lowered, thereby increasing the ener- gy of structure (1) while leaving that of (2), (3) and (4) relatively constant. On the other side of stoichiometry, however, the number of Al atoms stays constant and, therefore, the optical structure stays constant. Both of these conclusions are confirmed by the experimental data [23,24]. Another of the band presented here is given by a comparison of the optical structure in CoAl with that of NiAl [25]. Referring to figure 4, since CoAl has one less electron, the Fermi level for CoAl decreases from Ef to Ef' which is below D2 so that structure (1) is eliminated. The experimental data shows that this is indeed the case, i.e., the 2.5 eV structure is missing in Co Al, whereas the 4.0 eV peak is still present [25]. [6] [7] [9] [10] [11] [12] [13] confirmation Stru Cture 4. References [1] Hume-Rothery, W., J. Inst. Metals 35, 309 (1926). [2] Jones, H., Proc. Roy. Soc. (London) A144, 225 (1934): A147, 396 (1934); A49, 250 (1937). [3] Konobejewski, S. T., Ann. Phys. 26, 97 (1936). [4] Cohen, M. H., and Heine, V., Advances in Physics. N. F. Mott, Editor (Taylor and Francis Ltd., London, 1958), Vol. 7, p. 395; Hume-Rothery, W., and Roaf, W., Phil. Mag. 6, 55 (1961). [5] Jan, J.-P., Canad. J. Phys. 44, 1787 (1966); Jan, J.-P., Pearson, W. B., and Saito, Y., Proc. Roy. Soc. (London) A297, 275 (1967). [14] |15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] Sellmyer, D. J., Ahn, J., and Jan. J.-P., Phys. Rev. 161, 618 (1967). Johnson, K. H., and Amar, H., Phys. Rev. 139, A760 (1965); Amar, H. Johnson, K. H., and Wang, K. P., Phys. Rev. 148, 672 (1966); Johnson, K. H., in Energy Bands in Metals and Al- loys, L. H. Bennett and J. T. Waber. Editors (Gordon and Breach Science Publishers Inc., New York, 1968), p. 105. Arlinghaus, F. J., Phys. Rev. 157, 491 (1967); Intern. J. Quan- tum Chem. IS, 605 (1967). Muldawer, L., Phys. Rev. 127, 1551 (1962); Muldawer, L., and Goldman. H. J.. in Optical Properties and Electronic Structure of Metals and Alloys, F. Abeles, Editor (North-Holland Publishing Company, Amsterdam, 1966), p. 574. Harrison, W. A., Pseudopotentials in the Theory of Metals, (W. A. Benjamin Inc., New York, 1966). Deegan, R. A., J. Phys. C. (Proc. Phys. Soc.) (London) l, 763 (1968). Hansen, M., Constitution of Binary Alloys (McGraw-Hill Book Company Inc., New York, 1958). Liberman, D., Waber, J. T., and Cromer, D. T., Phys. Rev. 137, A27 (1965). Slater, J. C., Phys. Rev. 18, 385 (1951). Slater, J. C., Wilson, T. M., and Wood, J. H., Phys. Rev. 179, 28 (1969). Connolly, J. W. D., Phys. Rev. 159,415 (1967). Snow. E. C., and Waber, J. T. Phys. Rev. 157, 570 (1967). Sommers, C. B., and Amar, H., Bull. Am. Phys. Soc. Ser. II, 13, 57 (1968) (unpublished). Jan. J.-P., and Vishnubhatla, S. S., Canad. J. Phys. 45, 2505 (1967). Herman, F., and Skillman, S., Atomic Structure Calculations (Prentice-Hall Inc., Englewood, New Jersey, 1963). Connolly, J. W. D., these Proceedings, p. 27. Slater, J. C., and Koster, G., Phys. Rev. 94, 1498 (1954). Rechtien, J. J., Kannewurf, C. R., and Brittain, J. O., J. Appl. Phys. 38, 3045 (1967). Jacobi, H., and Stahl, R., Z. Metallkunde 60, 106 (1969). Sambongi, T., Hagiwara, R., and Yamadaya, T., J. Phys. Soc. Japan 21, 923 (1966). Jacobi, H., Vassos, B., and Engell, H. J., J. Phys. Chem. Solids 30, 1261 (1969). Bradley, A. J., and Taylor, A., Proc. Roy. Soc. Al59, 62 (1937). 25 Discussion on “Electronic Densities of States and Optical Properties of CsCl Type Intermetallic Compounds” by J. W. D. Connolly (Pratt and Whitney Aircraft) and K. H. Johnson (MIT) W. E. Spicer (Stanford Univ.); The rise in the density of states above the Fermi level in the nickel-aluminum alloys is quite striking. There is nothing like it in either nickel or aluminum. I wonder if you can say anything about the physics producing this? Is it mixing? J. W. D. Connolly (Pratt and Whitney Aircraft): The conduction electrons in that region are definitely com- binations of s and p type electrons from aluminum. W. E. Spicer (Stanford Univ.); But there is no way of telling why it comes in the alloy and not the aluminum? K. H. Johnson (MIT): The energy bands in that re- gion are fairly densely clustered and to some degree parallel. When that happens, the density of states can become relatively large. You don’t have this sort of behavior in aluminum or nickel separately. As to why they cluster, it seems to be true generally for the cesi- um-chloride system of alloys we have studied. It probably results more from the crystal symmetry than from the component atoms. J. W. D. Connolly (Pratt and Whitney Aircraft): You might think of it in terms of the hybridization effect tending to push the bands up. S. J. Cho (National Res. Council): I did a similar cal- culation for the ordered palladium-indium system recently. I have found very similar structures near the Fermi surface with one peak right above the Fermi level from the hybridization as you said. 26 photoemission data. 1. Introduction One of the primary computational difficulties in the theoretical determination of the physical properties of a crystalline solid is the evaluation of a three-dimen- sional integral over a complicated (usually nonanalytic) region of momentum space. A simple example of this type of integral is found in the expression for the elec- tronic density of states, as a function of the energy e. d n (e) de > Thºe) dk (l) where Tn(e) is that region of k space where En(k) < e. n(k) is the energy of an electron in the n" energy band. he problem of calculating these integrals arises from he nature of En(k). This is usually available only nu- merically at a limited number of k points, as the results f an ab initio energy band calculation, such as by the PW or KKR methods. Since it is a costly procedure or an ab initio calculation to provide enough En(k) alues for an accurate numerical evaluation of the in- egral, a straightforward solution is to apply some inter- olation scheme to obtain values of En(k) between hose provided. A modified LCAO technique to do this as suggested fifteen years ago by Slater and Koster The Calculation of Densities of States by LCAO Interpolation of Energy Bands with Application to Iron and Chromium J. W. D. Connolly Advanced Materials Research and Development Laboratory, Pratt and Whitney Aircraft Corp. Middletown, Connecticut 06457 The LCAO (linear combination of atomic orbitals) interpolation method is described as a means of calculating the density of states curves of a crystalline solid. This method is shown to be more straightforward and convenient to use than the composite (LCAO-OPW) techniques that have recently been proposed for transition metals. A computer program is described which determines the LCAO in- teraction integrals from an ab initio energy band calculation by a nonlinear least squares procedure, and then uses these parameters to sample the Brillouin zone at a large number of points in order to cal- culate the density of states curve to a high degree of accuracy. As examples of the application of this program, the results of calculations on chromium (in both the nonmagnetic and antiferromagnetic states) and iron (nonmagnetic and ferromagnetic) are presented and compared with the recent Key words: Chromium; electronic density of states; interpolation method; iron; photoemission. [1]. This method is sometimes called the “tight-bind- ing approximation,” although this is really a misnomer, since the method is quite general and applicable to many types of crystals having wide bands which would not normally be considered to be tightly bound. For ex- ample, it has been recently shown by Dresselhaus and Dresselhaus [2] that this method is capable of satisfac- tory results for germanium and silicon. It was this belief that this method was applicable only to narrow band electrons which led to the proposal of using composite schemes [3,4], in which the narrow band electrons are treated by the LCAO approximation and the wide band electrons by an OPW (orthogonal- ized plane wave) or pseudopotential approximation. These approaches were designed to handle the energy bands of transition and noble elements and are able to reproduce their energy band structure (as determined by APW calculations) quite well. It is a purpose of this paper to show that such composite schemes are un- necessary for the description of the electronic structure of transition and noble elements, and that the electrons which are described by OPW wave functions in the composite schemes can equally well be described by combinations of s- and p-type atomic orbitals. The LCAO method has the advantages of being more 27 straightforward and convenient, without sacrificing either speed or accuracy. The method has been applied to the calculation of the density of states curves for many materials. The resultant curves for two of them, ferromagnetic iron and antiferromagnetic chromium, are presented in section 4 of this paper. 2. The LCAO Method In a periodic potential, the one-electron wave func- tion can be expressed as a linear combination of Bloch sums (Pn of atomic orbitals; (Pn(k, r) -w: lin (r-Ri) exp (ik-R) (2) where the sum is over the N lattice vectors R in the crystal, k is the reciprocal vector and r the position vector. The functions lin(r-Ri) are atomic orbitals cen- tered on an atom at lattice vector Ri. In actual practice, it is more convenient to use Löwdin functions [5] as basis functions. These are combinations of atomic or- bitals which are orthogonal to each other, CD, (r– R;) - X. lim (r- R;) (A-1/*) mī; ni m, R, (3) where Amj,ni is the overlap matrix between the atomic orbitals. This eliminates the need to consider the over- lap integrals between basis functions, leaving to be determined only the interaction integrals between the Löwdin orbitals, i.e., a ſºr-Roºr-R) () where 3% is the one-electron Hamiltonian operator. It can then be shown [1] that the electron energies are simply the eigenvalues en of the matrix, Hmn (k) — X. e *(R, - R;) O. mj; ni R. J (5) where Ri is the position of the atom in the unit cell on which the orbital (bn is located. The size of the H-matrix defined in (5) is determined by the number of atoms per unit cell and the number of atomic orbitals taken on each atomic site. For example, for a transition element with one atom per unit cell, the electronic structure can be adequately described by a 9 × 9 H-matrix, cor- responding to one s function, three p functions and five d functions. For a crystal with two atoms per unit cell like antiferromagnetic chromium or hexagonal cobalt, the size of the matrix is doubled to 18 × 18. Symmetry further simplifies the problem by reducing the number of independent interaction integrals, Omj;ni. The relationships between them can easily be generated by applying operations of the symmetry group to the integrand in definition (5). Therefore, the higher the symmetry of the crystal, the less are the number of integrals to be determined. For example, there are only four independent (d-d) integrals between nearest neighbors in the bec structure. Tables of these integrals can be found in reference 1. The integrals are now treated as adjustable parame- ters to be determined from an ab initio calculation by a nonlinear least squares procedure described in the next section. 3. Determination of the Interaction Integrals In the least squares procedure, we are required to minimize the following expression; Sjuk [en (k: a) - E), (k)|* (6) }} where the En(k) are the calculated (ab initio) energy band eigenvalues at wave vector k, as from an APW or KKR calculation, and en(k;0) are the eigenvalues of the Hamiltonian matrix of eq (5) which is dependent on a number M of parametric integrals oi, which we denote by a vector ov. The eigenvalues en and the correspond- y $º ing eigenvectors cy") satisfy the following equation: X. Hij(k, a)cy)(k) = e, (k, a cº")(k) (7) j The minimization of expression (6) involves the solu- tion of the set of equations; & X. ſ dk |-1. o:) –E (k) ðCYi en (K, ov) = 0, i = 1, 2 . . . M (8) These are a set of nonlinear equations which cannot be solved directly, and must be solved by iteration. We first assume an initial approximate set of parameters oo and that en is approximately linear in a. This leads to set of equations: X. Aſ {o, -o)) = b, i = 1, 2 . . . M (9 j where ðe, Öe Ajj = |al. |; ; l] > 00: 60.j o-oo bi = X. ſ dl, |(E,d) – en) ; and ! Equation (9) is to be solved by matrix inversion for O. until convergence is achieved. 28 The key quantities in the definition of the A-matrix and the b-vector are the derivatives of the eigenvalues en with respect to the parameters. These are con- veniently found by an application of the Hellman-Feyn- man theorem (see, for example, ref. 6), which in terms of our variables takes the form; Öen — - X. (n) > * * * * C 00. l, ºn (10) In the LCAO method the quantities 6 Him/60; are par- ticularly simple, as can be seen from eq (5) and the eigenvectors Co.") can be easily determined at the same time as the en, so that there is no difficulty in evaluating the required derivatives. The Hellman-Feynman theorem is also a useful tool for calculating derivatives with respect to other varia- bles, such as the k-vector. The first derivative with respect to k is the velocity function, and its zeros are critical points which contribute discontinuities to the energy derivative of the density of states curve. There is also a formula for the second derivative [6] which although not as simple as the first derivative is still easi- ly evaluated in terms of derivatives of the Hamiltonian matrix elements, i.e., 6°e, — a. C(n)|D C(n) ðcy” 2. l lm }}l 92H ÓM1, 9/M. Dºº- º -> | ! ) m. C * * (11) Ó Cy” 00: 0CV }*S where Mir = Hi, - Eöl, and mº' is the “partitioned inverse” of M, in which the n" eigenvalue is partitioned off. The second derivatives with respect to the k-vector are of course important in the evaluation of the elec- tronic effective masses. A summary of the calculational procedure is as fol- lows: (1) The parametric interaction integrals are deter- mined by the least squares procedure described by eqs (9), using the derivatives defined by eq (10). The number of ab initio eigenvalues En(k) used in this procedure is small, on the order of 10°. (2) These integrals so determined are used in eq (5) to define the H-matrix, which is diagonalized to give the electronic energies en(k) at a large number of k-points, typically on the order of 104 — 105. nonlinear (3) The density of states curve is calculated from the energy function en(k). This is done by means of an ordinary histogram method. Others have suggested the use of secondary in- terpolation, either linear [7] or quadratic [8], in order to save computational time. However, the linear scheme does not give the correct behavior around critical points and the quadratic scheme is in error near band crossings. The errors introduced may be small, but they are as yet unknown, and until a more satisfactory interpolation scheme is developed, we feel that the histogram method is adequate for our purposes, and not likely to introduce any errors other than statistical. 4. Application of the Method to the Density of States Curves for Chromium and Iron As a first example, the method was applied to the Fe energy bands calculated by Wood [9], who used the APW method. The calculation of the Fe density of states based on these bands has been done before in two different ways, (1) by Mattheiss [10] who used a linear interpolation method and (2) by Cornwell et al., [11] who used the LCAO technique. The resultant curve, shown on figure 1, is qualitatively similar to these two previous calculations. The discrepancy between the three calculations is never more than 10% over the whole energy range, which gives us a reasona- ble degree of confidence in the accuracy of the method. The LCAO integrals calculated by the procedure described in section 3 are close to those of reference 11. The d integrals do not differ by more than .01 Ry. The discrepancy in the s and p integrals is larger, between .01 and .l. Ry, but these do not contribute strongly to density of states, so that this discrepancy is not reflected in a large discrepancy in the curve. Taking a total of 84 eigenvalues from reference 9 (representing 6 energy bands on a cubic k-point mesh of spacing Tl2a), a fit was obtained whose rms devia- tion was approximately .003 Ry. For all the calculations done on transition metals, the rms deviations were al- ways of this order or better. (See table 1 for the details of the other calculations.) The LCAO matrix for the bec structure is of order 9, since there are 5 d-type basis functions, 3 p-type and one S-type. All the interaction integrals are included up to the second nearest neighbors for a total of 27 different parameters. The density of states curve is a smoothed histogram over 1,785 nonequivalent points in 1/48 of the Brillouin zone (representing a cubic mesh of spacing T18q, and 29 60 I I I I –I I I I I T NON-MAGNETIC IRON 50 H. DENSITY OF STATES (ELECTRONS/ATOM-RY.) tºº F4 40 H. º 30 H. sº 20 H. F5 * 10 H. 3 *Eº FERMI …’ O | I l | | i I I _l 0.] 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ENERGY (Ry.) FIGURE 1. The density of states for iron in its nonmagnetic state derived from the bands of ref. [9]. E3, E4, and E3 are the energies where the integral of the density of states equals 3, 4, and 5 respectively. EA is, therefore, the nonmagnetic Fermi level. E., and E5 would be the minority and majority spin Fermi levels assuming a rigid band exchange splitting. equivalent to 65,536 points in the entire zone). A calculation was also done on ferromagnetic iron, in which the same method with the same number of parameters was applied to the energy bands calculated by the KKR method by Wakoh and Yamashita [12]. This calculation has also been done using the self-con- sistent APW method [13] and found to give virtually TABLE 1. Parameters of the LCAO Calculations Fe Cr Ferromagnetic Non- Non- |Antiferro- magnetic O. £3 magnetic magnetic Number of parameters...... 27 27 27 27 48 Number of energy eigenvalues used in least squares fit....... 84 84 84. 154. 83 RMS deviation (Ry.).............. 0.0034 0.0039 || 0.0014 || 0.0012 || 0.0009 Ab Initio bands taken from Ref. No................. 9 12 12 15 15 Density of states at the Fermi level: Theoretical... 47 18 12 9 Experimental (states/ atom – Ry.)..................... a28 b13.4 bO.2 “Quoted in Ref. 12. * Quoted in Ref. 15. identical results. The resultant density of states curve is shown in figure 2. This was derived by doing two separate calculations on the two spin bands of reference 12, and superimposing the results. The densi- ty of states curve shown here is quite different from that shown in reference 12, due to the greater accuracy used in this calculation. The photoemission data for Fe [14], which should provide a measure of the density of states curve, shows three peaks at energies .5, 1.1 and 2.1 eV below the Fermi level. These agree approximately with structure in the theoretical curve at .5, 1.0, and 2.4 eV. However, the other structure in the theoretical curve is not seen. The discrepancies with the experimental data are most likely due to a transition probability which is variable over the Brillouin zone, which is neglected in the analy sis. This should be the subject of further investigation Also, the structure is much sharper in the theoretica curve than in the experimental data, which may be reflection of the effect of lifetime broadening. The second example of the application of the LCA method is a calculation on paramagnetic and antife romagnetic chromium using the bands of Asano an Yamashita [15]. The calculation has been repeate using the self-consistent APW method [13] and foun to give virtually identical results. The paramagneti bands were fit using the 9 × 9 LCAO bec matrix wit the 27 first and second nearest neighbor integrals. I 30 60 FERRO-MAGNETIC IRON DENSITY OF STATES 50 H. (ELECTRONS/ATOM-RY.) 40H- 30|- 2OH- | 10H- | | /". FERMI LEVEL | O | I l I l I l | l —0.7 —0.6 –0.5 —0.4 –0.3 –0.2 –0.1 O 0.1 0.2 ENERGY (Ry.) FIGURE 2, The density of states for iron in its ferromagnetic state derived by superimposing the two spin bands of ref [12]. 60 50 H. CHROMIUM DENSITY OF STATES (ELECTRONS/ATOM-RY.) 40H — NON-MAGNETIC --- ANTI-FERROMAGNETIC | 3OH | 20H \_ NON-MAGNETIC 1 OH FERMI LEVEL ANTI-FERROMAGNETIC FERMI LEVEL O —r-T | | | –0.8 —0.7 —0.6 —0.5 —0.4 –0.3 –0.2 –0.1 O 0.] 0.2 ENERGY (Ry.) FIGURE 3. The density of states for chromium in its nonmagnetic and antiferromagnetic states, derived from the bands of ref [15]. Note the drop in the density of states at the Fermi level due to the formation of the antiferromagnetic gap. ne antiferromagnetic state, the LCAO matrix doubles states curves are shown in figure 3. The main feature to size to accommodate two atoms per unit cell, one of be noted is that the two curves are virtually identical for ach spin, the space group changes from body centered the occupied electronic states, with the exception of the simple cubic, identical to that for CsCl, and the region immediately around the Fermi level. Because of umber of integrals increases to 48. The two density of the formation of an antiferromagnetic energy gap in the 31 bands on either side of the Fermi energy, the density of states is decreased. In this case, the decrease amounts to 25%, which compares favorably to the 30% decrease seen in the experimental electronic specific heat coeffi- cients (cf. table 1). The optical density of states derived from the photoemission data [14] shows structure at 4, 1.2 and 2.3 eV below the Fermi energy. The 0.4 eV structure is not seen in the theoretical curve, but the two agree with the theoretical peaks at 1.2 and 2.2 eV below the Fermi energy. Again, the discrepancies between the experi- mental and theoretical curves may be due to a variable transition matrix and lifetime broadening effects. 5. Discussion and Comparison with Other Interpolation Schemes We have tried to show in the previous sections that the LCAO interpolation scheme is a straightforward, convenient and accurate way of representing an energy band structure, even when the electrons are not tightly bound. Alternate methods, proposed by Hodges et al. [3] and Mueller [4], have assumed that the loosely bound conduction electrons must be described by orthogonal- ized plane wave (OPW's). This unnecessary dichotomy leads to difficulties in the hybridization terms in the Hamiltonian matrix. These terms involve interaction in- tegrals between Bessel functions and atomic wave functions, which are not easily parametrized in terms of simple functions over k-space. In reference 3, this difficulty was circumvented by the use of drastic approximation, i.e., by assuming the atomic d-orbitals were extremely localized (effectively 6-functions) and that the one-electron potential was constant over the unit cell. These two assumptions have the effect of eliminating the integral, but both of them are completely unjustifiable. Another difficulty with the composite LCAO-OPW schemes is the large number of OPW's required to reproduce the conduction energy levels. Going just to nearest neighbors in bec reciprocal space involves tak- ing 13 OPW's, i.e., corresponding to k = (0,0,0) and 12 vectors of the type (1,1,0). Hodges et al., [3] found that for the foc structure (which is the only structure to which the composite methods have as yet been applied) it was necessary to go to second nearest neighbors in reciprocal space, which would make a total of 15 OPW's. This difficulty, which would have required the solution of an unreasonably large secular equation, was avoided in reference 3 by using those 4 OPW's which have the lowest free-electron energies. However, by omitting the other 11, the symmetry of the eigenvectors is destroyed, thereby lifting the degeneracy of some of the eigenvalues. The introduction of arbitrary func- tions, referred to as “symmetrizing factors,” which restores the proper degeneracies, still does not correct the errors in the eigenvectors. This is not too serious for the description of the energy band structure, but might lead to errors in the calculation of properties which de- pend on the wave functions. An accurate description of the OPW wave functions would also require a knowledge of the atomic core functions, since they determine the orthogonalization terms in the OPW. Besides being free of the above difficulties, the LCAO method has other advantages: (1) It is easily ex- tendable to more complicated structures with more than one atom per unit cell. Examples are the applica- tions of the method to the intermetallic alloy 3'NiAl [16] and the transition metal oxide ReO3 [17]; (2) Spin- orbit effects can be inserted in a simple way [18], by the introduction of only one extra parameter for each nonzero l-value used in the basis functions; (3) The wave functions are easily calculable, since all that is necessary are the atomic valence orbitals without any need for the core functions, so that it may be possible to determine properties like the charge density and transition matrix elements from the results of an LCAO calculation. 6. References [1] [2] [3] Slater, J. C., and Koster, G. F., Phys. Rev. 94, 1498 (1954). Dresselhaus, G., and Dresselhaus, M. S., Phys. Rev. 160, 64 (1967). Hodges, L., Ehrenreich, H., and Lang, N. D., Phys. Rev. 152 505 (1966); Ehrenreich, H., and Hodges, L., Methods in Com putational Physics 8, 149 (1968). Mueller, F. M., Phys. Rev. 153,659 (1967). Löwdin. P.-O., J. Chem. Phys. 18, 365 (1950). Löwdin, P.-O. J. Mol. Spectry. 13, 326 (1964). Gilat, G., and Raubenheimer, L. J., Phys. Rev. 144, 390 (1966) Raubenheimer, L. J., and Gilat, G., Oak Ridge Nationa Laboratory Report No. TM-1425 (1966). Mueller, F. M., Garland, J. W., Cohen, M. H., and Bennema K. H., (to be published). Wood, J. H., Phys. Rev. 126, 517 (1962). Mattheiss, L. F., Phys. Rev. 139, A1893 (1965). The Fe densit of states curve is shown in his figure ll. [4] [5] [6] [7] [8] Cornwell, J. F., Hum, D. M., and Wong, K. G., Phys. Lette [9] [10] [11] 26A, 365 (1968). [12] Wakoh, S., and Yamashita, J., J. Phys. Soc. Japan 21, 17 (1966). w [13] Connolly, J. W. D., Intern. J. Quant. Chem. S2, 257 (1968). [14] [15] Eastman, D. E., J. Appl. Phys. 40, 1387 (1969). Asano, S., and Yamashita, J., J. Phys. Soc. Japan 23, 7 (1967). [16] Connolly, J. W. D., and Johnson, K. H., these Proceeding p. 19. [17] Mattheiss, L. F., Phys. Rev. 181,987 (1969). [18] Friedel, J., Lenglart, P., and Leman, G., J. Phys. Chem. Soli 25, 781 (1963). 32 band model. The electronic properties of aluminum are in many ays among the simplest and best understood of all the ommon metals. In particular, the optical properties of luminum can be accounted for over a wide energy nge; below 0.1 eV the Drude theory works well [1] hile at higher energies interband transitions become mportant. The dominant optical structure is a peak in at ~ 1.5 eV [2] for which the various experiments ive values of 40 and higher [3]. Although the different easurements disagree on the magnitude of this main ak in e2, they agree on the energy at which eº is a aximum. The existence of optical structure at 1.5 is also consistent with the various band models for uminum [4,5}. In the intervening photon energy nge 0.1 < ha) < 1.5 eV, the Drude model has not been und to fit the data unless a frequency dependent laxation time T is introduced [1]. There is, however, theoretical justification for this procedure. This investigation was undertaken to determine nether these optical properties could be correlated th the extensive Fermi surface data that exist for alu- num [6,7]. The aim has been to achieve a quantita- e fit to the measured optical constants, and to better derstand the difficulties of the Drude model in the This work was sponsored by the Department of the Air Force. Also at Department of Electrical Engineering, MIT. *Present address: Victoria University of Wellington, Wellington, New Zealand. 417–156 O - 71 - 4 Optical Properties of Aluminum * G. Dresselhaus and M. S. Dresselhaus.” Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Massachusetts 02173 D. Beaglehole” Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742 The Ashcroft energy band model which provides a good representation of the measured Fermi sur- face of aluminum is used here to calculate the optical properties. New reflectivity measurements in alu- minum have also been carried out between 2p and 3000 A using a sensitive continuous frequency scanning technique. A Kramers-Kronig analysis of the reflectivity data yields a frequency dependent dielectric constant which is essentially in agreement with the results of the calculation. This comparison suggests that the optical properties of aluminum can be described in terms of a one-electron energy Key words: Aluminum (Al); dielectric constant; electronic density of states; interband transition; optical properties; pseudopotential. 0.1 to 1.5 eV range. Since there exist discrepancies in the literature concerning the magnitude of the peak in e2 at 1.5 eV and since the difficulty with the Drude model could arise from low energy interband transi- tions, careful reflectivity experiments were carried out, from which a Kramers-Kronig analysis gave the dielec- tric constant. The correlation of these reflectivity measurements with Fermi surface data was aided by the pseudopoten- tial energy band model of Ashcroft [8] which provides a good representation of the Fermi surface as deter- mined by the de Haas-van Alphen measurements [7]. From the Ashcroft Hamiltonian, the frequency depend- ent dielectric constant for aluminum was calculated using k-dependent momentum matrix elements and a finite, though constant interband relaxation time. It has been found that for other materials such as germanium [9] and copper [10], phenomenological band models, which provide good agreement with experimental data at or near the Fermi energy, can be made to fit the measured e(0) for photon energies within a few electron volts of the Fermi level. Such studies have shown that the dielectric constant is not only sensitive to the joint density of states but also to the k dependence of the mo- mentum matrix elements. For aluminum, there are two factors which make such a phenomenological approach attractive. First of all, the major optical structure occurs for low photon 33 energies, so that the bands near the Fermi surface are emphasized. Secondly, the electronic properties of alu- minum seem to be well described by a nearly free elec- tron band model; thus, a pseudopotential calculation requires a very small number of Fourier coefficients of the potential, while the Fourier expansion method requires only interacting s and p bands. For these reasons, aluminum appears to be a good candidate for studying the applicability of a one-electron theory to the optical properties of a metal. In the present experimental study, greater sensitivity was achieved by a continuous reflectivity scanning technique [11], which provided reflectivity data between 2p and 3000 A. These data are very similar to those analyzed by Ehrenreich, Philipp and Segall [2], except for the presence of a reproducible shoulder at 1.2 eV and a small kink at 1.45 eV. A Kramers-Kronig analysis of the reflectivity data was carried out to yield the frequency dependent el and e2 curves. To extend the frequency range of the reflectivity data to be used in the Kramers-Kronig analysis, the present measure- ments were matched to Bennett’s infrared data [1] and to Madden’s ultraviolet data [12]. Finally, the high energy reflectivity slope beyond 20.0 eV was adjusted to agree with ellipsometry measurements in the neighborhood of 2.0 eV [13]. The results of the Kramers-Kronig analysis of the reflectivity data are shown for eº and or = e200/47 as the uppermost solid curves in figures 1 and 2 respectively. In order to dis- play more clearly the low energy structure it is also con- 7O | . + 6OH- 6| T of q | ol Interbond O º - - - Drude 5OH- o \\ o o O O o Calculated | 40– 9 M “. . . \\ O 3OHo \ O O OW Wo 2OH,- º \ tob- \ N N & | T--——- O | 2 3 4 5 Energy (eV) FIGURE 1. Imaginary part of the dielectric constant ex versus photon energy. The total es curve is obtained from a Kramers-Kronig analysis of the experimental re- flectivity. The Drude contribution using the parameters mont= 1.5 and Topt = 0.5 × 10−'' s is shown as a dashed curve. The resulting subtraction gives an experimental interband e2 shown as a dark solid curve. The open circles represent a calculation of the interband dielectric constant based on the Ashcroft model using an interband t = 0.484 × 10−" s. to oxio” Totd .O x 10 - Interb and — – - Drude o o o o o Calculated 7.5 x 1015 5.0 x 10” bºº Energy (eV) FIGURE 2, Real part of the conductivity or versus photon energ The decomposition of the total experimental or curve into the Drude and interband coi tributions is shown. The open circles represent the calculated interband conductivity an all parameters are the same as for figure 1. venient to plot the conductivity ori. To analyze these e perimental results in terms of an energy band model, i is necessary to separate the total ex or or into intraban and interband contributions. The results of one suc separation are also shown in figures 1 and 2; it is this i terband contribution to ex and or that is directly co pared with a dielectric constant calculation. With th intraband-interband separation shown here, the exper mental interband peak in e2 at 1.5 eV has a magnitud of e2 = 52.0. An explicit dielectric constant calculation was ca ried out using the RPA expression for e(a) [14]. Suc a calculation involves an integration over the Brilloui zone of an expression depending on energy bands n an n' separated at wave vector k by honn, and couple through a momentum matrix element Pnnſſk). Becau of the difficulty in evaluating this integral, the comput tion must be carried out on a high speed electron computer and a special effort must be made to calc late the energy levels En(k) and momentum matrix el ments Pnnſſk) as rapidly and accurately as possibl This is efficiently accomplished through use of a mod Hamiltonian. The momentum matrix elements Pnn'ſ at every point k are calculated by differentiation of t model Hamiltonian with respect to k [9, 10]. Thus, th matrix elements are consistent with the energy ba curvatures as expressed by the effective mass su rule. - In this work, two model Hamiltonians were e ployed: the Ashcroft pseudopotential model [8] a the Fourier expansion band model [15]. The Ashcr model for aluminum is based on 4 plane waves, resu 34 ing in a (4 × 4) model Hamiltonian involving 3 parame- ters which are evaluated to yield an accurate represen- tation of the Fermi surface data. The model Hamiltoni- an for the Fourier expansion method also leads to a (4 × 4) matrix representing interactings and p bands, and the Fourier expansion coefficients here are evaluated from Fermi surface data; these data are most con- veniently expressed by the Ashcroft band model at the Fermi level. Because of the free electron character of the energy bands in aluminum, the Fourier expansion is not as rapidly convergent as it is for germanium [9] and copper [10] and third neighbor terms in the Fouri- r expansion were retained in order to achieve a good it to the Fermi surface data. In the case of aluminum, he major advantage of the Fourier expansion echnique in correlating very diverse experimental data ver a wide range of energy and wave vector is not sig- ificantly exploited, since very scanty information is vailable away from the Fermi surface. Thus, the Fouri- r expansion technique mainly serves to re-express the shcroft band model so as to treat the symmetry pro- erties of the Hamiltonian more correctly without in- reasing the size of the matrix. However, the proper use f symmetry is important in calculating matrix ele- ents at a high symmetry point such as W. Therefore, is of interest that the results of the dielectric constant lculation based on the Fourier expanded model amiltonian are in many ways similar to those based on e Ashcroft model Hamiltonian. In fact, this calcula- n demonstrates that the Fourier expansion technique n be made to work not only for energy bands amena- e to a tight binding treatment, but also to nearly free ectron energy bands. Because of the greater simplicity of the Ashcroft amiltonian, a detailed comparison with the experi- ental data is given in this paper only for e(a)) based on e Ashcroft model. In the case of the Ashcroft band del, the momentum matrix elements were also cal- lated by differentiating the model Hamiltonian. This thod is fully equivalent to taking matrix elements of momentum operator between the plane wave basis tes of the Ashcroft model. Because of the large con- butions to the calculated e2(a)) in the range 0.1 < ho 0.6 eV (see the open circles in fig. 1), it was found re convenient to deal with o (o) rather than with e(a)). e results calculated for ori are shown as open circles figure 2. A value of Tinterband =0.5 × 10−" sec for the erband relaxation time yielded good agreement with observed magnitude of the peak in e2 (or ori) at 1.5 . This peak in e2 arises from interband transitions in vicinity of the K or U points in the Brillouin zone. In ition, the calculation exhibits a low energy peak in e2 below 0.8 eV, arising from interband transitions oc- curring near the W point. Because of the large uncer- tainty in the experimental determination of the low energy interband contribution to 0-1, it is difficult to compare theory and experiment in this region. At slightly higher energies, a small shoulder is found in the experimental data at 1.2 eV and a small kink at 1.45 eV. In this work, the Monte Carlo dielectric constant calcu- lation was carried out to sufficient accuracy to show that the Ashcroft band model yields no such structures at 1.2 and 1.45 eV. The intraband contributions to the conductivity were treated in terms of the Drude model. A satisfactory overall fit to both ori(a)) and O2(a)) could be accom- plished using a constant relaxation time Topt in the Drude model, with Topt in the range, Top! = (0.5 + 0.2) × 107* sec; the range of values found for the optical mass was mix = 1.5 + 0.15. An example of the kind of fit that was obtained for the interband contribution to ori and O2 is shown in figures 2 and 3 respectively where the curves are derived from the experimental data and the open circles from the energy band model. Because of the very large amplitude of the intraband contribu- tion to O2, it is not convenient to display both the in- traband and interband contributions to O2 in the same figure. The quoted errors in Drude parameters reflect the range over which the parameters could be varied without significantly changing the quality of the fit to the experimental data. The availability of more accu- rate experimental data in the difficult energy range 0.1 < hoo - 0.6 eV in aluminum would serve to narrow the uncertainty in the Drude parameter determination. The strong interband absorption that appears below 1 eV in both figures 2 and 3 provides the reason for the failure 7.5 x 1015 Inter bond 5.ox ſolº o o O o O Colculated F 2.5xio” O Q) (ſ) CN b O -2.5.x ſolº –5. Oxio” Energy (eV) FIGURE 3. Imaginary part of the conductivity, O2 versus photon energy. The curve represents the experimental interband contribution whereas the open circles are corresponding calculated values. All parameters are the same as for figure 1. 35 of a model based only on intraband contributions to the conductivity in this energy range. The approximate equality of Topt and Tinterband is also of interest. Since the dominant contributions to the in- terband conductivity arise from states near the Fermi surface, it is largely the same states which participate in both the intraband and interband transitions. Thus, a temperature dependence of both Topt and Tinterband could be anticipated. It is also of interest to note that Tinterband for aluminum is significantly smaller than the corresponding room temperature value Tinterband = 2× 10 * sec found for silicon and germanium through a similar dielectric constant calculation [9]. One outstanding puzzle which remains in this analy- sis is the value of mio, - 1.5. Because of the nearly free electron character of the aluminum energy bands, a value close to unity would be expected for mºſ. One estimate for mº, based on a one-electron energy band model has been made by Ehrenreich et al. [2] yielding mji= 1.2 as an upper limit. Even with this maximum value of mºr, it is not possible to satisfactorily repro- duce the reflectivity data. In summary, the 4 plane wave Ashcroft energy band model for aluminum [8] has been used to calculate the optical properties in the energy range below 5 eV. Our recent experimental measurements are generally in good agreement with the calculated dielectric constant above 1.4 eV. In the range 0.8 × ha) - 1.4 eV, structure is found experimentally, but is not well correlated with any structure in the theoretical curves. In the region below 0.8 eV, the band model suggests additional struc- ture; however, the experimental data are not suffi- ciently accurate either to corroborate or to conflict with predictions of the band model. The free electron or in- traband contribution to e(a)) is satisfactorily fit by the Drude expression with constant Topt over the entire many useful discussions. it is concluded that the optical properties of aluminum can well be described by a one-electron energy band model for this material. [10] [11] [12] [13] [14] Ehrenreich, H., and Cohen, M. H., Phys. Rev. 115,786 (1959 [15] Dresselhaus, G., and Dresselhaus, M. S., “Optical Properti energy range of the dielectric constant calculation; however, the optical mass parameter mi, though con- sistent with other experimental determinations [2], is too large to be understood simply on the basis of a nearly free electron model. On the basis of this dielectric constant calculation, Acknowledgment The authors would like to thank Dr. W. J. Scouler for References [l] Bennett, H. E., Silver, M., and Ashby, E. J., J. Opt. Soc. Am. 53, 1089 (1963). Ehrenreich, H., Philipp, H. R., and Segall, B., Phys. Rev. 132 1918 (1963). Beaglehole. D. (unpublished). Beeferman, L. W., and Ehrenreich, H. (to be published). Hughes, A. J.. Jones. D., and Lettington. A. H., J. Phys. C. 2 102 (1969). Moore, T. W., and Spong, F. W., Phys. Rev. 125, 846 (1962) 126, 2261 (E) (1962); Kamm, G., and Bohm, H. V., Phys. Rev 131, 111 (1963). Priestley, M. G., Phil. Mag. 7, 1205 (1962). Ashcroft, N. W., Phil. Mag. 8, 2055 (1963). Dresselhaus, G., and Dresselhaus, M. S., Phys. Rev. 160, 64 (1967). Dresselhaus, G., Solid State Communications 7,419 (1969). Beaglehole. D., Applied Optics 7, 22.18 (1968). Madden, R. P., Canfield, L. R., and Hass, G., J. Opt. Soc. A 53, 620 (1963). Hass, G., and Waylonis, J. E., J. Opt. Soc. Am. 51, 719 (1961 [2] [3] [4] [5] [6] [7] [8] [9] of Solids.” Proceedings of the International School of Physi “Enrico Fermi,” J. Tauc. Editor (Academic Press, N.Y., 196 36 K. H. Johnson (MIT): Is it possible that the anoma- lous skin effect could account for some of the deviation from the Drude theory in the low energy region? M. Dresselhaus (MIT): If that were an important ef- ect, one would also expect significant departures from he Drude approach below 0.1 eV; but it works very ell there. It is only in the energy region about 0.5 eV, here we think there is a low energy interband transi- ion, that there is a breakdown in the Drude picture. . Powell (NBS): What is the reason for the difference n the magnitude of the interband e2 calculated here no that calculated by Ehrenreich, Phillipp and Segall n 1963? . Dresselhaus (MIT): For one thing, they used a dif- erent energy band model. But more important is the econd factor. In the early days, when people did this ype of calculation, they did not realize the importance f included k dependence in the momentum matrix ele- ments. In the meantime, we have learned how impor- nt this k dependence is and it is very important for luminum. I should like to add that when this calcula- on was first made by Ehrenreich et al. it represented n important contribution to the understanding of the tical properties of metals. . W. Pratt (MIT): I gather you choose the relaxation me to give you a best fit to the data? . Dresselhaus (MIT): I assume that you are talking out the interband relaxation time. The choice of that antity regulates the height of the peaks in the dielec- ic constants. You have the liberty of adjusting one laxation time and with this value of the interband laxation time you have to fit all the peaks in both ei ld e. You don't have as much freedom as you might ink. The relaxation time is the only adjustable trameter in the whole interband dielectric constant lculation. All parameters of the Ashcroft band model determined by the Fermi surface data. . W. Pratt (MIT): How sensitive is it to the choice of e relaxation time? . Dresselhaus (MIT): If you change the relaxation e by an order of magnitude, you will perhaps change Discussion on “Optical Properties of Aluminum" by G. Dresselhaus, M. S. Dresselhaus (MIT), and D. Bedglehole (University of Maryland) the height of the peaks by about a factor of 2. F. M. Mueller (Argonne National Lab.): You used the local pseudopotential model of Ashcroft. How impor- tant are non-local corrections far away from the Fermi surface? M. Dresselhaus (MIT): The dielectric constant of alu- minum in this energy range is not very sensitive to ener- gy bands far from the Fermi surface. Aluminum is one material for which you would not expect large non-local corrections to the pseudopotential near the Fermi sur- face. I should also like to say that if you use a model with the proper symmetry (which the Ashcroft model does not display), the electron bands far from the Fermi level change somewhat. However, with his values of the parameters, the details of the energy bands at very high energies do not seem to be very important for the deter- mination of the optical constants in the energy range below 5 eV. For aluminum, we find that if you have a good fit to the Fermi surface you also get a good fit for the optical data. I don’t say that this will hold for all other materials. Aluminum has this nice property and for this reason is an attractive material for testing the validity of a one-electron energy band model. F. Herman (IBM): We did a self-consistent OPW cal- culation for aluminum. We found remarkable agree- ment with the Ashcroft model for the entire region below the Fermi surface extending up to 5 volts above. For energies greater than 5 volts above the Fermi sur- face significant deviations arose. M. Dresselhaus (MIT): At high energies, where E(k) is free-electron like in the Ashcroft model, something is clearly wrong. However, the optical constants below 5 eV are not sensitive to the energy bands far from the Fermi energy. J. R. Anderson (Univ. of Maryland): We have completed dPIVA experiments in aluminum primarily on the 2nd band hole surface and determined slightly different parameters for the Ashcroft model. It appears as if we can exhibit 1.2 eV and 1.4 eV structure from our energy bands. M. Dresselhaus (MIT): Our calculations showed little or no theoretical evidence for such structure. States transitions across a small spin-orbit induced bandgap transition for light incident along the trigonal direction levels; magneto-reflection. The optical properties of arsenic have been studied ver most of the photon energy range 0.023 ha) < 2.1 eV [1,2]. Because of the difficulty in preparing suitable op- ical faces, these studies have been mainly confined to easurements on the trigonal face, which is a cleavage lane in arsenic. The experimental reflectivity results or this face are plotted in figure 1 as a function of log o) in order to clearly illustrate the dominant structure t infrared frequencies. The data in this figure below .32 eV were taken by Riccius [2] and above 1 eV by Dardona and Greenaway [1]; data have not been re- orted in the region 0.32 < ha) < 1 eV. These measure- ents are also incomplete with respect to crystal orien- ation because the rhombohedral symmetry of arsenic esults in anisotropic optical constants. Although the optical measurements are still in- omplete, very extensive Fermi surface studies have een carried out [3-10]. Based on these Fermi surface easurements, Lin and Falicov have constructed an ergy band model [11] which characterizes the Fermi urface very well. Their Fermi surface for the holes ound the T point in the Brillouin zone is shown in ure 2. The large turnip-shaped pockets of this figure On the Optical Properties and the Density of in Arsenic * R. W. Brodersen” and M. S. Dresselhaus” Department of Electrical Engineering and Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02:139 The infrared reflectivity of arsenic is calculated and correlated with Fermi surface, magnetoreflec- tion and optical reflectivity measurements. These infrared properties are strongly affected by interband . The unusually large intensity of this interband is due to the simultaneous occurrence of a strong interband momentum matrix element and a large density of states. By considering this interband transi- tion explicitly, good agreement is obtained with the experimental data of Riccius. Key words: Arsenic (As); electronic density of states; Fermi surface; interband transition; Landau are designated as the O. carriers while the 6 necks con- necting the turnips contain the y carriers. To achieve the charge compensation characteristic of semimetals, an equal number of 8 carriers is required, and the Lin- Falicov model places these in 3 nearly ellipsoidal pockets about the L points of the Brillouin zone. Additional information on the energy bands near the Fermi surface has been obtained from a series of mag- netoreflection experiments [12,13]. These experiments ‘Work supported in part by the Advanced Research Projects Agency under Contract No. -90. **NSF graduate fellow. ***Also at MIT Lincoln Laboratory, which is supported by the U.S. Air Force, and a visit. scientist, Francis Bitter National Magnet Laboratory, Massachusetts Institute of chnology, Cambridge, Mass., with support from the U.S. Air Force Office of Scientific search. HOO 8OH- §§ - >- E 6OH- > H. O X-e Lil —l Li- Lil Dr. 40H 2O | | | | | | | | | | | | ––––– O.O2 Of O. 2 4 2 HO ENERGY (eV) FIGURE 1. Reflectivity data of arsenic for the trigonal face versus. photon energy. The data between 0.02 < ho < 0.32 eV is from Riccius (ref. [2]) and between 1 < ho < 10 eV from Cardona and Greenaway (ref. [1]). 39 indicate that spin-orbit interaction might be important in understanding certain features of the infrared pro- perties of arsenic [13]. In the magnetoreflection ex- periment, interband Landau level transitions are ob- served across a small bandgap of 0.172 eV attributed to the spin-orbit splitting of two bands which could other- wise cross. The symmetry properties of these transi- tions together with the values of the observed bandgap and cyclotron effective mass (H || trigonal direction) all contribute to the identification of this series of interband Landau level transitions with the y carriers of the hole Fermi surface. The energy extrema for both the conduction and valence bands are along the Q or binary axis, but close to the T point [11]. Because of the low symmetry on the Q axis, the energy extrema for these strongly coupled bands need not occur at the same Q point as is indicated on figure 3. The magneto- reflection data associated with this small bandgap are well explained by the Lax two-band model [13]. Since the dominant structure in the optical reflectivi- ty of figure 1 occurs at ~0.18 eV, it is of interest to seek a connection between this edge and the interband transition across the 0.172 eV bandgap. The present work explores this connection. For many small gap materials (e.g. InSb, Ge), the frequency dependent dielectric constant is only slightly affected by interband transitions across bandgaps in the infrared. The reason for this is closely related to the small density of states usually associated with these small bandgaps. In semiconductors like InSb and Ge, the strong interband coupling and small bandgaps result in small effective masses (large band curvatures) and a small density of TRI GONAL TT AXIS º e BISECTRIX TU AXIS *** * *...” % BINARY TW AXIS C. POCKET FIGURE 2, Hole Fermi surface of arsenic as determined by Lin and Falicov (ref. [11]). states. Thus, only a small volume of the Brillouin zone is involved in these low photon energy transitions and only a small effect on the optical properties results. At larger photon energies, large volumes of the Brillouin zone participate in interband transitions and these in- terband transitions have a much larger influence on the optical properties. The situation in arsenic is somewhat different from that in most small bandgap materials. Magnetoreflec- tion measurements show that strong band coupling oc- curs for momentum matrix elements in the binary- bisectrix plane; however, these two bands are expected to be only weakly coupled through momentum matrix elements in the trigonal direction [14]. The strong coupling in the binary-bisectrix plane produces a large oscillator strength for interband transitions when light is incident along the trigonal direction. On the other hand, the weak coupling in the trigonal direction results in large effective mass components mea for both th valence and conduction bands. In fact, instead o repelling one another, the energy bands in thi direction are nearly parallel for a small range of values around the critical point. It is this feature of th energy bands that gives rise to a large increase in th joint density of states and makes arsenic different fro other small gap materials. The unusual occurrence o both a large oscillator strength and a large density o states serves to emphasize this low frequency inter band transition in the optical properties of arsenic. The joint density of states is thus of great importanc in determining the magnitude of interband contribu tions to the dielectric constant. Fermi surface measure ments provide some information on the density o states for the occupied bands. The electrons whic have a total concentration of 2.1 × 10*/cm” are con tained in 3 nearly ellipsoidal surfaces around the points in the Brillouin zone. Using the cyclotron effe . t \ NO SPIN SPLITTING SPIN SPLITTING Q ; O.4 * X- >– 0.2 Q § (D HOLES LLI Or O *T. T. — E 2 # F LL u! —O.2 Q2 –04|ST | | H-> T O. O.2 —-W k, Q AXIS k, Q AXIS (d) (b) FIGURE 3. Energy bands along the binary or Q-axis near the carriers. (a) The energy bands calculated by Lin and Falicov, not including the spin-orbit in action. (b) An enlarged view of the region near the accidental degeneracy with spin-o coupling taken into account. 40 2 O | | | | | | | C | | | | | | -O.2O -O16 -O 12 —O.O8 is for the hyperboloidal y necks. tive mass parameters obtained from de Haas-van Alphen data [8], and assuming an ellipsoidal Fermi surface, the density of states curve shown in figure 4 is obtained. A qualitative density of states curve for the oles can be constructed by approximating the turnip- haped hole constant energy surfaces by ellipsoids. The est ellipsoidal fit is determined from the de Haas-van lphen data by taking the best extrapolated values of the extremal periods as the principal ellipsoids, even hough the former are not 90° apart [8]. Finally, the ensity of states for the minority holes, the y carriers, an be found by approximating the constant energy sur- 'aces by hyperboloids as shown in figure 2. The boun- ary between the O. and y carriers provides a cutoff for he y-carrier hyperboloids. An explicit value of k, for —O.O4 EN | | | | O O.04 O.O8 O.12 O16 Fe ERGY (eV) FIGURE 4. Density of states for the various carrier pockets versus energy measured from the Fermi level. Curve a is for the electrons at the L points, curve b is for the o hole pockets and curve c this cutoff is found from the pseudopotential calcula- tion of Lin-Falicov [11]. The density of states curves for the hole carriers are shown in figure 4. Although the number of carriers in the y necks is only ~2% of the total hole carrier concentration of 2.1 × 10*/cm3, their relatively large density of states near the Fermi surface results in a greater importance than their relatively small numbers would seem to indicate. In the present work an attempt is made to correlate the infrared measurements with the Fermi surface data, the magnetoreflection data and the optical data beyond 1 eV. The frequency-dependent dielectric con- stant is calculated using the expression of Ehrenreich and Cohen [15]: •) € & T == — 1 - €interbandº In 0 (...) × ſak this expression f(k) is the Fermi distribution func- on, Tnn is an interband relaxation time, and the oscilla- r strength fagnºn is related to the momentum matrix ements Pan' coupling bands n and n' by * 2 ſabºn — ( ) mohonºn here honn is the energy separation between bands n d n'. The integration is over all k states in the Bril- uin zone; because of the presence of the Fermi func- m, m' Pó, P8 }\'h * h h’ X." f ( k)fagnºn €og F 6 core —- Eintraband –H €interband i (t) –H CO), m' –H Tnn' j \-l – 1 (o-o-ti) ( ) & (1) Thn' tion, the summation over band indices n,n’ involve only transitions from occupied to unoccupied states. The in- traband contribution eintraband is treated by a Drude model for the O., 3 and y carriers. An explicit evaluation of the intraband term then depends on the various car- rier densities, optical effective masses, relaxation times and core dielectric constant. Most of these quantities are known from other measurements on arsenic. As a first approximation to the arsenic infrared reflectivity calculation, all interband transitions were 41 treated as frequency independent through a core dielectric constant. This would be a good approxima- tion if all the important interband transitions occurred at photon energies much greater than 0.32 eV, the upper limit of the available infrared data. With this very simple treatment, the core dielectric constant is esti- mated from the optical reflectivity at high energies. From the reflectivity data of figure 1, we would esti- mate ecore to be in the range 60 - E > H. wº- O Lil —l º # 99T \\\\ –-T____- - < ...” N-se” 4O | | | | | | | | | | | O OO4 O.O8 Of 2 Of 6 O.2O O.24 O.28 O32 ENERGY (eV) FIGURE 5. Reflectivity of arsenic for a trigonal face versus photon energy. Curve (a) is the line drawn through the data of Riccius (ref. [2]). Curves (b), (c), and (d) show the calculated reflectivity including only the Drude and core contributions to the dielectric constant for several values of ecore. the three calculated curves in figure 5 include this 2.5% reduction. The most sensitive parameter in the least squares fitting procedure used to construct figure 5 is the core dielectric constant. It determines the photon energy of the reflectivity minimum and the magnitude of the reflectivity above =0.25 eV. The best least squares fit was found for a core dielectric constant of 100, which is the value used in curve (c) of figure 5. On the other hand, to yield a reflectivity minimum at ha) = 0.19 eV a value of ecore = 110 is required and the resulting reflec- tivity curve is shown in figure 5 as curve (d). These values of ecore are considerably larger than that sug- gested by the high frequency reflectivity data and in- dicate that other interband transitions have been neglected that are, in fact, important at infrared frequencies. Furthermore, the three calculated curves of figure 5 show that by a judicious choice of ecore it is possible to fit some particular feature of the reflectivity but that no values of the Drude parameters can be found to fit the shape of the reflectivity curve near the reflectivity minimum. It is clear from this figure that the interband transition across the spin-orbit induced bandgap at the Q point must be included explicitly in order to yield the sharp structure observed experimen- tally. In order to calculate einterband associated with the in terband transitions across the small bandgap at Q several additional parameters must be evaluated; the are the energy bandgap Eg, the interband momentu matrix element coupling these bands Păn, the unoccu pied conduction band effective mass tensor, the Ferm energy and the interband relaxation time Tinterband Many of these parameters are either known from othe measurements or can at least be estimated. For exam ple, the magnetoreflection experiments not only pro vide a value for En = 0.172 eV, but also show that th two strongly coupled bands at the Q point can b described by a Lax model [13]. Therefore, the inte band momentum matrix elements of eq (1) and the e fective mass tensor for the unoccupied conductio band can be found from Fermi surface effective mas data. This type of approach is expected to yield reliabl values for the large momentum matrix elements and f the light mass components of the unoccupied ban However, only estimates can be obtained for the sm momentum matrix elements and the heavy mass co ponents [16]. The effective mass tensor for the valence band known from Fermi surface measurements [8]. Th tensor, or more conveniently the valence band inver effective mass tensor oft, can be related to the mome 42 tum matrix elements with Q point symmetry through the two-band model, yielding oft, - 1–2|Pă, a |*|mob, (2a) oft, = 1-2|P, ... [*|mob, (2b) o!, H 1–2 |Pă, dº |*|moE, (2C) oft, -2 Re (P., P. c.)/mob, (2d) oft, - ofty = 0 (2e) in which the momentum matrix elements employ a su- perscript to denote the direction of the momentum operator, subscripts v and c to denote valence and con- duction bands, respectively, along with indices 1 and 2 to denote the two spin states. In these equations, all quantities are evaluated at the band extrema and the x, ty and z directions correspond to the binary, bisectrix and trigonal axes, respectively. Since both oº, and Cº. depend upon the same matrix element Pi.e. (and similarly for off, and oft. which depend on Ph, º], an experimental determination of the effective mass ten- sor of the occupied valence band tests the validity of he two-band model in both the strong and weak oupling directions. Fermi surface data for the y necks [8] indicate that the Lax two-band model is applicable n both strong and weak coupling directions. From eqs (2), the momentum matrix elements about he Q point are evaluated. However, completely nalogous equations can be written for the conduction and inverse effective mass parameters oft under the ransformation c <> v, and the – sign in eqs (2a), (2b) nq (2C) into a + sign. In this way, all momentum matrix lements as well as the conduction band effective mass nsor can be estimated. Explicit values used for these omentum matrix elements are |Pă, oil” = 10 mob, § 02 |*= 30 mob, and P㺠|*= 0.8 moby. The evaluation of the dielectric constant involves an tegration over k space. This integration was carried ut in the volume of k space for which the two-band odel is valid. Contributions from other regions of the rillouin zone are treated through a core dielectric con- ant ecore. In the z direction, the limit of integration is ken as the boundary of the y neck with the O. pocket, mit of integration is taken as the midpoint between ad- cent y necks, leading to a cutoff energy of ~0.3 eV. er the volume of the k space integration, the non- rabolic enhancement of the momentum matrix ele- ents as given by the two-band model is never greater an 15%. Therefore, the k dependence of these mo- ntum matrix elements can be neglected. Further- re, the small displacement in k space of the two- ich is at ~5.2 × 10−4 a.u. [11]. In the x-y plane, the band extrema was neglected in accordance with the Lax model approximation [13]. An estimate for the remaining parameter Tinterband is available from the width of the magnetoreflection resonances, yielding Tinterbana = 10 ” sec at low tem- peratures [13]. The shape of the calculated reflectivity curve is rather insensitive to the precise value of Tinterband provided that the condition (otinterband * 1 is satisfied [17]. This condition does seem to be satisfied according to the results of this reflectivity calculation. Because of the scanty data available near the reflectivi- ty minimum only an estimate for Tinterband can be made from the reflectivity data. To carry out the integration over k space explicitly, nonparabolic effects in the energy were also neglected. This is a valid approximation because of the small size of the volume of integration and the small range of band energies that are involved. Thus, the valence band energy E, in the vicinity of the band extremum at Q can be expressed in terms of the principal axis coordinate system of the hyperboloidal constant energy surfaces alS E, H – h2 (ork}, + o-yºkº, -oºkº, ) (3) 1710 2 in which the k vector is measured relative to the critical Q point and the inverse effective mass tensor of is writ- ten in the principal axis coordinate system. For sim- plicity, a similar expression was assumed for the con- stant energy surfaces in the conduction band, charac- terized by the same inverse effective mass components or and Oy in the light mass directions, but a different o," parameter in the heavy mass direction. However, the constant energy surfaces for the conduction band are ellipsoids not hyperboloids. With these simplifica- tions, the integration of eq (1) can be carried out ex- plicitly yielding an analytic, though complicated, ex- pression in closed form. The real and imaginary parts of the dielectric constant einterband obtained in this way are shown in figure 6. With this determination of einterband, the infrared reflectivity of arsenic was calculated, yielding good agreement with the experimental data [2]. In this case, a reasonable value of ecore = 80 produced a reflectivity minimum at 0.19 eV and a much better fit to the reflec- tivity lineshape was achieved. It is in fact einterband that provides the difference between this value of ecore and the larger values of ecore that were required to fit par- ticular features of the experimental data in figure 5. Once again, the measurements could be brought into better agreement with the Drude model at very low 43 2O | J | O.16 | | | O.32 | | O.48 O.64 O ENERGY (eV) FIGURE 6. Real and imaginary parts of einterband versus photon energy. In this figure Tinterband = 5 × 10 ” s and values for the other pertinent parameters are given in the text. photon energies through correction for an oxide layer reducing the reflectivity by ~ 3%. A least squares fit- ting procedure was also tried, but it was found that the values for these parameters determined or estimated from other experiments provided the best fit, with the exception of oº. For this parameter the least squares fitting procedure yielded oº, a 5 in rough agreement with the two-band model estimate. However, the deter- mination of oº, depends strongly on the shape of the reflectivity curve close to the reflectivity minimum. Since the available data in this region are rather scanty, the quoted value of oº, a 5 is only an estimate. The results of the reflectivity calculation including the interband transitions explicitly are shown in figure 7 as the solid curve; a comparison is also made in this figure with the measured reflectivity points [2]. It is thus found that the infrared properties of a trigonal ar- senic face are strongly correlated with Fermi surface [8]. magnetoreflection [13] and optical [1] measure- ments. The two-band model is also found to work sur- prisingly well for the spin-orbit split bands at Q. FIGUR The data points are from Riccius (ref. [2]). The curve is the calculated reflectivity including the interband transition associated with the y necks as well as the intraband and cor contributions to the dielectric constant. - [7] [8] [9] [10] [ll] [12] [13] [14] [15] [16] [17] {OO 8O 6O 49–1–1–1–1–1–1–1–1–1–1–1–1–1–1–1– O O.O4 OO8 O.H2 O46 O2O O.24 O.28 O.32 ENERGY (eV) E 7. Reflectivity of arsenic for a trigonal face versus photon energy. References Cardona, M., and Greenaway, D. L., Phys. Rev. 133, A168 (1964). Riccius, H. D., Proc. Ninth Intern. Conf. on the Physics o Semiconductors, Moscow, 1968, p. 185. Berlincourt, T. G., Phys. Rev. 99, 1716 (1955). Ketterson, J. B., and Eckstein. Y., Phys. Rev. 137, A177 (1965), and references quoted therein. Shapira, Y., and Williamson, S.J., Phys. Letters 14, 73 (1965 Datars, W. R., and Vanderkooy, J., Proc. Intern. Conf. on th Physics of Semiconductors, Kyoto, 1966; J. Phys. Soc. Japan 21, suppl. 657 (1966). Vanderkooy, J., and Datars, W. R., Phys. Rev. 156,671 (1967 Priestley, M. G., Windmiller, L. R., Ketterson, J. B., an Eckstein, Y., Phys. Rev. 154, 671 (1967). Ishizawa, Y., J. Phys. Soc. of Japan 25, 160 (1968). Fukase, T., J. Phys. Soc. of Japan 26,964 (1969). Lin, P. J., and Falicov. L. M., Phys. Rev. 142,441 (1966). Maltz, M., and Dresselhaus, M. S., Phys. Rev. Letters 20, 91 (1968). Maltz, M., and Dresselhaus, M. S., Phys. Rev. 182, 741 (1969 Maltz, M., Ph. D. Thesis, Massachusetts Institute of Technol gy, 1968 (unpublished). Ehrenreich, H., and Cohen, M. H., Phys. Rev. 115, 786 (195 The effective mass data in reference 8 is consistent with t two-band model even in the weak coupling (z) directio despite the possibility of a stronger coupling to bands outsi the two-band model. The magnetoreflection line widths were found to be relative insensitive to temperature between liquid nitrogen and liqu helium temperatures. See reference 14. 44 Discussion on “On the Optical Properties and the Density of States in Arsenic" by R. W. Brodersen and M. S. Dresselhaus (MIT) N. W. Ashcroft (Cornell Univ.); How sensitive are your results to the value of m” used? In principle an ef- fective mass derived from dB v A or other low tempera- ture galvanometric data should not be used in the trans- port problem. Corrections due to electron-phonon in- teraction enhancements are quite substantial. R. W. Brodersen (MIT): The values of m” for the cy and 3 carriers used in our calculation affect only the Drude contribution and not the interband transition; thus the reflectivity is not strongly dependent on the exact values of these masses. The effective masses of the y carriers are important and these have been calcu- lated from both dBiv A and magnetoreflection data. The m* values obtained for the y carriers from these two measurements agree to within experimental error, showing that the mass corrections for this carrier due to the electron-phonon interaction are quite small. 45 Density of States and K. J. Ferromagnetism in Iron * Duff Scientific Laboratory, Ford Motor Company, Decarborn, Michigan 48121 T. P. Dcus Department of Physics, University importance of intra-atomic exchange and itinerancy to x-ray emission. 1. Methodology of Band Structure Calculation The aim of the present work is to explore the mag- netic properties of iron via a new band structure calcu- lation in which particular care is given to the exchange interaction, which in turn requires that the wave func- ions of the electronic states be as realistic as possible. or this purpose a variational method was chosen, and the trial wave function [1] was taken in the form 5 19 l, a X. \mu}(r) + X. Piubew (F) 171 = 1 i = 1 (1) he yn and pºli are the expansion coefficients deter- ined from the variational procedure. The functions uq"(r) are tight-binding (TB) wave functions formed from atomic 3d wave functions calculated for the 3d'4s onfiguration [2]. The uopw' are orthogonalized plane ave (OPW) functions, and the sum over i extends to second nearest neighbors in reciprocal lattice space. The presence of the OPW functions in (1) serves hree purposes: (1) the diffuse 4s/4p . . . states are well Approximated by OPW states; (2) the OPW functions an contribute some d component to the d wave func- ions thus making the radial part of the d wave function ully state dependent; (3) hybridization is built into the wave function. If yn are large and pºli are small, a pure *Supported by a National Science Foundation Grant at University of California, Riverside, alifornia 92503. of Utah, Salt Lake City, Utah 841 12 The band structure of ferromagnetic iron has been calculated by a variational method using a basis of tight-binding functions and orthogonalized plane waves. Exchange matrix elements are evaluated without approximation by a local potential. Correlation effects are explicitly included. Histograms for the density of states are constructed and compared with photoemission and optical reflection and x-ray emission data. The calculation leads self-consistently to the observed magnetic moment. The relative the origin of iron’s ferromagnetism is discussed. Key words: Electronic density of states; ferromagnetism; iron; optical reflection; photoemission; 3d state results; if the pi are large and yn are small we have a 4s/4p . . . state; if yn and pºli are comparable, the wave function is a hybrid. The secular equation to be solved is |H – ES] = 0 (2) where H and S are nondiagonal Hermitian matrices of the Hamiltonian and of unity respectively and each is of the form For the submatrix labelled TB, matrix elements are of the standard TB form; two center integrals only were retained. Similarly the OPW submatrix consists of standard OPW matrix elements. For a local operator V(r) the hybrid matrix elements take the form hybrid OPW (3) x ſuºrouſ-R) (4) and these were calculated by expanding the function ud (r-Rn) about the origin and carrying out summa- tions as far as necessary. Because of the radically different spatial behavior of the 3d and 4s wave functions, different methods of in- corporating correlation effects were used for each. For the 4s states the “screened exchange plus coulomb hole” approach of Hedin [3] was used. Here, in the cal- 47 culation of matrix elements, a screened coulomb in- teraction e "12/r2 is used, and in our case o was a static, wave number independent screening constant. For an electron gas of uniform density the coulomb hole potential is a constant and has the value where v(q) is the Fourier component of the coulomb potential. For our choice of dielectric function the cou- lomb hole energy reduces to — Oſ3. Correlation among d electrons has been shown by Kanamori [4] and by Hubbard [5] to introduce lo- calization properties into otherwise itinerant electrons, (5) (n : + n | – 1) V, - (n f – where Vo is a spherically averaged exchange interaction due to the other d electrons. Two complete band struc- tures were calculated corresponding to assumed values of A of .4 Ry and .55 Ry. 2. Results of the Band Structure Calculation Energies were calculated at 110 points in 1/48 of the Brillouin zone (BZ). The energy bands for A = .55 Ry are depicted in figures 1 and 2. The most striking fea- ture is the greater width found here and the appearance of a definite spin dependent component of the width. For example, at T we can separate out a non-spin com- ponent of the doublet-triplet splitting of .12 Ry (cf. 0.10 O,6 04 O.2 OO –O.2 –04 E--- – O.6 –O.8 - || O – | .2 - |.4 T N FIGURE 1. and to considerably weaken the exchange from the value calculated from Bloch wave functions. We adapt an expression given by Hubbard for the effective self exchange interaction #A - Weſt - — V. V(;7. --A)” – ºnAV. (6) where Vc is the usual self coulomb energy, A is the band width of the d electrons and n is the average occupation per atom of a single band, that is, n is taken as 1/10 (n + n | ). This expression was used for the one center self exchange energy, and the total one center coulomb plus exchange energy weighted according to population was 1) Vol.4 + (1 – n ||5) V.It (7) Ry for APW or KKR methods [6]), whereas the spin contributions are 0.11 Ry and 0.06 Ry for majority and minority spin respectively. Both the greater width and the spin dependent width found here are due to the in- clusion of two-center exchange matrix elements for d- states. Clearly, assumptions made about the efficacy of screening of interatomic exchange have important con- sequences for the band width. The density of states histograms for majority spin, minority spin and total electrons are given in figures 3, 4, and 5 respectively. Peaks in the total density of states occur at about 0.19 Ry, 0.39 Ry and 0.54 Ry below the Fermi level, and the occupied width of the d levels is about 0.8 Ry. The optical density of states as H Calculated bands for ferromagnetic iron. 48 P FIGURE 2, Calculated bands for ferromagnetic iron. H. Cºo º -1. [] -1.2 -1.0 -0.6 -0.6 -ou -0.2 0.0 ENERGY (#70BERGs) FIGURE 3. Density of states histogram for majority spin electrons. measured by Blodgett and Spicer [7] indicates three beaks, one of which is now attributed to surface con- amination [8]. A second peak of small amplitude ap- ears just under the Fermi surface and has no counter- art in the theoretical density of states presented here. 417–156 O - 71 – 5 The remaining experimental peak coincides with the total density of states maxima between —0.4 and – 0.6 Ry. Eastman [8] likewise shows a peak of about this width in his experimental optical density of states, but his peak lies closer to the Fermi surface than the 49 5. F. T #" - *-* | 2. | | | O I n y I I I I -1. [] - 1, 2 — 1 . 0 -0. 8 –0. 5 –0. 11 –0. 2 0, 0 ENERGY (RY DBERGS) FIGURE 4. Density of states histogram for minority spin electrons. Cºo I I T I- T- —r- I Uſ) ( C CI LL CO C c – . X— -r CIT >. CID H.- E 5: (ſ) 2 C CC H. ( ) R- LL — LL " e. | LL] T' | 2. | | I- T- | I- I — 1 . H - 1 - 2 — 1 - 0 –0. 8 – 0. 6 - 0. LA –0. 2 0, 0 ENERGY (RY DBERGS) FIGURE 5. Total density of states histogram. present theory depicts. However, some latitude exists for adjustment of the theoretical position of the Fermi energy, as discussed below. Unfortunately, the optical experiments are not defini- tive in the overall width of the occupied portion of the d bands, although they do give some support to the idea of wider bands than had previously been obtaine theoretically. Support for this point of view also come from x-ray emission spectra. Figure 6 represents sche matically the results of Tomboulian and Bedo [9] (thei fig. 5, M3 emission band). They assumed the band widt to be given by the interval ED of figure 6, i.e., about 50 CN > N. S. H l A L'E I D l;2 l;5 l;7 50 53 eV FIGURE 6. Schematic representation of the M3 x-ray emission spectra reported by Tomboulian and Bedo [8]. -jº & C w r— T —I- I- I 2^ P’ Cºo 2’ 2^ * (ſ) * wº- 2^ Lt. É >m (VI © O 2^ # H. CI 2^ (a) Th". 2 we 2– cº es 5 * 2^ 2^ E H. 2^ > T_ up C) Lu º 2^ c; º: >. 2^ Cºd >. 2^ g c. 2^ -jº E & 2^ Gä I 2^ C) 2. LL to gº. Lº z CT — T} >. 2^ e; (ſ) /* CT -0.08 -0.05 -0.ou -0.02 0.00 0.62 0.bu p. bs 0.08 R I GID BRND SHIFT (RY DBERGS) FIGURE 7. The solid line graphs the magnetic moment as a function of rigid band displacement of majority spin states with respect to the minority spin states. The dotted line gives the rigid band shift calculated from eq6 as a function of the band width parameter A. eV. If we accept at least part of the previously ignored spectra, the band width could be as large as 11 eV (AD on fig. 6). Moreover, if the kink at B is interpreted as a point of superposition of two curves, one for spin up and the other for spin down, the energy interval from A to B would give the magnetic splitting at the bottom of the band of about 5 eV in good agreement with our theoretical value of 4.5 eV at the point H, which also happens to be the point where the magnetic splitting is largest. Further experimental clarification of this is desirable: 3. The Magnetic Moment and the Origin of Ferromagnetism As remarked above, two band structures were calcu- lated, with the band width A of eq (6) equal to 0.4 Ry and 0.55 Ry. The second value of A seems more ap- propriate to the wider d bands found here. In each case the magnetic moment was obtained from the density of states histograms. The moments were 2.06 pºp/atom and 2.19 pºp/atom respectively. Thus, within the limitations of assuming Hubbard’s formula, a first principles 51 derivation of the magnetic moment has been achieved. To explore the role of correlation further, the band width A was regarded as an adjustable parameter and the corresponding values of the effective exchange cal- culated. Using this to produce a rigid band shift of the spin states relative to one another, an approximate esti- mate of the dependence of the magnetic moment on A is found. The solid curve of figure 7 is the magnetic mo- ment which results from given rigid band displace- ments, and the dotted curve graphs the rigid band shift as a function of A. The magnetic moment is charac- terized by a steep rise to about 2.18 pºp/atom and then a broad plateau to 2.25 pºp/atom followed by another steep rise. The plateau is produced when both spin states have minima of their density of states at the Fermi surface, and conversely the steep sections are characterized by a maximum of the density of states of either spin (or maxima for both spins) at the Fermi sur- face. Over the plateau region very large changes in A produce only minor changes in the magnetic moment, so in this range the choice of A or the accuracy of Hub- bard’s formula is not critical. However, its dominant role in producing the ferromagnetic moment is revealed by the rapid decrease in moment for A less than about .5 Ry. For A = 0, the magnetic moment would be 1.3 pºp/atom, implying that the remaining 0.9 pºp/atom comes from itinerancy of the d electrons. We therefore suggest the following mechanism for the origin of ferromagnetism in iron. Hund's rule at a given site is responsible for initially polarizing the elec- trons at that site. This is amplified by some itinerant ferromagnetism of the d states, and the itinerancy cou- ples the moments on different sites. Finally we note from figure 7 that there is latitude for Some arbitrary displacement of the two density of states curves without altering the magnetic moment ap- preciably. Thus the total density of states at the Fermi surface can be adjusted over a wide range from a small value ( = 5 electrons/atom Ry) on the plateau to a large value ( = 20 electrons/atom Ry) on the upper parts of the steep section. Thus no reliable prediction of the electronic specific heat or susceptibility can be made, particularly in view of the uncertainty in the orbital con- tribution to the magnetic moment. 4. References [1] For a discussion of the adequacy of the model of overlappings and d bands see F. M. Mueller, Phys. Rev. 153,659 (1967), V. Heine, Phys. Rev. 153, 673 (1967), and J. Hubbard, Proc. Phys. Soc. 92,921 (1967); pseudopotential formalisms based on this model have been given by L. Hodges, H. Ehrenreich and N. D. Lang, Phys. Rev. 152, 505 (1966) and Walter A. Harrison, Phys. Rev. 18l. 1036 (1969); a first principles calcu- lation of this type has been performed by R. A. Deegan and W. D. Twose, Phys. Rev. 164,993; combined tight-binding-OPW calculations not utilizing d states have been reported by W. Schneider, L. Jansen and L. Etienne-Amberg, Physica 30, 84 (1964), and A. Barry Kunz, Phys. Rev. 180,934 (1969). [2] We thank Dr. T. Gilbert of the Argonne National Laboratory for supplying us with the wave functions for the 3d'4s configuration. [3] Hedin, L., Phys. Rev. 139A, 796 (1965). [4] Kanamori, J., Prog. Theoret. Phys. 30, 275 (1963). [5] Hubbard, J., Proc. Roy. Soc. (London) A276, 238 (1963); A277, 237 (1963); A281, 401 (1964). [6] Wakoh, S., and Yamashita, J., J. Phys. Soc. Japan 21, 1712 (1966); DeCicco, P. D., and Kitz, A., Phys. Rev. 162, 486 (1967). [7] Blodgett, A. J., and Spicer, W. E., Phys. Rev. 158, 514 (1967). [8] Eastman, D. E., J. Appl. Phys. 40, 1387 (1969). [9] Tomboulian, D. H., and Bedo, D. E., Phys. Rev. 121, 146 (1961). 52 Calculation of Density of States in W, Ta, and Mo I. Petroff and C. R. Viswanathan Department of Electrical Sciences and Engineering, School of Engineering and Applied Sciences University of California, Los Angeles 90024 Density of states curves were calculated for tungsten, tantalum and molybdenum from correspond- ing energy band structures obtained by a nonrelativistic APW calculation. The Fermi energy and the density of states at the Fermi energy were obtained for each material. The calculations were part of a study intended to calculate theoretical photoemission yield curves which could be compared with ex- perimental results. Key words: Electronic density of states; Fermi energy; molybdenum; photoemission; tantalum; tungsten. 1. The Band Structure tial V(r) Apw which is spherically symmetric around each The APW method [1,2] consists in solving Schroed- atom site up to a radius Rs, while constant for r > Rs. 2 x- This pattern is repeated in each Wigner-Seitz cell. The Hill = Eli (1) augmented plane waves are functions li such that: inger's equation by expanding the wave function li in terms of aug- mented plane waves in the presence of a crystal poten- | =eiki'r for r > R, and (2) CC l l e 2* a ſe --- \ | . F & ; I ji(k, Rs) (l-lml)! |m| |m| * - l, 2. 2. (2l + 1) i ul (Rs) uſ(r) (l-H |m|)! P!" (cos 0)P," cos 0,) exp im (d (bka) m = — l (3) for r < Rs. The eigenvalues E at various points in the Brillouin zone are obtained by setting up the system of equations for obtaining the expansion coefficients, and then evaluating the secular determinant of this system as a function of E. The eigenvalues are those values of E for which the determinant goes to zero. All the three materials have a bec crystal structure. The energy eigenvalues were determined at 55 points in 1/48 of the Brillouin zone as shown in figure 1. The points are distributed uniformly on a cubic mesh throughout the zone. Each point is located at the corner of a cubic subzone having edge dimensions T/4a with the edges being oriented parallel to the coordinate axes kº, ky and kº. There are 1024 points in the entire zone. 2. The Density of States Density of states curves based on the 1024 points at hich the energy eigenvalues are actually calculated how poor resolution. The contribution to the density of FIGURE 1. Brillouin zone for body-centered cubic structure. 53 states function from points of high symmetry where the energy eigenvalues converge from several directions is not apparent unless the band structure is known at in- tervals that are smaller than those used in solving for the eigenvalues. Additional points in the band structure were obtained by reducing the mesh by a factor of 4 by means of interpolation. The resulting number of points in the Brillouin zone at which energy eigenvalues are available increases from 1024 to 65,536. The interpolation scheme is carried out in the wedge representing 1/48 of the Brillouin zone, which contains the original 55 points for which the band structure was calculated. Values at the midpoints of the edges of the cubic subzones are obtained by linear interpolation. Values at the center of the cube faces are obtained by interpolating between previously obtained midpoints. Only one pair of opposite edges is required for the inter- polation since a two-way interpolation using midpoints at all four edges of the cube face would give the same result. Values at the cube centers are obtained by inter- polating between previously obtained cube face cen- ters. Again, only one pair of faces is needed. At the wedge boundaries where the symmetry planes cut the cubic subzones along face or body diagonals, as in the 2N case of the NPT, NPH, or THP planes, reflection pro- perties in the appropriate symmetry plane are used to complete the cubes in order to apply the interpolation scheme described. In order to reduce the Brillouin zone mesh by a factor of four, the interpolation procedure must be applied twice. This results in 1785 points in 1/48 of the Brillouin zone. Reflection properties are again applied to extend the results throughout the en- tire zone. The density of states is plotted in units of electrons per atom per rydberg. Each energy eigenvalue can ac- commodate two electrons, and the number of eigen- values in a band is equal to the number of cubic sub- zones into which the Brillouin zone has been subdi- vided. In k-space, there are two electrons per atom per band. The following relation is thus established: 2N 3C F no. of subzones electrons per atom per AE (4) where N is the number of eigenvalues in some energy interval AE, and x is the corresponding number of elec- trons per atom in the same interval AE. To obtain x in units of electrons per atom per rydberg the right side of eq (4) should be divided by AE: (5) 3C = electrons per atom per rydberg AEX no. of subzones 45 H. ao! 9 ..........” Dr. — — . ......“” uſ 35 H m .......“ Q || || ........" O & 3O H. ſr. LL] ſh- s 25 F ...' S ... <ſ .." * or 20 F ..” lſ LL] - • * L- Q- ...” T (ſ) TJT *4 : 15 H. O Cr. - H. Sº IOH — lil 5 H- .' | º EF º O … I | |TF) L | | | | | O.3 O.5 O.7 O.9 |..] 1.3 |.5 |.7 |.9 2.] ENERGY (RYDBERGS) FIGURE 2, Density of states curve for tungsten. The dotted line is the integrated density of states, with corresponding units to the right. The zero of energy corresponds to the constant potential for r > Rs. To find the Fermi level, the density of states curve is integrated. This curve shows the number of electrons per atom as a function of energy. The Fermi level is that energy at which all the valence electrons for the given material are accounted for. In tungsten and molyb- denum this energy will correspond to six elec- trons/atom, and in tantalum to five electrons/atom. The density of states histograms are presented i figures 2, 3 and 4. In each case the integral of the densi ty of states is graphed in the same figure. The energ range considered includes the bands occupied by th valence electrons, and the unoccupied states above th 54 |6 — 4 § — |2 Lil § > & O - IO = É 0- i # > ſl. O 8 % º O Or É 5 % Lil 3. H. 4 O LL — LL] 2 O O.9 |. |.3 |.5 ENERGY (RYDBERGS) FIGURE 3. Density of states curve for tantalum. |6 — |4 § - || || || || .............…“ – 12 CD 9 - Dr. ă Cr- – |O H- Lil <ſ Cl- - Cº. ă #. H. (ſ) 2 <ſ 3. É H. 0. O LL] Cſ) —l 2 Lil O Dr. H. O Lil — LL] | | |. | |.3 ENERGY (RYDBERGS) FIGURE 4. Density of states curve for molybdenum. ermi level; a total span of approximately 1.8 Ry. The TABLE 1. Results obtained from the density of states esulting Fermi energy and density of states at the calculation. The Fermi energy is given in rydbergs as ermi energy for each material are shown in table 1. measured from the bottom of the lowest band. 3. Discussion • - e. Fermi Density of The density of states curve for tungsten is in good Material energy StateS at greement with the corresponding results obtained by Fermi energy attheiss [3] for this metal. The calculation by Rydbergs Mattheiss extends to 1.3 rydbergs on the energy scale. Tungsten................................ 0.4988 7.41 electrons The curve for tungsten has lower peaks and higher 3-> atom Ry inima than the curves for tantalum and molybdenum, Tantalum. 0.4077 | 17.9 electrons here high peaks alternate with deep minima. In other atom Ry ords, in tungsten, the electronic states do not group Molybdenum 0.4978 5.0 electrons emselves into “bands” as much as in the other two " " " ` e atom Ry etals. 55 The group VI elements tungsten and molybdenum exhibit three peaks below the Fermi level. Tungsten shows a broad peak above the Fermi level and in molyb- denum the peak splits into two peaks. For the higher energy states, two peaks are present in the vicinity of 1.8 and 2.1 Ry. These peaks are much more apparent in molybdenum than in tungsten. There are three peaks in tantalum at positions cor- responding to the first three peaks in tungsten and molybdenum, but in this case the second peak is broad and flat and almost merges with the third peak which is narrow and sharp. The Fermi energy falls within the range of the third peak slightly below the T35 level as predicted by Mattheiss [3] on the basis of a rigid-band model. There is an asymmetric broad peak above the Fermi level beginning with a maximum at about 0.9 Ry and tapering off over a range of 0.4 Ry. The two peaks at higher energies are also present and are very similar to the corresponding peaks in molybdenum. In all three cases, there is a low density region beginning beyond the top of the d-band and extending over a span of approximately 0.4 Ry. It should be noted that in the case of the group VI metals, the Fermi energy coincides with a slowly vary- ing part of the density of states curve, such that a small variation in the band structure should not affect the magnitude of the density of states at the Fermi energy to a great extent. In tantalum, the group V metal, the Fermi energy coincides with a peak in the density of states curve, and a small variation in the band structure could have a drastic effect on the density of states at the Fermi level in view of the steepness of the curve in that region. Relativistic as well as nonrelativistic APW calcula- tions of the energy band structure of tungsten are found in the literature in connection with Fermi surface stu- dies [3,4]. It appears that the relativistic and non- relativistic calculations tend to lie on either side of the experimental results [4]. A basic problem in this respect is that in addition to relativistic effects there is an uncertainty in the APW potential which is due to the exchange term. The contribution of the exchange in- teraction to the crystal potential is evaluated by means of an approximation. This is necessary in order to con- vert the potential into a one-electron potential. A 30% reduction in the magnitude of the exchange contribu- tion to the potential was shown by Mattheiss [3] to produce significant changes in the band structure of tungsten, comparable in magnitude with relativistic corrections. In molybdenum, relativistic effects are ex- pected to be less pronounced because it is a lighter metal, and a comparison of the theoretical photoemis- sion with experimental results should provide a more direct indication of the role of exchange effects. The modified tungsten band structure obtained by Mattheiss [3] by reducing the exchange contribution to the potential by 30% results in density of states curve which is very similar in character to the curve obtained from the unmodified calculation except for a general displacement toward higher energies. The height of the peaks relative to each other remains the same, as well as their relative widths. When the band structures for tantalum and molybdenum are calculated without reducing the corresponding exchange potentials, the differences in the resulting band structures are not of such a nature as to produce relative shifts along the energy axis in the density of states curves. When the results for tungsten, tantalum and molybdenum are compared to each other, no displacement along the energy axis is found, but the relative magnitudes and widths of the peaks vary considerably from one materi- al to the next. 4. Acknowledgments The authors are grateful to J. H. Wood for providing APW computer programs which were used in the ener- gy band calculations. 5. References l] Slater, J. C., Phys. Rev. 51,846 (1937). 2] Wood, J. H., Phys. Rev. 126, 517 (1962). 3] Mattheiss, L. F., Phys. Rev. 139, A1893 (1965). [ [ [ [4] Loucks, T. L., Phys. Rev. 143, 506 (1966). 56 Adjustment of Calculated Band Structures for Calcium by Use of Low-Temperature Specific Heat Data” R. W. Williams and H. L. Davis Metals and Ceramics and Solid State Divisions, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830 extremely sensitive to exchange. 1. Introduction The first band theoretical calculation on metallic cal- cium was made by Manning and Krutter in 1937 [1]. They carried out a Wigner-Seitz cellular calculation in an attempt to explain a number of electronic properties of metallic calcium, including the conductivity. This early band calculation demonstrated that calcium’s metallic behavior is probably due to the overlap of ans- band with a d-band. Their calculations further implied hat the density of states, as a function of energy, hould be rather s-like up to the vicinity of the Fermi nergy where the d-bands would then contribute a nar- ow, sharp bump. It is interesting to note from these onclusions that metallic calcium would have some d- ike electronic character which does not appear in the ree atom, thereby implying certain transition-metal- ype behavior for metallic calcium. Also, there have been several, more recent band tructure calculations pertaining to metallic calcium. Some of these attempts have employed different calcu- ational techniques such as the OPW method and the »seudopotential method [2-5]. However, there are asic disagreements between all of these calculations, ither when compared to experiment or to one another. or example, there is no agreement among the *Research sponsored by the U.S. Atomic Energy Commission under contract with Union arbide Corporation. The electronic band structure of calcium has been studied theoretically by employing the Korringa- Kohn-Rostoker method. The crystal potentials used in our calculation were obtained by means of a stan- dard superposition of free-atom charge densities. Ek vs k curves and the density of states at the Fermi energy were calculated for various potentials, with the measured low-temperature electronic specific heat coefficient, y, being used as an empirical aid to adjust the exchange portion of the crystal potential. The important feature of the potentials used is that they all give band structures which have definited- band character in the vicinity of the Fermi surface. These d bands or their corresponding d scattering resonances vary rapidly in energy for small changes in the exchange, resulting in values of y which are Key words: Calcium; de Haas-van Alphen; electronic density of states; low-temperature specific heat; pseudopotential; transition-metal behavior. presently available calculations concerning detailed features of calcium’s Fermi surface, with which experi- mental, de Haas-van Alphen results, the most direct verification of metallic band structure calculations, could eventually be compared. At present such dif- ferences between the calculations have not been resolved by comparison with experiment, since the only available de Haas-van Alphen data [6-8) is sketchy. Furthermore, the only data available was apparently obtained from microcrystalline-aggregate-type sam- ples, and can be misinterpreted. At the same time, no reliable calculations have appeared for the low-tem- perature specific heat coefficient, y, which is a mea- sure of the density of states at the Fermi energy. This is surprising, since such calculations would allow some comparison with experiment. This point is especially valid for calcium, since possible effects, which could enhance the band structural density of states, are ex- pected to be small. Thus, a comparison between calcu- lated and experimental y should provide a more direct test of a calculated band structure for calcium than would be possible for other metals where enhancement effects are expected to be larger and cause a greater degree of quantitative uncertainty. In an attempt to clarify some of the present circum- stances mentioned above, we have decided to carry out a rather complete calculational study on metallic calci- um with the aim of obtaining an experimentally verifia- 57 ble band structure. To this end, due to possible uncer. tainties connected with the available de Haas-van Alphen data, we have decided to use the low-tempera- ture specific heat coefficient as a means of experimen- tally testing our band structure calculations. Then, pro- vided our calculations would lead to a reasonable agree- ment between calculated and measured y, it was hoped that they would provide a quantitative prediction of cal- cium’s Fermi surface, which could be eventually verified by an exhaustive de Haas-van Alphen study. While our projected study is not yet complete in all phases of desired detail, we have found many interest- ing features caused by the presence of a d-band in the vicinity of the Fermi energy. Also, we feel these fea- tures are of general enough interest that they should be presented at this stage of development of our study. 2. Discussion of Calculational Details All of the band structure calculations reported here were performed by use of computer codes based upon the nonrelativistic form of the Korringa-Kohn-Rostoker (KKR) method of band theory [9–11]. Throughout, ex- clusive use has been made of muffin-tin potentials. The resulting accuracy then obtainable with the KKR method is governed by the highest & value, denoted by ^mar, present in the spherical harmonic expansion of the wave functions inside the muffin-tin spheres. Since previous work has shown for the transition metal region of the periodic table that use of £mar = 2 introduces cal- culational errors of at most a few thousandths of a ryd- berg in energy eigenvalues [11-13], we have used this value of £mar in the present work. All results were then obtained by direct numerical solution of the one-parti- cle, band-theoretic eigenvalue problem, with the restrictions mentioned above, without recourse to any interpolative scheme. The necessary one-electron potentials were obtained by use of the heuristic prescription outlined by Mattheiss [14]. In this prescription a potential is ap- proximated as V(r) = Wes.(r) + Ver(r). Both the electro- static coulomb contribution, Wes.(r), and the exchange contribution, Ver(r), are calculated using free-atom charge densities. At a given lattice site, Wes.(r) is equated to the sum of, (1) the Coulomb potential cen- tered at the site due to the free-atom charge density, and, (2) the contributions from the tails of the same Coulomb potential when centered on neighboring sites. The exchange contribution is obtained using a Slater- type approximation: Ver, (r)=-68|3p(r)/87)]”. p(r) is the spherically symmetric lattice superposition of the atomic charge densities and obtained analogously to the calculation of Wes.(r). 6 is a parameter which, in present terminology, equals one for full Slater exchange [15]. This prescription, as stated, still con- tains two unspecified quantities. First, what charge densities are to be used? Also, what atomic configura- tions are to be used? Second, given the charge densi- ties, what value of 8 is to be used? Determining the an- swers to these questions is presently an active research field; however, such is not the main topic of the present undertaking. Rather, in a practical attempt to avoid these questions, we have adopted the approach of ar- bitrarily using the analytical Hartree-Fock wave func- tions of Synek et al. [16] for the atomic configuration 3p°4s”, and then allowing 8 to vary as a parameter in at- tempting to determine the most experimentally plausi- ble calculated band structure. To obtain the Fermi energy for a given potential, KKR constant-energy-search techniques [13] were used to obtain points on constant-energy surfaces within the Brillouin zone of the foc lattice. Such points allow numerical evaluation of the volume contained by a constant-energy surface, and hence the integrated density of states for an energy is directly obtained. The Fermi energy is obtained by adjusting the energy until the proper integrated density of states is found. For all constant-energy surfaces calculated in this investiga- tion, of the order of 25,000 points were calculated on each surface. This number of points enables the in- tegrated density of states, for a given energy, to be ob- tained to an accuracy of about four significant figures. Such accuracy is not redundant but required in order to obtain an accurate value, for a given potential, of the density of states at the Fermi energy, n(EF), which we obtain by direct numerical differentiation of the in- tegrated density of states information. This point is especially valid in the present study where n(EF) fall in an energy region where the density of states i rapidly changing. Of course, the calculated low-tem perature electronic specific heat coefficient, for a give potential, directly follows from n(EF). 3. Results and Discussion Our first calculation was made with 8 = 1.00 and th Ek vs k curves for various symmetry directions ar given in figure 1. The notable feature of figure l is th nearly filled s-p band intersecting with a d-like band i the vicinity of the Fermi energy. This is qualitativel the same as has been proposed previously by Mannin et al. [1]. Our resulting density of states is 13.2 mi lijoules/mole-K”. This value of y is very much highe than the reported experimental value of 3.08 milli 58 enhancement of y is expected to be small and since the y obtained from a band structure calculation should be less than or equal to the y obtained from a specific heat measurement, it is apparent that some adjustment of he crystal potential is required. Consequently, we car- ied out a series of calculations of y for several different otentials. These potentials were generated by chang- ing the exchange contribution, and the resulting values fy as a function of 8 are given in table 1. As is noticed y this table, y is extremely sensitive to the assumed otential. We give the Ek vs k curves for 8 = 0.88 in figure 2. his particular value of 8 at least gives a reasonable alue of y. This figure should be compared to the Ek vs curves for £3 = 1.00 given in figure 1. The essential eatures are the same, namely, a narrow d-like band in- ersecting an s-p band in the vicinity of the Fermi ener- gy. However, it is seen from a comparison of the two igures that there has been a very rapid rise in the d ands from 8 – 1.00 to 3- 0.88. Furthermore, there has een a broadening of these same bands. The Fermi nergy of both sets of energy bands lies slightly in the ump of the d band density of states. Since the bump oves in energy and changes in width as 8 is varied, is results in a rapid change of the density of states at º | i .*. Of H _^ ! * | 2- 2 | e - e - *- * > -* _2~" L.-: as-yet-->isis:=: - e - © "º ę P- Cº. |-8 = 8- :::: .-sas-s- *-*—s- -e-e-"T 2.” Se *T*-*------—-—. º `-- e-º" ūj -:= - | ...” | `-----. :* --> <- "-----...T.I.I.I.I.I:::---. P –5.--~ T---~. O H- ...----------- “...:---- ...~:=:::::2.--—” -º-º-º: 2~ "--ºº:::::::--------- `--~~~~ | CALCIUM EXCHANGE = {OO | | - l T W X U W L K FIGURE 1. Energy bands for calcium calculated using the potential with exchange parameter 8− 1.00. The energies are expressed in rydbergs measured relative to the constant value of the potential between the muffin-tin spheres. ..~" ...-----"TTTTT T.I...... ------- _.---"T"T F--— _.--~"T _.--~" | ...--" 2." “S. ***-. ! T**::::... ...--------------------- --~ 2. fºL. — —. _-T O3 -----º. 2’ `s. `------T "-----... “..... _^ T*---. --~~. L." -—- T*-------. I.I.I.:* `s. . -----~~~ >< --~~ `'s*~2-2--- O2 H .------- ~. Tº TTTT -i-.------- —--~~ _2~ .*. •’ ">. .--~" º º :* 2° ſº-L-T ºs L. LL] ./ A O. H. Z : ./ CALCIUM EXCHANGE = O.88 O | T W X U W L K FIGURE 2: Energy bands for calcium calculated using the potential with exchange parameter 3- 0.88. The energies are expressed in rydbergs measured relative to the constant value of the potential between the muffin-tin spheres. joules/mole-K” [17]. Since the electron-phonon the Fermi energy with 8. To illustrate further this effect we have treated the d band as a d scattering resonance and computed its movement in energy as a function of exchange. The results are given in table 2. Note how TABLE 1. Results of the calculations for the low- temperature specific heat coefficient, y, as a function of exchange parameter, 8 £3 Calculated y (mjoules/mole-K") 1.00 13.20 0.90 6.14 .89 3.62 .88 2.02 TABLE 2. The numerical results showing the change in energy of the d resonance as a function of 8. AEsd equals the difference in energy in rydbergs between the bottom of the s-p conduction band, T1, and the d resonance, Ed 1.00 0.174. 0.90 .280 0.88 .300 B= AEsa 59 3O 2O H. V EFFECT = h? ſ! + | Vc hº. J. (A + 1) RY + an r2 EXCHANGE = {.OO -- EXCHANGE = O.88 s –2O - — 40 | FIGURE 3. Rs is the radius of the muffin-tin sphere. rapidly the d resonance is moving in energy away from the bottom of the s-p band as a function of exchange. We can also consider the graph of the effective potential, shown in figure 3. The 6 – 2 portion of the ef- fective potential for 8 = 1.00 shows a “well” and, thus, a tendency to “bind” the d electron. However, for 3 = 0.88 it is seen that there is a tendency to push the bot- tom of the “well” up, or “unbind” the d electron. Physically speaking, for these potentials metallic cal- cium attempts to “bind” a d electron but is only par- tially successful. If one considers scandium, which has one more conduction electron, he sees that it has a free atom configuration 4s”3d". This configuration should not be far different from that of metallic scandium. Scandium apparently manages to “collect” this d electron. 4. Summary It has been shown that consideration of the low-tem- perature electronic specific heat coefficient, y, can pro- The effective potentials, as a function of radial distance, for 3 = 1.00 and 0.88. The effective potential is the sum of the crystal potential and the centrifugal potential. r vide insight into the problems that exist in doing a first- principle band calculation for metallic calcium. While the calculations are as yet incomplete, it is felt that the d electron, or the extreme sensitivity of the location o the d resonance to small changes in the effective crystal potential, may have a rather important role i the electronic properties of calcium. For example, on might expect very interesting effects from calcium con ditions of pressure or alloying, depending on how the band is “filled” or the d resonance is moved about. 5. Acknowledgments The authors are grateful to Dr. J. S. Faulkner and D Jan Linderberg for many helpful comments and discu sions concerning this work. 6. References [1] Manning, M. F., and Krutter, H. M., Phys. Rev. 51, 761 (1937 [2] Harrison, W. A., Phys. Rev. 118, 1190 (1960). 60 [3] [4] [5] [6] [8] [9] Harrison, W. A., Phys. Rev. 131, 2433 (1963). Altmann, S. L., and Cracknell, A. P., Proc. Phys. Soc. 84, 761 (1964). Vasvari, B., Animalu, A. O. E., and Heine, V., Phys. Rev. 154, 535 (1967). Berlincourt, T., Proc. 7th Int. Conf. Low Temperature Physics (University of Toronto Press, Toronto, 1960), p. 231. Condon, J. H., and Marcus, J. A., Bull. Am. Phys. Soc. 6, 145 (1961). Condon, J. H., and Marcus, J. A., Phys. Rev. 134, A446 (1964). Korringa, J., Physica 13, 392 (1947). [17] Kohn, W., and Rostoker, N., Phys. Rev. 94, llll (1954). Ham, F. S., and Segall, B., Phys. Rev. 124, 1786 (1961). Segall, B., Phys. Rev. 125, 109 (1962). Faulkner, J. S., Davis, H. L., and Joy, H. W., Phys. Rev. 161, 656 (1967). Mattheiss, L. F., Phys. Rev. 133, A1399 (1964). Slater, J. C., Phys. Rev. 81,385 (1951). Synek, M., Rainis, A. E., and Roothaan, C. C. J., Phys. Rev. 141, 174 (1966). Griffel, M., Vest, R. W., and Smith, J. F., J. Chem. Phys. 27, 1267 (1957). 61 Discussion on “Adjustment of Calculated Band Structures for Calcium by Use of Low-Temperature Specific Heat Data” by R. W. Williams and H. L. Davis (Oak Ridge National Laboratory) W. Kohn (Univ. of California): The Rapporteur (R. E. R. E. Watson (Brookhaven National Lab.): I believe Watson) mentioned some work in which the magnitude of the exchange was estimated from the atomic data. I don’t know the details of this but I can understand in principle the rationale of such a procedure. But the Rapporteur also mentioned other work in which this parameter was adjusted. That I have never understood. May I elaborate on my lack of understanding here: If you are interested in electrons near the Fermi surface and if you are courageous enough to say that in some way they should be like free electrons then there is a definite value of the exchange there. Of course, they are not very much like free electrons but the fact they are not very much like free electrons affects everything about them, not just the exchange constant. So I would very much like to have some explanation of what is the rationale of saying, if we don’t get a good fit with our simplified theory we are going to adjust that one parameter rather than, for example, trying to do some serious work on correlation effects and try to fit the data that way. H. Davis (Oak Ridge National Lab.): From my point of view, we consider the resultant potential as a model potential and nothing more. You look at the phase shifts. Let’s say you are trying to compare an electron calculation of Fermi surface parameters with single particle data such as the de Haas-van Alphen effect. You adjust to get proper phase shifts at that energy and these are the basic parameters of the theory. The result is just a model potential. that the screening is responsible for about half the up- ward d-shift in gold. F. Herman (IBM): I would like to comment on Profes- sor Kohn’s question. My own feeling really is that the adjustment of the Slater exchange is a passing phase in- fluenced by pseudism. You try to adjust things to make theory and experiments agree. My own present feeling is the important criterion in a first principle calculation is not really the comparison between the energy eigen- value spectrum and experiment because if you think very carefully about the question you will be very hard put to provide a theoretical basis for demonstrating that there is any direct connection between an energy eigen- value spectrum and the optical excitation spectrum using the approximate Hamiltonian. My own feeling at the present time is that if one really wants to do a first principles calculation one should use that exchange ap- proximation which would give the lowest total energy for the crystal in the Hartree-Fock spirit. Now people are just beginning to try to calculate total energy in band calculations. I have a paper which will appear shortly in the International Journal of Quantum Chemistry which discusses this question and tries to reconcile the problem by indicating that what one should do is use the Gaspar-Kohn-Sham exchange ap- proximation, perhaps also including inhomogeneous corrections, and after getting an energy eigenvalue spectrum this way, introduce corrections which hope- fully give better estimates of levels whose differences correspond to optical excitations. 62 Normal Volume and as 1. Introduction Several attempts [1-4] have been made to estimate the contributions of low energy electron-phonon and electron-electron interactions to the Fermi surface pro- erties of the noble metals by comparing calculated and effective masses met with experimental values. his approach can be understood within the context of the Landau Fermi liquid theory, in which the effective ass enhancement from combined electron-phonon no electron-electron interactions is given approxi- ately by [5,6] m* = m (1 + o-e-H oph). (1) experimental he coefficients o'e and Oph correspond to specific Lan- au coefficients for the electron-electron and electron- honon interactions and me” is the result of a one-elec- ron calculation of the effective mass in which effects f the static periodic lattice potential are included. The xperimental and band values of the electronic specific eat are related by the same combination of enhance- ent factors. Although the theory assumes an isotropic ermi surface, it is possible to gain an estimate of the ffects of anisotropy by comparing the values of the mped enhancement factor O = Ove -F Oph (2) termined by applying eq (1) to the cyclotron effective asses associated with the selected orbits about dif- rent sections of the Fermi surface. This work was supported by the U.S. Atomic Energy Commission. *Present address: University of Colorado, Boulder, Colorado. Fermi Surface Properties of the Noble Metals at q Function of Pressure * W. J. O'Sullivan,” A. C. Swifendick, and J. E. Schirber Sandiq Laboratories, Albuquerque, New Mexico 87107 We present the results of nonrelativistic KKR calculations of the Fermi surface properties of Cu, Ag, and Au at normal volume and as a function of pressure. In particular we compare electronic specific heats, effective masses and the associated pressure shifts with the corresponding experimental results for the noble metals. In contrast to the results of previous calculations we find that the Herman-Skill- man-Mattheiss crystal potential is an excellent effective potential for both Cu and Ag. Key words: Crystal potential; effective masses; electronic density of states; electronic specific heat; noble metals; pressure effects. Using this approach, Mueller and Zornberg [1] determined a value of O in Cu of about 0.25, while Faulkner, Davis, and Joy [2] (FDJ) and Dresselhaus [4] inferred a significantly smaller value. Christensen [3] has estimated an o' for Ag of about 0.05, by compar- ing band effective masses calculated using the APW method with experiment. In this paper we present the results of nonrelativistic KKR calculations of several band effective masses and the linear electronic specific heat coefficients for the noble metals at normal volume and as a function of pressure. Estimates of the total enhancement factors for the noble metals are derived from comparison between the calculated results and experiment. 2. The Band Calculations The band effective masses and the electronic specific heat coefficients for the noble metals were cal- culated with the KKR method using a maximum angu- lar momentum contribution, lmar =3. The potentials used in these calculations were derived from Hartree-Fock-Slater atomic charge densi- ties obtained from a slightly modified version of Her- man and Skillman’s [7] program. The superposition procedure described by Mattheiss [8] was used to ob- tain the spherically symmetric Coulomb and exchange potentials within the “muffin-tin” spheres. For both the atomic and crystal exchange potentials the unmodified Slater free-electron exchange approximation was used. The effect of lattice dimension change on the crystal potential is incorporated in the variation of the con- 63 tributions from neighboring atoms as the lattice varies in size, and in the scaling of the muffin-tin sphere radius with the lattice. Since we choose the muffin-tin radius so that the muffin-tin spheres make contact, the latter effect is determined unambiguously. The results of the energy band calculations and the calculated Fermi surface dimensions for the noble metals at normal volume and as a function of pressure will be published [9,10] elsewhere. Germane to this paper, however, is the fact that the calculated energy band structure for Cu is in essential quantitative agreement with that calculated by Burdick [ll] with the Chodorow potential, and the experimen- tal and calculated Fermi surface cross-sectional areas agree to about 1%. A comparison of our calculated bands for Ag with the bands calculated by Christensen [3] using Dirac-Slater wave functions, suggests that our d-bands are about 0.03 Ry too high. However, the calculated and experimental Fermi surface cross-sec- tions agree to ~ 1% except in the case of the “neck” cross-section for H|| [111] where the calculated area underestimates the experimental value by about 10%. The effective masses reported here were calculated from h? / 3A * – T | -- and the electronic specific heat coefficients, y, were derived from y = }Tºkºp (EF), (4) where p(EF) is the calculated density of states at the Fermi energy. In these calculations, p(EF) was deter- mined by a straight line fit of the derivative with respect to energy of the number of occupied states. The calculations were carried out over an energy range of EF + 0.002 Ry, with a fixed energy increment of 0.0002 Ry. A mesh density corresponding to 561 k vectors in 1/48th of the Brillouin zone was used. This represents the same mesh used by Faulkner, Davis and Joy [2], and with it one can determine the Fermi energy for a noble metal to about 0.0002 Ry. 3. Comparison of Experimental and Calculated Results The Fermi surface topology of the noble metals is simple enough that measurements of the effective masses corresponding to a small number of orbits con- stitute a reasonable sampling of the entire surface. The Fermi surfaces of the noble metals consist of essen- tially spherical electron surfaces centered at T with in- terconnecting necks in the [111] directions. With the magnetic field in the [ll]] direction a “belly” (B(111)) and “neck” (N(111)) orbit are observed. With the field along [110] a hole orbit made up of a combination of necks and spheres resembling a “dogbone” (D(110)) is observed. A similar fourfold symmetric hole orbit known as the “rosette” is observed for fields in a [100] direction in addition to another belly orbit about the spherical body of the surface (B(100)). The calculated results for the band effective masses associated with all but the rosette orbit are compared with experiment in table 1. In table 2 we list the calcu- lated electronic specific heat coefficients for the noble metals along with the corresponding experimental values. The enhancement factor for Cu is about 0.10+0.02 if we accept the lower value for the dogbone effective mass measured by Koch, Stradling and Kip [12]. In any case, o for Cu is about 0.1, in agreement with the observations of FDJ [2] and Dresselhaus [4]. The con- sistency of the values for O in Ag is, of course, for- tuitous. We estimate o in Ag to be 0.03+ 0.02, in essen- tial agreement with the results of Christensen [3]. The combination of electron-electron and electron-phono TABLE 1. A comparison between the calculated an experimental values of the effective masses for specifi noble metal orbits Cu m*/mo m*/mo O. (calculated) (experimental) N(111)...... 0.412 0.46+ 0.02 "........................ 0.12 B(111) ...... 1.28 (1.410+ 0.005, 1.385 + 0.005)".. .10 B(100) ...... 1.24 1.370 + 0.005 "..................... .10 D(100) ...... 1. 11 1.290+0.005%)1.225+0.005)"... .15(0.10) Ag N(111)...... 0.38 0.39 + 0.02 "........................ 0.03 B(111) ...... .902 || 93+0.01 "........................ .03 B(100) ...... .896 || 93+0.01 °........................ .03 D(110)...... 1.00 | 1.03 + 0.01 “........................ .03 — Au N(111)...... 0.321 0.29"(0.44)"........................ < O(0.37 B(111) ...... .844 | 1.14+0.03/......................... 0.28 B(100) ...... .842 | 1.08+0.03/......................... .28 D(100)...... ,830 1.00+ 0.03.J......................... 20 " Reference 12. " Reference 16. * Reference 14. * Reference 15. * Reference 17. J Reference 18. TABLE 2. Calculated and experimental linear elec- tronic specific heat coefficients for the noble metals. The estimated relative error in the calculated y(m.J/mole-degº) values is about l percent. Calculated | Experiment “ O' Cu...…. 0.641 0.698 0.09 Ag................................ .624 .644 .03 Au......… .564 .727 .29 * Reference 19. effects in Ag is much reduced over that in Cu. It is dif- ficult to assess the effects of our neglect of relativistic contributions in the case of Au. The enhancement fac- tor we find for Au is about 0.28. Although effective pressures of up to 100 kbar were “applied” in these calculations, no significant varia- tions in the calculated band effective masses were ob- served, with one possible exception. The calculated ef- fective mass for the N(111) orbit in Ag decreases with increasing pressure. Calculated values for (0 lm y/0 lm V).T were obtained, although the large uncertainty in the alculated derivatives makes them of doubtful value for omparison with experiment. These results are listed in table 3. The experimental values for Cu were deter- ined by inserting measured values for the electronic art of the linear thermal expansion coefficient in the elation (Ó ln y/6 lin V)1 - 38. WB/(yT) (5) where Be is the electronic linear thermal expansion 20efficient, V is the molar volume and B is the bulk modulus. The uncertainty in the calculated volume erivatives of the specific heat could be reduced if we ere to increase the k space mesh density used in the olume calculations. While our calculated values for Ó ln y/6 lin V seem o agree with the free electron value of 2/3 and the mea- ured values of Carr et al. [13], our treatment neglects ny volume dependence of the enhancement factor. In conclusion we find that the Herman-Skillman tomic calculation with full Slater exchange provides n excellent starting point for detailed energy band and ermi surface properties both at normal volume and as function of pressure. TABLE 3. Calculated and experimental values for (0 ln y/6 ln V).T 417–156 O – 71 – 6 Metal Calculated Experimental Cu.…. 0.65 (+ 0.25) | * 0.63 (+0.06) *.43 ° 1.21 (+ 0.13) a 1.12 (+0.05) Agº… .83 (+0.30) Au.…. .38 (+ 0.25) * Reference 20. * Reference 21. " Reference 13. * Reference 22. 4. References [1] Zornberg, E. I., and Mueller, F. M., Phys. Rev. 151,557 (1966). [2] Faulkner, J. S., Davis, H. L., and Joy. H. W., Phys. Rev. 161, 656 (1967). [3] Christensen, N. E., Phys. Stat. Sol. 31, 635 (1969). 4] Dresselhaus, G., Solid State Comm. 7,419 (1969). [5] Ashcroft, N. W., and Wilkins, J. W., Phys. Letters 14, 285 (1965). [6] Peris, D., and Nozieres, P., The Theory of Quantum Liquids, (W. A. Benjamin, Inc., N.Y., 1966), p. 338. |7] Herman, F. and Skillman. S., Atomic Structure Calculations, (Prentice Hall, Inc., Englewood Cliffs, N.J., 1963). [8] Mattheiss, L. F., Phys. Rev. 133, A1399 (1964). [9] Schirber, J. E., and O’Sullivan, W. J., Proc. Colloque Interna- tional Du C.N.R.S., Sur Les Proprietes Physiques Des Solides Sous Pression, Grenoble, France, to be published. [10] O’Sullivan, W. J. Switendick, A. C., and Schirber, J. E., Phys. Rev., to be published. [ll] Burdick, G. A., Phys. Rev. 129, 138 (1963). [12] Koch, J. F., Stradling, R. A., and Kip, A. F., Phys. Rev. 133, A240 (1964). Carr, R. H. McCammon, R. D., and White, G. K., Proc. Roy. Soc. (London) A280, 72 (1964). [14] Joseph, A. S., Thorsen, A. C., and Blum, F. A., Phys. Rev. 140, A2046 (1965). [15] Howard, D. G., Phys. Rev. 140, 1705 (1965). [16] Joseph, A. S., Thorsen, A. C., and Blum, F. A., Phys. Rev. 140, [13] A2046 (1965). [17] Shoenberg, D., Phil. Trans. Roy. Soc. (London) A255, 85 (1962). [18] Langenberg, D. N., and Marcus, S. M., Phys. Rev. 136, A1383 (1964). [19] Martin, D. L., Phys. Rev. 141, 576 (1966). [20] Davis, H. L., Faulkner, J. S., and Joy, H. W., Phys. Rev. 167, 601 (1968). [21] Shapiro, J. M., Taylor, D. R., and Graham, G. M., Canad. J. Phys. 42,835 (1964). [22] McLean, K. O., and Swenson, C. A., private communication. Discussion on “Fermi Surface Properties of the Noble Metals at Normal Volume and as a Function of Pressure" by W. J. O'Sullivan, A. C. Swifendick, and J. E. Schirber (Sandia Labs.) K. H. Johnson (MIT): The deviation you find in the Fermi surface of gold at the hexagonal face of the zone is, I believe, due to relativistic effects which you have not considered. We have carried out fully relativistic band calculations on gold, and we find the relativistic contributions very important. W. J. O’Sullivan (Sandia Labs.): We are of course aware of this. In the paper we make no bones about our calculations on gold being more than qualitative. We just constructed a potential, carried out the calculation, and accepted the results. We made no attempt to adjust the potential in order to provide better agreement with Fermi surface data. Christensen [1] has included relativistic effects to some extent in silver by construct- ing a potential from the relativistic atomic Hartree- Fock-Slater wave functions. He also gets very good agreement with the Fermi surface data. R. E. Watson (Brookhaven National Lab.): [In the course of reviewing the papers, the Rapporteur (R. E. Watson) had noted that there appears to be a subtle fea- ture in the shape of the noble metal Fermi surface necks which has not been reproduced by either numeri- cal or analytic band descriptions to date. The following comment refers to this.] F. M. Mueller (Argonne National Lab.): Details of the noble metal Fermi surfaces will be affected by whether the potential outside the muffin tin sphere is kept con- stant or not. A non-constant potential will affect things because the belly orbit is largely of s-like symmetry and one would anticipate that it will sample the outside dif- ferently than the p-like levels. That is, the outside potential is different for states of odd symmetry than for states of even symmetry, and you go continuously from s-like to p-like states as you go around toward L. I think this might be the source of the Fermi surface deviation. It is something left out of the calculations u until now. J. Waber (Northwestern Univ.); The relativistic effects to which Dr. Keith Johnson alluded, are perhaps more important than may generally be realized. There is an interaction between two forces. The direct relativistic effect causes the s and p electrons to be attracted in toward the nucleus and concomitantly, there is a move- ment of the d bands outward radially because th nuclear charge is screened more effectively by the s and p electrons. I think that to a large extent this is on reason why one sees the d-bands in copper near th Fermi surface, but one finds them well below the Fermi level in silver. Finally, because of two relativistic ef fects, the d-bands are near the Fermi level in gold; the are driven up in energy by this indirect relativistic ef fect. In addition, they are separated and broadened b spin-orbit coupling. These comments will illustrate th importance of including relativistic effects in studyin metals with large atomic number, like gold. [1] Christensen, N. E., Phys. Stat. Sol. 31,635 (1969). 1. Introduction This work is concerned with the band structure hanges caused by compression and with the con- quent effects upon the components of the density of ates (and associated band charge) as characterized by he quantum number 6. The present report is prelimi- ary inasmuch as only twelve metals, all of them cubic, ve so far been investigated. They are: Li (boc), Cs ce), Ca (foc), Sr (foc), Ba (bcc), La (fec), Ce (fec), U cc), Pu (fec), Pb (fec), W (bcc) and Fe (bcc). Because the large number of graphs that would be required if mplete results were to be presented for all twelve of ese metals, only the results for the five underlined ove will be given in detail. However, comparisons ll be drawn between each of the others and the most propriate one of the five selected for detailed presen- ion, and the analyzed charge distributions for all will given in tables. Slater's augmented plane wave PW) method [1] was used in a self-consistent nner to obtain the bands. The present investigation not greatly concerned with details of the band struc- e in the vicinity of the Fermi surface, but rather with overall or gross features that are pertinent to the in- atomic interactions, elastic properties, and the elec- nic transitions which occur under compression. us, the nonrelativistic nature of the calculations is ely to be of only minor consequence except, perhaps, U and Pu where the spin-orbit splitting of the 5f ork performed under the auspices of the U.S. Atomic Energy Commission. Calculated Effects of Compression Upon the Band Structure and Density of States of Several Metals” E. A. Knnetko University of California, Los Alamos Scientific Laboratory, Los Alamos, New Mexico 87544 Energy bands were obtained self-consistently by the augmented plane wave method for the follow- ing metals: Li, Cs, Ca, Sr., Ba, La, Ce, U, Pu, W and Fe. The density of states and the electronic charge were resolved into S-, p-, d-, and f-like components. Under compression the charges associated with the higher values of £ increase, mainly at the expense of the s-like component, and a more compact overall distribution is thereby achieved. The present results indicate that such “electronic transitions” are of general occurrence and probably play a significant role in determining the compressibility of metals. Key words: Alkali and alkaline earth metals; augmented plane wave method (APW); cerium; com- pressibility; electronic density of states; lanthanides. levels is of the order of 0.1 Ry and the velocity and Dar- win corrections are far from negligible. Comparison of the present results for Pb with those of Louck's relativistic APW calculation [2] gives credence to this notion. There have been several investigations into the ef- fects of compression upon the electronic structure. Ham [3], using the quantum defect method, calculated bands at different interatomic distances for the alkali metals. Sternheimer [4], using Wigner-Seitz boundary conditions, numerically solved the Hartree-Fock equa- tion for Cs in an investigation of a possible s to d electronic transition thought to cause the isomorphic collapse observed at high pressure. Recently Berggren [5], using the statistical atom model, found increases in d-character resulting from compression of the metals K through V. Royce [6] made what is probably the most extensive investigation of the effects of compres- sion upon electronic structure; he compared the depen- dence of various calculated atomic orbital radii upon the atomic number, Z, with the Z dependence of cor- responding empirical radii at different stages of com- pression. The observed compressibilities were thereby correlated with the degree of stability of the electronic structure under pressure. In particular, the low com- pressibilities of the transition metals seemed to result from the stability of the d-orbitals. In the present work no attempt to obtain a quantita- tive correlation between the stability of the electronic bands and compressibility was made. Nevertheless, as 67 TABLE 1. Lattice constants, exchange scale factors and charge distributions as analyzed into s, p, d and f Components. Linear | Unit Charge inside APW sphere Plane wave charge outside APW sphere Total charge per atom Ex- Metal compres-| cell change and sion dimen- potential structurel 9% sion S p d f Total S D d f S D d f scale A factor Libc.c... 0.00 || 3.502 ().350 | ().294 || 0.015 ().001 || 0.340 (). 170 || 0.155 0.014 || 0.001 || 0.520 0.449 || 0.029 || 0.003 || 0.820 Libc.c... 5.0 3.326 ().336 || 0.302 ().015 0.001 || 0.346 || 0.17() (). 160 || 0.014 || 0.001 || 0.506 || 0.462 0.029 || 0.002 Cs bec...] 0.0 6.13 ().406 || (). 122 (). 107 | ().001 || 0.364 || 0.213 || 0.092 || 0.053 || 0.001 || 0.619 || 0.213 || 0.160 0.002 || 0.690 Cs bec...] 5.0 5.82 ().384 (). 115 (). 125 ().001 || 0.375 0.219 || 0.093 || 0.062 || 0.001 || 0.602 || 0.208 || 0.187 | ().003 Ca foc...] 0.0 5.565 (),635 | (). 419 ().397 | ().007 | ().545 ().214 || 0.212 (). 113 || 0.006 || 0.849 || 0.63] | 0.509 || 0.013 || 0.717 Ca foc...] 5.0 5.287 | ().596 || 0.371 (). 467 | ().007 | ().559 || 0.223 (). 199 (). 132 0.006 || 0.818 0.569 0.598 ().013 Sr foc . . . (). () 6.07.26 0.650 ().322 || 0.459 || 0.008 || 0.560 || 0.230 (). 178 || 0.1452| 0.007 || 0.879 0.500 0.604 || 0.015 0.703 Sr foc ...| 5. () 5.7690 (). 624 ().287 | ().512 ().008 || ().569 || 0.226 (). 168 || 0.168 || 0.007 || 0.850 || 0.455 || 0.680 0.015 Ba bec...] 0.0 5.010 || 0.490 0.081 0.691 0.014 0.763 || 0.371 0.044 0.287 0.014 |0.807 || 0.325 0.978 || 0.028 0.690 Ba boo... 15.0 || 4.008 || (). 133 0.062 | 1.083 || 0.059 || 0.654 0.105 || 0.025 | 0.48] | 0.043 || 0.238 0.088 | 1.565 || 0.103 6 La foc. 0.0 5.285 ().286 (). 113 | 1.34] | ().547 | (). 669 (). 161 0.066 || 0.389 || 0.053 || 0.447 | (). 179 | 1.730 || 0.600 || 0.693 3 La foc. 5.0 5.021 0.230 0.126 | 1.657 || 0.384 || 0.65l 0.098 || 0.059 0.462 0.032 || 0.328 0.185 2.120 0.416 'y Ce foc. 0.0 5. 1612 ().28] | (). 103 | 1.294 | 1.653 (). 662 (). 155 || 0.057 || 0.375 0.075 0.436 (). 159 | 1.669 | 1.730 || 0.696 cy' Ce foc. 10.0 4.660 (). 150 (). 109 | 1.673 | 1.280 0.775 0.09() 0.04] 0.547 || 0.120 0.240 || 0.150 2.210 | 1.400 y U bec. 0.0 3.524 ().057 | ().079 || |. 159 || 3.757 || 0.895 ().040 || 0.009 || 0.539 || 0.318 0.097 | ().088 | 1.695 || 4,092 || 0.690 y U bec. 5.0 | 3.418 || 0.054 || 0.108 | 1.060 | 3.829 0.923 || 0.034 0.005 || 0.529 || 0.340 |0.088 || 0.113 | 1.589 4,169 6 Pu foc. 0.0 4.637 ().057 | ().068 ().847 | 6.309 || 0.624 || 0.03] | 0.009 || 0.325 | 0.257 | ().088 0.078 || 1 || 72 | 6.567 || 0.690 6 Pu foc. 5.0 4.405 || 0.059 ().053 ().81] | 6.407 || 0.575 0.037 || 0.013 || 0.304 || 0.22] | 0.095 || 0.066 | 1. 115 6.627 Pb foc...] ().() 4.9057| 1.428 | 1.550 ().224 || 0.063 ().705 || (). 164 || 0.395 || 0.103 || 0.043 | 1.592 | 1.945 || 0.350 || 0. 106 || 0.69 Pb foc...] 5.0 4.6603| ] .351 | 1.513 ().256 || 0.076 ().795 || 0.183 || 0.42] | 0.14() 0.05] | 1.53() | 1.934 || 0.396 (). 127 W bec. ...| 0.0 3.15 ().350 ().352 || 3.914 || 0.069 | 1.3643| 0.232 || 0.247 || 0.816 || 0.069 |0.582 || 0.600 || 4.730 (). 138 || 0.70 W bec...] 5.0 2.84 ().306 ().322 || 3.783 0.106 || | .46() 0.2] 1 || 0, 188 || 0.956 (). 104 || 0.517 | 0.510 || 4.730 0.210 Fe bec...| 0.0 2.8606 (). 397 || 0.375 6.286 || 0.03 || || 0.902 || 0.200 ().243 || 0.434. 0.033 0.597 || 0.618 6.719 0.063 || 0.73 Fe bc.c...] 5.0 2.7] 78 ().378 ().379 || 6.23() ().036 || 0.967 || 0.203 || 0.254 || 0.47] 0.039 || 0.580 || 0.632 6.701 || 0.075 will be seen, considerable insight concerning the mechanisms underlying compressibility, beyond what has been provided by the work of Royce and particu- larly with respect to metals of the rare-earth type, seems to have been gained. 2. Method The calculations for either 256 or 128 k-vectors in the reduced Brillouin zone, depending upon whether the structure was foc or bec, were made through use of a modified version of an APW program developed by Wood [7] and adapted to the self-consistent field method by DeCicco [8]. The self-consistency criterion was 0.002 Ry. For most of the metals the core char was held fixed through the calculations, but in met where the band charge contains about a tenth, or mo of an f electron, it was found imperative to use a “sof core. The charge distribution in such a core is atomi and the configuration used in generating it is deriv from the analyzed charge distribution appearing in t previous iteration (see below). This treatment is nec sitated by the sensitivity of the core states as well as the band states to even slight charges in f-charact The sensitivity is also illustrated in another way. generating a potential for the (N + 1)-st iteration, o about 10% of the potential from the N-th iteration c be used without causing instability where f-electr CS °o = 5.82 CS °o = 6.13 i : : Energy bands along [002] direction for bec Cs in the normal state and under 5% linear compression. Energies are not absolute. FIGURE 1. y-Ce do=5.1612 O.5O 3. i 2 5' ; FIGURE 2: Energy bands along [002] direction for foc y-Ce and o'-Ce, for which lattice constants are 5.16 A and 4.66 A, respectively. Energies are not absolute. re involved. In most metals a feedback of 50% causes o difficulty. The number of iterations necessary to chieve self-consistency is variable, from three to as any as ten, and depends upon several factors which ill not be mentioned here. Starting potentials were obtained from a superposi- ion of free atomic charge distributions by using the Löwdin technique [9]; the distributions were calculated rom Hartree-Fock-Slater solutions obtained by using variant of the program written by Herman and Skill- man [10]. The local exchange potential was a scaled ‘ersion of the Slater approximation [11], in which the cale factor, O. was used as a variational parameter in inimizing the total energy of the free atom. For all toms through Lw, O lies between 1.0 and about 0.68 12]. The values of O. used for the metals in this study re given in table 1. Even in free atoms the approximation of the local change by a single-parameter expression is ques- O25 ; 5 * 5 t 5 㺠5 FF > O.O E. H. •- >- N F | É | - 4' : I # do-4.90 | \l a o-466 4 Lil -O.25H - - | - | | | T A X T A X FIGURE 3. Energy bands along [002] direction for foc Pb in the normal state and under 5% linear compression. Energies are not absolute. i FIGURE 4. Energy bands along [002] direction for 6-Pu in the normal state and under 5% linear compression. Energies are not absolute. tionable. In a crystal the use of such an exchange, which is optimized for the free atom, is subject to even greater criticism because of the moderate sensitivity of O to changes in potential near the “edge” of the atom when the latter self-energy correction [13] is switched on or off [12]. Nevertheless, its use seems justifiable if only because of its improved character; the virial theorem is satisfied in the case of the free atom [14] for a value of O. only slightly smaller than that which minimized the Hartree-Fock total energy. The use of the original Slater exchange (O. = 1.0) causes the 4f band in both La and Ce to lie several tenths of a Ry below the bottom of the 6s band whereas the optimized exchange places this band at a physically realistic posi- tion, i.e., above the 6s band. Since the APW’s used have associated values of £ as high as 12, it is possible to resolve the band charge, and the density of states as well, into components cor- responding to different values of Č. The plane wave 69 8-Pu co-4405 TS, 2 5 2 5 ; O O FIGURE 5. Energy bands along [002] direction for bec Fe in the normal state and under 5% linear compression. Energies are not absolute. .5 |- .O K- O. 5H TS-4- O 2-t - O.2 O -O.2 O.2 RYDBERGS FIGURE 6. Density of states in toto and as resolved into s, p, and d components for charge inside the APW sphere, shown for the normal and compressed bec Cs crystal. y-Ce a'-Ce 3H- o,-5.1612 do-466 - O.2 O.4 O RYDBERGS FIGURE 7. Density of states in toto and as resolved into s, p, d, and f components for charge inside the APW sphere, shown for normal (y) and compressed (o") fcc Ce. O 6 O O ſ: % O 5 7 — O. 4– O. 2H- º ~! N. y ...” | 4 t f z - ets -----S-T-Yi ſe -O.4 O 9. 8 O.4 –O. RYDBERGS FIGURE 8. Density of states in toto and as resolved into s, p, d, an f components for charge inside the APW sphere, shown for norma and compressed Pb. * | T | II- | | | H | * # Fe Fe 3 is do-2,8606 oc-2.7177 º ſ *N N (ſ) | ſ s: go 10 | (ſ) | § u_0.5 d |-4 O >- FF E FF % .” S 9. --> Z-S-2--~4. ~~~ g 0-63 -O.2 O -O4 -O.2 O RYDBERGS FIGURE 9. Density of states in toto and as resolved into s, p, and components for bec Fe in normal and compressed crystals. charge can also be analyzed as to its s, p, d and character, as will be discussed later. Smoothing of the density of states curves wa achieved by averaging over five histograms, the origi of each having been shifted slightly relative to those the others. 3. Results The energy bands for both the normal lattice co stants and for those decreased as indicated are show along the [002] direction in figures 1 through 5, for C Ba, Pb, Fe and Pu, respectively. The correspondi densities of states in toto as well as resolved into s, p, f and plane wave components are given in figures through 9. Table 1 contains the resolved band charg for all the metals listed. In addition, table 1 includes t s, p, d and f content for the plane wave charges. T 70 analysis was achieved by assuming that the Č components in the plane waves are mixed in the propor- tions as are the charge distributions on the APW sphere surface, where the external plane waves are joined onto the interior wave functions. The effects of compression upon the band charges, as shown in table 1, may be summarized as follows: (1) The components for which 4 × 0 generally grow at the expense of the s-like charge, the d component gaining the most except in Li, U and Pu; (2) In La and Cef character is weakened while in U and Pu it is strengthened; (3) Where thef character is rather trivial, i.e., amounts of less than about 0.1 electron per atom, it remains so; (4) The total d character in W and Fe is very stable as indicated by the small absolute changes. The density of states in figures 6 through 9 show how the total as well as the resolved charges are distributed in energy for both normal and compressed states. The densities shown for the s, p, d and f components pertain only to the wave functions inside the sphere inasmuch as it was not possible to analyze the plane wave func- tions. For Cs we see in figure 6 that the d character falls off rapidly below the Fermi energy. This reflects the in- teraction between the 6s and 5d bands which is partly due to the symmetry requirement that the bands ac- tually touch at point H(002) as well as at other points on he zone boundary. The increase in d character under compression is caused by the slight band broadening pparent in figure 1. Referring to Li in table 1, it is clear that, because the nteraction is between the 2s and 2p bands, the very mall d character is unchanged by the compression; ere it is the p-like charge which gains. The changes alculated for the alkaline earths Ca and Sr are omewhat similar to that occurring in Cs. In both of hese metals there is a comparatively large d 3omponent which is enhanced by pressure, the mounts of the other types of charge being roughly imilar to the corresponding ones in the alkalis. In Ba he f-character becomes significant at a much higher :ompression, due to the broadening of the 4f band. In La the 4f band lies between the 6s and 5d bands ot shown), the symmetry requirement at (002) causing trong interactions to occur between all three bands. onsequently there is over a half of an f electron in the and charge. Compression causes the f-like charge to iminish. A similar effect occurs in Ce (fig. 2). Here the and 6s bands are closer together than in La and there no double degeneracy at (002) to tie them together. s shown, when Ce is compressed the 5d band roadens so as to strongly perturb the lowest member of the 4f band and by distorting this branch, impart d- character at the expense off-character. The situation with respect to band interaction in La is similar [15]. It is possible that this mechanism will remain valid even in the presence of magnetic interactions which are not included here. On the other hand, in both U and Pu the f-character which is very strong increases under compression. The bands for Pu are shown in figure 5. Fe has a very stable band structure, as is quite obvi- ous in figures 4 and 8 and in table 1. A similar result is found for W. In both of these metals, most of the charge is d-like and very stable. The relative changes in the other charge components with pressure follows the general pattern. - Pb under compression gains some d and f-character at the expense of the very large s and p-character. The s and p components in the normal metal (including that in the plane waves) are only slightly smaller than in the free atom. Consequently, self-consistency was reached very rapidly for this metal. The instability which does occur seems to be attributable to the proximity of the 6d and 5f bands. At a linear compression of 15% (not shown), the density of states is drastically altered and looks much like a serrated parabola, as might be found in a metal having free electrons [16]. 4. Discussion and Conclusions As a metal is squeezed, the charge distribution in each atomic cell necessarily becomes more compact. This occurs in two ways: (1) by simple distortion of the radial wave functions, and (2) by redistribution of charge among the various component distributions cor- responding to different values of the quantum numbers n and Č. The results presented herein clearly demon- strate the general features of charge redistribution under compression. How each individual component is affected is determined by details of the band structure. In general, those bands which contain the valence elec- trons and those which lie close enough in energy to per- turb them differ by no more than 1 in their values of n + 6. Thus the degree of compactness of the cor- responding atomic-like charge distributions increases with 6. In Ce, for example, the values of n + 4 for the 4f, 5d, and 6s bands are 7, 7, and 6, respectively. In metals beginning at about K, the presence of d- bands begins to be felt. Such bands are either occupied to some extent or act perturbatively to produce d- character in other bands. Probably the d-bands are responsible for the much greater compressibilities ob- served for K, Rb and Cs than for Li and Na [17]. In Pb, one of the most compressible metals, the high distorta- 71 TABLE 2. Calculated exchange splittings and relativistic corrections for atoms of Ce and U. Ce(6so.) (6sſ?) (4f6)* U(7so) (7sp) (6dB) (5fg)” 4f 6S 5f 6d 7s Spin-orbit splitting (Ry)"............................................... ().0295 ().0914 0.0549 Velocity correction (Ry)"............................................... + .030() — ().0839 — ..] I 27 — 0662 – 0.2432 Darwin correction (Ry)"................................................ — .0011 — ()465 — .0022 — .0007 +. 1328 Exchange splitting (Ry)".............................................. — . 1540 — 0104 — .2019 — , 1297 — ,05332 * Perturbation calculations by Herman and Skillman [10]. bility of the very plentiful s-like charge plus the availa- bility of d and f-like states probably act together to produce the large compressibility. Only two of the transition metals, Fe and W, have so far been considered. However, the stability of their band structures is probably characteristic of transition metals, apparently from the large d- component in the charge distribution, and is illustrated in their rather small compressibilities. This stability probably results from the difficulty of significantly al- tering charge distributions that are already quite com- pact due to their large d-character. The alteration off-character by compression requires special consideration. In metals lying below La the f- character is very small, of the order of 0.01 electron per atom or less, and is usually increased rather trivially by compression. Inasmuch as there are no f-bands in- volved in these metals, this behavior merely reflects the use of APW’s having associated values of £ as high as 12. The behavior of the two lanthanides, La and Ce, is diametrically opposed to that of the actinides, U and Pu, regarding the way f. character is altered by com- pression. Considering first the case of 4f electrons, it seems paradoxical that, under compression, charge flows out of a very compact f-like distribution into a more extended d-like one. This is deceptive, however. stemming The effective radius of the atom is quite sensitive to the amount of 4f charge. As the charge is expelled, the atom shrinks. The empirical radii of rare-earth atoms are roughly 20% greater in the divalent than in the trivalent form, because of the difference in screening by 4f electrons. The effect of the change in f-character upon the atomic radius, as calculated for Ce metal, is large enough to account for a major part of the observed compressibility. This was shown by calculating for a " Calculated by the author in Hartree-Fock-Slater approximation with optimized statistical exchange potential [12]. free atom, the effect of a change in configuration from (4f).73 (5d).57 (6s),44 (6p)" it to (4f),40 (5al).21 (6s).24 (6p)"" which correspond to the calculated charge dis- tributions in table 1. The radius of the outermost 5d maximum is reduced at the rate of 14.8% perf-electron removed. Thus, the band structural changes and the high compressibilities of La and Ce are apparently caused by the softness of the atom, which in turn stems from instability of the 4f orbitals under compression. To extend this mechanism to the other lanthanides is quite tempting, but is inadvisable inasmuch as little is known concerning band structures in those metals. The results presented here for La and Ce are only tentative, inasmuch as exchange splitting and relativistic effects are not included. As table 2 shows the exchange splitting of the 4f state in the Ce atom is of far more consequence than the relativistic cor rections. Assuming that all of the corrections show will be of similar magnitude in the metal, there shoul be a separation between the 4fo and 4f6 bands of abou 1 eV. The perturbing, or hybridizing, effect of the 5 band should be similar to that indicated by the presen results for Ce in spite of the large exchange splitting ef fect, though the band change distributions will b somewhat different. U and Pu gain f-character under compression, as i indicated by these calculations. The relative change i small and the d-character loses correspondingly. In ac tinides, at least through Am, the 5f orbital is not so we buried as the 4f orbital in a lanthanide. This is illu trated as follows: If we denote by RS the APW spher radius (approximately equal to the effective atomi radius in a metal) and by rf the radius of the calculate principal maximum of the f-orbital, RS/rf is 4.87 for C 2.90 for U, and 3.2 for Pu. In U and Pu the 6d and principal maxima lie outside the APW sphere, where in Ce the 5d orbital has its maximum just inside the sphere. Consequently, (1) the 5f screening in actinides is not nearly so effective as that of the 4f charge in the lanthanides, and (2) the Coulomb interaction energy in- volved in squeezing the 6d and 7s charge into an atomic cell is so large that the corresponding bands lie well above the 5f bands in both metals. These calculations indicate that the 6d and 7s interactions with the 5f band diminish under compression because both the 6d and the 7s rise relative to the 5f. The compressibility of 6-Pu is about three times as large as that of y-U [18], so that this property does not seem to reflect band structural changes here as well as in many other metals. The effects of exchange and relativity in the lower ac- tinides U and Pu will be somewhat different than in Ce, inasmuch as the 5f band is several times broader than the 4f band in Ce and the spin-orbit splitting is also much greater (table 2). The actinides beyond Am are chemically similar to the lanthanides, presumably because the 5f shell is more deeply buried in them than in the lower actinides. Thus, one might anticipate that the calculated compressive effects among the higher actinides would resemble those obtained for La and Ce. These results and conclusions compare with those of Royce [6] as follows: (1) In both cases the transition metals show a high degree of stability under compres- sion; (2) The present work suggests that the f-> d rather than the d -> f transitions are probably responsi- ble for the high compressibilities of lanthanide metals; (3) Rather than an absence of 5f electrons in U and Pu, as suggested by Royce, the present results show large 5f band charges, which, because of their larger spatial extension relative to that of the 4f charge in the lantha- nides, are actually augmented under compression at the expense of d charge. The present work also indicates that charge redis- ribution under compression is of general occurrence and must be considered in conjunction with orbital sta- bility in discussing compressibility. 5. Acknowledgments The author is grateful to the following: J. H. Wood and A. M. Boring for enlightening discussions, E. C. Snow for generously making available his recent im- provements in APW programming, Mrs. A. Lindstrom for her assistance in the calculational work, and to Wil- liam N. Miner for editing the manuscript. 6. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] Slater, J. C., Phys. Rev. 51,846 (1937). Loucks, T. L., Phys. Rev. Letters 14, 1072 (1965). Ham, F. S., Phys. Rev. 128, 82 (1962). Sternheimer, R., Phys. Rev. 78,235 (1950). Berggren, K. F., Phys. Letters 27A, 125 (1968). Royce, E. B., Phys. Rev. 164,929 (1967). Wood, J. H., Phys. Rev. 126, 517 (1962). De Cicco, P. D., Phys. Rev. 153,931 (1967). Löwdin, P. O., Phil. Mag. 5, 1 (1956). Herman, F., and Skillman, S., Atomic Structure Calculations (Prentice-Hall Inc., Englewood Cliffs, N.J., 1963). Slater, J. C., Phys. Rev. 81,385 (1951). Kmetko, E. A., Phys. Rev. (to be published 1970). Latter, R., Phys. Rev. 99,510 (1955). Berrondo, M., and Goscinski, O., Quantum Theory Group, University of Uppsala, Report No. 198 (1967), Phys. Rev. (to be published 1969). Kmetko, E. A., Bull. Am. Phys. Soc., Ser. II, 14, 358 (1969). Kmetko, E. A., to be published. See for example K. A. Gschneidner, Jr., Solid State Physics 16, 275 (Academic Press, N.Y., 1964). Private communication of unpublished results by J. F. Andrew and S. Marsh of this laboratory. [11] [12] [13] [14] [15] [16] [17] [18] 73 Discussion on “Calculated Effect of Compression Upon the Band Structure and Density of States of Several Metals” by E. A. Kmetko (Los Alamos Scientific Laboratory) D. J. Fabian (Univ. of Strathclyde); Willens and co- workers have studied the effect of pressure on soft x- ray emission from copper. They conclude that the variation of deformation potential throughout the band will give a particular effect at certain critical points, such as the Fermi surface and Van Hove singularities. Would Dr. Kmetko like to comment on this? E. A. Kmetko (Los Alamos Scientific Lab.): I believe the effect could be quite serious at the Fermi level. However, I was specifically directing my attention to the overall change in hybridization which occurs under pressure. The main effect there on copper would be similar to iron. Very little overall change would occur in dehybridization. 74 CHAIRMEN: E. A. Stern . R. Cuthill J RAPPoRTEUR. G. W. Pratt, Jr. Optical Properties and Electronic Density of States” M. Cardond” Brown University, Providence, Rhode Island and DESY, Hamburg, Germany The fundamental absorption spectrum of a solid yields information about critical points in the opti- cal density of states. This information can be used to adjust parameters of the band structure. Once the adjusted band structure is known, the optical properties and the density of states can be generated by numerical integration. We review in this paper the parametrization techniques used for obtaining band structures suitable for density of states calculations. The calculated optical constants are compared with experimental results. The energy derivative of these optical constants is discussed in connection with results of modulated reflectance measurements. It is also shown that information about density of empty states can be obtained from optical experiments involving excitation from deep core levels to the conduction band. absorption. 1. Optical Properties and One-Electron Density of States The optical behavior of semiconductors and insula- tors in the near infrared, visible, and ultraviolet is determined by electronic interband transitions. An ad- ditional intraband or free electron contribution to the optical properties has to be considered for metals. We shall discuss here the relationship between the inter- band contribution and the density of states. The inter- band contribution to the imaginary part of the dielectric constant can be written as (in atomic units, h = 1, m = l, e = 1): ––– F" ei (a)) 4To |. |Voor dS. (1) where weſ=0o-op is the difference in energy between he empty bands (e) and the filled bands (f). The spin ultiplicity must be included explicitly in eq (1). The *An invited paper presented at the 3d Materials Research Symposium, Electronic Density ºf States, November 3–6, 1969, Gaithersburg, Md. - 'Supported by the National Science Foundation and the Army Research Office, Durham. * John Simon Guggenheim Foundation Fellow. A detailed comparison of the calculated one-electron optical line shapes with experiment reveals deviations which can be interpreted as exciton effects. The accumulating experimental evidence point- ing in this direction is reviewed together with the existing theory of these effects. A number of simple models for the complicated interband density of states of an insulator have been proposed. We review in particular the Penn model, which can be used to account for response functions at zero frequency, and the parabolic model, which can be used to account for the dispersion of response functions in the immediate vicinity of the fundamental absorption edge. Key words: Critical points; density of states; dielectric constant; modulated reflectance; optical oscillator strength tensor Fºſ is related to the matrix ele- ments of p through Feſ=2< flple -- eſp|f> oef-'. The Bloch functions are normalized over unit volume. Degenerate statistics have been assumed in eq (1) and spatial dispersion effects have been neglected. It is customary to take the slowly varying oscillator strength out of the integral sign in eq (1) and thus write: e(0)=#| FN (o) (2) where F is an average oscillator strength and Nº the combined optical density of states. Structure in ei(0) (eq.(1)) appears in the neighborhood of critical points, where V, oof-0. Such critical points can be localized in a small region of k space or can ex- tend over large portions of the Brillouin zone over which filled and empty bands are parallel (sometimes only nearly parallel). Once the critical points which cor- respond to observed optical structure are identified in terms of the band structure through various devious and sometimes dubious arguments, their energies can 77 be used to adjust parameters of semiempirical band structure calculations. Four different parametric techniques of calculating band structures have been used for this purpose: the empirical pseudopotential method (EPM) [1], the k. p. method [2], the Fourier expansion technique (FE) [3], and the adjustable orthogonalized plane waves method (AOPW) [4]. Once reasonably reliable band structures are known it is important to calculate from them the imaginary part of the dielectric constant ei(a) and to compare it with experimental results so as to confirm or disprove the initial tentative assignment of critical points and thus the accuracy of the band structures. Rich struc- ture is obtained in both experimental and calculated spectra and hence a rather stringent test of the accura- cy of the available theoretical band structure is in prin- ciple possible. In order to calculate numerically the integral of eq (1) it is necessary to sample eigenvalues and eigenfunc- tions at a large number of points in the Brillouin zone. The amount of computer time required for solving the band structure problem with first-principles methods (OPW, APW, KKR) at a general point of the Brillouin zone makes such methods impractical for evaluating eq (1). The parametric methods (EPM, k p, FE, but not AOPW) require only the diagonalization of a small matrix (typically 30 × 30) and hence it is possible to sample the band structure at about 1000 points with only a few hours of computer time. Cubic materials, in particular those with Ta, O, and Oh point groups, are simple in this respect: symmetry reduces the sampling required for the evaluation of eq (1) to only 1/48 of the Brillouin zone. Hexagonal and tetragonal materials have relatively larger irreducible zones and hence a larger number of sampling points is necessary if the resolution of the calculation is not to suffer. Once the band structure problem has been solved for all points of a reasonably tight regular mesh, the bands and matrix elements at arbitrary points can be obtained by means of linear or quadratic interpolation. The method of Gilat and coworkers [5] has become rather popular for the numerical evaluation of eq (1) [4,6]. In the case of a cubic material the Brillouin zone is divided into a cubic mesh and the band structure problem solved at the center of these cubes (sometimes a finer mesh is generated by quadratic interpolation from the coarser mesh [6]). Within each cube of the mesh the bands are linearily interpolated and approxi- mated by their tangent planes. The areas of constant energy plane within each cube corresponding to a given aleſ are added after multiplying them by the correspond- ing oscillator strength and thus the integral of eq (1) is obtained. The real part of the dielectric constant e, can be ob- tained from ei by using the Kramers-Kronig relations. It is also possible to obtain e, and e, simultaneously by calculating the integral: Feſoef co)-1-#| || ſº-win); ". (3) with m) small and positive. For m -> -- o the imaginary part of eq (3) coincides with eq (1). Equation (3) can be evaluated with a Monte Carlo technique. Points aré generated at random in k space within the Brillouin zone and the average value of the integrand for these points calculated. The process can be interrupted when reasonable convergence as a function of the number of random points is achieved [7,8]. We show in figure 1 the results of a calculation of ei from the k p band structure of InAs with the method of Gilat and Raubenheimer [6]. The band structure problem, including spin-orbit effects, was solved at about 200 points of the reduced zone (1/48 of the BZ). We have indicated in this figure the symmetry of the critical points (or of the approximate regions of space) where the structure in e, originates. The experimental ei spectrum, as obtained from the Kramers-Kronig anal- ysis of the normal incidence reflectivity [9], is also shown. The agreement between calculated and experi- mental spectra is good, with regards to both position and strength of the observed structure, with the excep- – 25 (U#v, Užw-U%,Uže) I º THEORY n As 2 O H (Tav-Tsc) ſº — — — EXP, El + A | Hºlº-tºe) U |5 H. º ! Lev -éc s r |-4v., L5w-l-4c, L5c J’ Eot Ao | (Lev Lac, L5c) | O (T7v-Tsc) | —H(A$v-Ağc) | (A4v,A5v Age- (T7v - T7c) 5 – Fo / | !. - (T8v - T7c) E. & S- (A 6v - Asc) (T7 v-Tsc) '(X7 v- X7c) O | | | | | | O 2.O 4,O 6,O 8.0 ENERGY (eV) FIGURE 1. Imaginary part of the dielectric constant of In/As as calculated from the k p method ( experimentally (...) [9]. The group theoretical symmetry assign- ments were made with the help of the calculated isoenergy plots. ) [6] and as determine 78 tion of the position of the E2 peak. This is to be at- tributed to an improper assignment of the E2 peak when the 6 adjustable band structure parameters were deter- mined. The E2 peak had been attributed, following the tradition, to an X critical point while it is actually due to an extended region of k space centered around the U points [8]. It should be a simple matter to readjust the band structure parameters to lower the energy of the calculated E2 peak by about 0.5 eV; in view of the large amount of computer time required to recalculate the energy bands this has not been done. The structure calculated around 6 eV, due mostly to spin-orbit splitting of the L3 levels, has not yet been observed experimentally. The conventional experimental determination of ei from normal incidence reflection data [9] suffers from considerable inaccuracy: to the experimental error produced by possible improper surface treatment and contamination one has to add the uncertainty in the high-energy extrapolation of the experimental data required for the Kramers-Kronig analysis. Some of these difficulties are avoided by comparing the calculated reflectivity spectra (obtained from e with Fresnel's equation) with the experimental results. This is done in figure 2 for GaSb: the experimental data [10] have not been Kramers-Kronig analyzed because of the mall range of the energy scale. Two calculated spectra ave been plotted in this figure: one obtained from the p band structure [6] and the other obtained from a O 9 GO Sb — — PS EU DO POTENT | A L k p • e • e F XP E R | MENT O 8 H (29.7 °K.) -- | * GURE 2, Reflectivity of GaSb calculated from the k p [6] and rom a pseudopotential band structure [11]. Also, experimental reflectivity [10]. non-local pseudopotential calculation with 14 adjusta- ble parameters [11]. The discrepancy between experi- mental and calculated curves at high energy, a common feature of many zincblende-type materials [12], has two origins: the measured reflectivity should be low because of increased diffuse reflectance at small wavelengths while the calculated one should be high because of the finite number of bands included in the calculation. In this region where er—l is small, the con- tribution to e, of transitions not included should lower the calculated reflectivity. During the past few years a lot of activity has been devoted to the measurement and analysis of differential reflection spectra obtained with modulation techniques [13-15]. The wavelength (or photon energy) derivative spectra [14] should permit an accurate analysis of the line shapes of the spectra of figures 1 and 2. We show in figure 3 the temperature modulated reflection spec- trum (thermoreflectance) of GaSb [15]: it has been shown that for the III–V materials [15] this spectrum is very similar to the photon energy derivative spec- trum, difficult to obtain experimentally. The cor- responding photon energy derivative spectrum ob- tained from the calculation of figure 2 is also shown in figure 3. The calculated and experimental shapes of the E1, E1 + A peaks show discrepancies of the type at- tributed in section 2 to exciton interaction. Derivative spectra for other germanium- and zincblende-type materials have been calculated by Walter and Cohen [12] and by Higginbotham [16]. The methods to calculate band structures from first principles, without or with only a few adjustable parameters (one [17] or three [4]) have recently achieved considerable success. However the calcula- — THERMOREF. 80°K — — CALCULATED |.O H. O. 5 º- O - — O.5 k- FIGURE 3. Measured thermoreflectance spectrum of GaSb [15] compared with the energy derivative of the spectrum of figure 2 [16]. 79 tion of energy bands at one general point of the BZ requires a lot of time so as to make density of states cal- culations prohibitive. Moreover, the evaluation of the matrix elements required for eq (1) is difficult with first principles techniques. It is nevertheless possible to use first principles calculations at a few high-symmetry points of the Brillouin zone to adjust the parameters of semiempirical band structures from which the large number of sampling points required for the evaluation of eq (1) can be obtained with relative ease. The k p technique has proved particularly useful in this respect [2,18,191. Matrix elements of p can be easily evaluated from the eigenvectors in the k p representation. Spin- orbit interaction can also be easily included. This k p procedure has been applied to the relativistic OPW band structure calculated by Herman and Van Dyke for gray tin [19]. Figure 4 shows the reflectivity of gray tin calculated by this procedure with the method of Gilat and Raubenheimer together with experimental results [20]. Comparison with other experimental results for the germanium family suggests that the high-energy end of the measured spectrum is too low, probably due to surface imperfections in the delicate crystals, grown from mercury solution, which were used for this experi- Inent. The k. p fitting procedure has also been applied to a first principles relativistic APW calculation of the band structure of PbTe by Buss and Parada [7]. Figure 5 shows the reflectivity of PbTe obtained by this method O.6 O. 5 O 4 \ O.3 H. J | | | | | | | | with a Monte Carlo sampling technique and figure 6 the absorption coefficient, both compared with experimen- tal data [7,21,22]. In both cases the semiquantitative agreement between experimental and calculated data is remarkably good in view of the absence of the ad- justable parameters. The calculated reflectivity is, at high energies, considerably higher than the experimen- tal one, as discussed earlier for other materials. The El peak of the experimental reflectivity spectrum appears split in the calculated spectrum, possibly because of in- accuracies in the first-principles band structure. The calculated E1 structure appears due mostly to transi. tions along the X direction. The experimental El structure has been assigned [23] to the lowest gap along X. The calculated E2 peak corresponds to an ex- tended region of the BZ without definite symmetry, as inferred from electroreflectance measurements [23]. FIGURE 5. Reflectivity of PbTe calculated from the APW-k ‘p ban |.O 2.O 3.O 4.O 5.O ENERGY (eV) FIGURE 4. Reflectivity of gray tin calculated from a first principles OPW band structure fitted with the k p method [19]. Also experimental results [20]. FIGURE 6. Absorption coefficient PbTe calculated from the AP CA L CULAT | ON C A R D ON A 8: G R E E N AWAY O. 3 H. • * * * * Z E MEL, JEN SEN 8. \ - S C HOO LA R | | | | O | 2 3 4 5 eV structure [7], compared with experimental results [21]. F loº E O 2 / `ss -- ~~ C / – ~ `ss H 6 `ss O. O.5 × O - / ` - Or / — CAL CULAT |ON * > 9 — — — — CARD ONA 8, GREEN AWAY CO —x — SCAN LON <ſ —l | | | 4 |--|-- | 2 3 eV k p band structure, compared with experimental results [21, 22]. 80 We have so far discussed optical constants for cubic materials. While calculations for materials with lower symmetry require more computer time, one has the extra reward of being able to predict the experimentally observed anisotropy. Figure 7 shows the two principal components of ei for trigonal Se as calculated by San- drock [24] from the pseudopotential band structure. The similarity between calculated and experimental results [25], also shown in figure 7, is especially re- markable in view of the method used to determine the pseudopotential parameters: they were determined from the pseudopotential parameters required to fit the optical structure of ZnSe. Only a small adjustment was performed so as to bring the calculated fundamental gap (1.4 eV) into agreement with the experimental one (2.0 eV). The dielectric constant of antimony (trigonal) for the ordinary and the extraordinary ray has also been calculated by a similar procedure [26]. The reasonable agreement obtained between experi- mental and calculated optical constants suggests the use of the corresponding band structure to determine the individual density of states D(@): the main work, that of diagonalizing the Hamiltonian at a large number of points, has already been done. The programs required to calculate individual density of states are very similar to those used for the evaluation of eq (1): oeſ must be replaced by the single band energies and 3OH 25 H. 2OH- GURE 7. Imaginary part of the dielectric constant of trigonal selenium for both principal directions of polarization of the electric ield vector E as calculated from the pseudopotential band struc- ture (histograms) [24] and as determined experimentally [25]. 417–156 O - 71 - 7 Feſ must be removed. As an example we show in figure 8 the individual density of states of the 3 highest valence bands (six including spin) and the 3 lowest con- duction bands of gray tin [19]. Direct information about the individual density of states can be obtained by a number of methods discussed in this conference. We mention, in particular, optical techniques involving transitions from deep core levels to the conduction band or from the valence band to temporarily empty core levels (soft x-ray emission) [27]. If the sometimes questionable assumption of constant matrix elements is made, the corresponding spectra represent the con- duction (for absorption spectra) and the valence (for emission spectra) density of states because of the small width of the core bands. We show in figure 9 the densi- | O H. VALENCE BANDS |--CONDUCTION BANDS –- Tº KSV S O S. •-ºx Xsv X- L; 25v g | Lev § O 5 H E Asc º, 3 C) X5 La.sv Lic O ——1–1–1– H–––– –6.O –4.O –2O O 2.O 4.O 6.O ENERGY (eV) FIGURE 8. Individual density of states for gray tin, obtained from the OPW-k p band structure [19]. The top of the valence band is at 0 el'. The lowest valence band is not included. - - - - - - D calculated (Ge) D Calculated [Ga Sb) Ei (02 measured i 0. l, £ . an –03 i 0. 5 }- // * * A w / - 0. 2 # - * *s / - 0.1 0 0 —ew—- FIGURE 9. Conduction density of states calculate for Ge [4] and for GaSb [6] together with the function eia)” obtained from experimental data in the vacuum uv [28] (the horizontal scale for the eia)” curve has been shifted by 29.5 eV). ty of states of the conduction band of Ge calculated by Herman, et al. [4] and the corresponding density of states for GaSb as obtained by the k p method [6]. The densities of states for both materials are very simi- lar because of the similarity of their band structures. We also show in figure 9 the quantity eia)” obtained by Feuerbacher et al. [28] for Ge in the region of the M4.5 edge. The origin of energies has been shifted so as to make a comparison with the conduction density of states possible: eia)” should be proportional to D(@) under the assumption of constant matrix elements of p. While the rich structure of the calculated density of states is not seen in the eia)” curve, this curve is reproduced quite well if the density of states is broadened so as to remove the fine structure. The required lifetime broadening of about 1 eV is not un- reasonable for the M4.5 transitions. Using eq (2) with Nd replaced by the conduction density of states we obtain an average oscillator strength at the maximum of eja)*F = 0.15. This oscillator strength corresponds to the 20 4d electrons per unit cell and hence it should be divided by 20 to obtain the average oscillator strength per d- band. If one reasons that the transitions from 10 of the 20 d bands to a given conduction band are forbidden because of the spin flip involved while transitions from 5 of these 10 bands are forbidden or nearly forbidden by parity, one finds for the average oscillator strength of each one of the 5 allowed bands F = 0.03, which cor- responds to a matrix element of p = 0.13 (in atomic units): this value is quite reasonable in view of the fact that the typical valence-conduction matrix element is 0.6. The small value of this matrix element explains why the d core electrons are negligible in the k p analysis of the valence and conduction masses. 2. Exciton Effects We have devoted section 1 to a comparison of experi- mental optical spectra with calculations based on the one-electron band structure. Exciton effects, i.e. the final state Coulomb interaction between the excited electron and the hole left behind, are known to modify substantially the fundamental edge of semiconductors and insulators [29]. Exciton-modified interband spec- tra seem also to occur in metals at interband edges which have the final state on the Fermi surface [30]. Experimental evidence for these effects is reported at this conference in the paper by Kunz et al. We shall now discuss the question of exciton effects above the fundamental edge of insulators and semicon- ductors with special emphasis on the zincblende fami- ly. As mentioned in section 1 the gross features of these spectra are explained by the one-electron theory. The exciton interaction is responsible, at most, for small details concerning the observed line shapes. It is generally accepted [31,32] that the exciton interaction suppresses structure in the neighborhood of M3 critical points: the Coulomb attraction with negative reduced masses is equivalent to a repulsion with positive masses. Such a repulsion smooths out critical point structure: no M3 critical point has been conclusively identified in the experimental spectra. The E1 and El-H A critical points of figures 1–3 are of the M1 variety. Hence the line shape of the corresponding ei spectrum should be characterized by a steep low-energy side and a broader high-energy side. Figure 10 shows the shape of the El peak observed at low temperature by Marple and Ehrenreich [33] and by Cardona [34]. In order to avoid effects due to the overlap of the E1 and the E1 + A peaks it has been assumed that they have exactly the same shape but shifted by 0.55 eV. The contribution o C dTe 4 H_2. ſ THEORY (KANE) \ — — MARPLE W. e e e e CARD ON A \". 2 º \lº-. O e e o e | | | | | | | Iº e a 3.3 3.4 3.5 3.6 eV FIGURE 10. Contribution of the El gap to ei in CdTe as measure at low temperatures by Marple and Ehrenreich [33] and by Cardon [34]. Also calculation by Kane [32] using the adiabatic approxim tion. only E1 has been extracted from the measured spectrum and displayed in figure 10. It is clear fro this figure that the El peak is steeper at high energie than at low energies, against the expectations for an peak. Also in figure 10 we show the results of a calcul tion by Kane [32] of the effect of Coulomb interactio on the E1 line shape for Cate, using the effective ma approximation. The solution of the effective mas Hamiltonian with non-positive-definite mass is mad easier by the fact that the negative mass (along the direction) has a magnitude much larger (about te times) than the two equal positive masses. It is possib to use the adiabatic approximation [31], i.e., to sol the two-dimensional hydrogen atom problem with t 82 third coordinate as a parameter and then solve the adiabatic equation for the third coordinate. The agree- ment between the calculated and the experimental line shapes of figure 10 is excellent. Attempts have been made to calculate the dielectric constant including exciton interactions at an arbitrary point of k space, independently of the stringent restric- tions of the effective mass approximation [35,36]. Such calculation is possible if one truncates the Coulomb in- teraction between electron and hole Wannier packets to extend to a finite number of neighboring cells. The extreme and simplest case of a 6-function (Koster- Slater) interaction can be solved by hand [31,35] and gives around an M1 critical point the shapes of er and ei shown in figure 11: for an Mi critical point the Koster- Slater interaction mixes the Mi one-electron line shape with the Mill. The high energy side of the ei peak becomes steeper, in agreement with figure 10. The line hape observed for the E1–E1 + A peaks in the reflec- ivity spectrum is composed almost additively of the ei nd er line shape: at the energies of these peaks dR/de: no dB/der are almost equal. We also show in figure 11 he line shapes expected for the reflectivity spectra of he E1–E1 + A peaks and for the corresponding dif. erential spectra (dB/day). We show in figure 12 the hoton energy derivative spectrum of these peaks in gTe [37]: the observed line shapes disagree with hose expected from the one-electron theory (equal ositive and negative peaks) but agree with those pre- icted in the presence of a Koster-Slater interaction fig. 11). Similar results have been found for other incblende-type materials [37]. 3. Simplified Models for the Density of States As seen in section 1 the optical density of states, and us the dielectric constant, is a complicated function f frequency and its calculation requires lengthy nu- erical computation. For some purposes, however, it an be approximated by simple functions. In the vicini- of a critical point of the Mi variety, for instance, the ngular behavior of the dielectric constant can be ap- oximated by: ecº irt 1(0) - og)*-H constant (4) xciton effects are neglected. Exciton interaction can included, within the Koster-Slater model, by mul- lying eq (4) by a phase factor e” with d, small and sitive. s shown in figure 1, ei for the zincblende-type terials has a strong peak (E2) in the neighborhood of ONE ELECTRON | KOSTER-SLATER EXC |T ON 2^ €r der er Yº da) dCU ei dei e; da) da) /* d R M\ da, _* / L” R cer + ei / R = €r + ei dR | da) FIGURE 11. Modification in e, and ei introduced by the Koster- Slater exciton interaction in the neighborhood of an M1 critical point. Also, effect on the reflectivity under the assumption of an equal contribution of Aer and Aei to the reflectivity line shape. Hg Te 77 ok i –2. O –3.O 2.2 2.4 eV FIGURE 12. Photon energy derivative spectrum of the reflectivity of HgTe in the neighborhood of the E1 and E. --A1 structure [37]. which most of the optical density of states is concen- trated. The corresponding transitions occur over a large region of the BZ, close to its boundaries. In order to represent this fact, Penn [38] suggested the model of a non-physical spherical BZ with an isotropic gap at its boundaries. The complex energy bands of the material are then replaced by those of a free electron with an isotropic gas og at the boundary of a spherical BZ. This gap should occur in the vicinity of the E2 optical struc- ture. While this model represents rather poorly the rich 83 structure of ei (fig. 1), it is expected that it should give a good picture of er at zero frequency. The reshuffling of density of states involved in the case of the isotropic model should not affect er(a) = 0) very much because of the large energy denominators which appear in eq (3) for Q) = 0: the lowest gap oo, usually much smaller than (og, accounts only for a very small fraction of the optical density of states. Penn obtained with this model the static dielectric constant for a finite wavevector q. The result can be approximated by the analytic expression [38]: e(a) = 0, q) --(+)- || 1:4 º' (5) (Og Og H. F. g Ø — -: *(*) with .9% = 1 400 F 3 \of In eq (5) (op is the plasma frequency obtained for the density of valence electrons and of and kf the cor- responding free electron Fermi energy and wave number. The dimensionless quantity & is usually close to One. Figure 13 shows eq (5) for Si compared with the exact results of the Penn model [39]. These results are obvi- ously independent of the direction of q. A small depen- dence on this direction is found from a complete pseu- dopotential calculation by Nara [40] (see also fig. 13). The function e(o,g) is of interest for the treatment of dielectric screening. — Srinivasan's Result ––– Nara's Result 12- Along [ll] –––Penn's Interpolation Formula 10- E(0,0). 8- S|L|C0N 6- lº 2- ~~~ --> 0 I I I |Tº TFTDT 09 tº FIGURE 13. Static dielectric constant e (o, q) obtained by Srinivasan [39] for Si with the Penn model compared with the interpolation formula of eq (5) and with the results of a pseudopotential calcu- lation by Nara [40] for q along (111). Equation (5) yields for q = o the electronic contribu- tion to the static dielectric constant: 2 2 --- (*) --(+) COg COg (6) The experimental values of eo agree reasonably well with the results of eq (6) using for on the energy of the E2 peak [34]. Equation 6 has gained recent interest as the basis of Phillips and Van Vechten's theory o covalent bonding [41,42,43]. These authors use eq (6 and the experimental values of eo to define the averag gap op. With this gap and the corresponding gap of th isoelectronic group IV material they can interpret wide range of properties such as crystal structure [42] binding energy [43], energies of interband critic points [41], non-linear susceptibilities [44], etc. As a example we discuss the hydrostatic pressure (i. volume) dependence of eo for germanium and silico According to Van Vechten [41], on for C, Ge, Si, an o!—Sn is proportional to (ao)7° where ao is the lattic constant. If one makes the assumption that this la gives also the change in on with lattice constant for given material when hydrostatic stress is applied o can calculate the volume dependence of eo [4] Neglecting the one in eq (6), a valid approximation f Ge and Si, one finds: e, dW dI/ dI/ =2|[0.83–0.50] = 0.66 ( Equation (7) explains the sign and the small magnitu observed for (1/eo)(deo/dV). The experimental values this quantity are 1.0 for Ge and 0.6 for Si [41,45]. According to eq (6) the average gap (on determin the electronic dielectric constant for a = 0. As t lowest gap oo is approached (@o < 0), usually), exhibits strong dispersion. This dispersion is due, in t spirit of eq (3), to the density of states in the vicinity alo. For the purpose of calculating the dispersion of immediately below oo, the density of states can be proximated by that of parabolic bands with a reduc mass equal to the reduced mass p at alo. These ba are assumed to extend to infinity in k space: t unphysical contribution to er for |k| – 20 should small for a soo, because of the large energy denomi tors of eq (3). We thus obtain for a cubic material following contribution of the alo gap to the scalar diel tric constant below alo (under the assumption of a c stant matrix element of p equal to P) [46]: 84 Aer =2(2p) *on” |P ºf (a)/a)0) = Cº. f(0)/00) with (8) f(x) = 2 – (1 + x) 1/2 – (1–3) 1/2. Equation (8) represents quite well the behavior of er immediately below on for the lead chalcogenides [47] and a number of other semiconductors [48]. As an ex- ample we show in figure 14 the observed dispersion of er below on at room temperature [49] together with a fit based on eq (8) [48]. For the sake of completeness we have included in the fitting equations not only the effect of a), (Ed) but also that of its spin-orbit-split mate Eo-H Ao (also represented by an expression similar to eq (8)), the dispersion due to the E1 and El-H A gaps, and that due to the main on gap assuming on = E2. Thus the fitting equation with three adjustable parameters C6, C7, Cº. is [48]: cºſ(x)+(*)"fe.) er(0) = 1 + | %0 +}ſ ) %0s | ( ) 2\ 00s (9) +c(hº)-(+)h(x)}+cº tº 1S here C00s oot Ao, x,--*-, x = * CO os CO 1 a) is = 0) - A1, x,--º-, 3:2 = — CO 1s COg a: (x)=1 +. he fitting values of C6 (6.602) and C (2.791) are in [ualitative agreement with those calculated from the and parameters [48]. The parabolic model density of states can also be sed to interpret the strong dispersion in the piezo- irefringence observed near the lowest direct gap of A | 3 H. Go As | 2 H O \U IT || H | O | | | | | O O.4 O.8 | 2 | 6 eV GURE 14. Experimental results for e, in GaAs below the funda- mental edge at room temperature [49] (circles) and fitted curve based on a model density of states. T} | C > -C \ *E 3 O – GG AS – [loo) STRESS C X _º J| – || - | |>. w – 2 – l | ---|- | | | O O. 4 O 8 | 2 eV FIGURE 15. Piezobirefringence in GaAs for an extensive stress along (100) (room temperature). The circles are experimental points. The solid line is a fit based on the model of eq (5) [48]. Ge, GaAs [48], and other III–V semiconductors |50,51]: uniaxial stress splits the top valence band state (Ts) and a birefringence in the contribution of Eo to er results because of the selection rules for transi- tions from the split bands. The main contribution to this piezobirefringence is expected to be proportional to f"(x), which diverges like (a)— alo) 'º for a) -> aJo. Such behavior can be seen in the experimental results (circles) of figure 15 obtained for GaAs at room tem- perature. Included in this figure is the corresponding fit based on the model of eq (9) [48]. The long-wavelength, non-dispersive contribution to the piezobirefringence of figure 15 can be interpreted, at least qualitatively, in terms of the Penn model of eqs (6) and (7). Equation (7) yields two contributions to the change in er one due to the change in plasma frequency (i.e. carrier density) with stress and the other due to the change in the isotropic gap. The first contribution should not exist for a pure shear stress. For a hydro- static stress the second contribution can be written in tensor form as: — Aeo = 5e (10) 6 () where e is the strain tensor. We postulate that eq (8) remains valid for pure sheer stress. This crude generalization has a clear physical meaning in terms of the Penn model. The spherical BZ becomes ellipsoidal under a sheer stress and the energy gap at an arbitrary point of the BZ boundary kf becomes anisotropic. The gap at kp is assumed to become larger as kº becomes larger (kº is the distance between atomic planes per- 85 pendicular to kº). Equation (8) gives the right sign for the long wavelength contribution to the piezobirefrin- gence of figure 15 but a magnitude about five times larger. The agreement becomes better if the contribu- tion of the E., edge to the long wavelength piezobirefrin- gence, of opposite sign to that predicted by eq (10), is subtracted from the experimental results. 4. Third Order Susceptibility and Model Density of States It has been recently suggested [52] that the third order susceptibility of Ge, Si, and GaAs at long wavelengths is related to the Franz-Keldysh effect (i.e. the intraband coupling by the field) of interband critical points [53]. We discuss now the Franz-Keldysh con- tribution of the on, Eo, and El gaps to Xº1. 4.1. Average Gap (og In the spirit of Penn's model [38] we represent the long-wavelength dielectric constant by eq (6) with 3 = 1. The corresponding imaginary part of the dielec- tric constant is, for a > 0), [47]: e = A,” (a) – on) 7% A – offo/* (11) The Franz-Keldysh effect for the one-dimensional ab- sorption edge of eq (11) can be expressed in terms of the Airy functions Ai and Bi [54]. One must mention, however, that the isotropic gap problem in the presence of an electric field would only be completely equivalent to the one-dimensional problem if the field experienced by every electron were along the direction of the cor- responding k. The fact that the field 3 is the same for all electrons, regardless of k, can be taken into account by using an average field: with (cos” (8) where 8 is the angle between k and 3. The long- wavelength expression for Xº, thus is [54]: + Co.) Xº1 = 2 lim g–26-1/2 (c(e. {} (z) 207ta) 3 — 0 6) → 0 (13) where the one-dimensional electro-optic function G. (m) is given by [54]: G. (m) = 27Ai(m)Bi(m) - H(m) m-1} g2\113 {} = (...) In eq (14) H (m) is the unit step function and p the re- duced mass of the Penn model, given by p = 09(2kr) ". The Fermi momentum of the valence electrons is related to the plasma frequency op through kfº = (3/4)Top”. The limit for 3 -> 0 in eq (14) is easily found using the asymptotic expansions of Ai (m) and Bi (m) for m—--|-oo [55]. By subsequently performing the limi for o-> 0 one finds: (14) 287 /3 \2/3 goº/* xº-#(; ) (–1). (15 {} Or, for ease of evaluation, with Xº, in e.s.u., and th energies in eV: Xº1 = 1.45' 10" (eo-1) (16 We list in table I the values of on, ay, and eo — 1 fo Ge, Si and GaAs. The values of X%, calculated with e 1111 (16) are then listed in table II. This table shows agree ment in sign and magnitude between the values of x} predicted from og and the experimental ones. An i crease in the polarizability with field (x} = 0) is to b expected for the Franz-Keldysh effect since the i traband coupling by the electric field produces decrease in the energy gap. We shall consider now the contribution to x} of th interband coupling by the electric field across th isotropic gap oy. This coupling produces an increase i energy gap, and thus its contribution to X\}, is neg tive. This contribution to X} is readily found from e (6): 3 1 do (3) = —— - - g AX}, (€o on d(3*) 57T __3(€0-1) (#)" off" (l - 57 4. 0% In eq (17) we have made use of the second-order pertu bation expression: dog | < v|r|c > |* _2k} —- = 3 d(**) COg (0; (l Equation (18) is in agreement with the results of ref. [44]. Comparison of this equation with eq (15) shows that the magnitude of the interband contribution to Xº, is smaller than the intraband contribution. Its con- sideration does not change the sign of X}}, as found with eq (16) but introduces a numerical factor of the order of unity. In view of the uncertainties of this type of calculation we shall henceforth neglect this factor. 4.2. Lowest Gap E0 We use for the contribution of an isotropic Mo critical point to the real part of the dielectric constant the result of eq (8). A calculation similar to that performed above yields for an Mo critical point the following Franz- Keldysh contribution to x}}, [52]: P2p ||2 a) \* AX} = 0.06 wº (1+1.85 (...) + . . .) (19) or, transforming Axºn to e.s.u. and oo to eV (Pº and po are left in atomic units for ease of computation): 2, 1 1/2 AX}} (a)) = 6 e 10-100 ºf 9/2 (0. 2 x (1+1.85 (*) + . . COO .) (20) e have included in eqs (19) and (20) the first term in he dispersion of Axºn since it may be possible to ob- erve it experimentally in small band gap materials. his dispersion is given exactly by the function: 2 –2.5 — 2.5 #((1+...) +(-) –20.”) (21) COO C00 quation (21) is not immediately valid for the Eo edge ecause of the degeneracy of the valence band. How- ver, one can apply it to the Eo edge if one neglects the ield coupling between degenerate valence bands and ses appropriate average values of P* and pºo. Each one f the three valence bands can be assumed to have a ass equal to three times the conduction band mass nd a corresponding matrix element equal to #p” [48]. ence eq (20) must be used with the matrix element P2 nd poiâme if the three valence bands are to be in- uded. The spin-orbit splitting A of the valence band taken into account, if Aoss Eo, by replacing on by its Ao Q. ** * verage value E. H. Since P* is almost the same for l materials of the germanium family, we can replace by a typical value = 0.4 (in atomic units). The values of po and alo–Eo-H * for Ge, Si and GaAs are listed in table I. Using these numbers, eq (20) yields the values of x} (0)=0) listed in table 2. While this contribution Values of the parameters required for the evaluation Frequencies in TABLE 1. of the Franz-Keldysh contributions to Xº1. eV, as in Bohr radii, po in units of the free electron mass. P” has been taken equal to 0.4 for all materials. Ge Si GaAs COg 4.3 4.8 5.2 (Op 15.5 17.35 15.5 eo – 1 11 15 10 Mo 0.03 0.04 0.05 000 0.9 4. 1.5 do 10.7 10.3 10.7 C01 2.2 3.3 3.0 TABLE 2. Contribution of the various Franz-Keldysh effects discussed here to X\}, and X%. In units of 10−" e.s.u. Also, experimental values of the bound carrier contributions to : ) X}}}, and X}}s. En Eo El Experi- contribu- | contribu- | contribu- Iſl º tion tion tion e X}} 0.26 0.67 0.20 1.0 Ge X}}. 0.26 0.67 0.26 1.5 X}} 0.22 0.00 0.027 0.06 Si X:#; 0.22 0.00 0.036 0.08 X}} 0.12 0.087 0.045 0.12 GaAs X}} (). 12 0.087 0.060 0.10 is zero for Si and is not excessive in GaAs (it may, there- fore, be assumed as included in the average gap calcu- lation given above), it is dominant in Ge. In first approx- imation it may be added to the average gap calculation: excellent agreement with the experimental results is then found. For InAs, with alo = 0.5 eV and p = 0.02, we find from eq (20) Axºn (a)=0)=7. 10-1", which is of the order of the free-carrier contribution for the samples with the lowest electron concentrations measured (N = 2 - 10% cm−") [57]. This is contrary to the statement found in the literature that for these carrier concentrations in InAs x}}, is dominated by the free-carrier contribution [56,57,58]. 87 The interband contribution of E, to X\}, for a = 0 is easily obtained from the expression (see eq (8)): –3/2 (ſ Aer (a) = 0) = 2u.o.) 312 P2 (200) *P*% (1) (22) If one assumes that the repulsion produced by the field affects po in the manner predicted by the k p expres- sion with constant matrix elements of p(Lo Cº (99), the corresponding interband contribution to X; i vanishes. If, on the other hand, one assumes po to be field inde- pendent one also finds a negative contribution to X\}, but smaller than the Franz-Keldysh contribution, and hence we shall neglect it. 4.3. E1 Critical Points The E optical structure is usually attributed to Mi critical points along the {111} directions. While Mi critical points are known to yield no contribution to Xºl [52], there are Mo critical points of the same sym- metry slightly below the M1 critical points. This com- bination of Mo and M1 critical points with a very large longitudinal mass, can actually be approximated by two-dimensional minima [48]. The contribution of one of to the wavelength dielectric constant is (we assume four, and not eight equivalent {111} directions): these two-dimensional minima long- - 4 V3p. 1 P2 •) Qo0); Aér (23) where a, is the lattice constant and Pº the appropriate square matrix element. We have tried to calculate the ºn of these two-dimen- sional critical points in a way similar to that used above, Franz-Keldysh contribution to X but we have run into difficulties when evaluating the limits of the two-dimensional electro-optic functions for m–2 + Co. In view of this we have instead evaluated the effect of the three-dimensional M, critical points, with the longitudinal effective mass replaced by the value required to give at long wavelengths a contribution to er equal to that in eq (23). Under these conditions, and because of the large lon- gitudinal mass, only fields transverse to the critical point axis contribute to Xºr. When summing the con- tributions of the four equivalent valleys, it is found that the effect becomes anisotropic: the ratio of the third order susceptibility for 3 along {lll) (xiē), to that for 3 along {100} is 4/3. This argument is independent of the specific model chosen for the {111} transitions, pro- vided u > 0, . It gives the type of anisotropy (X% D- X}) observed for Ge and Si, but not for GaAs [57]. The Franz-Keldysh contribution of E, to xº, as found by the procedure sketched above, is: Ax} =0.52+. CloG) (24) (3) or, with an in eV and Xºn in e.s.u.: t P2 Ay!?), - 2.7 - 10-8 - X iſ 1 CloGo? (25) (a, in Bohr radii and Pºin atomic units.) The matrix element P should have approximately the same value as for the Ed gap. In order to take care of the spin-orbit splitting A of E, we substitute on by E. -- A/2. The approximate values of a, and a for Ge, Si and GaAs are listed in table I. The values calculated for the Franz-Keldysh contribution to E to X; and x}, are listed in table 2. While the calculated anisotropy has, for Ge and Si, the sign observed experimentally, its magnitude is far too small to explain the experimental anisotropy, especially after the Eo and the En contributions are added. There is a possibility that the E. contribution of eq (25) may have been underesti- mated. Exciton quenching effects [59,60], not included in our calculation, may increase this contribution. We cannot offer even a qualitative explanation of the sign of the X" anisotropy observed for GaAs. It would be interesting to determine, through measurements of other III-V or II — VI compounds, whether it is con- nected with the lack of inversion symmetry in these materials. The interband contribution of the E edges can be evaluated in a manner analogous to that used for the Eo and the on gaps. We also find that this contributio negative and smaller than the Franz-Keldys contribution. is 5. Acknowledgments I am indebted to Drs. Buss, Kane, Phillips, Va Vechten, and Aspnes for sending preprints of thei work, prior to publication and to the staff of DESY fo their hospitality. I would like to dedicate this work to the memor of Rolf Sandrock, of whose untimely death I heard while this paper was in print. 6. References [1] Cohen, M. L., and Bergstresser, T. K., Phys. Rev. 141, 78 (1966). [2] Cardona, M., and Pollak, F. H., Phys. Rev. 142,530 (1966). [3] Dresselhaus, G., and Dresselhaus, M. S., Phys. Rev. 16 649 (1967). 88 [4] Herman, F., Kortum, R. L., Kuglin, C. D., and Shay, J. L., in “II–VI Semiconducting Compounds,” D. G. Thomas, ed., (W. A. Benjamin, New York, 1967), p. 503. [5] Gilat, G., and Dolling, G., Phys. Letters 8,304 (1964); Gilat, G., and Raubenheimer, L. J., Phys. Rev. 144, 390 (1966). [6] Higginbotham, C. W., Pollak, F. H., and Cardona, M., Proceedings of the IX International Conference on the Physics of Semiconductors, Moscow 1968 (Publishing House Nauka, Leningrad, 1968) p. 57. [7] Buss, D. D., and Parada, N.J., private communication; see also article by D. D. Buss and V. E. Shirf in these proceedings. [8] Kane, E. O., Phys. Rev. 146, 558 (1966). [9] Philipp, H. R., and Ehrenreich, H., Phys. Rev. 129, 1550 (1963). [10] Cardona, M., Z. Physik 161, 99 (1961). [11] Zhang, H. I., and Callaway, J., Phys. Rev. 181, 1163 (1969). [12] Walter, J. P., and Cohen, M. L., Phys. Rev., to be published. [13] Seraphin, B. O., and Bottka, N., Phys. Rev. 145,628 (1966). [14] Shaklee, K. L., Rowe, J. E., and Cardona, M., Phys. Rev. 174, 828 (1968). [15] Matatagui, E., Thompson, A. G., and Cardona, M., Phys. Rev. 176,950 (1968). [16] Higginbotham, C. W., Ph. D. Thesis, Brown University, 1969. [17] Eckelt, P., Madelung, O., and Treusch, J., Phys. Rev. Letters 18, 656 (1967). [18] Brinkman, W., and Goodman, B., Phys. Rev. 149,597 (1966). [19] Pollak, F. H., Cardona, M., Higginbotham, C. W., Herman, F., and Van Dyke, J. P., Phys. Rev., to be published. [20] McElroy, P., Ph. D. Thesis, Harvard University, 1968. [21] Cardona, M., and Greenaway, D. L., Phys. Rev. 133, Al685 (1964). [22] Scanlon, W. W., J. Phys. Chem. Solids 8,423 (1959). [23] Aspnes, D. E., and Cardona, M., Phys. Rev. 173, 714 (1968). [24] Sandrock, R., Phys. Rev. 169,642 (1968). [25] Tutihasi, S., and Chen, I., Phys. Rev. 158,623 (1967). [26] Lin, P. J., and Phillips, J. C., Phys. Rev. 147,469 (1966). [27] Wiech, G., in “Soft X-Ray Spectra,” D. J. Fabian, ed., (Academic Press, New York, 1968) p. 59. [28] Feuerbacher, B., Skibowski, M., Godwin, R. P., and Sasaki, T., JOSA58, 1434 (1968). [29] See for instance R. S. Knox, “Theory of Excitons,” (Academic Press, N.Y., 1963). [30] Mahan, G. D., Phys. Rev. Letters 18, 448 (1967). [31] Velicky, B., and Sak, J., Phys. Status Solidi 16, 147 (1966). [32] Kane, E. O., Phys. Rev. 180,852 (1969). [33] Marple, D. T. F., and Ehrenreich, H., Phys. Rev. Letters 8,87 (1962). [34] Cardona, M., J. Appl. Phys. 36,2181 (1965). [35] Inoue, M., Okazaki, M., Toyozawa, Y., Inui, T., and Nanamura, E., Proc. Phys. Soc. Japan 21, 1850 (1966). {36] Hermanson, J., Phys. Rev. 150,660 (1966). [37] Shaklee, K. L., Ph. D. Thesis, Brown University, 1969. [38] Penn, D., Phys. Rev. 128,2093 (1962). [39] Srinivasan, G., Phys. Rev. 178, 1244 (1969). [40] Nara, H., J. Phys. Soc. Japan 20, 778 (1965). [41] Van Vechten, J. A., Phys. Rev. 182, 891 (1969). In this reference a correction factor of the order of unity is added to eq (6) so as to take the polarizability of the core d electrons into a CCOUnt. [42] Phillips, J. C., Phys. Rev. Letters 20, 550 (1968). [43] Phillips, J. C., Covalent Bonding in Molecules and Solids (University of Chicago Press, Chicago) to be published. [44] Phillips, J. C., and Van Vechten, J. A., Phys. Rev. 183, 709 (1969); Levine, B. F., Phys. Rev. Letters 22, 787 (1968). [45] Cardona, M., Paul, W., and Brooks, H., J. Phys. Chem. Solids 8, 204 (1959). [46] Korovin, L.I., Soviet Phys. Solid State 1, 1202 (1959). [47] Cardona, M. in “High Energy Physics, Nuclear Physics, and Solid State Physics,” I. Saavedra, ed., (W. A. Benjamin, New York, 1968). [48] Higginbotham, C. W., Cardona, M., and Pollak, F. H., Phys. Rev. 184, 821 (1969). [49] De Meis, W. M., Technical Report No. HP-15 (ARPA-16), Har. vard University (1965). [50] Shileika, A. Yu., Cardona, M., and Pollak, F. H., Solid State Communications 7, 1113 (1969). [51] Yu, P. Y., to be published. [52] Van Vechten, J. A., and Aspnes, D. E., Phys. Letters, in press. [53] Aronov, A. G., and Pikus, G. E., Proceedings of the IX Interna. tional Conference on the Physics of Semiconductors, Moscow, 1968, (Publishing House Nauka, Leningrad, 1968), p. 390. [54] Cardona, M., “Modulation Spectroscopy,” F. Seitz, D. Turn- bull, and H. Ehrenreich, eds., (Academic Press Inc., New York, N.Y.). [55] Antonsiewicz, H. A., in “Handbook of Mathematical Func. tions,” (M. Abramowitz and I. A. Stegun, eds.) (Dover Pub. Inc., New York, N.Y., 1965) p. 448. [56] Wolff, P. A., and Pearson, G. A., Phys. Rev. Letters 17, 1015 (1966). [57] Wynne, J. J., Phys. Rev. 178, 1295 (1969). [58] Patel, C. K. N., Slusher, R. F., and Fleury, P. A., Phys. Rev. Letters 17, 1011 (1966). [59] Hamakawa, Y., Germano, F. A., and Handler, P., J. Phys. Soc. Japan, Suppl. 21, 111 (1966). [60] Shaklee, K. L., Rowe, J. E., and Cardona, M., Phys. Rev. 174, 828 (1968). 89 M. S. Dresselhaus (MIT): Introduction of a finite relaxation time does not always help to achieve good agreement between the experimental and calculated dielectric constant curves. For example, in the first slide you showed on indium-arsenide, you have the ex- perimental es curve which is higher than the calculated curve. For a situation like that, introduction of a finite relaxation time can only make matters worse. M. Cardona (Brown Univ.); I wanted to mention this point and the fact that you had done calculations of dielectric constants with a finite phenomenological scattering time. Such procedure may sometimes im- prove the agreement between theory and experiment. M. S. Dresselhaus (MIT): It will not improve this par- ticular fit in InAs. In general, I think that for most of the curves that you have shown, the agreement between theory and experiment would be improved with a finite relaxation time. The second point I would like to make is the following: I am not exactly con- vinced that in calculating the differential reflectivity, it is correct just to simply differentiate the reflectivity with respect to frequency because the various dif- ferential techniques emphasize different features of the energy band structure. For example, if you do a thermal reflectance measurement on a metal, it is those bands that are very close to the Fermi level that are emphasized; for different bands emphasized in a piezo-reflectance measurement. example, are M. Cardona (Brown Univ.); This is correct and a very good point. Of course, one type of modulation experi- ment one does is simply wavelength or photon energy modulation. Your comment does not apply to this type f measurement which, however, is very difficult from he experimental point of view. It is much simpler to odulate the sample temperature (thermo-reflectance) or the electric field applied to it (electro-reflectance). An electro-reflectance spectrum, of course, should not e compared with the calculated derivative of the eflectivity. The thermo-reflectance spectra of zinc- lende semiconductors, however, should be very imilar to the frequency modulation spectra since roadening is much smaller than frequency shift and e temperature coefficients of all gaps are practically Discussion on “Optical Properties and Electronic Density of States" by M. Cardona (Brown University) the same. For metals, of course, transitions involving portions of the Fermi surface are greatly enhanced in temperature modulation spectra. J. Tauc (Bell Telephone Labs.): It is a very poor ap- proximation to compare the x-ray spectra with the band state densities. Even if one neglects the many electron effects it is necessary to compare the x-ray spectra with the densities of approximately projected states accord- ing to the symmetry of their ground states. Such calcu- lations by J. Klima (private communication) on Ge gave a very good agreement with experiment. M. Cardona (Brown Univ.); The question was whether one could improve agreement between theory and ex- periment for the 3d -> conduction transitions in ger- manium by including the appropriate matrix elements. This can be done, at least in part, by projecting the p component of the conduction band wave functions. The answer is yes. Actually, I notice that the paper by G. Weich and E. Zöpf contains some rather nice work along these lines. B. H. Sacks (Univ. of Calif.): Is there any indication that the application of pressure (either uniaxial or hydrostatic) to GeO2 could preferentially shift the band edges so as to give a direct gap material for use in ul- traviolet laser? G. W. Pratt (MIT): There is a great possibility of this. M. Cardona (Brown Univ.); In connection with Dr. Pratt's statement that 200 points is a very small number of points for an e2 calculation, I would like to point out that with such a coarse mesh, one can get tremendous resolution if one derives from it a much finer mesh by quadratic interpolation. Such interpolation can cut down very considerably on the computer time required for good resolution. J. Janak (IBM, New York): I would like to refute what Dr. Cardona just said. We have done some calculations for palladium and we find that to get the error down using quadratic interpolation, we have to go to about 3000 points in the reduced zone. That is to get an error of 5 millirydbergs. That is a particular case, it is true, but it is not always true that a few hundred will do. 91 and Liberman is displayed. 1. Introduction The purpose of this paper is (1) to present the elec- tron density of states calculated using our self-con- sistent orthogonalized plane wave (SCOPW) model for compounds in the isoelectronic sequences, Si-Alp and Ge-GaAs-ZnSe and (2) to show the effect on the density of states of using different exchange approximations. Slater [1], Kohn-Sham [2] , Gaspar [3], and Liberman [4]. In the past few years, a great deal of success has been attained in calculating the energy band structures of Groups III-V, II-VI and IV compounds using the first principles self-consistent unadjusted OPW model developed here at ARL. The SCOPW programs used to calculate the electronic band structure have given very good one-electron band energies for compounds such as CaS [5], ZnS and ZnSe [6], GaAs [7], Si [8], AlAs [9], and AlF [10]. These unadjusted band ener- The exchange approximations considered are those of | where k, = ko + Ku, ko locates the electron within the irst Brillouin zone, Ka is a reciprocal lattice vector, Ra s an atom location, lie is a core wave function and Q0 is he volume of the crystalline unit cell. The coefficients §u are determined by requiring lik,(r) be orthogonal o all core state wave functions. The variation of Bu to Theoretical Electron Density of States Study of Tetrahedrally Bonded Semiconductors D. J. Stukel, T. C. Collins, and R. N. Euwend Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio 45433 The electron density of states has been calculated using a self-consistent orthogonalized plane wave (SCOPW) model for compounds in the isoelectronic sequences Si-Alp and Ge-GaAs-ZnSe. The valence and conduction band density of states are presented. The location of the core states is also given. The effect upon the density of states of using the exchange approximations of Slater, Kohn-Sham, Key words: Aluminum phosphide (AIP); electronic density of states; exchange potential; gallium arsenide (GaAs); germanium; self-consistent orthogonalized plane wave model (SCOPW); semiconductors, tetrahedrally bonded; silicon; zincblende; ZnSe. gies fit the known experimental facts very well when Slater’s exchange is used [11]. Our SCOPW programs are described in section 2. The treatment of exchange is also discussed as is the method of calculating the density of states. The results are presented in section 3. 2. Methods of Calculations 2.1. Self-Consistent OPW Calculations The orthogonalized plane wave (OPW) method of Herring [12] is used to calculate the electron energies. In the OPW model [5,6], the electronic states are di- vided into tightly bound core states and loosely bound valence states. The core states must have negligible overlap from atom to atom. They are calculated from a spherically symmetrized crystalline potential. The valence states must be well described by a modified Fourier series V,(r) = S B | e”,” –X ei", "a X. Aſ v.0-R)] minimize the energy then results in the valence one- electron energies and wave functions. The dual requirements of no appreciable core over- lap and the convergence of the valence wave function expansion with a reasonable number of OPW's deter- mines the division of the electron states into core and 93 valence states. For Al, Si and P the 3s and 3p states (for Zn, Se, Ga. As and Ge the 4s and 4p states) are taken as the valence states. Very good convergence is obtained for ZnSe, GaAs, Ge, Si and Alp when 229 OPW's are used in the series expansion. The calculation is self-consistent in the sense that the core and valence wave functions are calculated al- ternately until neither changes appreciably. The Cou- lomb potential due to the valence electrons and the valence charge density are both spherically sym- metrized about each inequivalent atom site. With these valence quantities frozen, new core wave functions are calculated and iterated until the core wave functions are mutually self-consistent. The total electronic charge density is calculated at 650 crystalline mesh points covering 1/24 of the unit cell, and the Fourier transform of p(r)" is calculated. The new crystal potential is calculated from the old valence charge dis- tribution and the new core charge distribution. Then new core-valence orthogonality coefficients, Ağ, , are calculated. The iteration cycle is completed by the cal- culation of new valence energies and wave functions. The iteration process is continued until the valence one-electron energies change less than 0.01 eV from iteration to iteration. The appropriate charge density to use for the self- consistent potential calculation is the average charge density of all the electrons in the Brillouin zone. In the present self-consistent calculations, this average is ap- proximated by a weighted mean over electrons at the T, Å, L and W high symmetry points of the zincblende Brillouin zone shown in figure 1. The weights are taken to be proportional to the volumes within the first Bril- louin zone closest to each high symmetry point. The adequacy of this approximation has been tested and the The zincblende Brillouin zone. FIGURE 1. error in the energy eigenvalues has been shown to be less than 0.1 eV [8]. 2.2. Treatment of Exchange The nonlocal Hartree-Fock (HF) exchange term is so complicated that approximations are necessary in crystalline calculations. The best known approxima- tions to the HF exchange terms are due to Slater, Kohn- Sham-Gaspar, and Liberman. Slater's and Kohn-Sham- Gaspar's approximations (energy independent) result in different constant coefficients multiplying the densi- ty to the one-third power whereas Liberman’s approxi- mation yields a coefficient which is a function of r and the energy of the state being considered. a. Energy Independent Exchange Approximation The first simplified one-electron operator which replaces the exchange operator in the HF equation was suggested by Slater [1,13]. If one multiplies and di- vides the exchange term in the HF equations by di”(x1)(bi(x) one has [14] | (bf(x2)(bf (x,) (bj(x1)(b)(x2) 2 (b? (x.1) bi (x1) I 12 s dT2 j= 1 }*) — Vrd (x1) In this expression Slater then made the free electron gas approximation that the di's are plane waves and ob- tained 1 / 3 W. (olº--8; 00)}"F(ſ) T l 1 — mº Tim —- l |H| 2 n — F(m) 1 – m p(r) is the electron density of the system. Here it has been assumed that all states are filled for |k| < |kp|, the Fermi momentum, and all states are empty for |k > |kp|. The wave vector, k and the Fermi wave vector kF were taken to be kg (r) = [E], - V(r)] /* (1) where V(r), the total electronic potential, includes both Coulomb (nuclear and electron) and exchange contribu tions, KF (r) = [37°p(r)] /8 (2) and E, is the energy of the state being calculated. This definition of kr is derivable from phase space con siderations. Slater then averaged F(m) over the occu 94 pied states of the free electron gas and obtained the value 3/4. Hence 3 1/3 W.0–6.00) (3) If one makes Slater's approximation in the exchange term of the HF expression for the total energy and then varies the total energy, one obtains the Kohn-Sham- Gasper exchange approximation — 2 Wºrks - 3 V rs (4) b. Energy Dependent Exchange Coefficient Liberman investigated Vr/r)|ped with k and kf given by eqs (1) and (2). Hence V 84-3 "F ...--8; 00)}"F(n) (5) Slater, Wilson and Wood [15] modified Liberman’s ap- proximation by using •) - 3 1/3 ) 1 | wo-(e-rºot | # p(r) | | (6) instead of eq (2) so that m = 1 at the top of the Fermi dis- tribution. We found it advantageous to calculate kp both ways and always use the larger value. This ap- proach gives results slightly closer to the HF results for atomic calculations. The Fermi energy, EF, that was used in the SCOPW crystalline calculation was taken at the middle of the fundamental gap, defined by the top of the valence band and the bottom of the conduc- tion band. 2.3. Density of States Calculations To calculate the density of states, one must know the energy levels at an arbitrary point in the Brillouin zone. Since the SCOPW energy levels are only calculated at the T, X, L and W high symmetry points, one must somehow interpolate these energies throughout the Brillouin zone. We have determined the band structure in the remainder of the zone by fitting a pseudopoten- tial-type interpolation model to the SCOPW energy levels calculated at the high symmetry points. Using this interpolation scheme, the band energies and band- energy gradients at each of 155 course mesh points in the irreducible sector (1/48th) of the reduced zone are evaluated. A fine mesh consisting of 512 points is then centered at each coarse mesh point. The band energies at each of the fine mesh points are calculated from a knowledge of the band energies and band-energy gradients at the coarse mesh points. The density of states is then calculated by summing over all bands | 20 — — — Philipp and Ehrenreich (Exp.) SC-OPW | OO H. 8 O H. 2 "E -) £ 60 H S. (N \U •ol 2O H. O O |.5 3.O 4.5 6.O 7.5 9.O Energy (eV) FIGURE 2: The theoretical SCOPW (solid line) and the experi- mental (ref [18] dashed curve) ég curves for GaAs. (valence or conduction) at each of the 512 X 155 fine mesh points. We use the same technique to calculate the SCOPW e2 curves [16]. The joint valence–conduction density of states is combined with pseudopotential transition matrix elements to produce the resulting e2 curve. A typical comparison of experimental and SCOPW e2 curves is shown in figure 2. We have found that the SCOPW e2 peaks usually match experiment to within 0.2 eV. However, the shape of the e2 peaks has not matched experiment at all closely. The lack of agree- ment as to shape may be due to the use of pseudopoten- tial matrix elements and to exciton assisted transitions which we have not taken into account. 3 Discussion of Results 3.1. Energy Independent Exchange Approximation q. Valence Bands The general structure of the SCOPW results is very similar in appearance to the adjusted non-self-con- sistent OPW results of Herman, Kortum, Kuglin and Shay [17]. A typical SCOPW band structure (ZnSe) is shown in figure 3. Density of states results for the valence bands are presented in figure 4. One has three regions for all the compounds studied. This is true for both Slater's and Kohn-Sham’s exchange terms. The first region consists of the “s-like” bands which start around 12 eV and has a width which varies in these materials between 1 to 4 eV. This region has two main features; one is associated with the Lºy or Liv 95 i L3v Liv T- i. REDUCED WAVE VECTOR FIGURE 3. ZnSe calculated energy bands using the SCOPW model plus the position of the Zn 3d core energy. The SCOPW high symmetry point values are indicated by heavy dots. A pseudopotential interpolation scheme was used to generate the lines. symmetry point and the other is the high energy struc- ture associated with the K point region of the Brillouin ZOI) 62. The second region is relatively narrow ranging from 1 to 2 eV in width over the five compounds, and is located from – 8 eV to — 3.5 eV. Its structure consists of a large peak which has in some compounds a TABLE 1. shoulder or small peak on the low energy side. This large peak arises from the K point region. The source of these energy states is the “light-hole p-like” bands (Wiv – Liv — Tisw-X3; in fig. 3). This band is nearly flat throughout the Brillouin zone except near the T point. There, it rises in energy very fast and becomes degenerate with the top valence band. However, there are so few states outside region two that no appreciable structure is found in region three from this band. The third region is the one which contributes most to the calculations and to investigations involving the band gap and the imaginary part of the dielectric con- stant ex. The width of this region is well defined as can be seen in figure 4. The width varies in these com- pounds from 4.9 eV to 2.7 eV. This region has three distinct features. The low energy structure in this re- gion comes again from the K point. The middle peak is located at the energy of the X4, or X.50, and the high energy shoulder or peak is associated with the L30. The electron states which form this region are derived from the “heavy-hole p-like” bands (top valence bands in fig. 3). This region contains twice as many electrons as re- gion one or region two. The most striking differences in going from the Group IV compounds to the III-V and then to the II-VI is that the width of the regions becomes smaller and there is a gap between the “s-like” and “p-like” bands in III-V’s and II-VI's. For example, in Ge, region one has a width of 3.8 eV; in GaAs, this region’s width is 2.4 eV; and in ZnSe, this region has a width of only 1.2 eV (see table 1). Thus, as one goes from a covalent bonding The band widths of the valence bands plus the energy regions where the “s-like” band (region 1), the “light-hole p-like” bands (region 2), and the “heavy-hole p-like” bands (region 3) contribute to the density of states using Slater's and Kohn-Sham’s approximations in the SCOPW model. The energy of the top of the valence band is set equal to zero. Total Compound Exchange band Region 1 Region 2 Region 3 approximations width (eV) (eV) (eV) (eV) Si..................… Slater.......................... | | .75 — | 1.75 to — 7.68 – 7.68 to – 6.24 — 1. 16 to 0.00 Kohn-Sham................... | 2. ().5 — 12. ().5 to — 7.78 — 7.78 to — 6.56 — 4.86 to 0.00 Ge.............................. . Slater.......................... ||| 9 | — ||.91 to — 8.17 –8. 17 to — 6.21 — 3.69 to 0.00 Kohn-Sham................... 12. 12 – 12. 12 to — 8.26 – 8.26 to – 6.14 — 4.44 to 0.00 Alſ’............................. Slater.......................... | 1.46 — | 1.46 to — 9.0:1 – 5.4.2 to — 4.48 – 3.44 to 0.00 Kohn-Sham................... | 1.66 — ||.66 to — 8.95 – 6.09 to — 4.67 — 4.07 to 0.00 GaAs.......................... Slater.......................... | 1.8] — | 1.81 to — 9.42 – 6.22 to — 5.45 — 3.27 to 0.00 Kohn-Sham................... | 1.94 — | 1.94 to — 9. 14. – 6.58 to — 5.88 — 4.06 to 0.00 ZnSe... . . . . . . . . . . . . . . . . . . . . . . . . Slater.......................... | 1.82 — 1 1.82 to — 10.59 | — 4.47 to — 3.76 — 2.71 to 0.00 Kohn-Sham................... | 1.83 — | 1.83 to — 10.30 | — 4.86 to — 4.08 — 3.06 to 0.00 Si VALENCE BANDS SLATER'S EXCHANGE }* q = 5.431 Angstroms AIP VALENCE BANDS }* SLATER'S EXCHANGE q = 5.420 Angstroms Xıy. L 1 —1– l -60 ENERGY (eV) - |2.O - ENERGY (eV) (ſ) 3 Ge VALENCE BANDS * GG As VALENCE BANDS |- h- SLATER'S EXCHANGE, s xm. SLATER'S EXCHANGE s as 5.6575 Angstrons (ſ) o = 5.6532 Angstroms (ſ) Ui- 5 O H _^4v. X- : E o |-iv- £ 2 – – LA, H. X Wry Lal C IV -— O N 9 # ſ 3 – |-2'v O |- 0. OC H. H. O O Uil UAJ —l —l [1] hº Xıy. la! ul Wy Lil > > g º # TF, I # ſ DC I L f l —1– l | 1 I | 1– —w- - || 2.O – 9 O – 6.O – |2.O - – 6.O –3.O O.O ENERGY (eV) ENERGY (eV) % ZnSe VALENCE BANDS º X- SLATER'S EXCHANGE º, q = 5.650 Angstroms u- C – X- H (/) 2 uſ |- O C 2 C – Cº. H. O Lu I |-- Wiv LL ſ > H. Xiv <[ _ # / Y # |Iv i 1 1–1– 1 I - |2.O – 9 O – 6.O – 3.O O.O ENERGY (eV) FIGURE 4. The density of states of the valence bands of Si, Ge. AlF, GaAs, and ZnSe calculated using the SCOPW model with Slater’s exchange. 417–156 O – 71 – 8 97 Si VALENCE BANDS KOHN-SHAM'S EXCHANGE q = 5.43; Angstroms - 2. O Al P VALENCE BANDS KOHN-SHAM'S EXCHANGE q = 5.420 Angstroms Xsv |-v. – 9 O – 6.O –3.O O.O - || 2.O – 6.O ENERGY (eV) ENERGY (eV) Ge VALENCE BANDS KOHN-SHAM'S EXCHANGE q = 5.6575 Angstroms GG As VALENCE BANDS KOHN - SHAM'S EXCHANGE q = 5.6532 Angstroms W 3. V Li */ º – 6.O ENERGY (eV) ZnSe VALENCE BANDS q = 5.650 Angstroms XIv L V 2- KOHN- SHAM'S EXCHANGE º FIGURE 4. º- X3v. - |-Iv. Jiv | | l l l I –– l l – 2.O – 9 O – 6.O ENERGY (eV) The density of states of the valence bands of Si, Ge, AlF, GaAs, and ZnSe calculated using the SCOPW model with Kohn-Sham exchange. 98 Si CONDUCTION BANDS SLATER'S EXCHANGE |- d =543| Angstroms |-3c — | OO 3O 6.O ENERGY (eV) Ge CONDUCTION BANDS SLATERS EXCHANGE |-- a = 5.6575 Angstroms oã- 3.O 6.O ENERGY (eV) Al P CONDUCTION BANDS SLATER'S EXCHANGE d =5.42O Angstroms º- GOAS CONDUCTION BANDS SLATER'S EXCHANGE a=5.6532 Angstroms ZnSe CONDUCTION BANDS SLATERS EXCHANGE T q = 5.65O Angstroms W !C L3 c –– I.O 4.O 7 O ENERGY (eV) FIGURE 5. crystal to a more ionic bonding crystal the widths of the valence “s-like” and “p-like” bands decrease. How- ever, the overall bandwidth from the bottom of the “s- like” band to the top of the “p-like” bands changes less than 1 eV. In comparing the SCOPW results obtained using later's exchange approximation to those obtained sing Kohn-Sham’s approximation and to experiment, one notes that the fundamental band gaps of all of the ompounds calculated except Ge match experiment hen Slater's approximation is used (see table 2). This ENERGY (eV) 7 O 5.5 8.5 ENERGY ( e.V.) The density of states of the lower part of the conduction bands of Si, Ge, AlP, GaAs, and ZnSe calculated using the SCOPW model with Slater’s exchange. 2. 5 is also true when comparing the peak positions of the calculated e2 curves with experiments. In this case, Ge using Slater’s approximation also matches closely the experimental results. In no case are the results ob- tained using the Kohn-Sham exchange approximation closer to experiment than those obtained using Slater's. Another observation is that the fundamental band gaps obtained using superposition of overlapping free atomic potential (non-self-consistent) results with Kohn- Sham’s exchange is close to the SCOPW results using Slater’s exchange in some crystals. Thus, if one does 99 Si CONDUCT |ON BANDS KOHN- SHAM'S EXCHANGE AngSiroms O = 5.43| –l l l 3.O ENERGY (eV) 6.O Ge CONDUCTION BANDS KOHN-SHAMS EXCHANGE }*- q = 5.6575 Angstroms { 3C W2 c Wic I l 3.O ENERGY (eV) 6.O A|P CONDUCTION BANDS GOAS CONDUCTION BANDS — ZnSe CONDUCTION BANDS KOHN - SHAM'S EXCHANGE KOHN - SHAM'S EXCHANGF KOHN- SHAM'S EXCHANGE (ſ) - | H q = 5.42O Angstroms iſ H as 5,6532 Angstroms 3H d=5,650 Angstroms <ſ H. H. H. s: s $5 H 's T ‘5- X- t H £ % - É |- Xsc ÉH º C O S-2 O # - 3 H z- O OC 3. Or. H H 5 O Fisc O uſ – * | Lill— I Lil U. g ; Tic g H |- º Fº F. H. —l II. s ul Ł |X|c § T Cº. ſ | I l I |- 1– t if. | l | O.O 3.0 6.O |O 4 O 7 O 2.O 5.O 8.O ENERGY (eV) ENERGY (eV) ENERGY (eV) FIGURE 5. not carry his calculations to self-consistency, he could be lead to the incorrect conclusion that Kohn-Sham’s approximation matches experiment better than Slater’s approximation for these materials. The most noticeable difference between the results using Slater's approximation and those obtained using Kohn-Sham’s approximation is that region three has become wider for Kohn-Sham’s approximation. In going from Slater's approximation results to those of Kohn-Sham’s, the width of region three increases from 0.35 eV to 0.79 eV while the overall energy difference The density of states of the lower part of the conduction bands of Si, Ge Alp, GaAs, and ZnSe calculated using the SCOPW model with Kohn-Sham’s exchange. between the bottom of the “s-like” band to the top of the “p-like” band changes by an amount varying between 0.1 and 0.3 eV for all the crystals studied. Also region one’s width shows a smaller increase in going from Slater’s results to Kohn-Sham’s. b. Conduction Bands The general structure of the SCOPW results are again similar to the non-self-consistent results of Her- man, et al. [17]. One does not have the separation into 100 TABLE 2. The fundamental band gaps calculated from the SCOPW model using different exchange approxi- mations. Energy (eV)| Energy (eV) | Energy (eV) Compound Slater Kohn-Sham experiment exchange exchange Si (A1c - T250)................ 1.10 0.10 a 1.12 Ge (L1c - T250)............... 1.27 1.01 b 0.66 (T2'e - T250)............... 1.18 1.73 b 0.80 AlF (X1c - T59).............. 2.14 0.97 GaAs (T1c - T59)............ 1.6l 1.97 ° 1.54. (X1c - T150)............ 2.57 1.46 ZnSe (T1c - T59)............ 2.94. 2.68 d 2.83 * A. Frova and P. Handler, Phys. Rev. Letters 14, 178 (1965). * G. G. MacFarlane, T. P. McLean, J. E. Quarrington and V. Roberts, Proc. Phys. Soc. (London) 71, 863 (1958). * M. Cardona, K. L. Shaklee and F. H. Pollak, Phys. Rev. 154, 696 (1967). * M. Aven, D. Marple, and B. Segall, J. Appl. Phys. Suppl. 32, 226 (1961). well-defined regions as in the valence band. For exam- ple, the first four electron conduction states at the T point are lower in energy than any of the first four con- duction states at the W point. However, calculations for the five crystals give like results. In the first 6 or 7 eV of the conduction band, there are three main peaks; the lowest peak comes from the K point region; the middle peak from the L point region, and the third peak again from the K region (see fig. 5). The onset of the conduction band changes from crystal to crystal in these examples. The reason for this is that the density of states associated with the Tyc or Tic is very small compared with the Lic or Aic minimums of the conduction band. Thus, one sees that the density of states at the band edge for ZnSe or GaAs using Slater’s exchange approximation does not appear in figure 5 using this scale. The main difference between the SCOPW results using Slater's and Kohn-Sham’s approximation is that the X point and L point eigenvalues move to lower ener- gy relative to the T point energies. This, of course, is similar to the behavior found in region three of the valence bands. This can cause quite a change in the description of the calculated crystal. For example, the GaAs results using Slater's approximation gives a direct fundamental gap of 1.61 eV at the T point. When Kohn-Sham’s approximation is used, both the Xic and Lic energy states are lower than the Tic. This gives an indirect fundamental gap of 1.46 eV between T15, and X1c. c. d-States At this time a wealth of information is being obtained experimentally about the energy location and interac- tion with other states of the so-called “core” electrons in these materials. It is therefore interesting to examine the top d-state (which is the highest energy “core” state in Ge, GaAs, and ZnSe) that these calculations predict. In GaAs and ZnSe the non-self-consistent results (the potential is formed from a supposition of free atomic potentials) result in the d-states above the “s-like” valence band close to the “p-like” valence band. For example, in ZnSe, the Zn 3d state is 5.3 eV below the top of the valence band. However, the SCOPW results (using Slater’s exchange) show this state to be below the “s-like” band at an energy of 12.6 eV below the top of the valence band. In the case of GaAs, the SCOPW results give the location of the Ga 3d state some 16 eV. below the top of the valence band, and in the case of Ge, the SCOPW results give the location of the Ge 3d state 33 eV below the top of the valence band. In fact, the relative location of the Ge 3d state is close to the As 3d state which is 34 eV below the top of the valence band. A point of speculation as to the effective number of electrons per atom, meff, versus energy is added. Philipp and Ehrenreich [18] found a break in the meſ curve of III-V compounds which was not found in the smooth curve of Ge. Their interpretation of the cause of this effect was that they had reached the onset of the d band. In looking at the results of our model, we could hypothesize that this break was caused by the fact that the “s-like” and “p-like” valence bands are separated. Thus, the break in the meſſ curve could be caused by the onset of the “s-like” valence band. 3.2. Energy Dependent Exchange Approximation In all of this work, the effort has been made to simu- late a Hartree-Fock crystal. That is, we have made ap- proximations to the exchange term. And as pointed out in the previous section, the Slater approximation of the exchange term gives SCOPW results which are very close to experimental results except for Ge. However, this does not insure that the results are the same as a true Hartree-Fock calculation. In fact, if one looks at a simple metallic model [19], the Hartree results match experiment for metals. When one adds the exchange term, the bands become too wide. Thus, one does not expect a Hartree-Fock crystal to match experiment for these compounds. In the atomic case, the authors [11] have been able to match Hartree-Fock eigenvalues by using Liberman’s approximation for the exchange term. Ex- 101 TABLE 3. Si, Ge, AlF, GaAs and ZnSe calculated using Liber- man’s approximation in the SCOPW model. The energy of the top of the valence band is set equal to The band gap and valence band widths of Z62/"O. Total Band Com- band s-band width p-band width gap pound width (eV) (eV) (eV) (eV) Si............ 18.5 — 18.5 to — 12.3 – 12.3 to 0.0 0.9 Ge........... 17.1 – 17.1 to — 12.5 – 12.5 to 0.0 1.5 AlP.......... 17.5 – 17.5 to 14.6 –9.3 to 0.0 2.8 GaAs ....... 16.8 — 16.8 to 14.0 – 9.9 to 0.0 2.8 ZnSe........ 15.5 — 15.5 to 14.1 –9.2 to 0.0 5.0 tending these calculations to the crystal case, one finds the results of the SCOPW model given in table 3. In comparing the band gaps with the experimental results given in table 2, it is easily seen that Liberman's ap- proximation results are not as good as the results using Slater's approximation. However, comparing valence band widths, one finds that Liberman’s results give bands which are too broad compared to experiment, and this is what is expected from a Hartree-Fock description of these crystals. 4. Conclusion It is concluded from this study that the SCOPW results using Slater's approximation for the exchange give a good description of the electron density of states. This good description was a result of the SCOPW model. That is, there was no empirical adjustment (or any other kind) made after the fact. The inputs to the model consist of lattice constants and the number of electrons. It is also noted from this study that although the SCOPW results using Liberman’s exchange approxi- mation are further away from matching experiment, the results appear to be closer to the Hartree-Fock descrip- tion for these materials. Thus, if one wishes to improve upon the Hartree-Fock method, such as adding Cou- lomb hole and screened exchange terms as investigated by Pratt [20], Hedin and Lundqvist [21,22], it is better to start from Liberman’s exchange approximation rather than Slater's exchange approximation. 5. References [1] Slater, J. C., Phys. Rev. 81,385 (1951). [2] Kohn, W., and Sham, L. J., Phys. Rev. 140, All33 (1965). [3] Gaspar, R., Acta Phys. Acad. Sci. Hung. 3, 263 (1954). [4] Liberman, D. A., Phys. Rev. 171, 1 (1968); Sham, L. J., and Kohn, W., Phys. Rev. 145, 56 (1966). [5] Euwena, R. N., Collins, T. C., Shankland, D. G., and DeWitt, J. S., Phys. Rev. 162,710 (1967). [6] Stukel, D. J., Euwema, R. N., Collins, T. C., Herman, F., and Kortum, R. K., Phys. Rev., March 1969. t [7] Collins, T. C., Stukel, D. J., and Euwena, R. N., (Phys. Rev. in press). [8] Stukel, D. J., and Euwenna, R. N., (to be published). [9] Stukel, D. J., and Euwena, R. N., (Phys. Rev. in press). [10] Stukel, D. J., and Euwema, R. N., (Phys. Rev. in press). [11] Stukel, D. J., Euwema, R. N., Collins, T. C., and Smith, V., (Phys. Rev. in press). [12] Herring, C., Phys. Rev. 57, 1169 (1940). [13] Slater, J. C., Quantum Theory of Atomic Structure, Vol. II, Ch. 17, App. 22. [14] See also the derivation of P. O. Löwdin, Phys. Rev. 97, 1590 (1955). [15] Slater, J. C., Wilson, T. M., and Wood, J. H., Phys. Rev. 179, 28 (1969). [16] Euwena, R. N., Stukel, D. J., Collins, T. C., DeWitt, J. S., and Shankland, D. G., Phys. Rev. 178, 1419 (1969). [17] Herman, F., Kortum, R. L., Kuglin, C. D., and Shay, J. L., II-VI Semiconducting Compounds, 1967 International Conference, D. G. Thomas, Editor (W. A. Benjamin Inc., N.Y., 1967), pp. 503–551. [18] Philipp, H. R., and Ehrenreich, H., Phys. Rev. 129, 1550 (1963). [19] Lundqvist, S., International Conference of Optical Properties of Solids, July 1969, (International Center for Advanced Studies, Chania, Crete, Greece). [20] Pratt, G., Phys. Rev. 118,462 (1960). [21] Hedin, L., Phys. Rev. 139, A796 (1965). [22] Hedin, L., and Lundqvist, S., Quantum Chemistry Group, Upp- sala, Sweden, Technical Report T III (1960). 102 Discussion on “Theoretical Electron Density of States Study of Tetrahedrally Bonded Semiconductors” by D. J. Stukel, T. C. Collins, and R. N. Euwerma (Aerospace Research Laboratories, Wright Patterson Air Force Base) F. Herman (IBM Res. Center, San Jose): I would like to comment on the first paper by Stukel, Collins and Euwena. I really think that the reason they find the results they do is that they have not asked the right question. There is no theoretical basis for there to be a connection between an energy eigenvalue spectrum based on an approximate Hamiltonian and an optical excitation spectrum. If one does calculations using dif- ferent exchange approximations and simply compared the eigenvalues with experiment there is simply no reason to expect agreement. This point prompts one to either use an empirically adjusted first principles method or else to change the question which is “How does one take an energy eigenvalue spectrum and from that with suitable modification find a spectrum that does correspond to an optical spectrum?” T. C. Collins (Aerospace Res. Lab.): What I would like to point out is that there is theoretical basis for using eigenvalues. First of all, the eigenvalues using Slater’s [1] exchange term match experiment within 0.2 eV. We do have one failure, germanium, which is off by half a volt in the indirect gap. All the rest fit to within that limit. There is no empirical adjustment needed or used in these calculations. In these calculations one wants binding or excitation energies of an electron. One generally sets up the equa- tion for the total energy. At this point you can substitute p'ſ" for the exchange term and vary. In this case you come out with 2/3 the value for the coefficient [2] in front of the exchange term in the effective Hamiltonian. Or, you can vary the total energy, then substitute pºſ” for the exchange term. In this case you get Slater’s value. Now where do you go to find binding energies? The way to find binding energies is to calculate the total energy of the N electron system and the N-1 electron system, then take the difference between the two. In the calculations, where does one substitute the exchange approximation? Making the approximation before the difference is taken one finds the eigenvalues obtained with the 2/3 approximation match the value for the binding energy. Likewise, if the substitution is made after the difference is taken, the eigenvalues ob- tained with Slater's value of the exchange approxima- tion match the binding energy. So essentially your eigenvalues, in either case, represent binding energies. We have also tried a “better” approximation, that of Liberman [3], for the exchange term. We derived the wave functions using all three values for the exchange term and calculated binding energies with Liberman’s approximation — we still got bad results. The only thing that matches experiment are the eigenvalues obtained using Slater's pºſ” value substituted for the exchange term. The same thing happens in the atom. For example, we looked at Kr. Eigenvalues obtained using Slater’s value for the exchange matched experiment where Har- tree-Fock eigenvalues did not. But, when we took the N and N-1 total energy difference the Hartree-Fock an- swers were in very close agreement with experiment. This was true all the way down the line from the 1s value to the 4p value. So it is just “magic” that p1/8 values of Slater’s works. We are trying to find out why. One of our purposes in this study is to develop a method to get the proper term. E. T. Arakawa (Oak Ridge National Lab.): We have measured [4] the e2 absorption of Na and K. Although we have some scattering of points, it does appear that there definitely is a plasmon contribution in the absorp- tion starting right around 6 eV for sodium. [1] Slater, J. C., Phys. Rev. 81,385 (1951). [2] Kohn, W., and Sham, L. J., Phys. Rev. 140, All 33 (1965). [3] Liberman, D. H., Phys. Rev. 171, 1 (1968). [4] Sutherland, J. C., Hamm, R. N., and Arakawa, E. T., J. Opt. Soc. Am. 59, 1581 (1969). 103 Electronic Density of States in Eu-Chalcogenides S. J. Cho Division of Physics, National Research Council of Canada, Ottawa, Canada The spin-polarized energy bands and the electronic density of states in the Eu-chalcogenides have been obtained by the augmented-plane-wave (APW) method. The results show that the fibands are ex- tremely sensitive to the exchange potential used, and the f(? ) bands become the highest valence bands with a band width of the order of 0.5 eV. Our results have been compared with the recent photoemission spectroscopy data. The UPS data show too large f band width and too small relative peak intensities of the fibands, which disagree with our results. The 4f bands in the Eu and Gd could be located within 3.0 eV below their Fermi energies. Key words: Augmented plane wave method (APW); electronic density of states; europium-chal- cogenides; exchange potential; f bands; photoemission. In the last few years a number of authors have stud- ied the density of states N(E), mainly for the transition and noble metals, by various experimental methods: UV photoemission spectroscopy (UPS), ion neutraliza- tion spectroscopy (INS), soft x-ray spectroscopy (SXS), and x-ray photoemission spectroscopy (XPS). Such measurements can provide useful information for un- derstanding the electronic band structures in solids, and can be used as a tool to justify the theoretical ener- gy band calculations. As far as transition and noble metals (Ni, Fe, Cu, and Co) are concerned there ap- pears to be qualitative agreement between theory and experimental results obtained by various methods [1-4] (except for early UPS reports [5]). A probable exchange splitting AEer of the energy bands in Ni at or near the Fermi level Ef has been reported to be about 0.35 eV [6]. There are few theoretical and experimental studies on the density of states for the rare earth metals and their alloys. Müller [7] has studied room temperature reflectivity and transmission for Eu and Ba metals (up to 4 eV), which have isoelectronic structures. He has found almost identical optical behavior for Eu and Ba metals, and has concluded that the 4f electrons do not influence the optical spectrum, and that the 4f levels might be located far below EP. Schüler [8] has studied temperature dependent transmission of Gd and Lu thin films, from which he has found that the transmission maximum of Lu is located at 0.75 eV which is independ- ent of temperature. The transmission maximum of Gd is located at 0.6 eV at room temperature and is split into two peaks below the Curie temperature Tc, at 0.8 eV and 0.45 eV respectively. The extra peak is due to the spin-polarized exchange splitting of the bands, which is estimated to be about 0.4 eV for Gd. However he has not obtained any information on the possible f band positions. Blodgett et al. [9] have made UPS studies for Gd, and reported that a possible f(?) band location is about 6 eV below Ef with a work function of 3.1 eV, and that a possible AEer is less than 0.1 eV from their tem- perature dependent photoemission measurements. On the other hand several authors [10.11] have studied theoretically the energy bands in the rare earth ele- ments. However, they have found it difficult to locate the fiband positions properly because of its sensitive dependence on the exchange potential used. Recently both Busch et al. [12] and Eastman et al. [13] have studied UPS for the Eu-chalcogenides. Their results are reproduced in figure 1. In this work we have studied the spin-polarized energy bands in the Eu-chal- cogenides in terms of the augmented-plane-wave (APW) method. We have found that the f band posi- tions are extremely sensitive to the exchange potential used. In our work the p" exchange potential [14] for the magnetic Eu" ions has been reduced by a factor of 3/4, which has produced proper energy gaps and rela- tive f band positions for the Eu-chalcogenides. Accord- ing to our calculations the f(?) bands are located in between the anion p band and the 5d conduction band X3. The calculated f(?) band width is about 0.5 eV and 105 |.5H | OH- O.5H O * 3H (\] C >< * 2H (ſ) 2 O H- | H. O T 0. N. O P. p states N. EuSe % 2H- 3. hly - iO.2e H O 8.1 eV LL] — Lil | H. 4f sº-se 6.5eV U (x4) 2 p states O.4 – Gds q = 2.3 + 0.2ev h!/ = }O.2eV. 6.9 eV (x8) O.3H- 8.6eV(x2) 7.7eV (x4) }* CONDUCT O.2 /BAND O. H. O f | l | º I -IO –8 -6 –4 –2 or EF El = VR * hy + £e (eV) FIGURE 1. Experimental density of states curves for EuO, EuS EuSe, and GdS (ref [13]). the up- and down-spin f band separation is about 6.0 eV. We have obtained the electronic density of states for the Eu-chalcogenides for 256 points in the Brillouin zone, and the results for EuO are shown in figure 2. Normally a knowledge of energy structures at more than 256 points is required in order to obtain reliable N(E) curves in solids. In our case the valence bands are well isolated from each other and present work should give us fairly reliable information. On the other hand the conduction bands are quite complex and our results might not represent detailed structures which could ex- ist. According to our results for the conduction bands there are two peaks which are mainly derived from the 4O }* Ul U 2 -— f(#) 3O EuO AE = 0.067Ry % UP – SPIN 2O * |O * > % olº tº , i. tº % % % % DOWN – SPIN % % %. 3O H. % % f(#) D % D3 º 4O H. à º § FIGURE 2: Theoretical density of states curve for EuO (N(E)/Ry.| ) unit cell tºg and eg d bands for the up-spin electrons, and three peaks for the down-spin electrons which are due to the t2g and en d bands and the f(J, ) bands. Both calculated and experimental data are tabulated in table 1. We can immediately notice from figures l and 2 that the experimentally observed density of states of f electrons N(E, f) is considerably smaller than the den- sity of states of the p electrons N(E, p), that experi- mentalf band widths are larger than expected for the f" (Euºt) band width, and that the experimental p band width of EuO is relatively larger than the corresponding p band widths for EuS and EuSe. These experimental results are in contrast to the theoretical results of a ratio of N(E, f):N(E, p) = 7.6, of about 0.5 eV for the band width, and of an almost constant p band width of about 2 eV for all Eu-chalcogenides. It is not clearly known why the measured N(E, f) is so small. It might be related to the difficulty of releasin more than one f electron because it takes much large ionization energies for subsequent f electrons from th 106 TABLE 1. Theoretical and experimental data (eV) of the valence band structures EuO EuS EuSe GdS f band width................... (a) 0.57 0.54 0.70 (b) 1.6 1.1 (e) 2.0 1.3 1.5 1.0 p band width.................. (a) 2.12 2.19 2.33 (b) 2.0 1.3 (°) 3.0 2.3 2.4 2.9 fp separation................. (a) 1.41 0.44 ||— 0.15 (b) 1.7 — 0.4 (°) 2.5 0.5 — 0.8 < 0 Average fºp separation....... (a) 2.00 1.20 0.50 (b) 2.5 0.8 - (°) 3.5 2.0 1.8 1.9 p to conduction band...... (a) 3.6] 3.04. 2.37 (b) 3.8 2.4 (°) 4.3 3.1 3.1 3.0 * Present work. b Ref. 12. * Ref. 13. same Eu atom. Another possibility would be a small transition probability for the transitions from the f(?) bands to the vacuum level due to the small density of states at or near the vacuum level, or that transitions are nondirect [15]. In principle the dominant transitions from the flat |f(?) bands should be direct transitions. However, the f electrons have one of the heaviest effective masses and their velocity should be very small. Accordingly the f electrons involve multiple scattering with phonons and electrons before reaching the surface, or some of them are captured by existing Eu’t ions which are either impurity centers or created from the Eu3+ ions. In this case the observed N(E, f) should become con- siderably broader and different from the initial N(E, f) which we are attempting to measure. Another possible origin of the large f band width ob- served could be related to the FJ (fº) multiplets of Eu3+ ions (atomic FJ multiplets have a band width of 0.6 eV [16]). Both Busch et al. [12] and Eastman et al. [13] have interpreted such possible FJ multiplets as arising from the Eu3+ ions created from Eu2+ ions. We expect that there are about 0.1% concentrations of Eu2O3 impurities in the samples. Therefore if F multiplets are involved, they would be more likely from Eu” impurities rather than from Eu2+ ions created from Eu%t ions. However, we cannot rule out the possibility of Fj multiplets of Eu°t being created from Eu2+ ions. This problem could be resolved from the similar studies with excess Eu’t impurities in the sample. In any case experiments by both Busch et al. [12] and Eastman et al. [13] do not give us proper information on the N(E, f). Busch et al. have reported a linear variation of the f band position with incident photon energies, and East- man et al. have shown f band positions independent of photon energies. According to the above UPS experi- ments the possible Fj multiplets of Eu2+ ions are located just below the f"(*S*12) bands. The considerably largerf band width of EuO compar- ing with the corresponding values observed for EuS and EuSe seem to indicate that the possible F, multiplet width decreases with increasing lattice constant, or that there is a larger amount of scattering for EuO than for EuS and EuSe due to the smaller lattice constant of EuO. As we can see from table 1, not only the experi- mental data of EuO disagree with theory, but two ex- perimental results also disagree with each other. It ap- pears that the experimental data of EuO by Busch et al. show better agreement with theoretical results than the results by Eastman et al. On the other hand the 4p band width of EuSe by Busch et al. is too small. In the case of EuS and EuSe there is reasonable agreement between theory and experiments, except for the fiband width. It is interesting to note that the relative positions of the top of the fand p bands among EuO, EuS, and EuSe show good agreement between theory and the ex- periments (see table 2). Eastman et al. [13] have also studied UPS for GdS, (see fig. 1), in which they have found that the overall situation is not much different from EuS, except for a partially filled valence 5d band, and that the possible N(E, f) is further weakened and has no sign of 4f7->4fº 5d transitions. These experimental results are interest- ing because they tell us that any reflectivity or absorp- TABLE 2. Energy differences of the highest f and p bands among Eu-Chalcogenides (eV) p bands f bands EuO-EuSe...................... (a) 3.0 1.3 (b) 2.9 1.1 (e) 3.1 1.0 EuS-EuSe...................... (a) 0.4 0.06 (°) 0.7 0.04 * Present work. b Ref. 12. * Ref. 13. 107 tion peaks from the 4f bands in GdS are difficult to ob- serve. In the reflectivity or transmission experiments for Eu [7] and Gd [8] we have not observed any possi- ble interband transitions from 4f bands, which could be related to almost negligible transition probability from the f(?) bands. According to the photoemission measurements for Gd metal by Blodgett et al. [9], there is a large d band peak at or near Ef and a broad peak at about 6 eV below Ef. In addition there is a small peak at about 2.8 eV below Ef. They have not elaborated to discuss a small peak at 2.8 eV. The same authors [5] have also re- ported a large peak at about 5 eV below Ef for Co, Fe, and Ni, which has been found to be spurious. Referring to experimental data in table 1 for the Eu-chalcogenides and GdS, it is reasonable to expect that the possible f(?) band positions in Eu and Gd metals could be located at less than 3 eV below Ef, and that a small peak at about 2.8 eV below Ef in Gd observed could be the possible f(?) band position. AEer for Gd and the Eu-chalcogenides should be larger than the coresponding values of 0.35 eV for Ni because the magnetic moments of Gd and the Eu-chal- cogenides are more than 11 times that of Ni. The AEer of about 0.4 eV for Gd estimated from transmission data [8] is a reasonable value. We have also obtained the AEer values of about 0.4 - 0.5 eV for the Eu-chal- cogenides [17]. Because of the exchange splitting of energy bands below Tc, the up-spin electrons have lower energy than corresponding down-spin electrons by the amount of AEer. Therefore, it takes more energy to lift up-spin electrons than down-spin electrons under the same ex- perimental conditions. Accordingly, in principle, we should be able to observe such band width broadening of AEer in the temperature dependent studies of N(E). However, practically constant N(E) with variable tem- peratures reported for Fe and Co by XPS [1], for Ni by UPS [5] and INS [3], and for Gd by UPS [9] are in contradiction to above physical phenomena. At present various experimental methods to study V(E) have shown poor energy resolution. Therefore further experi- mental work could elucidate this problem. UPS data for Eu-chalcogenides [12,13], GdS [13]. and Gd [9] mentioned above are based on an assump- tion of equal transition probability from various occu- pied bands throughout the Brillouin zone, which is cer- tainly not a reasonable assumption for the case of f electrons because of the small number of transitions from f band density of states observed. It would be worthwhile to carry out more experimental studies by using other techniques such as SXS, INS, or XPS to see whether we can obtain more realistic information on the N(E, f). References [1] Fadley, C. S., and Shirley, D. A., Phys. Rev. Letters 21, 980 (1968). [2] Cuthill, J. R., McAlister, A. J., Williams, M. L., and Watson, R. E., Phys. Rev. 164, 1006 (1967). [3] Hagstrom, H. D., and Becker, G. E., Phys. Rev. 159, 572 (1967). [4] Eastman, D. E., J. Appl. Phys. 40, 1387 (1969). [5] Blodgett, A. J., Jr., and Spicer, W. E., Phys. Rev. 146, 390 (1966); 158, 514 (1967); Yu, A. Y.-C. and Spicer, W. E., Phys. Rev. 167,674 (1968). Wohlfarth, E. P., 1964 International Magnetism Conference (In- stitute of Physics, London, 1965), p. 51. [7] Müller, W. E., Phys. Kondens. Mat. 6, 243 (1967). [8] Schüler, C. C., Optical Properties and Electronic Structure of Metals and Alloys, p. 221, North-Holland, 1966. - [9] Blodgett, A. J., Jr., Spicer, W. E., and Yu, A. Y.-C., Optical Pro- perties and Electronic Structures of Metals and Alloys, p. 246, North-Holland, 1966. [10] Fleming, G. S., Liu, S. H., and Loucks, T. L., Phys. Rev. Let- ters 21, 1524 (1968). [ll] Dimmock, J. O., and Freeman, A. J., Phys. Rev. Letters 13, [ 6 | 750 (1964). [12] Busch, G., Cotti, P., and Munz, P., Solid State Commun. 7, 795 (1969). [13] Eastman, D. E., Holzberg, F., and Methfessel, S., Phys. Rev. Letters 23, 226 (1969). [14] Slater, J. C., Phys. Rev. 81,385 (1951). [15] Berglund, C. N., and Spicer, W. E., Phys. Rev. 136, AlO30 (1964). [16] Dieke, G. H., and Crosswhite, H. M., Appl. Optics 2,675 (1963). [17] Cho, S.J., (to be published). 108 Discussion on “Electronic Density of States in EU-Chalcogenides” by S. J. Cho (National Research Council of Canada, Ottawa) D. E. Eastman (IBM, New York); With regard to a S. J. Cho (National Res. Council); I have already “band” description of the 4f electrons in the Eu-chal- discussed this subject elsewhere in this Conference cogenides, there is considerable experimental evidence (see “Ultraviolet and X-Ray Photoemission from Eu- that the 4f electrons are very localized (with important ropium and Barium” by G. Broden et al.). correlation effects), e.g., a band description appears to be inadequate. 109 discussed. spectra. The dioxide of germanium exhibits a polymorphism which is rather unique in nature. The commonly occur- ring form is the O-quartz hexagonal structure which is relatively reactive and is the thermodynamically unsta- ble phase at ordinary temperatures. The form that does not occur naturally is the rutile tetragonal structure, and it has until recently been prepared only by conver- sion from the hexagonal phase. In the last few years, methods have been developed for the growth of single crystals of the tetragonal material, enabling the deter- mination of optical and electronic properties. This tetragonal form has proven to exhibit chemical proper- ties which are consistent with possible applications as masks and protective coatings for germanium semicon- ductor devices. It also exhibits interesting optical pro- perties in the near ultraviolet region of the electromag- netic spectrum. For these reasons, tetragonal GeO2 warrants some attention, and we have determined the energy band structure in order to aid the design and in- terpretations of optical and electrical experiments imed at a better fundamental understanding of the hysical nature of this material. The results of these reliminary investigations are reported herein. Tetragonal GeO2 possesses the rutile structure with space group D}} [1]. The unit cell is tetragonal, with c/a ratio of 0.65, and contains six atoms, two germani- m and four oxygen. The Brillouin zone is also simple etragonal. Energy Band Structure and Density of States Tetragonal GeO, F. J. Arlinghaus and W. A. Albers, Jr. Research Laboratories, General Motors Corporation, Warren, Michigan 48090 The electronic energy bands of tetragonal GeO2 have been calculated and correlated with optical properties of single crystals of this material. The agreement between theory and experiment is suffi- ciently good to warrant calculations of densities of states and conduction band effective masses preparatory to the determination of the dielectric constant as a function of energy. The calculated ener- gy bands, density of states, and the experimental optical absorption edge data are presented and Key words: Augmented plane wave method (APW); electron density of states; GeO2; indirect transition; optical properties; Slater exchange; vacuum ultraviolet reflectance A self-consistent APW (augmented plane wave) ener- gy band calculation was performed for GeO2. Sphere radii were chosen as follows: the oxygen spheres were made to touch, giving as large an oxygen radius as possible; then the germanium spheres were made to touch these. Thus chosen, the spheres fill 559% of the space in the crystal. The starting potential was derived from Get” and O- ionic potentials. The Get" ionic potential was calcu- lated by the Herman-Skillman procedure [2]; the O- potential is that of Watson [3]. An Ewald problem is solved to obtain the average potential for the region out- side the spheres [4]; exchange is included by means of the Slater p" approximation [5]. At each stage of the calculation, bands are obtained and a new trial potential obtained from the calculated charge distribution. Ten iterations were required be- fore the eigenvalues were stable to within 0.005 Ryd- berg. It was found necessary to include the higher core states (the oxygen 2s- and germanium 3d-bands). The band structure is shown in figure 1. The main features are the broad conduction band with a parabolic minimum at T and the profusion of valence bands, the higher ones quite flat, arising principally from oxygen 2p levels. The direct gap is at T; its magnitude is calculated to be 5.52 eV, for transition T5+ → Tit, allowed for light polarized L to the c-axis. The value of 6.04 eV is cal- 111 i FIGURE 1. Electronic energy band structure of tetragonal GeO2. culated for transition Ti -> Ti, allowed for polari- zation. A basic feature of the calculated bands is the presence of an indirect edge. The direct gap I st → TIt is 5.52 eV; however, the valence band state R1, on the top edge of the zone, is slightly higher than the Tst state, so that the indirect transition R → Tit is only 5.25 eV. This indirect edge is different from that of ger- manium, in that the valence band comes up away from the zone center; in all other cases of an indirect edge known at present, the conduction band comes down. The numbers resulting from these calculations are in good agreement with the somewhat limited experimen- tal data available to date. The fundamental optical ab- sorption edge has been studied in some detail in tetragonal GeO2 single crystals at room temperature. The resulting data are summarized in figure 2. The rather large dichroism observed at the absorption edge is consistent with the Ts" → T1" and the T1--> Tit transitions discussed above. The experimentally deter- mined energy gaps, assuming 0, the absorption coeffi- cient, proportional to (hu – Em)" at high absorption coefficients (see inset of fig. 2), are 4.99 and 5.10 eV [6]. These values are somewhat lower than those pre- dicted by the energy band calculations, but the agree- ment is considered quite satisfactory in view of the preliminary nature of the calculations. º 10 | hºm FIGURE 2, 3.5 4.0 4.5 5.0 - 5.5 fia (eV) Room temperature optical absorption edge of tetragona GeO2. 112 The data of figure 2 indicates considerable contribu- tions to the absorption coefficient at energies lying below the fundamental direct gaps. This is consistent with the possibility of an indirect transition as pre- dicted by the band calculations (R1 => lit). However, also possible are exciton and impurity state transitions in this same range of energy, and it has not been possi- ble to sort out the relative contributions in order to ascertain the existence of the indirect transition unam- biguously. We are currently studying the absorption edge at low temperatures in an effort to clarify this point. Preliminary vacuum ultraviolet reflectance spectra on single crystals of tetragonal GeO2 at room tempera- ture have been obtained by William Scouler at MIT’s Lincoln Laboratory. His results suggest a possible transition occurring in the region of 7.5–8.0 eV. The calculated band structure predicts a direct transition at the X-point (X1 -> Xl in fig. 1) at about 8.4 eV. Since the band calculations predict slightly larger direct gaps at T than experimentally observed, we feel justified in ten- tatively assigning the 7.5–8.0 eV structure in the reflectivity to the X-point transition. On the basis of the above discussion, we conclude that the band calculations presented here are reasonably representative of the true electronic energy structure of tetragonal GeO2. We have therefore deter- mined the density of states, effective masses, and mo- mentum matrix elements preparatory to a calculation of the dielectric constant as a function of energy. The density of states is shown in figure 3. This was obtained by summing states in 0.02 Rydberg energy intervals over a 64-point mesh in the Brillouin zone and smoothing the resulting data. Although more refined density of states could be obtained with a finer mesh, we feel that the basic structure of figure 3 will not be grossly altered by the inclusion of various other points of the Brillouin zone. Effective masses of the anisotropic conduction band electron have been calculated to be 0.42 m perpendicu- lar to the c-axis and 0.47 m parallel to the c-axis. These | | | ! | 1.2 H. - Tetragonal Geo, 1.0 H - - § 0.8H sºm # Cº. >- 3 0.6 H - # 0.4 H - 0.2 H. - 0 | | | | | | 0 10 20 30 40 50 60 DENSITY OF STATES (Arbitrary Units) FIGURE 3. Calculated density of states for tetragonal GeO2. values are consistent with rather crude estimates of the electron effective mass from the optical data. We conclude that the energy bands of tetragonal GeO2 presented herein constitute a good basis for the prediction and analysis of experiments related to the optical and electronic properties of this material. References [1] Gay, J. G., Albers. W. A., Jr., and Arlinghaus. F. J., J. Phys. Chem. Sol. 29, 1449 (1968). [2] Herman, F., and Skillman, S., Atomic Structure Calculations (Prentice-Hall Inc., Englewood Cliffs, N.J., 1963). [3] Watson. R. E., Phys. Rev. 111, 1108 (1958). [4] Slater, J. C., and DeCicco, P. D., M.I.T. Solid State and Molecu- lar Theory Group Quarterly Progress Report No. 50, p. 46 (Oc- tober 1963). [5] Slater. J. C., Phys. Rev. 81,385 (1951). [6] Smith. R. A. Semiconductors (Cambridge University Press, London, 1959). 113 energy less than 5 eV. approximation; tight-binding. 1. Introduction The Augmented Plane Wave (APW) method has een applied to PbTe by Conklin, Johnson, and Pratt 1] who have obtained the relativistic energy bands at ine points of high symmetry in the Brillouin Zone (BZ). n this work we have used these energy bands to calcu- te the electronic density of states in energy and the in- rband electronic contribution to the optical constants f PbTe, sed to test the validity of an energy band picture, is .3 V in PbTe at room temperature. This is smaller than e expected accuracy of even the most optimistic first rinciples calculation. However, the density of states rovides a less stringent test of the validity of an energy and picture, because the density of states depends pon general features of the bands over a large region k-space and not upon the details of the bands at any he point in the BZ. The optical properties, on the other na, do depend critically on the energy difference tween regions of high density of states and for this This work was supported in part by the National'Science Foundation and in part by the my Research Office (Durham). *Present address: Physics Research Laboratory, Texas Instruments Incorporated, llas, Texas. **Present address: Itek Corporation, Lexington, Massachusetts. The direct absorption edge, which is most commonly Calculation of the Density of States and Optical Properties of PbTe from APW-LCAO Energy Bands* D. D. Buss” and V. E. Schirfº Department of Electrical Engineering and Center for Material Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts O2139 The overlap integrals in the tight-binding secular equation for the relativistic p-bands in PbTe have been adjusted to give the best representation of the APW results at high symmetry points. The resulting LCAO bands have been used to calculate the density of states in energy and the optical constants of PbTe, The calculated density of states is found to have peaks which correspond closely to the four peaks measured by Spicer and Lapeyre. However, the assignment of peaks to bands is found to be dif- ferent from that proposed previously. The method of Dresselhaus and Dresselhaus has been used to ob- tain the oscillator strengths for optical transitions, and these are found to agree with previous calcula- tions. The interband electronic contribution to the optical constants has been calculated for photon Key words: Augmented plane wave method (APW); electronic density of states; k p method; LCAO; lead telluride (PbTe); optical properties; pseudopotential; random phase reason provide a more exacting test of the validity of an energy band picture. For both calculations, one must know the energ bands everywhere in k-space, and, because APW cal- culations are long and tedious, they can be performed only at a small number of points of high symmetry. Therefore, one is forced to use an interpolation scheme to obtain the bands in between points where the APW results are known. Three interpolation methods are in common use today. The Linear Combination of Atomic Orbitals (LCAO) or tight binding method was first proposed by Slater and Koster [2] and has been employed exten- sively by Dresselhaus and Dresselhaus [3]. The k. p. method has been in use for a long time but was first used to represent a large number of bands over the en- tire BZ by Cardona and Pollak [4]. More recently this method has been applied to PbTe [5]. The pseu- dopotential method is the most widely used of the three, and has been used extensively to study the opti- cal properties of the IV, the III-V, and the II-VI materi- als [6]. More recently the optical properties of PbTe and related compounds have been calculated using this method [7]. In this work an LCAO representation of the bands in PbTe was obtained by adjusting the band parameters 115 to give agreement with the APW energies as discussed in section 2 and appendix A. The density of electron states in energy was calculated from these bands, and the results are compared with photoemission data [8] in section 3. Section 4 describes the calculation of opti- cal properties and compares the calculation with ex- periment [9] and with previous calculations [5]. Oscil- lator strengths for optical transitions have been ob- tained directly from the energy bands using the method of Dresselhaus and Dresselhaus [3]. 2. LCAO Energy Bands Following the method proposed in reference 2, we have expressed the crystal wavefunctions as linear combinations of Bloch sums of Löwdin functions. The Löwdin functions resemble free atom wavefunctions, but they have the property that such functions located on neighboring atomic sites in a crystal are orthogonal. For this application Löwdin functions were chosen to resemble the Pb 6p and the Te 5p free atom wavefunc- tions. The former are located on the Pb atom sites R. and the latter are located on the Te atom sites R + T where T = aſ 2(100). The orbital portion of the Löwdin functions can be written as aa"(r — R) and aa"(r — R — T) where O = x, y, z. The orbital part of the Bloch sums is b; I (r) = #. [a]”(r–R) w * R = affe (r–R – T)e il; r |e ik R (1) CY where N is the number of unit cells in the crystal. (a!"(r–R), s|H, aft”(r-R"), s')=}\!”(a!"(r),s |L o aft”(r), s')ön, ſº (age (r–R-T) , S Hso a!"(r-R'-T), s') , - (a!" (r–R), s|Hola!" (r-R'-T), s') = (a!" (r-R-T), s|H, aft”(r-R"), s') = 0 (4 where s and s' designate spin coordinates (s, s'= ? , , ), and \" and \"" are constants. Thus Hso introduces two additional parameters into the tight binding Hamil- tonian, and, to the extent that the approximations made here are justified, \" and \" should resemble the spin- orbit parameters which pertain to the free atom. The 18 band parameters were obtained from the APW calculation in the following way. Reference 1 gives the intermediate results which were obtained by first solving the nonrelativistic problem and by sub- sequently including the Darwin and Mass-velocity cor- rections. The two spin-orbit parameters were set equal # X^{age(r), S | L • OF | agº (r) •) s') or, ſº When the crystal Hamiltonian is separated into a spin-independent part and a spin-orbit part H = H's -- Hso (2) the six Löwdin functions of eq (1) form a six-dimen- sional basis for Hsi. In this basis, matrix elements of Hi take the form of a Fourier series in k-space [3] in which the Fourier coefficient for the nth term in the ex- pansion involves matrix elements of Hsi between Löwdin functions which are nth neighbors, and, since Löwdin functions are localized, the expansion con- verges rapidly. In this work the series has been trun- cated with fourth neighbor terms, giving 16 independ- ent Fourier coefficients. When spin is included into the problem, the number of basis states doubles, but Kramer's theorem assures us that each level has at least a two-fold degeneracy. Every term in the Hamiltonian which does not explicitly involve electron spin is included in Hsi so that Ho-ji, or [(VW) X pl. (3) This term is quite important in PbTe [1], but since V is large only near the nucleus, it is reasonable to as sume that the only matrix elements of Hso which are im portant are those between Löwdin functions on th same lattice site, and in this work, all other matrix ele ments are taken to be zero. This simplification is equivalent to the approximation, suggested by atomic theory, that to zero, and the remaining 16 band parameters were ac justed to minimize the r. m.s. difference between th LCAO bands and the intermediate APW results give in table 1 [10]. The band parameters are defined in aſ pendix A and their values obtained are given in table These bands reproduce the 48 data points of table 1 : within an r. m.s. error of 0.014 Ry, and the size of t band parameters falls off rapidly with increasing ord as it must for the scheme to have physical meaning. The 16 nonspin-orbit parameters so determined we then fixed, and the two spin-orbit parameters were a justed to give the best r. m.s. fit to the final results 116 TABLE 1. APW results of Ref. 1 (see Ref. 10). The LCAO band parameters were adjusted to these energies I ((), (), ()) A (l, (), ()) X (2, 0, 0) X. (1/2, 1/2.0) Intermediate Final Intermediate Final Intermediate Final Intermediate Final –0.305 || 15 —0.262 I's –0.274 As –0.226 A7 —(). 136 X; –0. 132 X; –0.309 X. —().309 X5 —.3()5 —.262 —.274 —.3 l l AG –.215 X; —. 143 (7 – 410 X. —.387 X5 – 305 —.390 Tº –.40l Al –.405 At —.215 —.283 6. —,431 X. –.453 X5 –.684 || 15 —.659 I's —.697 Aſ —.700 At —.780 X. —. 747 7 —.607 X. —.599 X; —.684 —.659 —.742 A5 —.706 A7 —. 780 —.816 6 —.66l X. —.670 X5 —.684 –.735 lº —.742 —.788 At —.910 X. —.914. 6 —.723 X 3 —.723 X5 X (1, 1, 0) K(3/2, 3/2, ()) A (1/2, 1/2. 1/2) L(l. 1, 1) Intermediate Final Intermediate Final Intermediate Final Intermediate Final –0.274 X3 –0.274 X: —().200 K3 –0.200 K, –0.406 A3 –0.372 A. As | –0.432 L. –0.394 LT, L; –.452 X. –.426 X5 —.320 K, —.299 K3 —.406 —.397 A6 —.432 –.426 Li, —.63] X. —.624 X5 —.730 K, —.718 Ks —.631 A —.619 A6 —.546 L. —. 539 Lé --- —.81 | X3 —.81 | X.5 —.886 K3 –.886 Ks —.674. —. 709 At –.623 –.652 Li table 1. The spin-orbit effects were found to be ex- tremely well characterized by only two constants and the values of these constants (\"" = 0.06447 A* = 0.05396) are reasonably close to the free atom spin-orbit parameters [11]. The final LCAO bands along the three principle axes are shown together with the APW results in figure 1. The r. m.s. error between these bands and the final APW results is 0.020 Ry. 3. Density of States The density of states in energy was calculated from the bands of figure 1 in the following manner. The BZ was divided into cubes.4 Tſa on a side, and 24 of these cubes were found to lie at least partly within the reduced BZ, i.e., within the region having 1/48th of the BZ volume which is defined by the relations 0 < ke sky s kº s 2Tſa and kr + ky-- ks < 1.57|a. Within each box i, Ni points were generated at random, the energies were obtained at each point, and the average number of states in each energy range was determined. Finally the results of each box were weighted by how much of that box lay within the reduced BZ and summed. In all, the secular equation was solved at 4900 points. Convergence of the density of states result was guaran- teed by the fact that the answer after 2450 points dif- fered from the final answer by about 4%. A histogram of the density of states is shown in figure 2. The calculation is not valid above the dashed line because d-bands, which have been ignored in this cal- culation, become important here. The arrows in figure 2 indicate the energies at which photoemission experi- ments [8] predict peaks in the density of states. These peaks are labeled c, vi., v.2, v3 and are associated with corresponding peaks in the calculated density of states as shown in the figure. Spicer and Lapeyre [8] have identified the peaks c, v1 and v2 with the L point band edges of the 2nd conduc- tion band, the 2nd valence band and the 3rd valence band respectively (bands are numbered going away from the gap), and they have presented evidence that the v3 peak results from states elsewhere in the zone. In the present calculation, the contribution to the density of states from each region of the zone was cal- culated separately. It was found that the region around L does not make the major contributions to these peaks. Instead the bands proved to be flat over a rela- tively large region of k-space which includes the point of minimum gap on the A- and X-axes. This region can be thought of as being formed by six surfaces which are approximately planar and are perpendicular to the six A-axes. In this calculation the surfaces intersect the A- axis at Tſa(.7, 0, .0), the X-axis at Tſa(.8, 8, .0), and the A-axis at Tla (1.0, 1.0, 1.0) (the L point). In addition, the calculation reveals that the c, v1, and v2 peaks result from states in the principal conduction band, the prin- cipal valence band and the 2nd valence band respec- tively, and that they occur .08 Ry, .04 Ry, and .02 Ry away from their respective band edges. The v3 peak is found to result from the coincidence of maxima from two bands; the third valence band near T and the primary valence band near X. The fact that this peak receives contributions from two quite dif- 117 i x | T T |-4,-5 5 –|.O |→ |-|--| |→ O .4 .8 |.2 |.6 2.O |.6 |.2 .8 .4 O .4 .8 |.O DELTA k = # (£,0,0) SIGMA K =# (º, ö,0) LAMBDA K =# (€,ć,ć) FIGURE 1. The relativistic LCAO bands of PbTe on the three principle axes. The crosses give the APW results to which the bands were fit. i - O | | | | .O4 .O8 || 2 || 6 , 2 O .24 28 NUM BER OF ELECTRONS / UNIT CELL FIGURE 2: A histogram of the density of electron states calculated from the bands of figure 1. The arrows indicate the positions of the peaks found in ref. [8]. The calculation is not valid above the dashed line. ferent regions of k-space could explain why the peak does not disappear sharply with increasing photon energy even if the transitions causing this peak are direct [8]. In reference 8, the vi, v2, and v3 peaks were observed directly in the photoemission spectrum, but the c peak was inferred indirectly using the optical measurements of reference 9. This was done by associating a joint den- sity of states maximum with the 2.2 eV peak in the mea- sured reflectivity and thereby placing the c peak 2.2 eV above the vi-v2 doublet. Joint density of states maxim are, however, most closely related to maxima in th imaginary part of the dielectric constant, e2(a)), an peaks in e2(a)) can be shifted by as much as .4 eV fro peaks in the reflectivity [6]. The e2(a)) calculated fro the reflectivity in reference 9 and the eg(@) calculate in reference 5 show a shoulder at 1.8 eV and no struc ture at 2.2 eV. We interpret this shoulder as arising from transitions between the v1 and c density of states 118 maxima, which, in our calculation, are separated by 1.7 eV. This interpretation is confirmed by the observation that the v1 and c maxima both result from states in the same region of k-space and that this region of k-space contributes strongest to e2(a)) at this energy [5]. Furthermore, the v2 and c maxima do not form a joint density of states maximum because the states con- tributing to those peaks are in different regions of k- space, and hence no peak in e2(a)) occurs at 2.5 eV. The peak at low electron energy which begins to ap- pear in the high photon energy results of reference 8 is interesting. If it could be resolved by going to higher photon energy, and if it could be shown to result from a density of states maximum (and not from scattered electrons [8]), then it would give information about the S-bands in PbTe, A previous calculation has indicated a sharp peak in the density of states located 7.6 eV below the valence band edge [12]. This is too low to be resolved by the maximum photon energy used in reference 8 (11.5 eV), but perhaps the tail of this peak is being observed. 4. Optical Constants In order to calculate the optical properties of a material, it is necessary to know the oscillator strengths as well as the resonant frequencies for each transition. That is to say, at every value of k one must know the momentum matrix element coupling the occupied bands with the unoccupied bands as well as the energy of each band. 4Te” |p;(k) Momentum matrix elements can be calculated from the wavefunctions obtained from any energy band cal- culations. However, one can often estimate the momen- tum matrix elements directly from the energy bands. If two bands are quite close together in energy, it is often reasonable to assume that the bands interact (in the sense of a k p expansion) only with one another and that the band curvature is determined only by the mo- mentum matrix element coupling those two bands. When this is true, the Taylor expansion of H(k) through terms linear in k can be identified with the k p Hamiltonian matrix, and matrix elements of momentum can be related to the coefficients in the expansion. This method was first used by Dresselhaus and Dresselhaus [3]. and the conditions for its validity seem to be well satisfied in PbTe [13]. Following reference 3 we ex- press the v component of the momentum matrix ele- ment coupling bands i and j as 77) 6H1, n(k) p;(k)=; Sºk Ök, Un, j(k) (5) where U(k) is the unitary matrix which diagonalizes H(k). X U (k)H, (k) U.j(k) = E(k)ö, (6) l, m The interband contribution to the complex dielectric constant e(a)) = e(0) + ieg(@) is given in the random phase approximation by [14] {foſ Ej (k)] – foſby (k)]} “”)--a-XXiàº, k i,j This expression has been evaluated for intrinsic PbTe at temperatures sufficiently low that the Fermi func- tions foſB) are one or zero. Q is the unit cell volume, hoj (k) = Ei(k) – Ej(k), and T, taken to be 6 × 10−" sec (h/T = .005 Ry), gives a linewidth to each transition. The summation on the k values within the BZ is simpli- fied to a summation over the reduced BZ (sec. 3). This is accomplished by replacing |p\{k) |* in eq (7) by | 1 — g ; wºol-ix ºr (8) This substitution is valid because the bands have cubic symmetry. The magnitude squared pi;(k)|* of the matrix ele- ment between the principal conduction and valence [E] (k) =E (k) – ho-ih/TT (7) bands, calculated from the bands of section 2, is plotted in figure 3 along the three principal axes in k-space together with the results of reference 5. Analysis of the approximations involved in eq (5) suggests that this method should work best on the A-axis and worst at L. This conclusion is born out by figure 3. On the A-axis, the k p and the LCAO results agree to within 20%. APW calculations at L. give .331 (a.u.)” for this quantity [15]. The summation on the reduced BZ was carried out by dividing this region into boxes as in section 3. Within each box, a Monte Carlo integration was performed [3], and the results were weighted and added as in sec- tion 3. It was found that certain boxes contribute strongly to eſco) and the integration was performed more accurately in these boxes than in the boxes which con- 119 K. D. Calculation |.O --- sº —º- p - l *-4 y \ | ----- L.C.A.O. Colculation - | \ -- -º- – ſ | * .8 – | l -º- –4– - > | Q | | \ /\ , 6 H | -- | \ --- - t; | | | \ -- --- | \ -- \ / – fo. | \ \ / º J. c .4 H. | \ --- * — O | \ P Ö | \ as-" f N f 2 -- * --- A º: I * / – ſ - - / --- º/ 2 22 |- 21-T | T 4 .8 |.2 X 4 .4 .8 L DELTA K = # (£,0,0) SIGMA K = # (£,ć,0) LAMBDA K = # (£,č, ć,) FIGURE 3. The magnitude squared of the momentum matrix element coupling the principle conduction band with the principle valence band. The present calculation is compared with the k p results of ref. [5]. I | | 7O H. –sº 6O H — Present Colculofion -4 - - - - - K. p. Colculation 50 Kromers - Kronig Results H- z 40 s (ſ) 3 - K.) 3O Q # 20 O lau —l ul G | O O - |O –2O tha, (eV) FIGURE 4. The real and imaginary parts of the calculated interband electronic dielectric constant together with the results of ref. [5] and ref. [9]. tribute less strongly. To insure convergence, the number of random points Ni within the ith box was chosen to be sufficiently large that the final result for e2(a)) differed by a maximum of .1 or less from the inter- mediate result calculated with Niſ? points. 11,850 points were required to obtain convergence using this criterion. To facilitate the calculation, the eigenvalue problem was not solved at every random point. Instead it was solved on a mesh of points k = T/5a (l, m, n) (l, m, and n are integers), and the energies and momentum matrix elements were obtained between the mesh points by linear interpolation. The real and imaginary parts of e(a)) calculated in this way are shown in figure 4 together with the Kramers- Kronig results of reference 9 and the k p results of reference 5. The overwhelmingly dominant feature of this calculation of e2(a)) is the peak at 1.9 eV which 120 results from transitions between the v1 and c density of states maxima. Even though the present calculation is based on the same APW results as that of reference 5, the energy differences between the principal conduc- tion band and the principal valence band are slightly larger causing the present result to be smaller at low energies and larger at high energies. In addition the mo- mentum matrix elements are consistently larger. In spite of the apparent inability of the present calcu- lation to reproduce the experimental e2(a) [16], e1(a) = 0) = 36.2 is surprisingly close to the measured optical dielectric constant e. =31.8 [17]. This quantity is not as sensitive to small inaccuracies in the energy bands as are the optical constants at higher frequencies. In- stead it reflects an average gap and an average oscilla- tor strength. 5 Conclusion The calculation of density of states and optical con- stants was undertaken primarily to test in a general way the accuracy of the APW-LCAO energy bands and secondly to test the validity of eq (5) for approximating interband oscillator strengths. The good agreement between the calculated density of states and the photoemission data indicates that the bands are sufficiently accurate to justify unambiguous assignment of observed peaks in the valence band den- sity of states with features of the band model. We con- clude that agreement between the calculated peak in the conduction band density of states and that deduced from optical properties in reference 8 is largely for- tuitous, but that a reinterpretation of the measured op- tical properties is consistent with the calculated posi- tion of this peak. Based on the success of the density of states calcula- tion, the calculation of e(a)) is disappointing indeed. It is tempting to blame this lack of success on the oscilla- tor strengths, but this is not necessarily the case. The differences between the present calculation and that of reference 5 are due as much to small differences in the energy bands as to differences in the oscillator strengths and we have not been able to draw any con- usions about the validity of eq (5) beyond what is hown in figure 3. This calculation illustrates an important difference etween PbTe and the wider gap materials. In the ormer, even small changes in the bands represent arge percentage changes and can drastically affect oth the size and shape of the calculated e2(a)) whereas n the latter this is not the case [18]. Considerable efinements will have to be made in the band picture of PbTe before agreement as good as that obtained in better known materials is achieved. 6. Acknowledgments The authors are grateful to George Pratt, Gene Dres- selhaus and Mildred Dresselhaus for their advice and assistance in carrying out this work and to Sahrab Rabii for making available to us his energy band calculations. We are also grateful to Prof. Marvin Cohen and Yvonne Tsang for sending us the results of their calculations of the optical constants of PbTe prior to publication. 417–156 O - 71 - 9 7. References [1] Conklin, L. B., Jr., Johnson, L. E., and Pratt, G. W., Jr., Phys. Rev. 137, A1282 (1965). [2] Slater, J. C., and Koster, G. F., Phys. Rev. 94, 1498 (1954). [3] Dresselhaus, G., and Dresselhaus, M. S., Phys. Rev. 160, 649 (1967); Dresselhaus, G., Solid State Communications 7, 419 (1969). [4] Cardona, M., and Pollak, F. H., Phys. Rev. 142, 530 (1966). [5] Buss, D. D., and Parada, N.J., to be published. [6] See for example: M. L. Cohen, II-VI Semiconducting Com- pounds, 1967 International Conference (W. A. Benjamin Inc., 1967), p. 462; F. Herman, R. L. Kortrum, C. D. Kuglin and J. L. Shay, ibid., p. 503. [7] Lin, P. J., Saslow, W., and Cohen, M. L., Solid State Communi- cations 5, 893 (1967); Tung, Y. W., and Cohen, M. L., Phys. Rev. 180, 823 (1969); Tsang, Y., and Cohen, M. L., to be published. [8] Spicer, W. E., and Lapeyre, G. J., Phys. Rev. 139, A565 (1965). [9] Cardona, M., and Greenaway, D. L., Phys. Rev. 133, A1685 (1964). [10] The APW calculation finds the valence and conduction band edges at L to have L6 and Lé symmetry respectively (see table 3 of ref. 1) whereas experimental evidence in- dicates that the symmetry of these two levels should be reversed (see ref. 1). The constant potential in the plane wave region was chosen somewhat arbitrarily to be – 0.80138 Ry and it was found that increasing this potential had the effect of changing the relative ordering of the bands around the gap at L without significantly affecting the bands elsewhere. (J. B. Conklin, Jr., Ph. D. thesis, E. E. Department, M.I.T., June 1954, p. 71.) S. Rabii has recalculated the bands at L using a potential in the plane wave region of –0.59140 Ry (private communication) and his results are given in table 1 for the L point. This adjustment represents the only departure we have made from “first principles.” Moreover, the density of states and optical properties do not depend critically on the position of the band edge (see secs. 3 and 4) and are quite insensitive to the adjustment. [11] Free atom Hartree-Fock calculations for Pb and Te indicate that the 8Pº, level lies 0.09356 Ry and 0.06176 Ry above the *P12 giving spin orbit parameters of A" = 0.06238 and A" = 0.04.118 Ry, F. Herman and S. Skillman, Atomic Structure Calculations (Prentice-Hall Inc., 1963). [12] Schirf, W. E., S. M. Thesis, E. E. Department, M.I.T., 1966. 121 [13] The general applicability of this method will be discussed elsewhere. [14] Ehrenreich, H., and Cohen, M. H., Phys. Rev. 115,786 (1959). [15] Bailey, P. T., O'Brien, M. W., and Rabii, S., Phys. Rev. 179, 735 (1969). [16] Although the e(0) found in reference 9 is not directly measured, we assume it is correct for the purposes of evaluating the validity of our calculation. [17] Walton, A. K., and Moss, T. S., Proc. Phys. Soc. 81, 509 (1963). [18] See for example: Herman et al. cited in reference 6, p. 522. 8. Appendix A This appendix defines the parameters used in the tight binding secular equation which is discussed in section 2 and gives their value. The 6 × 6 secular equation obtained by taking matrix elements of the spin independent part, Hsi, of the crystal Hamiltonian between Bloch sums having the symmetry of eq (1) is expanded in the most general Fourier series consistent with this symmetry. The result of the expansion through fourth order terms is given in table 2. The Fourier coefficients in this table are regarded simply as adjustable parameters, but they can be related to overlap integrals of Hsi between Löwdin functions as shown in table 3. The 16 Fourier coefficients were adjusted until the r. m.s. difference between the eigenvalues of the LCAO secular equation and the nonspin-orbit energies of table 1 is a minimum. The values so obtained are given in table 4. When spin-orbit interaction is added, the basis states of eq (1) are multiplied by a spin state, | ? ) or || ). The 12 × 12 matrix equation which results from writing Hso in this basis is Fourier expanded as before, but the ex- pansion is terminated with zeroth order. Table 5 gives the form of spin-orbit matrix in terms of the two parameters \" and \"" which were adjusted holding the other 16 parameters fixed as discussed in section 2. TABLE 2. Most general Fourier expansion in k-space of the matrix elements of Hsi of eq (2) between the Bloch sums of eq (1). The abbreviation (o. --|3+) means (b. 1(r)|Hsib; (r)). Other matrix elements may be obtained by pairwise interchange of x, y, z. (x +|x+), (x -|x-) = X0 + X cr = Y, (c) + c2) + X2 (c., c, -- c.c.) + Y2cycz + X3c., cycz + X4C2r-F Yı (C2) -- c.22) (x +|y--) º (x–|y–) = – Z 2s rs, FF Yºsrs ſcº (x +|x-), (x -|x+) = Q0+ (), (c,c) + c, c.) + Recycz + (), cer + R4(C2) + c22) (x +|y-) = (x-y-H) =–S2s rs, ca – cos (ka a/2) so = sin (ka a/2) c2a = sin (kaa) TABLE 3. The Fourier coefficients of table l expressed in terms of tight binding overlap integrals Zeroth........... Åo, Qo #[(a.”(r)|Hºla!" (r)) + (a|*(r-t)|Hºla!" (r-t))] First............. Å (a!" (r)|Hºla!" (r-t)) Yi (a)" (r)|H, la"(r-t)) Second.......... Å2, Q2 2[(a!" (r)|Hºla!" (r-R)) = (a!" (r-t)|Hºla!" (r-R – t))] Y2, R2 2[(a!"(r)|Hºla!" (r-R)) + (a|"(r-t)|Hºla!" (r-R – t))] Z2. Sº 2[(a)"(r)|Hºla!"(r-R)) + (a|"(r-t)|Hºla!" (r-R,-1))] Third............ Å3 8(a) '(r)|Hºla!" (r-R – t)) Y3 8(a)"(r)|H|a'(r–R,-T)) Fourth.......... ÅA. Q. #[(a!"(r)|Hºla!" (r-2+)) + (a|"(r-t)|Hºla!" (r--T))] Y4. R. #[(a)"(r)|Hºla!"(r-2+)) + (a|*(r-t)|Hºla!" (r-t))] (l 2 T='' (100) R-4 (011) 122 TABLE 4. The values of the band parameters which were used to obtain the bands of Fig. 1. Zeroth Order First Order... - - - a s = - * * * * * - - - * * * - - - - - - Second Order........... Åo Im. —- 0.52918 Qo Im. - ().04498 A = –0.26730 Yi = 0.04363 (), ()()798 Y, - – (), ()()673 Zg = — (), ()()66 = — (), ()():5|8 g = – (),03403 S., - – () ()2059 Åg = 2 R() 2 - Third () roler Fourth Order... . . . . . . . . . Spin ()rbit... * - - - e. e - - a s • * * A 4 = ().() I 203 } := – ().()2() 12 ) = (), ()() 78 () = 0.037 12 R 1 = — () ()()27 | A "" = () ()()·4.17 A " " = (). ().539() TABLE 5. Matrix elements of Hso between Bloch sums. This matrix is of course Hermitian. b. ? b. ? b. J. b. J. \ y z 3 y Z 3 y Z 3 y () — i \ . () () — i \ () () () A () () }\ - b. ? () () i)\ - () () () () — i)\t () () — i\- () () () () — A i)\" () — A - i)\- () () — i A () () () A T () () \t b. t () () () — i)\ – () () — i)\t () — A - i)\ () —A ix: 0 () i)\" () 0 ix- 0 b. 1 A = }(\"" -- \!”) () () — i)\- () () () () () () () i)\t () by J. () () 0 123 Plasmon-Induced Structure in the Optical Interband Absorption of Free-Electron Like Metals” B. l. Lundqvist and C. Lydén Chalmers University of Technology, ** Göteborg, Sweden We have extended Butcher's method to include the spectrum of interacting electrons, which con- tains satellite structure. The calculated optical conductivity shows a weak additional absorption, start- ing at about the frequency (or + (op. where a) is the interband threshold frequency and ap the plasma frequency. Key words: Electronic density of states: free-electron like metals: interband absorption: optical properties; plasmon; pseudopotential: quasi-particle: satellite structure. The common description of optical properties of free- electron like metals in terms of Drude and interband contributions [1] is in qualitative agreement with ex- periment. In some cases there is even a quantitative agreement, as for instance the Drude absorption due to phonon-aided processes in the infrared [2]. What con- cerns the magnitude and shape of the interband absorp- tion as calculated in the one-electron approximation there seems still to be deviations from the experimental values [3]. Further, the slow increase of the effective number of electrons [4], difficulties to fit data to an ef- fective oscillator formula [5]. and difficulties to fit high- and low-frequency optical masses [3]. suggest the possible existence of further structure in the ab- sorption at photon energies above the ordinary inter- band transition range. A recent investigation of the one-electron spectrum of the electron gas, considering effects to the lowest order in the coupling to the density fluctuations, and in particular to the plasmons, indicates an important satellite structure in the spectrum [6]. Due to the in- teraction the quasi-particle branch gets reduced spec- tral weight and is accompanied by a pronounced O-(a)) = e 2h 3 X. () 127°m°o: —ha) where Mil, (k) is an interband dipole matrix element and E1(k) a band energy in the band indicated by 1. To get our rough estimate we have used the dipole matrix *Work supported by the Swedish Natural Science Research Council. **Mailing Address: Foch, S-402 20 Göteborg 5, Sweden. plasmon-induced satellite structure for both occupied and unoccupied states (see fig. 3 of the paper by Hedin, Lundqvist and Lundqvist in this volume). The energy separation between the two branches is somewhat larger than the plasma frequency op. As the source of this effect is the existence of plasma oscillations, we consider it reasonable to expect this kind of structure to be important for free-electron like metals as well. A satellite structure in the one-electron spectrum may produce structure in the optical absorption spec- trum. From a simple-minded point of view there should not only be the ordinary band-to-band transitions. which start at the threshold energy Eq. There might also be additional absorption processes, due to transi- tions from the occupied satellite branch, starting at about the photon energy ha) = Eg-H hop. The purpose of this note is to present an estimate of the relative im- portance of these kinds of absorption contributions in the ultraviolet region. We have made a numerical cal- culation of the optical conductivity of sodium. The optical interband absorption at the photon frequency a is in the one-electron approximation given by the conductivity [7,8]. ded"k |Min (k) |*6(e–Ei(k))6(e–H ha) – En (k’)). (1) element from the nearly-free-electron approximation [7]. | Mir (lk) |* = (e |GV, -> (IC) (i,j) ITV.E." (2) =#7 (1,1,0), (l 125 where V6 is the screened pseudopotential, G the smal- lest reciprocal lattice vector and e(k) the free-electron energy. Because the integration will include k-values at the Brillouin zone boundary, the expression of pdp reference 7 is modified by the 4|Volº-term in the denominator. To describe the continuous one-electron spectrum for interacting electrons [6], we replace the sharp ex- citation energies in the first equation by the spectral function and get [8] or (a)) = Tom * () kdh |(-k)| (e (p) – e ( This expression has been evaluated numerically using the values of the spectral function from reference 6 at the density of sodium (electron gas parameter rs = 4) and Vo- 0.323 eV [7]. The resulting absorption power is shown in figure 1 in order to illustrate the relative importance of the transitions between satellite and quasi-particle states. The highest value of or(a)) is roughly one half of Butcher's result [7]. The reason for this is that due to the dynamic interaction between the electrons the quasi-particles get reduced weight. This reduction is typically 30 percent. Estimations of some electron-hole scattering processes, e.g. virtual exchange of plasmons, show that there are mechanisms able to give a compen- sating enhancement of the absorption near the threshold ha) = Ea. [9–11]. The figure indicates that there is a weak contribution to the absorption power, starting at about ha) = E0 + hop ( = 8 eV for Na), which is caused by the plasmon- induced satellite structure in the one-electron spec- trum. As this contribution is of roughly the same size as the one from transitions between quasi-particle states, we expect it to be measurable. In the ultraviolet region another kind of structure has been proposed [12]. In a pseudopotential treatment, the screening in the effective potential Vo should be dynamic in nature. The plasmon resonance in the screening function gives rise to a structure, which O-10" [S-' ] H t º f i l —t— Eghue 1O fu) [e Vl O –——— O 5 FIGURE 1. The calculated optical absorption of sodium. () #HIVFſ. de/A (k, e)A (p, e -H ha)), (3) should be in the measurable range [12,13]. However, this additional absorption starts at a) = @p ( = 6 eV for Na) with a distinct onset, and it should be possible to separate it from the structure proposed in this note. There have been several papers on the optical ab- sorption of an electron gas, where the process leading to a final state consisting of an electron-hole pair and a plasmon is taken into account [14-19]. This intraband , absorption process is different from the interband process discussed in this note. Besides, as stressed by Hopfield [12], there is no optical absorption in a homogeneous electron gas in the long-wavelength limit due to the lack of momentum sinks. The same conclu- sion follows, when a proper set of diagrams in the per- turbation expansion of the optical conductivity is con- sidered [19,20]. The formula, which we have used for the conductivi- ty, does not satisfy the sum rule for oscillator strengths. Certain vertex corrections, describing the final state in- teractions, ought to be included. Preliminary estimates indicate that processes, where real plasmons are exchanged, give a positive contribution to the absorp- tion power in the photon energy region of interest here. We are presently performing numerical calculations of vertex corrections. The experimental information about the optical con- stants of sodium in the ultraviolet region available in the literature concerns only the real part of the dielec- tric function [21]. As this quantity is so dominated by the free-electron behavior, the weak absorption mechanism proposed in this note should have a negligi- ble effect on it. To conclude, we stress the desirability of accurate measurements of the optical absorption by free-electron like metals in the photon frequency range around and above the plasma frequency. References [1] Phillips, J. C., Solid State Physics 18, 56 (Academic Press. New York 1966). [2] Miskovsky, N. M. and Cutler, P. H., Solid State Comm. 7, 253 (1969): see also Nettel, S.J., Phys. Rev. 150, 421 (1966); and Haga, E., and Aisaka, T., J. Phys. Soc. Japan 22,987 (1967). 126 [3] Smith, N. V., Phys. Rev. 183, 634 (1969). [4] Ehrenreich, H., Philipp, H. R., and Segall, B., Phys. Rev. 132 1918 (1963). [5] Powell, C. J., Phys. Rev. (to be published). [6] Lundqvist, B. I., Phys. kondens, Materie 7, 117 (1968). [7] Butcher, R. N., Proc. Phys. Soc. 381A, 765 (1951). [8] Brust, D., and Kane, E. O., Phys. Rev. 176,894 (1968). [9] Mahan, G. D., Phys. Letters 24A, 708 (1967). [10] Watabe, M., and Yasuhara, H. (to be published). [ll] Young, C.Y., Phys. Rev. 183,627 (1969). [12] Hopfield, .J., Phys. Rev. 139, A419 (1965). [13] Foo, E. N., Phys. Rev. (to be published). , [15 Tzoar, N., and Klein, A., Phys. Rev. 124, 1297 (1961). | Matsudaira, M.J. Phys. Soc. Japan 17, 1563 (1962). DuBois, D. F., Gilinsky, V., and Kivelson, M. G., Phys. Rev. 129,2376 (1963). Esposito, R. J., Muldawer, L., and Bloomfield, P. E., Phys. Rev. 168,744 (1968). Bose, S. M., Phys. Letters 29A, 555 (1969). Ron, A., and Tzoar, N., Phys. Rev. 131, 12 (1963). DuBois, D. F., and Kivelson, M. G., Phys. Rev. (to be published). Sutherland, J. C., Arakawa, E. T., and Hamm, R. N., J. Opt. Soc. Am. 57, 645 (1967). 127 Theory of the Photoelectric Effect and its Relation to the Band Structure of Metals" N. W. Ashcroft and W. L. Schdich.” Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14850. We develop the theory of the external photoelectric effect in terms of quadratic response to the in- cident electromagnetic field. Electrons in the solid are in states determined by their interactions with themselves, the ions and the surface. We denote by Ho the Hamiltonian for this part. In the presence of the electromagnetic field we have a coupling term: l H. =–1 | dra (r, t) : J (r)em', (m = 0+) C where A(r, t) is the vector potential, and J(r) is the current density operator for the electrons. Let R be a point exterior to the solid. Then the expectation value of the operator measuring the external current density at R in the states of (H0 + H1) is, to second order: (Jo (R, 0)-(+) ſ' drug-10 | drag-rº 2. | dº ſº.6., 1946, 19. ((Ju(x1, T1).J., (R, t).J., (x2, Tg))). There is no linear response and no other terms to order A* giving a measurable result. We show that (Ja(R, t)) may be related to the expectation value of the time ordered product of the three current operators. This alternative description can be evaluated in the independent particle model (no scatter- ing) and leads to a compact formulation of photoemission. There need not be a simple dependence of (Jo (R, t)) or of its spectral reduction (Ja (R, t, e)) (corresponding to measured electron distribution curves) on the joint density of states. Rather (Ja(R, t, E)) depends on the density of bound states but is not at all simply related to the density of states above the vacuum level. This emerges quite clearly from an analysis carried out in the well known constant matrix element approximation. A careful examination of the terms appearing in the photoelectric current shows that it is not always correct to interpret photoemission in terms of a “volume effect” or a “surface effect.” The contri- butions from these two interfere. The usual explanations of the processes involved (i.e., the sequential operations of excitation, transport, and transmission) are also somewhat blurred. The effect of electron-electron scattering is well known to be important and will be discussed both in terms of its manifestation in the observed electron distribution curves and its ability to limit the con- tribution of the conventional volume effect. Key words: Electronic density of states; electron-electron scattering; electrons in a box; joint densi- ty of states; Kronig-Penney model; photoelectric effect. 1. Introduction photoemission similar in spirit to the theory of linear response. A general expression for the photocurrent is The external photo effect can be viewed, for moderate amplitudes of the incident photon field, as the quadratic current response of a system to an ap- plied electromagnetic field. This statement is based on the experimental fact that the photocurrent is propor- tional to the intensity of the incident field over at least 8 orders of magnitude of the latter [1]. We use this ob- servation to develop a quadratic response theory of *Work supported by the National Science Foundation under Grant GP-7198. **NSF Predoctoral 'ellow. 417–156 O - 71 - 10 derived in the next section, and in succeeding sections is analyzed in various approximations to determine its dependence on the underlying equilibrium electronic structure. Specifically we are interested in whether the photocurrent or its spectral reduction is proportional to a density of states and whether (or rather to what degree) inelastically scattered electrons retain useful information about single particle band structures. Some preliminary thoughts along these lines are given in section 4. 129 2. Quadratic Response Theory We consider a solid (or liquid) located in the half space x < 0. In the presence of the incident light the Hamiltonian of the system is written - H = Ho-H H with (1) Ho = h –H Ho Here h describes the kinetic energy of the electrons and the interaction between electrons and ions and electrons and surface, while He describes the interac- tion between electrons and electrons. For simplicity we shall invoke an adiabatic approximation in order to con- sider the ions as static. The interaction of the system with the electromagnetic field is represented by H1: u-- ſºry ºr t).J., (r)e", m = 0+ (2) where J.(r)=; X (p.6(r-x)+6(r-x)p.) (3) The scalar potential is zero by choice of gauge; we are thinking of a plane wave for the vector potential A(r,t). By using the Hamiltonian (1) we have assumed that ini- tially (in the distant past) there were no electromagnetic fields present. Note that we ignore contributions from the diamagnetic current, J% (r, t) =-ji. | dr S. 6(r-x)4,(r,t), (4) (Jo (R. 0)-ſºrſ. dT ſ 4.00-1000–1) |dºx | dºx2 S A, (x,1)A, (x,t) ((J.(x,t).J. (R, t).J. (x,t))) The double brackets refer to the ensemble average shown explicitly in eq (5). In deriving eq (8) we have neglected terms of the same or lower order in A on the assumption that Ja(R,t) gives a negligible contribution when it acts on an unperturbed state near (within several kBT) the ground state. This approximation fol- lows from the physical notion that those states favorably weighted by the thermal factor e-Bºº contain no electrons outside the metal. Equation (8) forms the | - i(a) – a ')'t ; X e 4C + () + () (Jo (R, t)) = | dº | dº X, A, (x)4, (x2) All , l’ (1) a) ' | (i.e.) E0 + ha) – Ho — ió' Jo (R) both here and below because it contributes only in or- ders higher than A*. (Similar considerations apply to possible terms in H1 proportional to A*.) In the photoemission process we are faced with the problem of determining the time independent elec- tronic current far away (say 1 cm) from the surface which is established long after (say 10−8 sec) the pertur- bation Hi has been applied to the equilibrium ensem- ble. Putting this another way, for R. & > 0 and t - 0 we seek (Ja(R,t)) where figuratively [2] | iH't -i Ht (J.(R,+))=; Se-hº (ne J.(R)e n), (5) and Hon }o = E}|n )0, B= 1/kh T (6) Here the n)0 are those many-particle eigenstates of the equilibrium ensemble which existed before the applica- tion of H1. To reduce (5) we work in the interaction representation and obtain to first order in the vector potential: — iPH t — i Hot e h = e " (1+}ſ. ds H1(s) + . . ) (7) This approximate evaluation is sufficient to determine the photocurrent to second order in the applied field. Substituting (7) in eq (5) we find the quadratic response to be: (8) basis for all further discussion; it represents a general expression for the photocurrent which we shall evalu- ate for specific systems in order to determine how the photocurrent depends on the electronic structure. 3. Independent Particle Model By performing the time integrations indicated in eq (8) we find: | J.G.) 6, 6' = 0+ E0 + ha)' – Ho + i ö (9) 130 To arrive at (9) we have assumed the electromagnetic field to have a time dependence given by cos Qt. If we now ignore the Coulomb interaction between electrons (i.e., neglect Ho), further simplifications follow im- (Jo (R, t)} = f" (en) mediately. Extracting the term for the observed steady photocurrent; i.e., omitting those terms that correspond to stimulated emission or that are time dependent, we find for the independent particle model | & #ſº ſº. 4,604.sº X f (en). }l , !!, l’ f" (er) (nju (x1)|u) where |n), u), and v) are single particle states with energies en, eu, and ep, the j’s refer to single particle cur- rent operators, and f-(e)= {1 + eff's-H)}-1 and f*(e)=l — f(e) are state occupation factors. The interpretation of this expression is that it describes those electrons that escape without any reduction in their energy or any lifetime effects. Note that the photocurrent is not simply related to a product of an excitation probability and a transmission factor. We have evaluated eq (10) for two specific examples of the independent particle Hamiltonian h. The first ex- ample is the model used by Mitchell [3] : free electrons in a semi-infinite box. The states |n), u), v) are easily evaluated and Mitchell’s results follow straightfor- wardly and need not be reproduced here [4]. The second example is the Kronig-Penney model [5], which, unlike Mitchell’s model, admits the possi- bility of band structure. We assume that the single par- ticle potential varies only in the x-direction and that its form is as shown in figure 1. (We eventually take the limit of Vo-> Co, b → 0 such that (2mVob/h”) → 2P/a.) To be specific, the parameters of the model, P., a, and wa, have been chosen to represent as nearly as possible the (111) planes of Na: the model possesses the same spacing in the x-direction and has the same Fermi ener- gy, first band gap, and photoelectric threshold. To limit the total current we introduce an effective depth into f b Vo | hwa | | | | | | | | | | | | | | | x=-#d x=-3a x=-#d x = -d x=O FIGURE 1. Portion of modified Kronig-Penney potential. The metal is in the half space x < 0 and the potential change at the surface is represented by a simple discontinuity. en + h() – e º – (vlj, (x2)|n) (10) ºf (uj.(R).) e n + h() – e º -i- ió the matrix elements: this is purely ad hoc and is similar to the cut-off procedure used by Fan [6]. This limita- tion is based on the physical observation that photoelectrons never emerge from a depth greater than a micron. That such a cutoff is automatically present can in fact be derived from eq (8), as will be shown below. Some of the results for this model are shown in figure 2. We have plotted energy distribution curves for a photon energy of 4 eV (1.71 eV above threshold). The temperature is assumed to be 300 K and the curves are parametrized by L which is approximately the number of planes from which electrons are allowed to emerge. Both the conventional “surface” and “volume” effects are present. In fact the equations of the model ex- plicitly show that these two effects may interfere with each other; i.e., the distributions are proportional to the : u- C L=5O 9 O -C C. 5 L= 40 # 3 5 E L=30 C) =y -*. 5 * # = § 3. %) + -5 L=5 3. E 92 5 O 9 O -C Cl– .# 4 Tä Tis To T2 Ta Te Ts Energy above threshold —- (electron volts) FIGURE 2: Energy distribution curves for modified Kronig-Penney model (no scattering). (See text for an explanation of the parameters.) 131 square of the sum of the two rather than to the sum of the squares. As the effective depth increases, the volume effect becomes dominant as expected. On the basis of our experience with specific models, we shall now discuss for the general case (but still within the limitations of the independent particle model) what information concerning the electronic den- sity of states we may expect to be determined from the spectral reduction of the photocurrent, which is essen- tially the expression in eq (10) but without the sum over the initial states |n). By limiting ourselves to the inde- pendent particle model, we are naturally assuming that at least the high energy part (say within 4 eV of the maximum) of the energy distribution curves is essen- tially unaffected by electron-electron scattering. This is consistent with phase space arguments about the degrading effect of electron-electron scattering and is the fundamental assumption used in the interpretation of the experimental energy distribution curves. Since we seek a density of states dependence we shall re- peatedly use below the constant matrix element ap- proximation. In examining the photocurrent expressions we find that there need not be a simple dependence of its spec- tral reduction on the joint density of states. This lack of dependence arises essentially from the fact that our for- mulation is based on the explicit consideration of a sur- effi (Jo (R, t)) F. lim (V) – Vi) (vº – Vº) (or - vº). xi− x 1 R’ — R x3- x2 | n + () — u — ió where; e.g., (ilºt(x1,s) lºſx2,0)) is the expectation value in the interacting ground state of the product of the Schrödinger operators lºt (x1,s) and l;(x2,0) in the in- teraction representation: iſ, (x2, o] = X. (x2|u)cº (12) ihs — ins lºt (xſ, s) = S (x|u)e" cºe " !! (13) l 3 8C2h 2 (#) | dº | des 4,604.s.) | e AL, l’ |ſ a cºws, own, 0)|In face; the evaluation of the formulae requires the deter- mination of wavefunctions in the presence of the sur- face, hence k-vectors cannot serve to enumerate eigen- states, nor need crystal momentum be conserved. The addition of lifetime effects (see below) further strengthens this conclusion. Since the usual derivation of a dependence of (J) on the joint density of states requires (at the very least) both these properties, it is not clear whether such a dependence could ever result from an evaluation of eq (10). But, for the limiting case where volume (direct) transitions are dominant such a dependence on the density of states might be found (though to extract it from our formalism would appear to be arduous). For the general case, however, it is im- possible (because of the presence of the surface) to ex- tract a simple dependence of (J) on the joint bulk den- sity of states. 4. Scattering Effects For convenience we go now to zero temperature to consider the effects of Coulomb scattering among the electrons. Equation (8) for the current can be related to a time ordered product of three current operators. The result can be evaluated in the Hartree-Fock approxima- tion which is a generalization of the independent parti- cle model to include lifetime effects. We find ſº ſº. 2T | 27 |ſ as cº-sº ove. o)| dn. 2T | — v-H iS' |ſ. ds e-itº (ili (R, 0) lit (x, 9) + ce) (11) Here the cut and cu are creation and annihilation opera- tors for the single particle eigenstates, u), associated with the Hamiltonian h. Note the similarity betwee eqs (10) and (11). The difference lies in the fact that eq (11) allows us to include the probability that an electron can propagate between any two points. Since an elec tron in the bulk material has a mean free path we no obtain a natural limitation of the volume effect, which in turn weakens the criterion of conservation of crysta momentum. We note, however, that within this approxi 132 mation we have not obtained a damping of the incident electromagnetic wave and hence the analysis is only ap- plicable to those materials whose electronic mean free paths (at the energy - Ef + h(?) are less than their photon absorption lengths. Furthermore, we have lost a clear representation of the transition matrix elements which appear straightforwardly in eq (10). We may reduce eq (11) to an expression similar to the result of Berglund and Spicer [7] for the current from those electrons which escape without scattering. To ac- complish this we are forced to make some approxima- (Ja (R, t)) - | * |. dr|A(r) : J (r, n)|*|T(r, R, n)|* which expresses the spectral reduction of the photocur- rent as an integral over the product of an excitation probability and a transmission factor. The assumption of the existence of such a formula is the starting point of the analysis of Berglund and Spicer [7]. tions whose physical interpretation and effect are dif- ficult to estimate. Specifically we must assume that the energy integrals over u and v may be carried out as in the independent particle model with the resulting con- servation of energy and that the effect of Jºds e-ins (lºt (xf, s)!/(x2, o]) is to require the spatial arguments to be identical (x2 = x1'). Finally, we must assume that the two remaining expectation values can both be written as products of an excitation amplitude J(x,n) (nonzero only for x inside the metal) and a transmission am- plitude T(x,R,n). We then find (14) Finally we consider the effects of inelastic electron- electron scattering. A multitude of terms arise in the reduction of the time-ordered product whose in- terpretation is difficult. One of the main contributions, however, can be interpreted as the “once-scattered” term of Berglund and Spicer [7]. Explicitly it is f" (e.g.) (Ja (R, t)) seat. =#| dº ſ dºx2 2. A, (xi) Av(x2) X. f-(e)(nli,(x)|w') €n + h () — 619/ Q m, w, w' {{, } S, t fº (es) ft (en) ft (e) ft (ec) ft (e) f (es) f C” (w (s |ri – r, |)); + h() – (eſ – es) – en — ( (Hºns) wº im" (ulja (R)|v) €n –– h () - (e. - es) - er + im" f" (ele) e en + h () – e –H im (wlji (x2) |n) (15) where the fº's are evaluated at zero temperature and e.g., ( (Hºns) wy-ſºrſ dºor)(ſr) Hººrn) As before |n), w) . . . are all single particle eigenstates associated with h. An interesting consequence of this expression which seems to have escaped previous notice is that energy need not be conserved in the inter- mediate state. Only if the matrix elements involving |w) and w') are independent of the energy of w and w') will energy be conserved. This implies that there is a potentially much greater oscillator strength for excita- tions that simultaneously create a pair (or a plasmon) than for single particle excitations. It also unfortunately implies that the spectral reduction of the photocurrent due to inelastically scattered electrons (i.e., scattered by other electrons) will bear no simple relation to the underlying electronic band structure. (16) 4-f 5. Concluding Remarks We conclude by mentioning the extensions of this work that immediately suggest themselves, some of which we are currently investigating. It is quite clear that more of the contributions to the inelastic com- ponent of the photocurrent need to be classified, analyzed, and numerically estimated. It must be the case, for example, that a class of terms exist which represent the damping of the electromagnetic wave into the material. It is also of considerable interest to have an accurate description of the behavior of electrons as they approach the surface and to learn, in particular, how this can be related, if at all, to the properties of 133 bulk electrons. In this connection we make two obser- vations. First, a soluble independent particle model somewhat more sophisticated than the Kronig-Penney model is evidently required. Second, we need an un- derstanding more complete that we presently have, of the physics of an interacting electron gas in the presence of a potential discontinuity, a problem long familiar in the context of low energy electron diffrac- tion. Finally, it is apparent that to facilitate direct com- parison with experiment the analysis in its final form must be extended to nonzero temperatures. 6. References [1] Elster and Geitel, Phys. Zeits. 14, 741 (1913); 15, 610 (1914); 17, 268 (1916). [2] To use simply —iht|h is technically incorrect since H depends explicitly on time; however, the substitution obtained in eq (7) is correct to first order in the vector potential. [3] Mitchell, K., Proc. Roy. Soc. 146A, 442 (1934). [4] This is true only to within a factor of two, by which Mitchell is in error. We agree with the correction of I. Adawi, Phys. Rev. A134, 788 (1964). [5] Kronig, R. de L., and Penney, W. G., Proc. Roy. Soc. Al30,499 (1931). [6] Fan, H. Y., Phys. Rev. 68,43 (1945). [7] Berglund, C. N., and Spicer, W. E., Phys. Rev. A 136, 1030 (1964). [8] We limit this statement to those electrons which are inelastically scattered by other electrons. Since we are only considering the spectral reduction of the photocurrent with respect to energy (and not with respect to direction) the effects of in- elastic phonon scattering are negligible. 134 Discussion on “Theory of the Photoelectric Effect and Its Relation to the Band Structure of Metals” by N. W. Ashcroft and W. L. Schaich (Cornell University) W. E. Spicer (Stanford Univ.); The suggestion that a This seems to agree with results in metals but not free electron final density of states rather than the band semiconductors. density of states should be used is quite interesting. 135 CHAIRMEN: E. T. Ardkawa C. J. Powell RAPPoRTEUR. S. B. M. Hagström Optical Density of States Ultraviolet Photoelectric Spectroscopy” W. E. Spicer Stanford Electronic Laboratory, Stanford University, Stanford, California 94305 The use of ultraviolet photoemission to determine the density of valence and conduction states is reviewed. Two approaches are recognized. In one, the photoemission as well as other studies are used to locate experimentally a limited number of features of the band structure. Once these are fixed, band structure calculations could be carried out throughout the zone and checked against other features of the photoemission data. If the agreement is sufficiently good, the density of states is then calculated from the band structure. The second method depends only on experimental data. Using this approach, features of the density of states are determined directly by the photoemission experiment without recourse to band calculations. In cases where bands are wide and k clearly provides an empirically im- portant optical selection rule, this is possible only for portions of the bands which are relatively flat. Suc- cessful determinations of this type are cited for PbTe, and GaAs. In metals with narrow d bands such as Cu, it has been found empirically that one may explain fairly well the experimental energy distribution curves in terms of transitions between a density of initial and final states (the optical density of states, ODS) requiring only conservation of energy. The ODS determined by such ultraviolet photoemission studies have more strong detailed struc- ture than the density of states determined by any other experimental method. Studies on a large number of materials indicate that the position in energy of this structure correlates rather well with the position in energy of structure in the calculated density of states. It is suggested, following the very recent theoretical work of Doniach, that k conservation becomes less important (and nondirect transitions more important) as the mass of the hole becomes larger. This is due to the change in k of electrons in states near the Fermi level as they attempt to screen the hole left in the optical excitation process. These electrons take up the excess momentum. One would expect the k conservation selection rule to play an increasingly important role as the mass of the hole decreases. This is in agreement with experi- In ent. Key words: Copper; copper nickel alloys; density of states; GaAs; Ge; nondirect transitions; optical density of states; PbTe; ultraviolet photoemission. 1. Introduction Photoemission can give a great deal of detailed infor- mation about the optically excited electronic spectra of solids. Adequate interpretation of photoemission data can produce detailed information on the electronic structure and, assuming that Koopmans’ theorem [1] holds, on the ground state density of states. The utility of photoemission lies in two factors: (1) The ability to determine the distribution in energy of * An invited paper presented at the 3d Materials Research Symposium, Electronic Density of States, November 3-6, 1969, Gaithersburg, Md. | Work supported by NASA, NSF, U.S. Army Night Vision Laboratories, U.S. Army— Durham, and the Advanced Research Projects Agency through the Center for Materials Research at Stanford University. electrons excited by monochromatic light, and (2) the ability to study the valence bands of solids over their entire widths. Difficulties arise in correcting for in- elastic scattering and electron escape probability and in interpreting the data so corrected. Correction for scattering and escape probability seems to have been rather successfully done in a number of cases [2,3,4,5,6]. There are still detailed questions open in in- terpreting the data; however, as will be shown in this paper, it is clear that considerable information on the density of states can be obtained from photoemission data independent of these questions. Let us look in more detail at the essence of optical ex- citation in solids and the photoemission experiment. 139 | P(E,hy) hy mºss ul $ à Ef / % FIGURE 1. Energy diagram for a metal. P(E, hy) is the probability of a photon of energy hy exciting an electron to final energy E. (b is the work function, Ei is the initial energy of the excited electron, Ef is the Fermi energy. Consider the probability, P(E,hv), of a photon, of ener- gy hy, exciting an electron to a final state of energy E (see fig. 1). The excitation spectrum in the solid is then given by the values of P(E,hv) for all values of energy. The external photoemission energy distribution N(E,hv) would correspond exactly to P(E,hv) if all excited electrons escaped without inelastic scattering. Thus, P(E,hv) → N(E,hv). (1) In contrast, the optical constants oo or e2 (from which attempts are often made to determine the electronic structure) are related to the integral of P(E,hv) over all possible final states e2 → ſ P(E,hv)dE. (2) e2 is the imaginary part of the frequency dependent dielectric constant and O is the optical conductivity, or = e2/o. For the relations in eqs (1) and (2), it can be seen that photoemission contains much more detailed infor- mation than do the optical constants. This is illustrated by figure 2a, b, and c. In figure 2a the imaginary part of the dielectric con- stant for Cu is plotted versus photon energy [7]. The arrows call attention to two values of photon energy, 5.0 and 10.2 eV. A maximum appears in e2 at hu = 5.0 eV. There has been considerable discussion [3,4,8,9,10] concerning the optical transition or transitions respon- sible for this peak. There is no measurable peak in e2 at 10.2 eV; rather, the curve is almost flat. In figure 2b and 2C, energy distribution curves, EDCs, are presented for hy equal to 5.0 and 10.2 eV. The striking thing about these curves is the large amount of struc- ture which is present in them. Whereas only one peak was present in the ex curve near 5.0 eV and none was present near 10.2 eV, several pieces of structure are present in the EDCs for each value of hy. From the energy at which the structure appears, the initial and final states involved in the optical transition can be quickly identified. In the present case, the elec- trons within 2 eV of the high energy cutoff, Emar, are excited from the almost free-electron-like conduction states lying within 2 eV of the Fermi level; whereas, the sharp structure lying more than 2 eV below Emar is due to excitation from the d states. By noting the manner in which EDC structure moves with hy, the relative importance of initial and final states can be determined and information can be ob- tained about selection rules and/or matrix elements. For example, it was possible to determine that the peak in figure 2b at about 2.7 eV was due to a direct transi- tion from states near the Fermi level with a threshold at about 4.4 eV [3]. Examination of band calculations showed that the transition must be centered near the L symmetry point. We will return later to the discussion of the interpretation of photoemission data. In fact, such discussion will provide the central theme for this paper; however, it is first useful to briefly review ex- perimental techniques and the effects of scattering on photoemission data. 2. Experimental Techniques As was suggested in the Introduction, a large amount of information can be obtained from the photoemission energy distribution curves. A second useful measure- ment is that of the spectral distribution of quantum yield. Let us briefly review the experimental methods for obtaining such data. In so doing, we will not attempt an exhaustive list of references, but rather will at- tempt to refer to recent articles representative of the various techniques. Because of his closeness to the work at Stanford, the author will draw particularly heavily on this work. For many years EDCs were obtained by measuring an I-V curve and differentiating it by hand. The most important modern advancement was the replacement of this tedious and demanding practice by various schemes which yield EDCs directly from the experi- ment. Most popular are methods which add a small al- ternating voltage to the retarding voltage so that the derivative is taken electronically [11,12]. By slowl (typically 1 volt/minute) sweeping out the retardin voltage, a complete derivative curve can be obtained. Recently |13,14], measurements have been made a the second harmonic of the alternating voltage to obtai 140 6 }- 5 }*- 40x1O’ (b) º, (c) Cu (Cesiołed) v = 5.0 eV 3.5H 2.OH- | .5 H 3 - 0–1–1–1–1–1–1–1–1–1 O || 2 3 4 5 6 7 8 9 PHOTON ENERGY (eV) l | |O || |2 ELECT RON ENERGY (eV) O. 5H 4.O O | I l | ſ I 4.5 5.0 6.O 7.O 8.O 9.O |O.O |||O ENERGY ABOVE FERMI LEVEL (eV) FIGURE 2: (a) es for Cu. (b) EDC obtained from Cu with Cs on the surface for hu = 5 eV. Note that this curve has several pieces of structure in it, whereas the e, curve had only one peak at 5 eV. (c) EDC for clean Cu. hu= 10.2 eW. Note that several pieces of structure occur in the EDC, whereas there is no strong structure near 10.2 eV in the EDC. the second derivative of the I-V curve. In this way weak structure in the EDCs can be detected and studied. A second approach is to take a I-V curve and then to either differentiate it electronically [15,16] or by means of a computer. The geometry and other details of the energy analyzer are also of considerable importance. Because of ease of construction, wide use has been made of a cylindrical approximation [11] to the more ideal spheri- cal geometry of the collector. This has given an energy resolution of between 0.1 and 0.3 eV, depending on the kinetic energy of the emitted electrons, the details of the emitter geometry, the uniformity of the collector work function, and other factors. Of particular im- portance for small electron kinetic energies are dif- ferences in work function between the face of the emitter and its sides. DiStefano and Pierce [17] have recently made an overall study of the factors limiting resolution. They conclude that a spherical collector with a spherical grid providing a field-free drift region should provide a significant increase in resolution pro- vided that effects of the earth’s magnetic field are properly minimized. Preliminary measurements with this geometry support these conclusions. In principle, the measurement of the spectral dis- tribution of quantum yield is much simpler than the energy distribution measurement. All that is needed is a standard detector of known response to which the emission of the sample under study can be compared. In the visible and near infrared spectral ranges, this is fairly easy to achieve because of the high light intensi- ties available and the large number of suitable detec- tors. It is considerably more difficult in the ultraviolet where light intensity may be low and there are con- siderable problems with detectors [18]. Groups at the National Bureau of Standards, Stanford University, and other laboratories are cooperating in an attempt to establish good standards on a national-wide basis. Another very necessary condition for successful photoemission experiments is the ability to provide emitter surfaces which are atomically clean. One must be able to provide such surfaces and insure that they do not contaminate in the course of study (pressures better than 10-8 or 10-9 Torr are usually necessary). Depend- ing on the material, surfaces may be provided by cleav- ing [19], evaporation [4,6,20], heating [21], sputtering [22], or a combination of these methods. In covalent semiconductors such as Ge, it is well known that care must be taken to preserve crystalline perfection; how- ever, in metals such considerations seem much less im- portant. In fact, for Cu and Ni, which have been studied both as single crystals and evaporated films, the evaporated samples have given to date as good or better results than have sputtered and/or heat-cleaned sam- ples [21,23]. This is despite the fact that some evaporated samples may have very small crystallite sizes (for example, about 100 Å in the case of Ni (6,20). The insensitivity to crystallite size is due to the escape length for photoexcited electrons often being much less than 100 Å. It is often useful to reduce the threshold for photoemission by placing a layer of cesium on the sur- face of a material. Ideally the cesium will only form a monatomic layer which reduces the work function without affecting any other properties of the solid. How- ever, since Cs may chemically combine, amalgamate, or interact in other ways with the material under study, one must take care. The best procedure is to obtain EDCs from clean material over a photon energy range of several eV or more before placing the cesium on the 141 surface. Then, after the cesium is placed on the sur- face, EDCs should be obtained from the same photon energy range. By comparison of the two sets of EDCs, an estimate can be obtained of any extraneous changes produced by the cesium. 3. Electron Scattering Phenomena As mentioned in the Introduction, one must un- derstand the effects of electron scattering in order to properly interpret photoemission data. Two principal scattering mechanisms are electron-electron and elec- tron-phonon scattering. In the first type of event, the scattered electron loses a large fraction of its original energy to a second electron, which is thus excited. The electron-electron event is characterized by a mean-free path which decreases rapidly as the primary electron energy is increased in the range E → 12 eV. The energy loss in the phonon-scattering event is much smaller than that in the electron-electron event and, since this energy loss varies roughly as the Debye temperature, it will be much smaller for the material containing heavier atoms than for those with lighter atoms. There is no evidence that the phonon mean-free path is highly dependent on electron energy as is the case for electron-electron scattering. Kane [24] has pointed out that the electron-phonon scattering will be enhanced for final states having low group velocity (i.e., states as- sociated with a high density of states). Eastman [25] has made the same observation for the electron-elec- tron event. However, it does not appear that massive distortion of the energy distributions are produced by these effects. There is a threshold for pair production in semicon- ductors and insulators of about the forbidden band gap energy (i.e., the electron must be above the conduction band minimum by this amount before it can produce a secondary). Thus, only phonon scattering is possible below this threshold. In a metal there is no such threshold. However, as mentioned previously, in both semiconductors above threshold and in metals the elec- tron-electron scattering length decreases quite fast with increasing electron energy. In figure 3 we present values [5,26,27,28] for Au obtained by several different methods. Note that the mean-free path drops by two or- ders of magnitude within a few eV. The electron-elec- tron scattering effects have been taken into account quantitatively in interpreting photoemission data [3,4,6}. In fact, photoemission measurements can be used to determine the electron-electron mean-free path. The solid curve in figure 3 was deduced from such measurements by Krolikowski and Spicer [4]. IOOOO - GOLD KROLIKOWSKI AND SPICER 6 C. R. CROWELL, et Gl o o O KANTER, –––– S. M. SZE, J. L. MOLL, AND \ T. SUGANO \ | \ \ \ \ \ \ \ |OOO |OO Iol-l-l-l-l-H- O || 2 3 4 5 6 7 8 9 |O || |2 ENERGY ABOVE FERM | LEVEL (eV) FIGURE 3. Electron-electron scattering length for Au as obtained by several workers [5,26,27,28]. More recently, Eastman [29] has developed a direct method for obtaining electron-electron mean-free paths from photoemission measurements. This is based on a variation of sample thickness. For electron energies below the threshold for pair production in semiconductors, photoemission has been used extensively by James and Moll [30] to study the scattering of electrons by phonons in GaAs. This is of particular interest because of its importance in the Gunn effect. DiStefano and Spicer [31] have developed special photoemission techniques to study the scatter- ing of hot electrons in alkali halides by phonons. We give the examples listed above to illustrate the degree to which scattering of excited electrons in the photoemission experiment has been studied and is un- derstood. This is not to say the processes are un- derstood in all detail. This is not the case; however, a good, first-order understanding does seem to exist. There are other possible scattering phenomena which are less well understood. These include scattering from: (1) Bulk imperfections (such as grain boundaries), (2) the sample surface, and (3) scattering from oxide or other “crude” layers on the surface [19]. 4. Interpretation of Photoemission Data: Direct and Nondirect Transitions The present author and his coworkers have sug- gested [2,19,32] that, for excitation from certain 142 quantum states characterized by low mobility holes, conservation of k may not provide an important selec- tion rule and that only conservation of energy need be considered in interpreting the photoemission data. Such transitions were called nondirect. The suggestion of prompted by the character of the photoemission data obtained from states of this character. Based on this data, it was further suggested that a measure of the density of states could be obtained directly from analy- sis of the photoemission data. Of course, such a strong departure from accepted theory was met with con- siderable skepticism. Recently, band calculations [25.33,34,35], as well as new photoemission data (much of which will be reported at this meeting), have shown that there are certain strong similarities between the experimental EDCs interpreted as nondirect and the EDCs calculated using band structure results and k conservation when broadening effects were included in the calculation. However, other important systematic differences do remain, which may have considerable significance. In this paper, I will place particular nondirect transitions was emphasis on this discussion since it is central to the ex- perimental determination of the density of states from uv photoemission. Before proceeding further with this discussion, it should be recognized and emphasized that there were a number of materials in which direct transitions were clearly identified and many in which only direct transi- tions were seen; for example, the column IV and III-V semiconductors [36]. It should also be recognized that the criterion of peaks “moving with hy” (or the criteria of peaks which are stationary independent of hu) has been considered a necessary, but not sufficient, condi- tion for identifying a nondirect transition [36.37,38]. In particular, abrupt appearance or disappearance or strong modulation of peaks has been taken as sug- gestive of direct transitions even when peaks “move with hu” [38]. PbTe [37], GaAs [36], Col'Te, CdSe, and CaS [38] provide examples of this. Another method for attempting to distinguish, experi- mentally, between direct and nondirect transitions is to examine the effects of reducing or destroying the periodicity of the lattice. Since k conservation is im- posed by the periodicity of the lattice, destroying that periodicity should remove any importance of k conservation as an optical selection rule. Examples will be given of cases where periodicity is reduced or destroyed by alloying, melting, or forming an amorphous solid. Brust [39] has recently pointed out the possibility of explaining these changes by introduc- ing an uncertainty in k rather than removing it completely as a selection rule. Neville Smith has played a key role in the develop- ment of calculations of photoemission from d bands at Stanford [33]. A paper describing some of his work is included in this conference as is work on indium and aluminum by Koyama and Spicer [40]. The group of Janak, Eastman, and Williams [41] has also completed calculations assuming direct transitions for Pd which they will report at this meeting. I will not attempt to summarize these papers; but rather I will attempt to emphasize certain points. The nondirect transition model was developed empir- ically since it appeared to give a good first approxima- tion to the behavior of experimental photoemission data in a number of cases, including Cu. This model has been described in detail elsewhere [2,3]. The essence of it is that the optical transition probability, P(E,hv), is given by the product of the optical densities of states (ODS) at energies E and E – hy: P(E,hv) → m (E) m (E - hv). (3) Here m(E) is the optical density of empty states at an energy, E; and m(E-hu) is the filled ODS at an energy hu below E. The term “optical density of states” is used since this density of states is obtained from the optical transitions as seen in photoemission. It is also ap- propriate since the optical density of states may be modified from the true density of states by optical matrix elements. Let us examine direct and nondirect models for Cu as well as the experimental data used most recently. Copper is most appropriate for a number of reasons. First, its band structure seems to be as firmly established as any of the noble or transition metals. Second, it possesses relatively narrow d bands which might provide nondirect transitions; and third, experi- mentally Cu has been studied as thoroughly or more thoroughly than any of the other noble and transition metals so that the experimental data now seems to be on a very good footing. Let us now examine photoemission from clean Cu for 6.0 s hy & 11.6 eV. In figure 4 we present EDCs for Cu from the work of Krolikowski and Spicer [4]. More Eastman [42] and Smith [43] reproduced these curves; thus, the experimental data seems quite reliable. This data has all of the charac- teristics which lead to the assumption of nondirect transitions. For one thing, the peaks superimpose when they are plotted against E– hy, i.e., against the initial state energy. Thus, it is apparent that the EDC struc- recently, have 143 4 xiól flay=9 OeV fia) = ||.2 eV |O2 eV 9.6 eV OH | l | | | | | | | -5 -4 -3 -2 - O -6 -5 -4 -3 | | | –2 - O |NITIAL ENERGY (eV) FIGURE 4. EDCs for clean Cu plotted versus the initial energy. The Solid curve indicates the experimental curve and the thin full curve gives the energy distribution calculated using the nondirect model for the values of photon energy indicated. The arrows indicate the position in energy of structure in the ODS. ture is due to the same structure in the initial ODS. Note also that the structure in the EDC varies very monatonically with photon energy. As we shall show later, a striking characteristic of the direct transitions calculations is the relatively larger amount of modula- tion which they predict in the peak strengths as a func- tion of photon energy. As described by Krolikowski and Spicer [4], the ODS was obtained from the photoemission and optical data. The ODS so obtained is presented in figure 5a and b. From this ODS, the thin full curves in figure 4 were obtained from this ODS using the nondirect, constant matrix element model. As can be seen, the agreement is rather good particularly since it is on an absolute basis. The notable difference is that the first peak broadens and the second peak appears to merge into it at higher photon energies. In figure 5a and b, the ODS obtained from the photoemission studies is compared to the density of states from two band calculations [44,45]. As can be seen, rather good agreement is obtained between the lo- cations of the major pieces of structure in the ODS and the calculated density of states. However, there is no such agreement between the relative strengths of the structure. This may be due to the effects of optical matrix elements, to difficulties in the band calculations (note the difference between the two calculated density of states), or to other effects. In figure 6, the results [33] of calculations based on the direct-transition model for clean Cu are presented. These calculated curves have strong similarities to the experimental data. However, in order to obtain such agreement it was necessary to include a Lorentzian broadening of 0.4 eV for the calculated curve. Other calculations [34,35] use broadenings of between 0.3 and 0.7 eV. If the broadening is not used, much too much sharp structure appears in the calculated EDCs and this structure is modulated much too strongly and fast. The use of the broadening function finds partial justification in several factors—the instrumental and lifetime broadening, the finite lifetime of the excited carriers, and the inaccuracy in the band calculations. However, it is important that we keep the broadening in mind since it tends to make the direct and nondirect calculations more similar and also since it may provide an empirical method of making correction for many body effects. In the limit of flat initial bands, the direct and nondirect models would be identical. As the bands become less flat, increased broadening will still tend to keep the agreement between EDCs calculated on the direct and nondirect models. Let us now examine the EDCs calculated by the direct method. In figure 6 we show the results of the calculations of Smith and Spicer and in figure 7 we compare the results of these calculations to experimen- tal data. Again, the comparison is on an absolute basis. Several things are noteworthy about these results: (1) The position in energy of peaks in the direct calcula- tions is constant on the E – hy plot, (2) the position of structure corresponds rather well with the position ob- served experimentally (the numbered lines correspond to the position of structure found experimentally and in the ODS), and (3) the modulation of peak heights and widths is much stronger than anything seen experimen- tally. If, in fact, such strong modulation was observed experimentally, this would have been attributed to the effects of direct transitions or matrix elements effects, despite the constant position in (E - hv) of the peaks. Such identification was made, for example, in the II-VI compounds [38], GaAs [36] and PbTe [37] where strong modulation was observed experimentally. The constant position in energy of the direct struc- ture in figure 6 and its agreement with experiment is 144 O ––– DENSITY OF STATES º (d) 8 OH ––––– BAND CALCULATION (D (b) (SNOW) (MUELLER) (3) | 49 – PHOTOEMISSION ODS | | OPTICAL DENSITY || (PRESENT WORK) | OF STATES # | 5 6.O- | à S | º: 3. OH- | s H | OH | tº É | | \ 2 : - *- y \ – * – O | | | _^l \v-TN / \ U –7 -6 -5 –4 -3 –2 - | o–H– | z ENERGY BELOW FERM | LEVEL -8 -7 -6 || -5 - 4 -3 - 2 - | O ENERGY BELOW FERM | ENERGY FIGURE 5. (a) Comparison of the ODS with the density of states calculated by Snow [44] using an 5/6 p"ſº exchange term. Snow’s density of states have been shifted by 0.2 eV to place the Fermi level to d-band energy in exact agreement with experiment. The absolute scale was placed on the ODS by placing 11 electrons within 5.5 eV of the Fermi level. Note that the four pieces of numbered structure coincide rather well in energy. The numbered arrows correspond to those in figure 4. (b) Comparison of the ODS with the density of states calculated by Mueller [45]. data than the direct model, suggests that many body ef- 3x10– - e º • e - @ fects may still be important in bringing in a range of k *. (3) 2 rather than a delta function in the optical absorption @ process. In another paper, presented at this meeting, Neville Smith [34] will show new experimental data which give N/\º – 4x16'. flaj =9 O eV Of —/ |- / | | l | | –6 – 5 – 4 –3 – 2 - || O |NITIAL ENERGY (eV) FIGURE 6. The EDCs calculated for Cu by Smith and Spicer assuming direct transitions. not surprising in retrospect in view of the agreement between the ODS and band calculations shown in , figure 5. It would appear, at least for the limited range of hu covered by this study, that the Cubands are suffi- ciently flat and that the broadening effects are suffi- 1—1–1—1–1 ciently large so that the k conservation condition does –5 –4 -3 –2 - O -6 -5 -4 -3 - 2 - O º º - * |NITIAL ENERGY (eV) not impose overwhelming constraints on the optical ex- e - - FIGURE 7. Comparison of the EDCs calculated for Cu using the direct citation process. The fact that the nondirect model transition model with the experimental EDCs. The full line gives gives better detailed agreement with the experimental experimental and the dashed line calculated EDCs. 417–156 O - 71 – 11 145 clear evidence of direct transitions in cesiated Cu. The transitions originate from states 2.8 to 3.8 eV below the Fermi level. It is in this region that the d bands have greatest curvature. Recognizing that this curvature should provide the most easily detectable evidence for direct transitions, Berglund and Spicer [3] looked especially for direct transitions in this region. Ap- parently poorer sample preparation conditions prevented them from seeing the transitions. The suc- cess of Smith is a tribute to him and to the advances in vacuum and preparation techniques made at Stanford and elsewhere in recent years. Smith has also made direct transition calculations of the EDC for cesiated Cu. These show the effects found experimentally; however, despite the inclusion of a 0.3 eV broadening factor, the predicted modulation is con- siderably stronger than that observed experimentally. There is perhaps a good analogy between the present situation in this matter and that with regard to x-ray emission spectroscopy for many years. The simple and popular view of the latter field was that one could al- ways explain the x-ray emission spectra just in terms of single particle transitions so that the valence band den- sity of states could be obtained directly from the emis- sion spectra if ‘‘atomic-like” matrix elements were properly taken into account. With the simple metals fair agreement was obtained between experiment and theory on this model, although certain nagging incon- sistencies remained. The situation has changed drasti- cally in the last few years since theorists have had suc- cess in treating the many body effects of the hole in the core state. I will not attempt to review this work since it will be discussed in some detail at this conference. However, there may be a parallel with regard to the uv photoemission work. At Stanford, Doniach [46] has been expanding his in- vestigation of many body effects in the x-ray photoemis- sion effect to include the many body effects associated with screening of the valence band hole in the uv opti- cal excitation process [32]. Preliminary results suggest that such effects exist, producing a spread in possible k in the optical transitions, and increase in importance as the effective mass of the hole increases. Thus, the flatter the valence band is, the larger the effect. If one looks at the Cu results with this in mind, one notes that the flatter the bands, the better the nondirect model works. In concluding this section, I would like to remark that the direct transition model is based on a rather idealistic assumption which applies best where the bands have good curvature; empirically, this model seems to work very well for a wide range of materials of this type. On the other hand, the nondirect, ODS, model should work best in materials with quite flat bands. It may never be completely correct (we must un- derstand the physics better before it is possible to pass quantitative judgment); however, its great simplicity may make it a good first approximation when it can be successfully applied, i.e., when the EDCs based on the ODS are in relatively good agreement with experiment. Certainly the success with Cu, Ni, and similar material, suggests that it may give us the best first approximation to the densities of states of these materials which can be obtained solely from experiment. There may be an intermediate range of bands and materials in which neither the direct nor the nondirect model applies with great accuracy. In this case, detailed understanding can only be obtained when theories such as that of Doniaeh are fully developed. In the meantime, it is probably well to keep open the pos- sibility of transitions occurring over a range of k and not just at a given value. It would be extremely nice if in the direct calculations, a broadening could be put in by a distribution in k before searching the zone rather than over energy after the vertical transitions have been tabulated. Experimentally, it is important to obtain data over a wider range in energy to test the selection rules with more rigor. Eastman [29] has already begun to do this with very interesting results. 5. Effect of Reducing or Destroying Crystal Periodicity: Liquid In, Alloys, and Amorphous Ge Another way of testing for the importance of k as an optical selection rule is to reduce or destroy the long- range order of a crystal. Clearly as the solid becomes increasingly disordered, any dependence of k must become less and less well defined, i.e., a single value of k can no longer be used to define a quantum state. Rather, if a description in terms of k is used, it must contain a distribution of k; a single k will be insuffi- cient. In the limit of complete disorder, k will lose meaning as a quantum number. 5.1. Indium Indium has been studied experimentally by Koyama [18] in the crystalline, amorphous and liquid forms. Note that, since it contains no d electrons, In would not be expected to fall within the class of nondirect materi- als. In addition, Koyama has made calculations based 146 |ND|UM : OPTICAL DENSITY OF STATES —- — CALCULATED DENSITY OF STATES —1– l | | | l | -- O |.O 2O 3.O 4.O 5.O 6.O 7.O 8.O 9 O ELECTRON ENERGY (eV) FIGURE 8. Comparison of the ODS for In with the density of states obtained from band calculations [40]. on direct as well as nondirect models. These calcula- tions will be described in detail in a separate paper of this conference [40]. Koyama's findings for crystalline indium are quite interesting: (1) Both the direct and nondirect transition models fit the experimental data fairly well (as they do for Al), (2) the EDCs for In are characterized by two broad peaks separated by a minimum which correlates [47] well (in either model) with a large band gap in the band structure of Ashcroft and Lawrence [48] (see fig. 8), and (3) the principal fea- tures of the EDC (the two peaks) were seen to persist in liquid indium despite highly increased electron scatter- ing. Since there seems, at least at present, to be less physical justification for the nondirect model in In than in Cu, it is tempting to assume for this material that direct transitions dominate in the crystalline material and that nondirect transitions occur in the liquid. Even then a question would arise as to why the density of states structure due to crystalline potentials persists into the liquid. (Shaw and Smith [49] have found theoretical evidence of such effects in Li.) Koyama [18] has suggested that this is due to the dominance of short-range interactions in determining the electronic structure and thus the density of states of both liquid and crystalline In. Clearly studies of In at higher resolu- tion and for a wider range in hu should prove very worthwhile. In any case for both Al and In, the density of states obtained by the nondirect analysis seems to be in fair agreement with the results of band calculations. As the direct transition calculations show, this may be due to the large range in k space from which direct transitions can take place and thus not be a true indication that k vector is unimportant (although, again, some uncertain- ty in k is probably important in bringing the direct and nondirect models into agreement). The sensitivity of the calculated EDCs to the electronic structure is illus- trated by the fact that, whereas Ashcroft and Lawrence's band structure for In agreed with experi- ment, other proposed band structures [49] did not give agreement with the ODS. Mosteller, Huen and Wooten [50] have recently stu- died the photoemission from Zn as a function of tem- perature and found that the quantum yield decreased significantly on cooling the sample from room to liquid No temperature. Based on this, they note the possibility that in Zn the ultraviolet optical transitions may be in- direct, i.e., phonons conserve k. Such temperature de- pendence has not been observed for other semiconduc- tors and metals such as Cu, Gd [51] and Cr [52] which have been studied as a function of temperature. The Zn results are mentioned here because of the similarity between the In and Al band structure and that of Zn and because In and Al have not been measured below room temperature. 5.2. Amorphous and Crystalline Ge In contrast to In, Ge provides a striking case of a material whose optical properties and uv EDCs change drastically when the long-range order is destroyed by forming amorphous Ge. Photoemission studies show clearly the direct nature of the transitions in crystalline Ge [36] in agreement with analysis of optical data [53]. Thus, differences between crystalline and amorphous Ge are of considerable importance. Figure 9 indicates eg for the amorphous and crystal- line material [54,55] and figure 10 indicates EDCs for 4 O ſ\ t \ | | | | ; : | i 3OH | | | | – CRYSTALGERMANUM € } DIRECT TRANSITIONS 2 / AMORPHOUS / GERMANIUM / 2OH | | | | l l | | | | \ \ \ \ \ —l- 6 2 3 4 5 PHOTON ENERGY (eV) FIGURE 9. eg for amorphous and crystalline Ge. 147 GERMANIUM 6 H (CLEAVED, INTRINSIC SINGLE CRYSTAL) (o) /*N 4 H / \ hv= |O.2eV S–Z \ \ *N S_2^ \ AMORPHOUS Ge (b) O }= 8 H 6 H 4 H. hy = | O.8 eV 9.8 eV 8.8 eV 2 H. O 4. 5 6 7 8 9 |O | | ENERGY ABOVE VALENCE BAND MAX. (eV) FIGURE 10. Photoelectron energy distributions for Ge surfaces. (a) Cleaved, intrinsic, single crystal. (b) Amorphous film. The vertical axis gives the number of electrons per absorbed photon per eſ’. The horizontal axis gives the electron energy relative to the maximum in the valence band. The sharp structure in (a) is due to direct transitions in specific regions of the zone. The single broad peak in (b) is due to a peak in the valence-band optical density of states. crystalline and amorphous Ge [54,55,56,57]. As can be seen, the changes in e2 and the EDCs which accompany the change in form of Ge are first order. The loss of sharp structure is clearly due to the loss of long-range order. In their studies of amorphous Ge, Donovan and Spicer have used a nondirect analysis with considera- ble success to treat data from the amorphous material. In figure ll the ODS obtained from these studies is compared to the density of states obtained from band calculations [58]. Brust is approaching the problem of amorphous Ge from calculated band structures by a method in which there is a spread in k associated with the optical transitions and thus is intermediate between the direct and nondirect models [59]. Because of its flexibility due to the possibility of assigning various values to the spread in k, this approach clearly has cer- tain advantages over the pure nondirect approach. 5.3. Cu-Ni Alloys A third example of the effect of disorder is in the al- loys such as those between the noble and transition metals. Here the lattice periodicity is not destroyed. Rather, atoms with two different potentials are ar- ranged at random, or almost at random (it appears that clustering effects are negligible [21]) within the periodic lattice. Since the potentials are quite different (for example the transition metal typically produces a virtual-bound state when dissolved in a noble metal), the effect on the periodicity should be considerable. Despite this, the effect on the ea and on the EDCs of the host metal does not appear to be drastic. The principal effect is in the production of a virtual-bound state under the proper circumstances. Such states have been and are being qualitatively studied through the use of photoemission [21,23,60,61,62,63]. In figure 12, the optical parameter oo is presented for the Cu-Ni alloys studied by Seib and Spicer [21,23]. Except for hy-2 eV in the Cu-rich alloys where the change is due to the formation of a virtual-bound Ni state, the changes are much less than those found in the crystalline to amorphous transformation of Ge. *As outlined in the Introduction, photoemission can give a more detailed look at the optical transition than can the optical data. Examination of EDC data from the alloys shows that the direct transition from the s-p- derived bands near the Fermi surface at L is not de- tectable in the alloys [21]. However, the transitions from the d states are much less affected. In fact, the EDCs from Ni and Ni-Cu alloys with up to 19 percent –––– CRYSTAL DENSITY OF STATES (THEORETICAL, HERMAN AND SHAY) — OPTICAL DENSITY OF STATES AMORPHOUS GE (EXPERIMENT) OO. 68 O. 4 O. 2 ENERGY ABOVE VALENCE BAND MAXIMUM (eV) FIGURE 11. Optical density of states for amorphous Ge as determined by photoemission compared with the electronic density of states for crystalline Ge calculated by Herman and Shay. The vertical axis is in units of states per eſ’ per atom for the crystal density of states and in arbitrary units for the optical density of states. The energy zero in both cases is taken at the maximum of the valence band. 148 19% Cu. - 81%. Ni 39% Cu. -6.1% Ni 49%. Cu-51% Ni 62% Cu-38%. Ni 77% Cu-23% Ni 87% Cu- 13 C | | | 6 8 |O |2 hv(eV) | | 2 4 O FIGURE 12. The optical constant ador for pure Ni and Cu and a series of Ni-Cu alloys. Cu (atomic present) are almost indistinguishable except for effects due to the change in work function. This is shown by the data in figure 13. Even for 39 percent Cu, the position of the two strong peaks in the EDC were unchanged [23]. Let us next examine the Cu-rich alloys. In figure 14 we present data for pure Cu and Cu containing 13 and 23 percent Ni [21]. As can be seen, the Cu d edge is lit- tle changed and the position in energy of structure from the d bands is similar to that in the pure material; how- ever, the relative strengths of the peaks are changed. The contrast in optical properties and EDCs between these alloys and Ge in its crystalline and amorphous forms is striking. For the alloys, the changes are rela- tively small whereas, for Ge, they are much larger. k conservation clearly plays the dominant role in deter- mining the optical transition probabilities in crystal- line Ge; thus, destroying the long-range order complete- ly changes the optical properties. The insensitivity of Cu and Ni to disruption of the long-range order suggests that the optical transitions from the d states of pure Cu and Ni are, on the average, much less strongly affected by the k conservation condition; however, the L transition from the s- and p-derived states is clearly a direct transition and this disappears in the alloys stu- died. 6. Methods of Determining the Density of States from Ultraviolet Photoemission Data Two extreme approaches can be taken in using photoemission data to determine the density of states of solids. One is to use the photoemission results to pro- vide input into band calculations. This approach is not necessary if first-principles band calculations give exact results. If this is not the case, the band calcula- tions can be adjusted to give agreement with the experi- mental data. Such correction is often necessary and, in addition to overcoming uncertainty in the potential used in the band calculation, the empirical correction may correct for departures from Koopmans theorem as, for example, suggested by Herman ||66|. One ap- proach is to parameterize the calculation and use ex- perimental data. de Haas-van Alphen data or optical data could also be used for adjusting the band calcula- tions. Since the de Haas-van Alphen data give experi- mental data only at the Fermi surface, it is not very sen- sitive to energy shifts from the Fermi level. Unam- biguous interpretation of structure in the optical con- stants, such as e2, has proven very difficult. Piezoreflec- tion has proven to be very powerful in Cu [10] but despite considerable effort, so far has not been success- fully applied to Ni [67]. A difficulty in piezoreflection also lies in estimating the absolute or relative strength of optical transitions whose symmetry is determined by these measurements. If first-principles band calculations were thought to be sufficiently good, the photoemission studies would 3x Oº h v = |O.O eV S Q) e- 2 ATN89%Ni-II*.co O / 5 I 2 H. 0– C LL CD Cr. O (ſ) OD <ſ ^. (ſ) 2 O | H. Or H. O til — U 3 uſ 2 O | | | | | | –6 –5 — 4 – 3 – 2 — O Ej = E-hy + 4 (eV) FIGURE 13. The EDCs at hy = 10 eV. obtained from pure Ni and a Cu- Ni alloy containing 11 atomic percent Wi. 149 4x10– 77%Cu –23% Ni S Gl) | 2 — hy= |O.2 eV 3 3 E T 0- C LL] £ 87%Cu–|3%Ni (ſ) OO $ 2 H (ſ) 2 O CC G LL — U 3. J 'T 2. COPPER O | | | | | | –6 –5 –4 –3 –2 – O E=E-hv4-4, (eV) FIGURE 14. EDCs from pure Cu and two different Cu-Ni alloys. Note that the Cu d edge and the position in energy of the d peaks is essentially unaffected by the alloy. simply serve as a check. For best results, this approach requires two conditions. First is a fairly accurate and well-advanced band theory. Without this, it is difficult to relate the photoemission data to the band structure in a meaningful way. Second is photoemission data which shows dramatic band structure effects such as the onset of the L transition in copper or the T transition in CdTe [38]. For materials like GaAs in which k conservation dominates the optical transition probability, Eden has developed a systematic method for comparing photoemission results and the results of band calculations. This will be reviewed briefly in the next section. A second approach is to attempt to obtain density of states information directly from the photoemission data. The more apparent the connection between the photoemission data (i.e., the optical transition proba- bility) and the density of states, the more efficient is this approach. As we will see in the next section, it is very difficult to obtain density of states information from photoemission data for a material such as GaAs where k conservation provides a dominant optical selection rule; however, in a case such as copper where k conservation does not play such a dominant role, the nondirect method of analysis gives a good mechanism for obtaining the principal features of the density of states from experimental data. The nondirect transition [3,4] model provides a sim- ple way to analyse the photoemission data to obtain an ODS. Once this is done, EDCs can be calculated and compared with experiment. In this way, the consisten- cy of the nondirect approach can be judged. Only where reasonable consistency is obtained can the non- direct approach be used in a meaningful way. However, even when clear evidence is obtained that some struc- ture is due to direct transitions, useful density of states information can apparently be obtained from the non- direct approach when EDCs calculated using the ODS reproduce closely enough the major strengths in the ex- perimental EDCs. (Cu [3,33,34] and Au [5] appear to be examples of this.) By major strengths, we mean at- tention should not be focused on relatively weak struc- ture which is clearly direct, but on the overall am- plitudes in the EDCs. 7. A Sampling of Experimental Data Since this paper is already lengthy, we will not at- tempt a comprehensive survey of the photoemission literature; rather, we will attempt to present only a few representative results which have not been presented previously in this paper in order to illustrate and ampli- fy the remarks made earlier. Photoemission measurements and the nondirect analysis has been made on a fairly large number of transition and noble metals other than those mentioned earlier. Eastman, in particular, has obtained the ODS for a wide range of transition metals [6.20,29]. In figure 15 we present the ODS obtained by Eastman for ten metals [68]. For the sake of comparison, the density of states from band calculations are also given [68,69]. Although the agreement between experiment and cal- culation is not perfect, it is encouraging, particularly when one realizes that the band calculations were not highly refined and in some cases were just obtained from the calculation for a different material using a rigid-band approximation. The agreement obtained sug- gests that there is a meaningful relationship between the ODS and density of states obtained from band cal- culations, as does the agreement found for Cu [3,4], Ag [3,62,70), Ni and other transition [6,52] and rare earth metals [71]. : In section 4, it was suggested that the narrower the bands the more valid the nondirect approach and thus the ODS of the correct density of states. If this is true, the situation within the transition metals should become less favorable as the atomic weight of the metal 150 PHOTOEMISSION THEORY PHOTOEMISSION THEORY SC SC T Ti -(.85 x Y) T(.9 x Hſ) | H. / M- | H. }= Z 2^ 2^ 2^ O H --1-1-1 O =5 16 ſ 1 1 O O #H l O ++ | | 1 O Y Y Zr Zr - LOUCKS ſ LOUCKS E O | H. º H sº H * / . / Sº, / O —l I | | 1 * 1 | 1 I I I uſ o Hº ++ === O oHº- O == O U Gd Gd Hf Hf É TKEETON 8 (Zr — º LOUCKS LOUCKS) H. |H. \ºme (ſ) | H. 5 / >- 2^ H O l | I | l ! I 1 -1-1-1 O 1–1–1–1–1– l l I 1 I op –5 eV O –5 eV O –5 eV O –5eV O 2 # V V Cr Cr 3 (Cr + RIG|D ASANO, <ſ /º - BAND) - et Ol CD | | }* O 2 Z | H .* | <ſ / ,” % _2^ / O O l == —l I I l O 1. ^1 1 | 1 O is 1. l | _l O I 4– l 1 f O Nb Nb Mo Mo (Cr-RIGID (Cr) | |- T BAND) |H Wºme / | | . | / 22' O " I l l I 1–1–1 i I e 1 I I I I 1. ^1 – l l ; 1 -5eV O = EF – 5 eV O = EF –5eV O= EF -5 eV O= EF FIGURE 15. Optical density of states obtained by Eastman as compared to the density of states obtained from band calculations. This figure is taken from ref. 68. increases since relativistic effects will broaden the rather good agreement with the EDCs obtained from bands. For example, the d-band width of Au is about soft x-ray photoemission work [72]. The photoemission twice that of Cu. Krolikowski and Spicer [5] have also results also have been found by Ballinger and Marshall studied clean Au in good vacuum for 5.4 shu s 11.6 [73] to correlate rather well with their band calcula- eV and in poor vacuum for hu values of 16.8 and 21.2 tions. On the other hand, work by Eastman at photon eV. From this work the ODS presented in figure 16 was energies of 16.8 and 21.2 eV in good vacuum gives obtained. As can be seen in figure 16, the ODS is in strong evidence that direct transitions are important in 151 VALENCE STATES - ~ * 2.5H % ow I SOFT X-RAY ° 3 / \o #otoÉMission - A N / \ --- O / O \ O / \O 2,OH- / \o al 2 (ſ) ~ #3 st as - H. : - —l U0 ºf }- LL) L. ' | 5 H- à o: ''< | – >- > OPTICAL DENSITY -- E 2 | OF STATE /6/ É UD O }* N/ / Lil 2 Or } - / / Li- tº E I.O / / O O }* uſ }- / }, a. [I } = / Á G - / F / O / 0- O.5H / 94.9 O Aerº - 9.9% [-acre; T / O °-----' | | | l | | | | | `- -12 -|| -IO -9 -8 -7 -6 -5 - 4 -3 -2 -| O ENERGY (eV) FIGURE 16. Comparison of the ODS and the soft x-ray photoemission results of Siegbahn, et al. [72]. The x-ray results have been shifted to lower energy by 0.6 eV to obtain the best fit. (It is difficult to set the absolute zero of energy in the x-ray experiment.) Au. This series of results suggest that quite useful den- sity of states information can be obtained from the rela- tively narrow bands of noble and transition metals by the ODS type of analysis even when direct transitions are important and that the broadening of the d band in going to Au does not make the ODS approach useless. Up to this point we have concentrated to a large ex- tent on materials for which the nondirect analysis can be used. In order to give perspective, let us now ex- amine GaAs in which k conservation has been found to provide a dominant optical selection rule as it has been found for Ge, Si, and other III-V compounds [36]. If structure in the EDCs is due to peaks in the initial or final density of states, this can be detected by plotting the EDCs against initial energy (E– hu) or final energy (E) respectively. This argument holds even if the transi- tions are direct. The distinction between direct and nondirect transitions is made on the basis of modula- tion of the strengths of the peaks with particular atten- | CLEAN Go As (d) hy = ||. 2 eV hy = 10.2 eV }*}*.}*}= | | i | | | I | A | …” I - => * tion being paid to evidence for them appearing or disap- pearing as photon energy is varied [36,37]. With this in mind, let us examine figure 17a and b where two typical EDCs for GaAs [36,74,75] are plotted versus final, figure 17a, and initial state energy, figure 17b. As can be seen, these EDCs are particularly strong in structure. Despite this, there is little tendency for the structure to fall at the same energy either on ini- tial (E-hv) or final, energy plot. This shows clearly that k conservation provides an important selection rule. As a result, it is difficult to obtain density of states infor- mation directly from such plots. Eden [74] and Eden and Spicer [75] have derived a reasonable way of analyzing such data. This is done by making a plot of the final state energy of structure in the EDC, E, of structure versus the photon energy. Such a plot is shown in figure 18 for cesiated GaAs. One can obtain from band calculations theoretical plots of the same type for the symmetry directions of the crystal. By su- perimposing the two plots, it is possible to make identifications of the structure in the EDC. Such identification is indicated in figure 18. Further details are available elsewhere [36,74,75]. To obtain informa- tion on the density of states, it is sufficient to note two features: (1) A horizontal set of points for E = 5 eV labeled, “Final States Near Lø.W.;” and (2) the 45° line between final state energies of about 4.5 and 8 eV labeled, “Transition II from Band 3 Minimum.” Since (1) is a fixed, final state, it would suggest a peak in the final density of states at about 5 eV. In figure 19 we present a band structure for GaAs by Cohen and Berg- stresser [76] along with the density of states calculated from it by Shay and Herman [77]. As can be seen, there is a very sharp peak in the final density of states at about 5 eV. The 45° line in figure 18 indicates a transition from initial states at a fixed energy E since E = E + hy. CLEAN Go. As (b) ſhvs 11.2 ev ſhv = 10.2ev —s A 6.O 8.O |O 12 Fi NAL ENERGY (eV) i l | l l l | | l - 6,O -4.O ~2.O O +].O | NITIAL ENERGY (E - hy) FIGURE 17. (a) EDCs from GaAs for photon energies of 10.2 and 11.2 eV plotted as a function of final state energy. (b) EDCs for GaAs plotted vs E - hv to refer the energy distributions to the initial states. Note that the structure in the EDCs does not coincide on either a final energy plot (fig. 17a) or an initial energy plot as in this figure. This gives clear evidence that the transitions are direct. 152 12.O I I I I I T I T I I / / / |||O H. / - / s’ / S / ! s |O.OH / ! - • / § § 3. 1,44; if if × 9.O H. CESIATED Go AS .4 S - 3 p PEAK / S : S SHOULDER / § ă so v VALLEY / .# s $- OH S º / , sº S S O % / s * S Z S p Lil S s S - a 70F / , sº a *, *, *, *s > / S tº s aş SS g E = h v LINE –-Z s S S S s ss $', P,” 3 6.OH- Z S & bº - * / SS S S Q LL 2^ s FINAL STATES NEAR L3,w >- s S S pop S S S S S S § 50H Tiss :* S S S S ^_ # s?] s sis O Lil 4 #. p / ~ 5 § 9 S 3 4 OH / T s sº ſº; ## - or / s $ s? 5 / , iii # , al / illº - LL 3. O |||}| }} § S S º Hill!. º, Ill!...: 20 / A TRANs TRANSITION I ‘sºo D p – ––––––––––––––––––––– jøroſſogºsºs ss - - - - - - - - - - - - - - - - - - - - - - |O —l I –– L | l l 1– l l | O 2O 3 O 4 O 5 O 6.O 7 O 8 O 9 O IOO || O |2 O PHOTON ENERGY (eV) FIGURE 18. A structure plot for the photoemission from cesiated GaAs. In such a plot the final energy of structure in the EDCs is plotted vs the photon energy. Such plots can be compared to predictions from band theories. They also provide at a glance certain information on the nature of the source of the structure in the EDCs, i.e., a horizontal line indicates transitions from flat portion of the valence band. Since the 45° line is located about 3.7 eV behind the E = hly line, the initial states must be located this distance below the top of the valence band. As can be seen in the density of states plot of figure 19, there is a sharp densi- ty of states peak at just about this energy. Thus the two density of states peaks which are perhaps strongest and sharpest can be identified directly from the photoemis- sion data; however, other strong structure which is not so narrow was not immediately detected from the photoemission data. This was because the curvatures were not sufficiently small so that a clear distinction could be made between the effects of initial and final density of states. As is reported in a paper by Buss and Shirf [78] at this meeting, work by Spicer and Lapeyre [37] on PbTe seemed to have been successful in determining peaks in the density of states which correlate well with their band calculations. This occurred despite the fact that direct transitions are clearly important in these materials. 8. Comparison of Density of States Determincitions Using Various Experimental Methods In addition to uv photoemission spectroscopy, three ther experimental techniques exist which can give irect information on the density of states of solids. In this section we will compare the density of states ob- tained by these methods for Cu with that obtained from OUIr measurementS. 8.I. Comparison with Results of lon Neutralization Spectroscopy In figure 20 the ODS for Cu is compared to the densi- ty of states obtained by Hagstrum [79) from Cu via the ion neutralization spectroscopy (INS) technique which he has developed. The peak between –2 and –4 eV is associated with the d states. As can be seen, the width of this peak is considerably greater than the d width in- dicated by the ODS or calculated band structure. In ad- dition, in the neutralization results even though the instrumental there is no detailed structure ion resolution is sufficient to resolve structure such as that seen in the ODS or calculated density of states. Hag- strum has noted [80] that since his technique depends on electrons tunneling from the surface of the metal, it is sensitive to the electronic structure just at the sur- face and that for d electrons this structure may be dif- ferent from that in the bulk of the material. If it is suggested that a change of the electronic struc- ture can take place at the surface, one must ask whether this can also affect photoemission studies. In principle, the photoemission is a bulk effect and thus would not be changed by variations in the electronic structure associated with the last atomic layer or so of the solid. However, the fast electron-energy depen- dency of the electron-electron scattering length (see fig. 3) and the low scattering length at high energies (as low as 10A in some materials) must be taken into account. Thus, as photon energy is increased up to 12 eV, the escaping electrons will come from regions closer and closer to the surface and it is possible that measurable changes in the EDCs might be due to changes in the electron structure at the surface. Comparison of the EDCs from cesiated [34] and uncesiated [4] Cu show that changes occur on cesiation in the relative strengths of the two leading d band peaks in Cu. Similar results are found in the Ni-Cu alloys [21]. These results are not understood, but are mentioned to indicate that the d band transitions appear to be sensitive to changes in the details of the conduction band electrons. If this is the case, changes of spatial distribution of conduction electrons at the surface might affect transitions from the d states. This could for example, contribute to the broadening of the first d peak from clean Cu which oc- curs as photon energy is increased (see fig. 4). The pur- pose of this discussion was to point out effects which might be important in photoemission but which have 153 #[40] [###| [000 [oolſ##0 |O 3. GALLIUM ARSENIDE BAND STRUCTURE COHEN AND BERGSTRESSER GALLIUM ARSENIDE DENSITY OF STATES SHAY AND HERMAN T (FROM BAND STRUCTURE AT LEFT) CRYSTAL MOMENTUM, K—— FIGURE 19. The band structure of GaAs calculated by Cohen and Bergstresser [76] and the density - of states calculated by Shay and Herman [77]. / / f £ 6| OPTICAL DENSITY OF STATES | 3 –––– DENSITY OF STATES DETERMINED BY 2^ 2. & ION NEUTRALIZATION \ _2^ § 5 H \-2^ E *- an or. St. 4F 2-> U / \ z or) / \ _2^ 3 H --~~ º Z 2^ —º / | § / ,” \ / 2^ _2^ Ł | 5 2 - _* --~~ | t _2^ | % | 2^ | # 2^ | | O 1– l l I l l | —l l —l- —l- —l- - || -IO –9 -8 –7 -6 –5 — 4 –3 –2 -] O ENERGY BELOW FERMI LEVEL (eV) FIGURE 20. Comparison between the ODS [4] and the results obtained by Hagstrum [79] through ion neutralization studies for Cu. not been established. If they do exist, it would appear that these effects are much smaller than the perturba- tion of the electron structure as seen at the surface in the INS experiments. 8.2. Comparison with Results of X-Ray Photoemission Spectroscopy Let us next compare the ultraviolet photoemission work with the x-ray photoemission data. The ODS for O DENSITY OF ELECTRON STATES Cu is compared in figure 21 with the results obtained by Fadley and Shirley [81] using the technique of x-ray photoemission spectroscopy (XPS). The XPS result is characterized by a single, almost symmetric, peak with a width at half maximum of about 3 eV. Since the total instrumental line width was about 1.0 eV, this width and lack of detailed structure does not appear to be in- strumental. If we make the reasonable assumption that the broad peak is due to d electrons, it is also signifi- — - - - - - XPS — ODS (ſ) > | 9 ºſ > |- qy *N. 2--S É / \s s / Y-–– (ſ) / / Lu l-’ > H sº H Lil Or. --~~~ TT | | | |- | | | –8 – 7 – 6 - 5 — 4 – 3 –2 — | O 46582 ENERGY BELOW THE FERMI ENERGY (eV) Ef FIGURE 21. Comparison between ODS [4] and results of the x-ray photoemission experiment of Fadley and Shirley [8]] for Cu. 154 cant that there is little evidence for the s- and p-derived states lying within 2 eV of the Fermi surface (see figure 5a and b). This effect can also be seen in the Au XPS data presented in figure 16. The s- and p-derived states can be clearly seen in the photoemission and INS work. The lack of any detailed structure in the excitation from the d states would also seem to be significant since such detailed structure does appear in the ODS as well as in the calculated band structure. However, it should be noted that substructure has been obtained in XPS results from Pt [8]], Ag and Au [72] (see fig. 16) and that the position in energy of this structure is in reasonable agreement with structure in the ultraviolet photoemission work. The reason for the lack of structure in the XPS for Cu is not clear at this time; however, it is interesting to note, as will be shown in the next section, that almost the same symmetric curve is obtained in soft x-ray emission spectroscopy as in the XPS results. 8.3. Comparison with Results of Soft X-Ray Emission Spectroscopy A fourth experimental method used to investigate the filled states solids is that of soft x-ray emission spec- troscopy (SXS). The results of such investigations [82,83] for Cu are compared in figure 22 with the ODS. As mentioned in the last section, the SXS curve is very similar to the XPS curve in that it contains a single al- most symmetric peak and shows no evidence of the s- and p-derived states lying between the Fermi level and the top of the d band. Cuthill, McAlister, Williams, and Watson [85] have reported structure in the SXS from Ni. However, it is not nearly as pronounced as that seen in the ODS of Eastman. There are some similarities between the ODS and the SXS results for Ni; however, the correlations do not seem to be strong. Cu-Ni alloys have been studied both by SXS [86] and ultraviolet photoemission [21,23]. It is interesting to note that in the photoemission and optical work it has been possible to clearly identify a Ni virtual-bound state in the Cu-rich alloy and that these virtual-bound states are much different than Ni states in pure Ni. For example, their width at half maximum appears to be less than half of that of pure Ni for Ni concentrations up to about 25 atomic percent in Cu. In contrast, in the x-ray work the spectrum obtained or Ni in Cu down to 10 percent concentrations was in- distinguishable from that of pure Ni [86]. These results uggest that interactions with the deep hole override alence band structure in determining the SXS from — ODS - - - - - - SXS M3 — — — SXS L3 / SS / `--- / _^ * / / / 2^ - - - - - T .* ...” – T | | –8 - 7 -6 -5 — 4 –3 – 2 - | O 46618 ENERGY BELOW THE FERMI ENERGY (eV) Ef FIGURE 22. Comparison between the ODS and results obtained from soft x-ray emission spectroscopy. The curve labeled M3 was obtained using M3 radiation [83] and that labeled L3 using L3 radiation [84]. Ni; if this is so, the SXS would yield more information on the interaction between the deep hole and the valence electrons than on the valence band density of State S. 9. Conclusions The ultraviolet photoemission work done to date shows that density of states data can be obtained from such measurements. Because of the high resolution available in such measurements (0.05 to 0.3 eV), more detailed information can presently be obtained than by any other experimental method used to determine ex- perimentally the density of states. In materials such as Cu where the most extensive work has been done, both experimentally and in theoretical calculations of the density of states, relatively good agreement is obtained between the position in energy of structure in the densi- ty of states. No other experimental method has given such clear-cut results or impressive agreement; how- ever, good agreement is not obtained in the relative strengths of structure in the experimental and theoreti- cal density of states. There are still fundamental questions which must be answered both with regard to the photoemission experiment and its interpretation and with regard to the band calculations and their rela- tion to optical excitation spectra. The photoemission data as well as calculations on Cu are probably the most complete available for any metal. The work of Smith [34] on Cu shows clear evidence of direct transitions from the regions of the d bands hav- ing large curvature. The calculations of Smith [34] and Smith and Spicer [33] show strong similarities between measurements and calculations based on direct transitions; however, the direct calculations pre- dict much stronger modulation of the intensities of peaks than is seen experimentally. It should also be 155 noted that a broadening of 0.3 to 0.4 eV is used in the calculations to bring them into closer agreement with experiment. It is suggested that the experimental data is consistent with a model (suggested by Doniach’s [46] theoretical work) which assumes that the delta function k selection rule be replaced by a selection distribution of k's, with the width of the distribution increasing as the curvature of the bands decrease (i.e., as the group velocity decreases). Thus, one would move in a continu- ous fashion from a completely direct transition model for a material with sufficiently wide bands to a non- direct-type of model for sufficiently narrow bands. The band widths at which such transitions take place would depend on the detailed characteristics of individual materials. It appears that some density of states information can be obtained from photoemission data even when the transitions are completely direct. This can occur because peaks in the valence band density of states may produce EDC peaks which move with photon ener- gy over a limited range of hy. Likewise, density of states peaks in the final states may produce peaks which fall at a constant energy over a limited range of hv. All of this is just a consequence of the fact that a large volume in k space must lie near a single energy to give a peak in the density of states. Such behavior has been pointed out at this meeting in, for example, GaAs and PbTe where the density of states peaks so identified have been found to correlate well with densi- ty of states peaks in the calculated band structure. However, other peaks in the density of states in GaAs were not identified. This may have been due to the fact that the hu range used was not sufficiently large or that too crude a method is being used to identify density of StateS Stru Cture. In a different type of approach, photoemission stu- dies can also be used in direct collaboration with band calculations by providing empirical data on the band structure. This data can then be used to refine the band structure and the density of states can be calculated from the refined band structure. 10. Acknowledgment It is a pleasure to acknowledge stimulating and fruit- ful conversations with Seth Doniach, Dean Eastman, Walter Harrison, Frank Herman, David Seib, Neville Smith and Leon Sutton as well as my other colleagues at Stanford University. I am particularly indebted to Seth Doniach, Dean Eastman, and Neville Smith for ac- cess to their work prior to publication. ll. References [1] Koopmans' theorem states that the one-electron energy eigen- value ej in the Foch equation for a solid is the negative of the energy to remove the electron in state (b; from the solid. The proof of Koopmans’ theorem depends on the spatial part of the wave function being of the Bloch type and on all other wave functions being unchanged when one electron is removed. If Koopmans theorem holds, it follows that the photon energ necessary to excite an electron from state (bj to state dº is just the difference between the one-electron energies of the two states. However, if the eigenfunctions of other states are modified in the excitation, Koopmans theorem will not hold, and many-body effects must be taken into account. See also: F. Seitz, Modern Theory of Solids (McGraw-Hill Book Company, Inc., New York, 1940), p. 313: J. Callaway, Energy Band Theory (Academic Press, Inc., New York, 1964), p. 117; J. C. Phillips, Phys. Rev. 123,420 (1961): L. G. Parratt, Rev. Mod. Phys. 31, 616 (1959). See, for example, refs. 3, 4, and 5. Berglund, C. N., and Spicer, W. E., Phys. Rev. 136, A1030; 136, A1044 (1964). [4] Krolikowski, W. F., and Spicer, W. E., Phys. Rev., in press. [5] Krolikowski, W. F., and Spicer, W. E., Phys. Rev., in press. [6] Eastman, D. E., J. Appl. Phys. 40, 1387 (1969). [7] Beaglehole, D., Proc. Phys. Soc. (London) 85, 1007 (1965). [8] Ehrenreich, H., and Philipp, H. R., Phys. Rev. 128, 1622 (1962). - [9] Mueller, F. M., and Phillips, J. C., Phys. Rev. 157, 600 (1969). [10] Gerhardt, U., Phys. Rev. 172,651 (1968). [11] Spicer, W. E., and Berglund, C. N., Rev. Sci. Instr. 35, 1665 (1964). [2] [3] [12] Eden, R. C., Rev. Sci. Instr., in press. [13] James, L. W., Moll, J. L., Spicer, W. E., and Eden, R. C., Phys. Rev. 174, 909 (1968). [14] James, L. W., Ph. D. Thesis, Stanford University (unpublished), 1969. [15] Allen, F. G., and Gobeli, G. W., Phys. Rev. 144, 558 (1966). [16] Wehse, R. C., and Arakawa, E. T., Phys. Rev. 180,695 (1969). [17] DiStefano, T. H., and Pierce, D. T., Rev. Sci. Instr., to be published. [18] Koyama, R. Y., Ph. D. Thesis, Stanford University (unpub- lished), 1969. [19] Spicer, W. E., Optical Properties and Electronic Structure of Metals and Alloys, F. Abelés, Editor (North-Holland Publishing Co., Amsterdam, 1965), p. 296. [20] Eastman, D. E., and Krolikowski, W. F., Phys. Rev. Letters 21, 623 (1968). [21] Seib, D. H., and Spicer, W. E., Phys. Rev., to be published. (Cu- rich alloys) [22] Calcott, T. A., and MacRae, A. U., Phys. Rev. 178,966 (1969). [23] Seib, D. H., and Spicer, W. E., Phys. Rev., to be published. (Ni- rich alloys) [24] Kane, E. O., Proc. Intern. Conf. Phys. Semiconductors, Kyoto, Japan, 1966, p. 37; Phys. Rev. 159, 624 (1967). |25 | Eastman, D. E., private communication. [26] Crowell, C. R., and Sze, S. M., Physics of Thin Films, G. Hass and R. E. Thun, Editors (Academic Press, 1967), Vol. 4. [27] Kanter, H., to be published. [28] Sze, S. M., Moll, J. L., and Sugano, T., Solid-State Elec. 7, 509 (1964). 156 [29] Eastman, D. E., to be published. [30] James, L. W., and Moll, J. L., Phys. Rev. 183, 740 (1969). [31] DiStefano, T. H., and Spicer, W. E., Bull. Amer. Phys. Soc. 13, 403 (1968). [32] Spicer, W. E., Phys. Rev. 154,385 (1967). [33] Smith, N. V., and Spicer, W. E., Optics Comm., in press. [34] Smith, N. V., Proc. of Electronic Density of States Symposium, Washington, D.C., November 3-6, 1969. [35] Janak, J. F., Eastman, D. E., and Williams, A. R., Proc. of Elec- tronic Density of States Symposium, Washington, D.C., November 3-6, 1969. [36] Spicer, W. E., and Eden, R. C., Proc. of 9th Intern. Conf. on the Phys. of Semiconductors (Moscow, USSR), Vol. 2, p. 65 (1968). [37] Spicer, W. E., and Lapeyre, J., Phys. Rev. 139, A565 (1965). [38] Shay, J. L., Proc. II-VI Conf., D. G. Thomas, Editor (Benjamin, Inc., New York, 1967), p. 651; Shay, J. L., Herman, F., and Spicer, W. E., Phys. Rev. Letters 18, 649 (1967); Shay, J. L., and Spicer, W. E., Phys. Rev. 161, 799 (1967); Shay, J. L., and Spicer, W. E., Phys. Rev. 169, 650 (1968); and Phys. Rev. 175, 741 (1968). [39] Brust, D., to be published. * [40] Koyama, R. Y., and Spicer, W. E., Proc. of Electronic Density of States Symposium, Washington, D.C., November 3-6, 1969. [41] Janak, J. F., Williams, A. R., and Eastman, D. E., to be published; Eastman, D. E., Janak, J. F., and Williams, A. R., to be published. [42] Eastman, D. E., private communication. [43] Smith, N. V., private communication. [44] Snow, F. C., Phys. Rev. 171,785 (1968). [45] Cohen, M. H., and Mueller, F. M., Atomic and Electric Struc- ture of Metals (American Society of Metals, Metals Park, Ohio, 1967), p. 75. [46] Doniach, S., private communication. [47] Koyama, R. Y., Spicer, W. E., Ashcroft, N. W., and Lawrence, W. E., Phys. Rev. Letters 19, 1284 (1967). [48] Ashcroft, N. W., and Lawrence, W. E., Phys. Rev. 175,938 (1968). [49] Shaw, R., and Smith, N., Phys. Rev. 178,985 (1969). [50] Mosteller, L. P., Huen, T., and Wooten, F., Phys. Rev. 184, 364 (1969). [51] Blodgett, A., Spicer, W., and Yu, A., Optical Prop. and Elec- tronic Structure of Metals and Alloys, F. Abelés, Editor (North- Holland Publishing Co., Amsterdam, 1965), p. 246. [52] Lapeyre, G. J., and Kress, K. A., Phys. Rev. 166,589 (1968). [53] Phillips, J. C., Solid-State Physics, F. Seitz and D. Turnbull, Editors (Academic Press, New York, 1966), Vol. 18, p. 55; Her- man, F., Kortum, R. L., Kuglin, D. D.,VanDyke, J. P., and Skill- man, S., Methods of Computational Physics, B. Adler, S. Fern- bach, and M. Rotenberg, Editors (Academic Press, New York, 1968), Vol. 8, p. 193. [54] Donovan, T. M., Spicer, W. E., J. of Non-Crystalline Solids, in preSS. [55] Donovan, T. M., Spicer, W. E., and Bennett, J., in preparation. [56] Donovan, T., and Spicer, W. E., Phys. Rev. Letters 21, 1572 | (1968). [57] Donovan, T., Ph. D. Thesis, Stanford University (unpublished), (1970). [58] Herman, F., Kortum, R. L., Kuglin, C. D., and Shay, J. L., Proc. Intern. Conf. on II-VI Semiconductor Comp. (W. A. Benjamin, Inc., New York, 1967), p. 271. [59] Brust, D., private communication. [60] Seib, D. H., and Spicer, W. E., Phys. Rev. Letters 20, 1441 (1968). - [61] Norris, C., and Nilsson, P. O., Solid State Comm. 6,649 (1968). [62] Walldén, L., Solid State Comm., 1,593 (1969). [63] Walldén, L., Seib, D., and Spicer, W., J. Appl. Phys. 40, 1281 (1969). [64] Smith, N. V., private communication. [65] Herman, F., Kortum, R. L., Kuglin, C. D., and Short, R. A., Proc. Intern. Conf. Phys. Semiconductors, (Kyoto, Japan, 1966), p. 7. - [66] Herman, F., Intern. J. of Quantum Chem. S, in press. [67] Gehardt, U., private communication. [68] Eastman, D. E., Solid State Comm., in press. - [69] Loucks, T. L., Phys. Rev. 144, 504 (1964); 159, 544 (1967); Keeton, S. C., and Loucks, T. L., Phys. Rev. 168,672 (1968); Fleming, G. S., and Loucks, T. L., Phys. Rev. 173, 685 (1968); Asono, S., and Yamashita, J., Phys. Soc. Japan 23, 714 (1967). Snow, F. C., private communication. Yu, A., Blodgett, A., and Spicer, W. E., Proc. Intern. Colloqui. um Optical Properties and Electronic Structure of Metals and Alloys, F. Abelés, Editor (North-Holland Publishing Co., Am- sterdam, 1966), p. 246. Siegbahn, K., Norling, C., Fahlman, A., Nordberg, R., Hamrin, K., Hedman, J., Johansson, G., Bergmark, T., Karlsson, S., Lindgren, I., Lindberg, B., ESCA Atomic, Molecular and Solid State Structure Studied by Means of Electron Spectroscopy (Almqvist and Wiksells Boktryckeri AB, Uppsala, 1967), Ser. IV, Vol. 20. Ballinger, R. A., and Marshall, C. A. W., to be published. Eden, R. C., Ph.D. Thesis, Stanford University (unpublished), (1968). [75] Eden, R. C., Spicer, W. E., to be published. [76] Cohen, M. L., and Bergstresser, T. K., Phys. Rev. 141, 789 (1966). [77] Shay, J. L., and Herman, F., private communication. [78] Buss, D. D., and Shirf, W. E., Proc. of Electronic Density of States Symposium, Washington, D.C., November 3-6, 1969. [79| Hagstrum, H. D., Phys. Rev. 150, 495 (1966). [80] Hagstrum, H. D., private communication. [8]] Fadley, C. S., and Shirley, D. A., Phys. Rev. Letters 21, 980 (1968). [82] Cauchois, Y., and Bonnelle, C., Proc. Intern. Colloquium Optical Properties and Electronic Structure of Metals and Alloys (North-Holland Publishing Co., Amsterdam, 1966), p. 83. [83] Bedo, D. E., and Tomboulian, D. H., Phys. Rev. 113, 464 (1959). [84] Figure 40 has been taken from ref. 81. [85] Cuthill, J. R., McAlister, A. J., Williams, M. L., and Watson, R. E., Phys. Rev. 164, 1006 (1967). [86] Clift, J., Curry, C., and Thompson, B. J., Phil. Mag. 8, 593 (1963). [70] [71] [72] [73] [74] 157 Discussion on “Optical Density of States Ultraviolet Photoelectric Spectroscopy” by W. E. Spicer (Stanford University) J. Dow (Princeton Univ.); You mentioned that Doniach had done some calculations including the Mahan singu- larity in the photoemission. Are there measurements which indicate that this many-body effect is present? W. E. Spicer (Stanford Univ.); The first place he looked for this was asymmetry in the x-ray line emis- sion from a deep level. He has been examining experi- mental data and he is encouraged at the asymmetry that is seen although the resolution is still a problem. H. Ehrenreich (Harvard Univ.); I just want to make a cautionary remark about the many-electron effects that have been discussed in connection with the x-ray emission and absorption problem by Mahan, Nozières, and Doniach recently. These effects are probably of im- portance only when the hole state is localized or when it occurs in a narrow band. Thus one might expect these effects to be of importance in transition metals as recently suggested by Doniach. They do not seem to be of importance in aluminum. Beeferman and I recently examined the many-electron contributions to the opti- cal absorption in aluminum in considerable detail and found them to be unimportant. Here, of course, the bands in which the excited electron and hole find them- selves are not flat. Therefore, our conclusions in no way contradict the proposal by Doniach. 158 The Density of States and Photoemission from l. Introduction Photoemission has become a powerful experimental tool for determining some features of the electronic states in a large spectrum of materials. In particular, by assuming a simple model, it is possible to deduce an “optical density of states” for some materials [1]. This model assumes that electronic transition probabilities are proportional to products of densities of states and that absorption occurs via “nondirect” transitions. In other materials [2], photoemission data has been used to document absorption processes which are due to “direct” transitions. In this paper results of recent work on experimental and model calculations for indi- um and aluminum are described. For these metals, it is found that either model gives a fair description of the photoemission properties. The data for indium is based on experiments performed by the authors [3]; the ex- perimental data for aluminum is taken from the work of Wooten, Huen and Stuart [4]. 2. Experimental Data Figure 1 shows a partial set of electron energy dis- tributions (normalized to the yield) for a sample of crystalline indium (single and polycrystalline samples had virtually identical spectra), plotted with the photon Indium and Aluminum” R. Y. Koyama” and W. E. Spicer Stanford University, Stanford, California 93405 Experimental photoemission data from indium and aluminum are briefly described and can be un- derstood in terms of a density of states model. In contrast to this, a direct transition model based on cal- culated band structures is found to yield photoelectron spectra which are fair reproductions of the den- sity of states. This suggests that for these two metals, conclusions drawn concerning the density of states are independent of the model used to explain the photoemission data. Key words: Aluminum (Al); direct and nondirect transitions; electronic density of states; indium (96); nondirect transitions; optical density-of-states; photoemission. *Work supported by U.S. Army Engineer Research and Development Laboratories, Fort elvoir, Virginia, Contract No. DA-44-009-AMC 1474 (7); and the Advanced Research Pro- ects Agency through the Center for Materials Research at Stanford University, Stanford, alifornia. **Present address: National Bureau of Standards, Washington, D.C. 20234. 0.021 0.018 H IND|UM 0.015 - ha)=10.5 108 3 0.012 || | | | à || || tº 0.009 H - 2% * - & S. %) 0.006 F /// 0.003 H - 0.000 —1– l 1—l l I –8 –7 -6 –5 – 4 –3 – 2 - 1 0 INITIAL ENERGY (eV) FIGURE 1. A partial set of experimental electron energy distributions referred to the initial states (electrons/photon-eW). |ND|UM • *, -- *s 2^ N 2" ~ ". 2~ / \ / ^ º /4- / \ / \ ". ..." / \ Z \ | .7 •. \ / \ *...", / ". \ 2^ \ | / / ... "... T \ || || / .* ". \| | / .* g V / / ...” **. gº ºn tº Nº. DS(E) / .” gº / ...” / .” 4- |EF | .." | -" — — — — — — OPTICAL DENSITY OF STATES * * * * * * CALCULATED DENSITY OF STATES | | | | l l I L | | 1– O | 2 3 4. 5 6 7 8 9 |O ELECTRON ENERGY (eV) FIGURE 2: The optical density of states function (deduced from the experimental data) and a calculated density of states for indium. 159 energy as a parameter. These photoemission curves are plotted with respect to the initial energies of the elec- evident that there is high probability of exciting elec- trons from 1.2 and 4.2 eV below the Fermi energy. Based on the density of states model with nondirect transitions, these distributions reproduce structure in the “optical density of states” (ODS), which is shown as the dashed curve in figure 2 [3]. These experimental results would lead to the conclusion that indium can be characterized by nondirect transitions and an optical density of states. Similar photoemission measurements have been made on evaporated films of aluminum by Wooten et al. [4]. Their data is reproduced in figure 3. These authors have concluded that the spectrum of emitted electrons is a good replica of the density of filled states. This again leads to the conclusion that aluminum too can be described by a density of states model deduced from empirical considerations. 3. Direct Transition Calculations The previous section indicated that indium and alu- minum can be characterized by a density of states model with nondirect transitions. They are typical of “nondirect” materials in that the photoemission spec- tra behave in a predictable and smooth fashion as the ALUM | NUM & | |. 3O eV * & |O,67 s. // |O.24 3 9.74 | 9.4| LL] 9. 8 $2 | | | | | —5.O –4.O –3.O –2.O - O O |N|TIAL ENERGY (eV) FIGURE 3. Experimental electron energy distributions curves for aluminum. [4]. , , ºw’ .." ..” & Jºe”." ALUMINUM ...” ; ...” I f | O 4 8 ELECTRON ENERGY |2i |6 ( e.V.) FIGURE 4. The calculated density of states for aluminum. photon energy is changed. When compared with materials that display “direct” characteristics, the photoemission from these metals are rather featureless. Yet, it is a valid question to ask what the expected form of the photoemission would be if the transitions were direct. If the electron energy structure of a material is known, various electronic properties can be calculated. The two primary quantities of interest here are the elec- tronic density of states and the photoelectron energy spectra. These have been calculated for indium [3] using the band structure of Ashcroft and Lawrence [5], and for aluminum [6] using Ashcroft’s [7] band struc. ture. Figures 2 and 4 show the results of the calculation “or the density of states. For indium, the experimentally deduced ODS (dashed curve of figure 2) does not show the sharp fea- tures of the calculated density of states (dotted curve of figure 2) (the sharp structure is caused by the interac- tion of the energy surfaces with the zone boundaries). However, the gross features such as the peak near the Fermi energy and the low energy bump are present in both curves. The symmetry of the indium lattice (BCT:c|a = 1.53 or pseudo-FCT:c|a = 1.08, as opposed to the higher foc symmetry of the aluminum lattice) required some simplification to be made in the calcula tion of the eigenvalues [3]. Therefore, this calculated density of states can only be considered as an approxi mation. The aluminum density of states shown in figure 4 i very nearly free electron like. There are only smal deviations at energies near X, L, and W of the Brilloui zone. By comparison, it is evident that the bands in in 160 |ND|UM fia) = IO 8 eV i ELECTRON ENERGY (eV) FIGURE 5. A comparison of the calculated (durect transition) electron energy distribution to the experimental distribution for induum. dium are perturbed significantly more than those for aluminum. By assuming that the transitions are direct, the photoelectron energy distributions were calculated for the two band structures [3,6]. It was assumed that all transitions were equally probable (subject to the con- straints on energy and momentum), and a simple escape function was used. Figures 5 and 6 display the calculated distributions at a single photon energy. For indium (fig. 5), there is a gross resemblance between the calculated and experimental distributions; there are two major groups of electrons in the distribution. It is of interest to note here that there is close replication of the structure in the calculated density of states (fig. 2) in the direct transition distribution of figure 5. For aluminum (fig. 6), the agreement between experiment and calculation is better. The calculated distribution shows stronger structure than the experiment, but all the structure is present. Also included in figure 6 is a calculated distribution based on figure 4 and the densi- ty of states model. Both of these calculated distribu- tions (direct and nondirect transitions) reproduce struc- ture in the density of states; the three bumps in the density of the states are clearly seen in the calculated distributions. 4. Conclusions For the energy range where these photoemission ex- periments are done (4.0 to 11.6 eV), the empirical densi- 'ty of states model can be used to deduce information about the electronic density of states. By the same 417–156 O - 71 – 12 fia) = || 29 eV ALUMINUM DIRECT MODEL DENSITY OF STATES MODEL- m(E,w) EXPERIMENT | l l | l O 2 4 6 8 |O |2 |4 ELECTRON ENERGY (eV) FIGURE 6. A comparison of the calculated (direct and nondurect transutuon) electron energy distributions to the experimental distribution for alumunum. token, calculations of photoemission spectra due to direct transitions also reflect structure in the density of states. Therefore, it seems that either model can be used to describe the photoemission properties of these two metals in the photon energy range studied here. At higher photon energies where there is no experimental data, there are predictable differences which would distinguish the photoemission due to direct transitions and those due to nondirect transitions. Further experi- mental work at higher energy would be useful in deter- mining which model is more applicable. 5. References [1] Cu, Ag–Berglund, C. N., and Spicer, W. E., Phys. Rev. 136, A1030 and A1044 (1964). Krolikowski, W. F., Ph. D. disserta- tion, Stanford University, 1967, unpublished. Co-Yu, A. Y. C., and Spicer, W. E., Phys. Rev. 167, 674 (1968). Ni–Blodgett, A. J. and Spicer, W. E., Phys. Rev. 146, 390 (1966). CaS – Shay, J. L., and Spicer, W. E., Phys. Rev. 169, No. 3,650 (1968). [2] GaAs, Gap, Si-Eden, R. C., Ph. D. dissertation, Stanford University, 1967, unpublished. Ge—Donovan, T M., and Spicer, W. E., Bull. Am. Phys. Soc. 13, 1659 (1968). CaTe–Shay, J. L., and Spicer, W. E., Phys. Rev. 161, No. 3, 799 (1967). [3] Koyama, R. Y., and Spicer, W. E., to be published. [4] Wooten, F., Huen, T., and Stuart, R., Optical Properties and Electronic Structure of Metals and Alloys. F. Abelés, Editor, (North-Holland Publishing Co., Amsterdam, 1966), p. 333. [5] Ashcroft, N. W., and Lawrence, W. E., Phys. Rev. 175,938 (1968). [6] Smith, N. V., Koyama, R. Y., and Spicer, W. E., to be pub- lished. [7] Ashcroft, N.W., Phil. Mag. 8, 2055 (1963). 161 Electronic Densities of States from X-Ray Photoelectron Spectroscopy " C. S. Fadley and D. A. Shirley Lawrence Radiation Laboratory, University of California, Berkeley, California 94720 In x-ray photoelectron spectroscopy (XPS), a sample is exposed to low energy x rays (approximately 1 keV), and the resultant photoelectrons are analyzed with high precision for kinetic energy. After cor- rection for inelastic scattering, the measured photoelectron spectrum should reflect the valence band density of states, as well as the binding energies of several core electronic levels. All features in this spectrum will be modulated by appropriate photoelectric cross sections, and there are several types of final-state effects which could complicate the interpretation further. - In comparison with ultraviolet photoelectron spectroscopy (UPS), XPS has the following ad- vantages: (1) the effects of inelastic scattering are less pronounced and can be corrected for by using a core reference level, (2) core levels can also be used to monitor the chemical state of the sample, (3) the free electron states in the photoemission process do not introduce significant distortion of the photoelec- tron spectrum, and (4) the surface condition of the sample does not appear to be as critical as in UPS. XPS seems to be capable of giving a very good description of the general shape of the density-of-states function. A decided advantage of UPS at the present time, however, is approximately a fourfold higher resolution. We have used XPS to study the densities of states of the metals Fe, Co, Ni, Cu, Ru, Rh, Pa, Ag, Os, Ir, Pt, and Au, and also the compounds ZnS, CdCl2, and HgC). The d bands of these solids are observed to have systematic behavior with changes in atomic number, and to agree qualitatively with the results of theory and other experiments. A rigid band model is found to work reasonably well for Ir, Pt and Au. The d bands of Ag, Ir, Pt, Au and HgC) are found to have a similar two-component shape. Key words: CaCl2; density of states; HgC); noble metals; rigid band model; transition metals; x-ray photoemission; ZnS. 1. Introduction The energy distribution of electronic states in the valence bands [1] of a solid is given by the density of states function, p(E). There are several techniques for determining p(E) at energies within -kT of the Fermi energy, Ef, where relatively small perturbations can excite electrons to nearby unoccupied states. However, because of the nature of Fermi statistics, an electron at energy E, well below Ef (in the sense that Er – E > kT), can respond only to excitations of energy Ef – E or greater. Because the valence bands are typically several eV wide, a versatile, higher energy probe is required to study the full p (E). The principal tech- niques presently being applied to metals are soft x- ray spectroscopy (SXS) [2,3] ion-neutralization spec- troscopy (INS) [4], and photoelectron spectroscopy (by means of ultraviolet [5] or x-ray [6,7] excitation). In each of these methods, either the initial or the final state involves a hole in the bands under study. Thus the measuring process is inherently disruptive. The actual initial and final states may not be simply re- lated to the undisturbed ground state [8], and only for this ground state does p(E) have precise meaning. Even if the deviations from a ground state description can be neglected, there are complications for each of the above techniques in relating measured quantities to p(E) [2,3,4,5]. Nevertheless, all four have been applied *An invited paper presented at the 3d Materials Research ymposium, Electronic Density of States, November 3–6, 1969, Saithersburg, Md. * Work performed under the auspices of the U.S. Atomic Energy ommission. with some success, and, where possible, experimental 163 results have been compared to the theoretical predic- tions of one-electron band theory. In this paper, we outline the most recently developed of these techniques, x-ray photoelectron spectroscopy. (XPS) [6,7|, and apply it to several metallic and non- metallic solids. In section 2, the principles of the technique are diseussed from the point of view of relat- ing measured quantities to a one-electron p(E). In sec- tion 3, we present results for the twelve 3d, 4d, and 5d transition metals Fe, Co, Ni, Cu, Ru, Rh, Pa, Ag, Os, Ir, Pt, and Au, making comparisons with the results of other experimental techniques and theory where ap- propriate. In addition, results for nonmetallic solids containing the elements Zn, Cd and Hg are presented, to clarify certain trends observed as each d shell is filled. In section 4, we summarize our findings. 2. The XPS Method The fundamental measurements in both ultraviolet photoelectron spectroscopy (UPS) and x-ray photoelec- tron spectroscopy (XPS) are identical and very simple. Photons of known energy impinge on a sample, ex- pelling photoelectrons which are analyzed for kinetic energy in a spectrometer. In UPS [5], photon energies range from threshold to ~20 eV, whereas XPS utilizes primarily the Ko x rays of Mg (1.25 keV) and Al (1.49 keV). For a given absolute energy resolution, an XPS spectrometer must thus be ~ 100 times higher in resolv- ing power. We have used a double-focussing air-cored magnetic spectrometer [9], with an energy resolution of Aeſe = 0.06 percent. Ae is defined to be the full width at half maximum intensity (FWHM) of the peak due to a flux of monoenergetic electrons of energy e. Conservation of energy requires that hy=E" – E!!--e-H do, (l) where hu is the photon energy, E! the total energy of the initial ground state, E" the total energy of the final hole state as seen by the ejected photoelectron, e the electron kinetic energy, and be the contact potential between the sample surface and the spectrometer. If E" corresponds simply to a hole in some electronic levelj, then the binding energy of an electron in level j is by definition Ep” = E" — E9, where the superscript v denotes the vacuum level as a reference. The Fermi level can also be used as a reference and a simple trans- formation yields hv= Eſ-H e-H bºp = – E + e-H bºp, (2) where Eºſ = – E is the Fermi-referenced binding ener- gy, and bºp is the work function of the spectrometer (a known constant). This transformation makes use of the relations Epº = Eiſ-H sample work function and dc = bºp — sample work function. Positive charging of the sam- ple due to electron emission can shift the kinetic energy spectrum to lower energies by as much as 1 eV for insu- lating samples but relative peak positions should remain the same. This effect is negligible for metals. Returning to eq (1), we see that the fundamental XPS (or UPS) experiment measures the kinetic energy spec- trum from which we attempt to deduce the final-state spectrum. This spectrum must then be related to p(E), as discussed below. In addition to p(E) modulated by an appropriate transition probability, there will be six major contribu- tors to lineshape in an XPS spectrum. Together with their approximate shapes and widths for the conditions of our experiments, these are: p 1. Linewidth of exciting radiation–Lorentzian, ~ 0.8 eV FWHM for the unresolved Mgko.1,2 doublet used as “monochromatic” radiation in this study. The use of a bent-crystal monochro- matic might permit narrowing this in future work [7]. 2. Spectrometer resolution – slightly skew, with higher intensity on the low-kinetic-energy side, ~0.6 eV FWHM for 1 keV electrons analyzed with 0.06 percent resolution. 3. Hole lifetime in the sample — Lorentzian, -0.1 to 1.0 eV for the cases studied here. 4. Thermal broadening of the state—roughly Gaussian, -0.1 eV. 5. Inelastic scattering of escaping photoelec- trons — all peaks have an inelastic “tail” on the low kinetic-energy side, which usually extends for 10 eV or more. 6. Various effects due to deviations of the final state from a simple one-electron-transition model. Contributions analogous to (3), (4), and (6) will be common to all techniques used for studying p (E). A UPS spectrum will exhibit analogous effects fro all six causes. In XPS, there is thus a present lowe limit of ~ 1.0 eV FWHM. Core levels with this widt are well described by Lorentzian peaks with smoothl joining constant tails [10] (see fig. 1), verifying that th major contribution to line width is the exciting x ray The corresponding lower limit for UPS appears to b 0.2 to 0.3 eV, so that XPS cannot at present be ex pected to give the same fine structure details as UPS ground The effects of scattering of escaping photoelectron [(5) above] can be corrected for in both UPS [5] an 164 Binding energy (eV) 3OO 29 O 28O 27O 26O r——H----|--|--|--|--|--|--|--|--|--|--|--|--|--|-- F-3ds/2 a 1.2 peaks # IO- - § | - S 3 - RU - O * | 1.1 eV F W H M - # | - =y O Inel O St IC a 3.4 peaks - C 5-Ntails - sºftwarºº - | * -- | | | | | | | | | L-L-----|--|--|--|--—— 950 96.O 97O 98O 99 O Kinetic energy (eV) Binding energy (eV) 8O 7O 6O 5 O 40 H-H-I-HTTTTTTTTTTT |5|- 4f 7/2 - f H 4fs/2 O Ir - Cl) F- (ſ) § - - N. 3 |OH - C 1 4 eV FWHM × H. -- Uſ) E F- -- -5 O - O F- 5 * T | | | | | | Lll-L-------|--|--|--|--|--|--|--|-- ||7O ||8O ||| 9 O |2OO |2|O Kinetic energy (eV) FIGURE 1. Core level photoelectron spectra produced by exposure of Ru and Ir to Mg 2-rays. The levels are Ru 3 d5/2–3 d5/2 and Ir 4fs/2–4f7/2. The peaks due to the Mgko'1, 2 and Mgkos, 4 x rays are noted, as well as the tail observed on each peak due to inelastic scattering. The analysis of these spectra into pairs of Lorentzian- based shapes is described in the text and reference 10. XPS [6]. This correction is particularly simple for XPS, however, because narrow core levels can be used to study the scattering mechanisms. As the kinetic energies of electrons expelled from core levels - 100 eV below the valence bands are very near to those of elec- trons expelled from the valence bands (i.e., 1150 eV ver- sus 1250 eV), it is very probable that the scattering mechanisms for both cases are nearly identical. Subject to this assumption [ll], we can correct an observed valence band spectrum, I,(e), by using an ap- propriate core level spectrum, Ic(e), as a reference [6,10]. If we construct a core level spectrum in the absence of scattering, Ic'(e'), from pure Lorentzian peak shapes, then Ic(e) and Ic'(e') can be connected by a response function, R(e.e'). Since XPS data is accumu- lated in discrete channels, Ic(e) and Ic'(e') can be treated as vectors with typically 100 elements and R(e.e') as a 100 × 100 matrix, these quantities being re- lated by Ic(e)= R(e.e') Ic'(e'). (3) If we now make certain physically reasonable assump- tions about the form of R(e.e'), the effective number of matrix elements to be computed can be reduced to sl()0. This permits a direct calculation of R(e.e'). The next step is to apply RT'(e.e') to the observed valence band spectrum, I,(e), to yield the corrected spectrum, I,' (e'). The Lorentzian widths in Ic'(e') are selected to be 0.6 to 0.8 times the observed widths so that no ap- preciable resolution enhancement is accomplished by this correction. In addition to inelastic scattering, we can also easily allow for the extra peaks present in any XPS spectrum due to the satellite x-rays of the anode, the most intense of which are Koga. In XPS spectra produced by bombardment with magnesium x rays, these satellites produce a doublet approximately 10 eV above the main (KO12) peak and with about 10 percent of the intensity of the main peak (see fig. 1). The details of this correction procedure are discussed elsewhere [10]. The application of this procedure to data on the valence bands of copper is illustrated in figure 2. The strong similarity between corrected and observed spec- tra indicates the subtle nature of this correction: the es- sential shape and position of the d-band peak is obvious in the uncorrected spectrum. By comparison, this rela- tively high information content in raw data is not found in UPS [5] or ion-neutralization spectroscopy [4]. An additional advantage of XPS is that the chemical state of the sample can be monitored via observation of core level photoelectron peaks from the sample and possible contaminants [6]. In this way it is possible to detect chemical reactions occurring in the thin surface -T-I-T-T — — — — — |3 © - |2 Cu, row dafo dnd l corrected curve "o IO 53 º 9 5 8 Inelastic 3 foil a 3.4 peaks 7 ...**** ee e “...”...";...'..." 6 • ** ... 5 1. –– i |→ T**** ****** -|5 -IO –5 O 5 |O E– Et (eV) FIGURE 2, Valence band photoelectron spectrum produced by exposure of Cu to Mg 2-rays, together with the corrected spectrum obtained after allowance for the effects of inelastic scattering and Mgko'8,4 x-rays in the raw data. A peak due to the 3d bands of Cu is the dominant feature of these spectra. 165 layer (~100 Å) responsible for the unscattered photoelectrons of primary interest. Furthermore, ex- perimental results for Fe, Co, and Ni indicate that UPS is more sensitive to surface conditions [6,12]. The relationship of corrected XPS spectra to p(E) can be considered in two steps: (1) a one-electron- transition model, in which the appropriate transition probability is expressed in terms of the photoelectric cross section, and (2) deviations from the one-electron- transition model. The cross section for photoemission from a one-elec- tron state j at energy E will be proportional to the square of the dipole matrix element between that state and the final continuum state, or; (E) or (l; | f | | (hv-H E)) |*, (4) where O (E) is the cross section and li(hv -- E) is the wave function of a continuum electron with energy hy + E. If there are no appreciable deviations of the final state from a one-electron transition model, the cor- rected kinetic energy spectrum will be related to p(E) by r(wº E-9) aſ g(E)0(e)/(nº +E")F(E') L(E–E")dE", (5) where G(E') is an average cross section for all states j at E", p'(hu + E') is the density of final continuum states, F(E') is the Fermi function describing thermal excitation of electrons near the Fermi surface and L(E – E") is the lineshape due to contributions (1), (2), (3), and (4) discussed above (essentially a Lorentzian). The factor p'(hu + E') can be considered constant over the energy range pertinent to the valence bands, as the final state electrons are ~ 1250 eV into the con- tinuum and the lattice potential affects them very little |6,13]. Therefore, the appropriate final state density will be proportional simply to e”. This function is only negligibly smaller for electrons ejected from the bottom of the valence bands (e = 1240 eV) than for those emitted from the top of these bands (e = 1250 eV). This constancy of p'(hv-H E") cannot be assumed in the anal- ysis of UPS data, however [5]. Any changes in or(E) from the top to the bottom of the bands will modulate the XPS spectrum in a way not simply connected to p(E). From eq (4) it is apparent that these changes can be introduced by variations in either ill, or liſh v -- E) across the bands. The differences in ſº, from the top to the bottom of the 3d band in transition metals have been discussed previously [2,14], but no accurate quantitative estimates of this effect on the ap- propriate dipole matrix elements have been made to date. It is thus possible that in both XPS and UPS, or(E) varies substantially from the bottom to the top of the valence bands because of variation in the initial-state wave functions. This question deserves further study. In XPS, there should be little difference in the final- state wave function, liſhv + E), between the top and bottom of a band, as a 1240 eV continuum state should look very much like a 1250 eV continuum state. The ef- fects of changes in final state wave function on G(E) need not be negligible in a UPS spectrum, however. Our discussion up to this point has assumed that the photoemission process is strictly one-electron; i.e., that we can describe the process by changing the occupa- tion of only a single one-electron orbital with all other orbitals remaining frozen. This assumption permits the use of Koopmans' Theorem [15], which states that binding energies can be equated to the energy eigen- values arising from a solution of Hartree-Fock equa- tions. Or, with some admitted errors [16], the one-elec- tron energies obtained from non-Hartree-Fock band structure calculations in which simplifying approxima- tions have been made can be compared directly to a measured binding energy spectrum. We illustrate the use of Koopmans’ Theorem in figure 3a, using a hypothetical level distribution for a 3d transition metal. There are, however, several types of potentially signifi- cant deviations from this one-electron model. We shall discuss these briefly. The final-state effects leading to these deviations can be separated into several categories, although we note that there is considerable overlap. In a more rigorous treatment some of these separations might not be meaningful, but we retain them here for heuristic pur- poses. The effects are: (1) Electrons in the sample may be polarized around a localized positive hole, thereby in- creasing the kinetic energy of the outgoing electron [8]. In this way, the entire I(e) spectrum would be shifted toward higher kinetic energy. Polarization might also occur to a different extent for different core levels, for different energies within the valence bands, and for levels at the same energy in the valence bands, but with different wave vector. The latter two effects could act to broaden I(e) relative to p(E). These polarization effects are schematically illustrated in figure 3b. Polariza- tions will only affect I(e) to the extent that the kinetic energy of the outgoing electron is al- tered, however (cf. eq (1)). Since both polariza- tion and photoemission occur on a time scale 166 localized d electron moment, giving rise to an 5p1/2-3/2 Vol. approximately 4 eV “multiplet splitting” in the 2P3/22 y bonds 3s photoelectron peak [17]. Also, it has been 2p1/2 predicted that nonlocalized conduction elec- (d) 2S º 3S trons should couple with a localized core or | | | valence hole yielding asymmetric line shapes Kinetic energy —- in electron and x-ray emission [18]. Both of the above effects would act to broaden I(e) spectra, | with the former being more important for (b) º systems with a d or f shell approximately half- filled. These effects are indicated in figure 3c. | | | | | | | | | | | | | It has also been predicted that the removal of | | a core or valence electron will be accompanied | by strong coupling to plasma oscillations [19]. (c) | This coupling would lead to broad sidebands | | | | | | | | separated from the one-electron spectrum by as much as 20 eV [19]. (3) It is also possible that not just one electron is fundamentally affected in the photoemission process, but that other electrons or phonons are simultaneously excited [20]. Electrons may be excited to unoccupied bound states or they may be ejected from the sample, and this effect is indicated in figure 3d. The only direct | observations of such electronic excitations dur- (e) | | ing photoemission have been on monatomic A A gases, where two-electron processes are found with as high as 20 percent probability [21]. FIGURE 3. Schematic illustration of various final-state effects on the Vibrational excitations have a marked effect on photoelectron spectrum of a hypothetical 3d transition metal: (a) the the UPS spectra of light gaseous molecules Koopmans’ Theorem spectrum, in which levels are positioned [22] but it is difficult to estimate their im- portance in solids. A classical calculation in- (d) --d | | | | | A | | | | | | | | | according to one-electron energies, with relative intensities determined by appropriate photoelectric cross sections; (b) the effect on spectrum (a) of polarization around a localized-hole final state; (c) dicates that for such heavy atoms as transition the effect on spectrum (a) of strong coupling between a localized hole metals, the recoil energy available for such ex- and the valence electrons (note the splitting of the 3s level); (d) the citations in XPS is s10-2 eV [7] . Also, the ob- effect on spectrum (a) of two-electron excitation during sº e * s servation of core reference levels with photoemission; and (e) the effect on spectrum (a) of phonon excitation linewidths very close to the lower limit of the during photoemission. technique (see fig. 1) seems to indicate that vibrational excitation does not account for more than a few tenths eV broadening and shifting to lower kinetic energy of features ob- of ~10-16 s, it is difficult to assess the im- served in the valence-band region. This effect portance of this effect. As the velocity of an is schematically indicated in figure 3e. XPS photoelectron is ~10 times that of a UPS photoelectron, the influence of polarization should be somewhat less on an XPS spectrum, however. For several reasons, then, XPS seems to be capable of giving more reliable information about the overall shape of p(E) than does UPS. However, the present XPS linewidth limit of 1.0 eV precludes determination (2) In addition to a simple polarization, a localized of anything beyond fairly gross structural features. hole can couple strongly with localized valence With these observations in mind, we now turn to a electrons [17] or with nonlocalized valence detailed study of the XPS spectra for several solids. We electrons [18]. In iron metal, for example, a 3s note also that the XPS method is applied to p(E) studies hole is found to couple in several ways with the in two other papers of these proceedings [23,24]. 167 3. Density-of-States Results for Several 3d, 4d, and 5d Series Metals 3.1. Introduction Figure 4 shows the portion of the Periodic Table rele- vant to this work. The twelve elements Fe, Co, Ni, Cu, Ru, Rh, Pd, Ag, Os, Ir, Pt, and Au were studied as metals, while the three elements Zn, Ca, and Hg were studied in the compounds ZnS, CaCl2, and HgC) to illus- trate the positions, widths, and shapes of filled core-like 3d, 4d, and 5d shells. Ultra-high vacuum conditions were not attainable during our XPS measurements, as the base pressure in our spectrometer is approximately 10 ° torr. Surface contamination of samples is a potential problem, because the layer of the sample that is active in produc- ing essentially inelastic photoelectrons extends only about 100 Å in from the surface |6,7]. This depth is not accurately known, however. Because the contamina- tion consists of oxide formation as well as certain ad- sorption processes with lower bonding energy for the contaminant, all the metal samples were heated to high temperature (700 to 900 °C) in a hydrogen atmosphere (10-8 to 10-2 torr) during the XPS measurements [6]. These conditions were found to desorb weakly bound species, and to reduce any metal oxides present. As mentioned previously, it is possible to do in situ chemical analyses of the sample by observing core- level photoelectron peaks from the metal and from all suspected contaminants [6]. For all metals, the most important contaminant was oxygen, which we moni- tored via the oxygen ls peak. Because core electron binding energies are known to be sensitive to the chemical state of the atom [7,25], the observation of core peaks for metal and oxygen should indicate 26 3.d64s?|27 3d 74s?|28 3d 84s?|29 3d'94s||3O 3d'94s? Fe CO N Cu Zn b C C f CC f CC f C C - - 44 4.d75s' |45 4d35s' |46 4d'9|47. 4d'O5s || 48 4d95s? RU Rh PC Ag CC h C p f C C f C C f C C - - 76 5.d66s?| 77 569 |78 5610|79 5d.196s' |80 5 d'06s2 OS Ir Pf Au Hg h c p f CC f CC f C C - - FIGURE 4. The portion of the periodic table studied in this work. The atomic number, free-atom electronic configuration, and metal crystal structures are given. Zn, Cd, and Hg were studied as compounds. The crystal structures are those appropriate at the temperatures of our metal experiments (700-900 °C). something about the surface chemistry of the sample. The intensities of contaminant peaks should also be a good indicator of the amounts present. Figure 5 shows such results for iron. At room temperature, the oxygen ls peak is strong, and it possesses at least two com- ponents. The iron 3p peak is also complex and appears as a doublet due to oxidation of a thin surface layer of the sample. As the temperature is increased in the presence of hydrogen, the oxygen peak disappears (the right component disappearing first) and the left com- ponent of the iron peak also disappears, leaving a nar- row peak characteristic of iron metal. Our interpreta- tion of the disappearing components is that the left ox- ygen peak (higher electron binding energy) represents Oxygen as oxide, the right oxygen peak (lower electron binding energy) represents oxygen present as more loosely bound adsorbed gases, and that the left iron peak (higher electron binding energy) represents ox- idized iron [6.25]. Thus at the highest temperatures in- dicated in figure 5, we could be confident that we were studying iron metal. Similar checks were made on all the other metal samples and oxygen can be ruled out as a contaminant for every case except Pa. (We discuss Pol below.) For example, the core level peaks for Ru and Ir shown in figure 1 do not indicate any significant splitting or broadening due to chemical reaction. The results presented in table 1 indicate similar behavior for all metals studied. The carbon ls peak was also ob- served and found to disappear for all cases at the tem- perature of our measurements. O|S o Fe3p i ----------------— 7|O 715 72O 725 | 205 ||85 ||95 Kinetic energy (eV) —- FIGURE 5. Oxygen 1s and 3p photoelectron peaks from metallic iron at various temperatures in a hydrogen atmosphere. Note that the Fe 3p component at lower kinetic energy (an “oxide” peak) disappears at high temperature along with the 01's peaks. Mgko. radiation was used for excitation throughout the work reported here. 168 TABLE 1. Summary of pertinent results for the fifteen solids studied. The reference core levels used for inelastic scattering correction are listed, along with their binding energies and widths. The widths of the d-band peaks are also given, along with the spacing of the two components in these peaks (if observed). Ref. core FWHM of FWHM of Separation of Solid Reference level binding core levels" | d-band peak || 2 components core levels energy " (eV) (eV) in d-band peak (eV) (eV) Fe.......... 3p ||2–32 (unresolved) "... 52 2.3 4.2 .......... Co.......... 3p ||2-83 (unresolved)"... 57 2.5 4.0 .......... Ni........... 3p1/2-312 (unresolved) "... 66 3.4 3.0 .......... Cu.......... 3p1/2-312 (unresolved) "... 75 4.2 3.0 | .......... ZnS........ 3p1/2-312 (unresolved) *.. 90 5.4 1.7 .......... Ru • * * * * g º e º e 3ds/2–5/2 & e º ºs º º is e º 'º e g g g g = < e º ºs s º 280 1.1 4.9 * * * * * * * * * * Rh tº $ e º º te e º e & 3ds/2–5/2 e e s s º e 9 e º e º e º ſº a s e e º e º 'º 307 1.3 4.4 & e º g g º żº º ºs & Pa.......... 3ds/2-52..................... * 335 1.3 4.1 ! .......... Ag * @ e º ſº º e º º is 3ds/2–52 * * * * * * * * * * * * * * * * * * * * * 368 1.0 3.5 1.5—1.8 CaCl2 tº gº e g g g 3ds/2–5/2 * * * g e º e º e g º ſº º e e g g g g tº * 408 1.2 2.0 * g g º e e g g g g Os tº gº gº e g is º ºs e := 4. 5/2–7|2 - - - - - - - - - - - - - - - - - - - - - . 50 1.3 6.5 * * * * * * * * * * Ir........... #: & º e º e º e º gº tº e º & E is e º e e º ºs . 60 1.4 6.3 3.3 Pt........... 4f12–72...................... 7] 1.5 5.8 3.3 Au * * g e º g º ſº e º 4fs2–72 * e º sº e º gº ºs e º ſº e º tº dº e º G & 8 & & 84 1.2 5.7 3.1 HgC) e tº gº e º e tº 4fs/2–72...................... 103 1.5 3.8 1.8 * Binding energy of the l-H 1/2 component, relative to the Fermi energy. * Equal widths assumed for both components in the least-squares fits for 3d and 4f levels. c The theoretical spin-orbit splitting for the 3p levels in this series range from 1.6 eV for Fe to 3.1 eV for Zn (Ref. 36). The partially resolved doublet in ZnS is found to have a separation of 2.8 eV, in good agreement. All metals were studied as high purity polycrystalline foils, except for Ru and Os, which were studied as pow- ders [10]. The nonmetallic samples (ZnS, CaCl2, and HgC)) were studied as powders at room temperature. Both considerations of chemical stability and observations of core levels indicated no significant surface con- tamination, although high purity for these cases was not of paramount importance. The results reported here were obtained with 1.25 keV Mgko radiation for excitation. However, no signifi. cant changes are introduced with AlKo radiation of 1.49 keV energy. We present below our experimental results for these d group metals, as well as the results of other experi- ments and theory. Statistical error limits are shown on all XPS results. Throughout our discussion, we shall speak of “p(E)” as determined by a certain technique, bearing in mind that no experimental technique directly measures p(E), but rather some distribution peculiar to the experiment (e.g., the UPS “optical den- sity of states” [5], or the INS “transition density func- tion” [4]), which is related to p(E) in some way (e.g., by our eq (5)). The location of the Fermi energy was determined by using eq (2). This determination was checked against photoelectron peaks from a Pt standard [10]. Our esti- mated overall accuracy in determining EP is +0.5 eV, so that precise comparison of features in XPS spectra with features present in the results of other experi- ments (all of which have roughly the same Ef accuracy) is not always possible. Finally, we note that the dominant feature in our results for all cases is a peak due to the bands derived from d atomic orbitals. The XPS method is not particu- larly sensitive to the very broad, flat, s- or p-like bands in metals, and such bands are seen with enhanced sensitivity only in studies using ion-neutralization spectroscopy [4]. 3.2. The 3d Series: Fe, Co, Ni, Cu and Zn Our results for Fe, Co, Ni, and Cu have been published elsewhere [6], but it is of interest to compare them with more recent results from theory and other experiments [3,12]. There are now enough data availa- ble that it is worthwhile to discuss and compare results for these iron group metals individually, as Eastman [12] has done. q. Iron (bcc) Hanzely and Liefeld [3] have studied Fe, Co, Ni, Cu, and Zn using soft x-ray spectroscopy (SXS). Their results for Fe, together with Eastman’s UPS results [12] and our own, are plotted in figure 6a. In comparing the three p(E) curves we note that their relative heights and areas have no significance: we have adjusted the heights to be roughly equal, in order to facilitate com- parison. Also the UPS curve is terminated at E, and is less reliable in the dashed portion, for E → E – 4 eV [12]. With these qualifications, the overall agreement 169 O × ~ T of) LL] ſº * =} Q | 3 Cſ) Cl– X LL Q- |- O +| + 2 l ! | i | ! l -|O –9 – 8 —7 –6 –5 – 4 –3 –2 –| E – Ef (eV) FIGURE 6. Results for iron metal. The XPS data were obtained at 780 °C and have been corrected for the effects of inelastic scattering and Mgko.3.4 x-rays. In (a) the XPS data are compared with UPS (ref. 12) and SXS (ref. 3) curves. In (b) the XPS data are shown together with a theoretical curve obtained by broadening the ferromagnetic density-of-states function of reference 26. Right ordinate is thousands of counts in the XPS data among these results from three different experimental methods is really quite good. The function p(E) appears to be essentially triangular, peaking just below Ef and dropping more or less linearly to zero at E - Ef-8 eV. Upon closer inspection however, the agreement is less impressive. The SXS results are somewhat nar- rower, but with more intensity above Ef, probably due to spurious effects [3]. There is little coincidence of structure, although the maxima for XPS and SXS coin- cide fairly well. A shift of ~1 eV of the XPS curve toward Ef or the UPS curve in the opposite direction would improve their agreement, but it is unlikely that the combined errors in the location of Ef location are that great. In figure 6b, the XPS results are compared to the one-electron theoretical p(E) calculated by Connolly [26] for ferromagnetic iron. The theoretical p(E) has been smeared at the Fermi surface with a Fermi func- tion corresponding to the temperature of our experi- ment (780 °C) and then broadened with a Lorentzian lineshape of 1.0 eV FWHM. It should thus represent a hypothetical “best-possible” XPS experiment in a one- electron model (i.e., eq (5) with or (E') and p'(hv + E") constant). The agreement between theory and experi- ment is good, particularly above EF-5 eV. The XPS (or SXS) results give somewhat higher intensity below Ef–5 eV than theory. We note that hybridization of the d bands can lead to significant broadening of the theoretical p(E) of Ni [14]. A similar sensitivity of the iron p(E) to the amount of hybridization could account for the discrepancy in width between XPS and theory. Our reason for comparing experimental results to ferromagnetic instead of paramagnetic theoretical pre- dictions is as follows: In experiments on ferromagnetic metals, no significant differences are observed between XPS [6, 17] and INS [4] results obtained above and below the Curie temperature (Tc, where long-range fer- romagnetic order should disappear). Furthermore, exchange-induced splittings of core electronic levels in iron are the same above and below T. [17]. It thus ap- pears that localized moments persist above To for times at least as long as the duration of the photoemission process. Local moments might be expected to affect the kinetic energy distributions of electrons ejected from valence bands and core levels [17] in much the same way, independent of the presence of long range order. Thus a comparison of experiment with a paramagnetic p(E) may be a priori irrelevant, in- as much as a ferromagnetic p(E) takes these effects into account in an approximate way. Eastman [12] has also noted that UPS results for Fe, Co, and Ni below To are in general in better agreement with ferromagnetic theoretical p(E)'s than with similar paramagnetic theoretical results. Accordingly, we shall compare our results only with ferromagnetic theoretical curves for Ni and Co in the next sections. b. Cobalt (fec) The experimental situation is illustrated by the three density-of-states curves in figure 7a. The comparison is quite similar to that for iron. Good overall agreement is apparent, with less agreement in detail. Eastman’s UPS curves [12] in both cases show structure near the Fermi energy that is missing from the SXS [3] and XPS results, and at lower energies the UPS curve tends to be higher than the others, especially in the dashed portion where it is less reliable [12]. In this region the XPS curve lies between the other two for Co as well as for Fe. One index of agreement among the three curves in the full width at half-maximum height, which is about 3, 4, and 5 eV for SXS, XPS, and UPS, respectively. In figure 7b, we compare our XPS results to a fer- romagnetic theoretical curve of Wong, Wohlfarth, and Hum [27] for hop Co (our experiments were done on foc Co, for which no detailed theoretical results are available). The theoretical curve has been broadened in an analogous fashion to that for iron. The agreement 170 | | | | -T- | I | T —||O Cobo |f (d) --- 9 º •- o LL 3: Q- |_ 8 -É -> O O Oſ) 0- × -- 7 ** ––––––––– *-* | | } | | | | H | | | T S- (b) G | -: Q Broddened | T | –9 – 8 –7 -6 –5 – 4 – 3 – 2 – O | 2 E- E + (eV) FIGURE 7. Results for cobalt metal. The XPS data were taken at 925 °C and have been corrected for inelastic scattering and Mgkos, 4 x-rays. In (a) these data are compared with UPS (ref. 12) and SXS (ref. 3) results. In (b) the comparison is with an appropriately broadened ferromagnetic theoretical curve from reference 27. is good for E > EF-3 eV, but the XPS results are somewhat high below that point. In fact, the overall agreement is probably best between theory and SXS (cf. fig. 7a). c. Nickel (foc) Experimental results for Ni are presented in figure 8a [3,12]. We note a slight decrease in the XPS results in the region E -< EF-4 eV relative to our earlier work [6]. This decrease is due to a more accurate allowance for a weak inelastic loss peak appearing at ~5 eV below the primary photoelectron peaks. The three sets of data show poor agreement, with the widths of the main peak decreasing in the order UPS, XPS, SXS. The SXS results are considerably narrower than the other two (FWHM =2 eV, 3 eV, and 5 eV for SXS, XPS, and UPS, respectively), but agree in overall shape with XPS. The SXS results in figure 8a were obtained from measure- ments of L x rays [3]. Similar work on M x rays (for which transition probability modulation may be a smaller effect [2]) shows somewhat more fine structure and a FWHM of ~3 eV [2], agreeing rather well with XPS. Nickel has also been investigated by INS [4] and a smooth peak of roughly the same position and width as the XPS peak is observed. Even with an allowance H35 H-I- | I-T- | Nickel --- 30 - (d) g º J s Q T 25 3 (ſ) Cl– >< - – 20 © |TF-H | (b. Broodened theo. G - - Q- * — -IO –9 -8 –7 -6 –5 -4 -3 –2 - O + 1 + 2 E-E, (eV) FIGURE 8. Results for nickel metal. The corrected XPS data are based on measurements at 870 °C. In (a) they are compared with UPS (ref. 12) and SXS (ref. 3) curves. In (b) they are compared to the ferromagnetic theoretical density-of-states function from reference 14, which has been broadened. for the poorer resolution of XPS, the two peaks appear- ing in the UPS results are not consistent with the XPS CUITVé. The various theoretical p(E) estimates for Ni have been discussed previously [2,12]. The FWHM of these estimates vary from ~3 to 4.5 eV, with the smallest width coming from an unhybridized calculation [14]. In figure 8b, we compare our XPS results to a hybridized, ferromagnetic p(E) for Ni [14] which has been broadened in the same manner as those for Fe and Co. It is clear that the XPS results are too narrow (though they would agree in width with the un- hybridized p(E) [14]), and that, allowing for our broadening, the UPS results are in best agreement with theory. In view of the considerable discrepancies between UPS and XPS, SXS, or INS, however, we conclude that Ni does not represent a particularly well-understood case, in contrast with Eastman's conclusions [12]. d. Copper (foc) The experimental curves from UPS [12,28], SXS [3], and XPS are shown in figure 9a. There is agree- ment in that all curves show a peak between 2.3 and 3.3 171 eV below Ef, but with UPS showing more detailed structure and a somewhat uncertain overall width [12,28]. The widths and shapes of XPS and SXS are in good agreement though shifted relative to one another by ~1 eV. (A more accurate Ef location has shifted our XPS curve relative to our previous results [6].) In recent UPS work at higher photon energy (hy = 21.2 eV), Eastman [29] has obtained results with more in- tensity in the region 2.5 to 4.0 eV below Ef and which agree very well in shape and width with XPS and SXS. For this case it appears that even a slight increase in photon energy in the UPS measurement causes the results to look a great deal more like those of XPS. Copper has also been studied in INS [4] and the results for the d-band peak are in essential agreement with XPS and SXS. In figure 9b, we compare a broadened version of the theoretical p(E) due to Snow [30] with our XPS results. The agreement is excellent, and would also be so for SXS if we permit a shift of ~1 eV in Eſ. The coin- cidence in energy of structure in the UPS curve with structure in the unbroadened theoretical p(E) has been discussed previously [28], but we note that the relative intensities of the various features noted do not in fact coincide with theory. I | | | | | i H | H 14 Copper (d) – 12 XPS SXS * rr, 9 e- —lſo t u | UPS 2 Q- s / 3 - /* - 8 o / 0- f × ; — O -9 – 8 -7 – 6 - 5 - 4 -3 - 2 - O +| + 2 E-E t (eV) FIGURE 9. Results for copper metal. The XPS data were obtained at 720 °C and have been corrected for inelastic scattering and Mgko.3, 4 x-rays. Curves from UPS (refs. 12 and 28) and SXS (ref. 3) are com- pared to the XPS results in (a). In (b) the XPS results are compared to a broadened theoretical curve based on reference 30. e. Zinc (as ZnS) Zinc has been studied by XPS only in compounds, because of the difficulty of obtaining a clean metallic surface. We present results for ZnS in figure 10. The 3d electronic states show up as a narrow intense peak with a FWHM of 1.7 eV and located – 13 eV below EP. (The separation of this peak from Ef may be too large, because of charging of the sample [25].) The valence bands are just above the d peak. The d states of metal- lic zinc have been studied also by SXS [3], and a peak of FWHM = 1.45 eV, at 8 eV below EP, was obtained. Thus XPS and SXS are in good agreement on the width of these core-like 3d states, which are only about 10 eV below Ef. 3.3. The 4d Series: Ru, Rh, Pd, Ag and Cd The corrected XPS spectra for the four metals Ru, Rh, Pd, and Ag are shown in figure 11. The metals are discussed separately below. d. Ruthenium (hcp) Our results for Ru are characterized by a single peak of ~4.9 eV FWHM. The high energy edge is quite sharp, reaching a maximum value at about EP-1.7 eV. The peak is rather flat, and there is some evidence for a shoulder at Ef-4.5 eV. The peak falls off more slowly with energy on the low energy side than near Er. The reference core level widths in Ru were quite narrow, as indicated in table 1, and spurious effects due to surface contamination are unlikely. There are no other experi- 7 – ZnS -- 5 }- –2O –|5 —|O –5 —O E – Ef (eV) FIGURE 10. Corrected XPS spectrum for ZnS, showing a narrow intense peak from the 3d levels, as well as the broad, flat valence bands. 172 –H–I-I-I-I H-H-I-H- - O ) E-E, (eV Corrected XPS spectra for the 4d metals Ru, Rh, Pd, and Ag. FIGURE 11. mental or theoretical results on Ru presently available for comparison with our data. b. Rhodium (foc) The XPS-derived p(E) can be described by a single triangular peak, very steep on the high energy side, and reaching a maximum at EF-1.3 eV. There is little evidence for structure on the low energy side, which falls off monotonically. The peak FWHM of ~4.4 eV is slightly smaller than that for Ru. No other experimental or theoretical results on Rh are available for com- parison. c. Palladium (fcc) Our corrected results for Po have much the same ap- pearance as those for Rh, but the Pa peak is slightly narrower with a FWHM of ~4.1 eV and the maximum occurs at EF-1.7 eV. The high-energy edge of the Pd peak is very steep, and most of the slope must be in- strumental. Therefore, as expected, the true p(E) for Pol is apparently very sharp at Ef. The results presented in figure 11 have been cor- rected for a weak inelastic loss peak at 6 eV, and also for the presence of a small peak at Eſ- 10 eV, arising from oxygen present as a surface contaminant. Sam- ples of Pa were heated in hydrogen to approximately 700 °C and then studied at this temperature with either a hydrogen or argon atmosphere. It was not possible under these conditions (or even by heating to as high as 900 °C) to get rid of the oxygen ls peak completely. For- tunately, the only effect of a slight oxygen contamina- tion on the valence band XPS spectrum of certain metals appears to be a sharp peak at Eſ- 10 eV (probably caused by photoemission from 2p-like oxygen levels). We have also observed this effect for slightly oxidized Cu. Thus we were able to correct our Pa results for this peak (which does not affect the region shown in fig. 11). A recently obtained uncorrected XPS spectrum for Pd is in good agreement with our results [31]. Palladium has also been studied by UPS [32,33], and the agreement with XPS is good in general outline. However, the precise shape of the UPS results below approximately EF-3.5 eV is uncertain [32,33]. In figure 12, we compare our results with the theoretical predictions of Freeman, Dimmock, and Fur- dyna [34]. The upper portion of the figure shows the fine structure of their p(E) histogram and in the lower portion we compare our results to the broadened theoretical curve. The agreement between XPS and theory is good, although the shape of the peak is somewhat different. d. Silver (fcc) Our results for Ag also appear in figure 11. They differ in several respects from the Pd curve. The d bands are filled and below Ef, giving rise to a narrow peak (FWHM = 3.5 eV) with its most intense com- ponent at EF-5.3 eV. The edges of this peak are quite sharp, in view of the instrumental contributions of XPS. |--|--|--|--|--|--|-- PC ſº (o) Q- l | ! | | f -l H | T | H ~~ T Broodened theo. J (b) Q- XPS ...” |--|--|--|-- , sº ©e I –5 O 5 E – E (eV) FIGURE 12. Comparison of Pd XPS results with theory: (a) theoretical density-of-states function from reference 34, indicating the complexity of the one-electron p(E), (b) XPS results and a broadened theoretical curve. 173 The 3ds/2 and 3ds/2 levels of Ag are also very narrow (see table 1), indicating no spurious linewidth contributions from instrumental or contamination effects. There is also strong evidence for a weaker component at ~Ef–6.6 eV. This two-component structure has also been verified by Siegbahn and co-workers in uncor- rected XPS spectra [7,31]. Very similar structure ap- pears in the d bands of several 50 metals and we discuss the possible significance of this below (sec. 3.5). Silver has also been studied by means of UPS [5,29,35], using radiation up to 21.2 eV [29] in energy. The results of these studies (in particular those attained at 21.2 eV) are in essential agreement with our own, in that they show a peak of ~3 eV FWHM at EF-5.0 eV. No theoretical p(E) predictions for Ag are available at the present time. e. Cadmium (as CdCl2) A corrected XPS-spectrum for CaCl2 is shown in figure 13. The 4d peak appears at ~Ef— 14.5 eV and the valence bands fall between roughly 5 and 10 eV below Ef. The 40 peak is very narrow (a FWHM of 1.7 eV, compared to 3.5 eV for Ag). As these d levels are quite strongly bound, we expect them to behave as core states, and perhaps to exhibit spin-orbit splitting (into da/2 and d52 components). There is no evidence for splitting of this peak, but its shape is consistent with a theoretical free-atom prediction of only a 0.8 eV spin- orbit splitting [36]. The analogous 5d-series levels in HgC) do exhibit resolvable spin-orbit splitting, however (sec. 3.4.e). T + -- CC Cl2 ºb 8 O e L I vºrº ©e-e e –2O —|5 — O –5 O E-E t (eV) FIGURE 13. Corrected XPS spectrum for CdCl2. The filled 4d states appear at E-E = –14.5 eV. The broader peak at E-Ef = -7e/ represents valence bands. 3.4. The 50l Series: Os, Ir, Pt, Au, and Hg The corrected XPS spectra for the metals Os, Ir, Pt, and Au are shown in figure 14. a. Osmium (hcp) Hexagonal Os gives a valence band spectrum similar to that of hexagonal Ru. As in the Ru case, the Ospeak rises sharply near Ef to a plateau beginning at Ef— 1.7 eV. The flat region of the Ospeak extends over approxi- mately 3 eV, and is broader than that for Ru. No com- parisons with theory or other experiments are possible as yet. The low energy tail of the Ospeak does not fall to the base line primarily because of spurious photoelectron intensity in the valence band region due to the proximi- ty of the very intense Os4f levels in energy (see table 1). These core levels appear to interact with very weak Mg . x rays whose energies are as high as ~1300 eV, giving rise to photoelectrons in the same kinetic energy region as valence bands interacting with the 1250 eV Mgkoi, 2 x rays. Similar problems were encountered with Ir, but they do not affect our conclusions as to peak shapes and structure. An additional problem was encountered in correcting for the Mgko.3, 4 x rays in both Os and Ir, as the low intensity 5p1/2 and 5p3/2 photoelectron peaks overlap the oa,4 regions of the reference 4f peaks. For example, this effect appears as a slight deviation of the data from the fitted function near a kinetic energy of 1202 eV in figure 1. However, the osa correction is a small one and could nonetheless be made with suffi- cient accuracy not to affect our fundamental conclu- SIOI) S. H- —H-T-I-T- * Os 33H Ir © I * "b 32- * ||F 3 # all 2, 3"| © C C = 10 © 3 3OH G C C gº 9|-- 29 He 28 | 8H H. 3 24- P+ O 3 vo 22}- [. E 3 9 Q 20– 3. 8 |8 7 ----|--|--|-- -—1–1–1–1–1–– –|O –5 O - |O –5 O E-E, (eV) E-E, (eV) FIGURE 14. Corrected XPS spectra for the 50 metals Os, Ir, Pt, and Au. 174 b. Iridium (fcc) The corrected XPS results for iridium are similar to those of Os in overall shape and width, but give evidence for two peaks, at approximately Ef-1.5 eV and Ef-4.5 eV. This two-peak structure is even clearer in the uncorrected XPS spectrum for Ir shown in figure 15. The higher-energy peak appears to be narrower, and, with allowance for this, we estimate the two peaks to be of roughly equal intensity. c. Platinum (fcc) Our corrected XPS results for Pt exhibit two partially-resolved peaks at Ef-1.6 eV and EF-4.0 eV, with the more intense component lying nearer Ef. The steep slopes of our spectra for both Ir and Pt near Ef are consistent with the Fermi surface cutting through the d bands in a region of very high p (E). The separations of the two components observed in the d bands are thus very nearly equal for Ir and Pt, but the relative intensi- ties are different. Theoretical results are available for Pt. The band- structure calculations of Mueller et al. [37] are shown in figure 16, together with our data. The theoretical p(E) is also shown after broadening, to facilitate com- parison. We note that both theory and experiment show roughly two major peaks but that the relative intensities are in poor agreement. The disagreement as to shape is the same as that observed for Pd in figure 12. (Relative intensities are arbitrary in both of these figures.) In ad- dition, the band-structure calculations give a total Ir f - 33H (uncorr) 32 H 3 - 3 O - 2 9 - 28 H 27 H----- -IO –5 E – Ef (eV) FIGURE 15. Uncorrected XPS spectrum for Ir, in which the two-peak structure is clearly shown. – O – 5 O E - Ef (eV) FIGURE 16. Comparison of Pt XPS results with theory: (a) the theoretical density-of-states function of reference 37, (b) the broadened theoretical curve is compared to our XPS results. width at half height of 8 eV, while the XPS data show a width of only 6 eV. Thus the overall agreement is only fair. d. Gold (foc) The d bands of gold are filled and should lie several eV below Ef, as our results in figure 14 indicate. Two peaks are again evident in the corrected XPS results for gold, and these have been verified in uncorrected XPS spectra obtained by Siegbahn and co-workers [7,31]. The statistical accuracy of our data is quite good, and we can say that the lower intensity peak at Ef–6.8 eV is narrower than the higher intensity peak at EF-4.1 eV. Apart from this, the shape of the d-band peak for Au is very similar to that for Pt. Gold has also been studied by means of UPS [29,38]. In experiments at photon energies up to 21.2 eV [38], a two-peak structure is found, with components at Ef—3.4 eV and Ef-6.1 eV. The component at — 3.4 eV is also observed to be split into a doublet [38], perhaps accounting for its extra width in the XPS results. Furthermore, a spectrum obtained with hu = 26.9 eV [29] (but not corrected for inelastic scattering) looks very much like our XPS results, again indicating that with increase in photon energy, UPS results converge rather quickly to those of XPS. 175 There are no theoretical p(E) estimates at present available for Au. e. Mercury (as HgC) In HgC), the filled 5d levels should be tightly-bound and core-like. Figure 17 shows a corrected XPS spec- trum for HgC), in which the 5d levels appear as a doublet whose components lie 13.6 and 12.0 eV below Ef. Valence bands overlap the high energy edge of the d peaks and extend to EF-5 eV. The intensity ratio of the two 5d peaks, as derived by least-squares fitting of Lorentzian curves to our data, is 1.4:1.0. The separation and intensity ratio are consistent with a d82–d5/2 spin- orbit doublet, as the free-atom theoretical prediction is for a 2.1 eV separation [36] and the intensity ratio should be given by the level multiplicities (i.e., 6:4 = 1.5:1.0). (We have verified that the intensity ratios for the 3ds/2–3ds/2 core levels of the 4d metals in table 1 follow this rule to within experimental accuracy (+0.1).) Thus the 50 levels of HgC) appear to be very core-like. Furthermore, the relative intensity of the two com- ponents in the doublet is similar to those observed in Pt and Au. We discuss the possible implications of this similarity in the next section. 3.5. Discussion of Results The XPS results for all 15 cases studied are presented in figure 18. In table 1 are given the binding energies and widths of the reference core levels used for correcting valence band spectra, as well as the i — |O –5 O E-Et (eV) FIGURE 17. Corrected XPS spectrum for HgC). The intense doublet at E-Ef = - 12 eV is due to the core-like 5d32 and 5d3/2 states. + Ru Rh Pd Ag Cdc1. | 2 | ! . i | \ || 2 || | | >. | . Os Ir P+ Au | -- - 5 E, -5 E; -5 E; -5 E; Energy (eV) ; FIGURE 18. Summary of the XPS results for the fifteen solids studied (cf. table 1). The peaks for ZnS, CdCl2, and HgC) lie at E-E = –13 eV, – 14 eV, and — 12 eV, respectively. width of the peak due to the d bands and (where ob. served) the separation of the two primary components in this peak. Within a 3d, 4d, or 50 series, the XPS results show systematic variation, giving somewhat wider d bands for Fe, Ru, and Os than for Cu, Ag, and Au, respective- ly, and even narrower core-like states - 10 eV below Ef for ZnS, CaCl2, and HgC). Much of this variation is no doubt connected with a one-electron p(E), but we note also that experimental spectra obtained from metals with partially filled d bands might be broadened by the coupling of a localized hole to localized d electrons [17] (see fig. 3c and sec. 2). The 4d bands studied are only slightly wider than their 3d counterparts; the 5d bands are considerably wider and show gross structure. Within two isomorphous series — Rh, Pa, Ag and Ir, Pt, Au, all members of which are face-centered cu- bic — there is sufficient similarity of the shapes of the d. band peaks to suggest a rigid-band model for p(E). If p(E) of Ag(Au) can be used to generate p(E) of Rh and Pa (Ir and Pt) simply by lowering the Fermi energy to allow for partial filling of the d bands, then this model would apply. The peaks for Rh and Pa are too wide to be represented by a Ag p(E), but the shapes of both could be very roughly approximated in this manner. The similarity of the two-peak structure for the three metals Ir, Pt, and Au gives more evidence for the utility of a rigid band model, especially as the uncorrected results for Ir (fig. 15) show a narrower peak near Ef (as though it were a broader peak cut off by the Fermi ener- gy). The application of this model to the prediction of the experimental p(E)'s for Ir and Pt is shown in figure 19. The predictions are reasonably good. In our opinion, this limited success for Ir, Pt and Au probably indicates some similarity in the d bands in these metals, but we 176 Ir, rigid band /N U Q- | | | | Au, | rigid Au, | bond expt. | | | | –|O –5 O — |O –5 O E-Ef (eV) FIGURE 19. An attempt to reproduce the shapes of the experimental XPS spectra for the 5 d metals Ir, Pt, and Au from a Au-like rigid band density of states. Vertical scales are arbitrary. Note that the Ir experimental curve does not fall to as low a value as Pt or Au at low energy due to spurious sources of photoelectron intensity (see text). do not take it as a verification of the rigid band model per se. The two-component structure observed in the d-band peaks of Pt and Au is very similar to the unresolved structure found in Ag. That is, a more intense com- ponent appears nearer Er. To estimate the intensity ratios of these components more accurately, we have least-squares fitted two Gaussian peaks of equal width to our data for these three metals. The ratios and separations so derived are: Ag–1.51:1.00, 1.8 eV; Pt— 1.60:1.00, 3.3 eV; and Au- 1.48:1.00, 3.1 eV. As our accuracy in determining these ratios is ~ +0.1, they could all be represented by a value of 1.50:1.00. A possible significance of this value is that it is the ex- pected (and observed) intensity ratio for a spin-orbit split d level (e.g., the 5al levels of HgC)). Thus, one might argue that as the 4d and 5d shells move nearer to the Fermi surface with decreasing Z, they must go continu- ously from core states to valence states, perhaps retain- ing some degree of simple spin-orbit character in the rocess. The observed separations are 1.5–2.5 times iarger than free-atom theoretical spin-orbit splittings 36], but the various perturbations of the lattice might e responsible for this. Speaking against such a simple nterpretation, however, is our observation (verified in PS results [29,38]) that for Au the component nearer } is broader. In fact, the UPS results for hu s 21.2 eV how this component split into two peaks [29,32]. In 417–156 O - 71 - 13 view of this, our intensity ratio estimates based on two peaks of equal width may not have fundamental sig- nificance, and the agreement of these ratios, particu- larly between Ag and Pt or Au could be somewhat ac- cidental. Nonetheless, the similarity in shape of our results for the d levels of Ag, Pt, Au, and Hg is rather striking. We have noted that for Cu, Ag, and Au, the recent UPS work of Eastman [29] at higher photon energies (21.2 to 26.9 eV) is in much better agreement with XPS results than previous studies using a range of lower photon energies [28,35,38]. It thus appears that as the photon energy is increased in a UPS experiment, the form of the energy distributions can be expected to ap- proach rather quickly that observed in XPS work. We feel that photoelectron spectra for which XPS and UPS show agreement ought to be much more closely related to p(E). Further UPS experiments at greater than 20 eV photon energies would thus be most interesting. 4. Concluding Remarks We have discussed the use of x-ray photoelectron spectroscopy (XPS) in the determination of densities of states. The application of this technique to the d bands of 12 metals and 3 nonmetallic solids seems to indicate that reliable information about the overall shape of p(E) can be obtained. The results show systematic behavior with changes in Z and crystal structure and agree qualitatively and in some cases quantitatively with theoretical predictions for both unfilled valence d levels and filled core-like d levels. Throughout our discussion, we have placed special emphasis on comparison of XPS with the closely re- lated ultraviolet photoelectron spectroscopy (UPS). It appears that UPS at the present time has an advantage in resolution, but that XPS results can be more easily corrected for inelastic scattering, are not significantly affected by final state density, and are less susceptible to the effects of surface contaminants. UPS results at photon energies =20 eV appear to be more reliable in- dicators of p(E) in the sense that they agree better with the rough outline predicted by XPS. The need for further work at higher resolution and at all photon ener- gies (including those in the relatively untouched range from 20 to 1250 eV) is evident. 5. Acknowledgments The authors wish to thank W. E. Spicer, D. E. East- man, S. Doniach, F. M. Mueller, A. J. Freeman and F. Herman for fruitful discussions relating to this work. 177 6. References [1] We shall use the term “valence bands” for those occupied elec- tronic states that are derived principally from atomic valence shell orbitals. [2] Cuthill, J. R., McAlister, A. J., Williams, M. L., and Watson, R. E., Phys. Rev. 164, 1006 (1967). [3] Hanzely, S., and Liefeld, R., this Symposium. [4] Hagstrum, H. D., Phys. Rev. 150,495 (1966); Hagstrum, H. D., and Becker, G. E., Phys. Rev. 159, 572 (1967); and Hagstrum, H. D., this Symposium. [5] Berglund, C. N., and Spicer, W. E., Phys. Rev. 136, A1030 and A1044 (1964); and Spicer, W. E., this Symposium. [6] Fadley, C. S., and Shirley, D. A., Phys. Rev. Letters 21, 980 (1968); and Fadley, C. S., and Shirley, D. A., J. Appl. Phys. 40, 1395 (1969). [7] Siegbahn, K. et al., Electron Spectroscopy for Chemical Analy- sis, (Almqvist and Wiksells AB, Stockholm, Sweden, 1967). [8] Spicer, W. E., Phys. Rev. 154, 385 (1967). [9] Siegbahn, K., Nordling, C., and Hollander, J. M., Lawrence Radiation Laboratory Report UCRL-10023 (1962). Fadley, C. S., Ph. D. dissertation, University of California, Berkeley, 1970 (Lawrence Radiation Laboratory Report UCRL- I9535 (1970)). We note that such level-specific effects as two-electron transi- tions (see ref. 21) and core-electron binding energy splittings (see ref. 17) may render invalid the assumption that the tail be- hind a core level photoelectron peak is due entirely to inelastic scattering. However, if such effects are significant, they will generally result in noticeable peaks in the tail. The only peaks observed for the cases studied here were weak and can be ex- plained as inelastic plasma losses. Therefore, we expect level- specific effects to have negligible influence on the results under discussion. They should be noted as a possible source of error in this correction procedure, however. Eastman, D. E., and Krolikowski, W. F., Phys. Rev. Letters 21, 623 (1968); and Eastman, D. E., J. Appl. Phys. 40, 1387 (1969). Baer, Y., Physik der Kondensierten Materie 9, 367 (1969). Hodges, L., Ehrenreich, H., and Lang, N. D., Phys. Rev. 152, 505 (1966). [15] Callaway, J., Energy Band Theory (Academic Press Inc., New York, 1964), p. 117. [10] [11] [12] [13] [14] [16] Herman, F., Ortenburger, I. B., and Van Dyke, J. P., to appear in Intern. J. of Quant. Chem., Vol. IIIS. [17] Fadley, C. S., Shirley, D. A., Freeman, A. J., Bagus, P. S., and Mallow, J. V., Phys. Rev. Letters 23, 1397 (1969). [18] Doniach, S., and Sūnjić, to appear in J. Phys., and Doniach, S., to be published. [19] Hedin, A. L., Lundqvist, B. I., and Lundqvist, S., Sol. State Comm. 5, 237 (1967) and this Symposium. [20] Nesbet, R. K., and Grant, P. M., Phys. Rev. Letters 19, 222 (1967). [21] Carlson, T. A., Phys. Rev. 156, 142 (1967). [22] Turner, D. W., Ch. 3 in Physical Methods in Advanced Inor- ganic Chemistry, M. A. O. Hill and D. Day, Editors, (Inter- science Publishers, Inc., London, 1968). [23] Broden, G., Heden, P. O., Hagström, S. B. M., and Norris, C., this Symposium. [24] Chan, D., and Shirley, D. A., this Symposium. [25] Fadley, C. S., Hagström, S. B. M., Klein, M. P., and Shirley, D. A., J. Chem. Phys. 48, 3779 (1968). [26] Connolly, J. W. D., this Symposium. (These results have als been presented in ref. 12.) [27] Wong, K. C., Wohlfarth, E. P., and Hum, D. M., Phys. Letter 29A, 452 (1969). [28]. Krolikowski, W. F., and Spicer, W. E., Phys. Rev. 185, 882 (1969). [29] Eastman, D. E., private communication, to be published. [30] Snow, E. C., Phys. Rev. 171, 785 (1968). [31] Siegbahn, K., private communication. [32] Yu, A. Y.C., and Spicer, W. E., Phys. Rev. 169, 497 (1968). [33] Janak, J. F., Eastman, D. E., and Williams, A. R., this Sym posium. Freeman, A. J., Dimmock, J. O., and Furdyna, A. M., J. Appl. Phys. 37, 1256 (1966) and private communication. Krolikowski, W. F., and Spicer, W. E., unpublished results. Herman, F., and Skillman, S., Atomic Structure Calculation (Prentice-Hall Inc., Engelwood Cliffs, New Jersey, 1963). Mueller, F. M., Garland, J. W., Cohen, M. H., and Benneman K. H., to be published; and Mueller, F. M., this Symposium. [38]. Krolikowski, W. F., and Spicer, W. E., to be published. [34] [35] [36] [37] 178 Discussion on “Electronic Densities of States from X-Ray Photoelectron Spectroscopy” by C. S. Fadley and D. A. Shirley (University of California, Berkeley) F. J. Blatt (Michigan State Univ.); Have you ever seen D. A. Shirley (Univ. of California): We have seen anything which resembles an Auger effect in Auger peaks. We can identify them by using exciting photoemission? radiation of two different energies. 179 Direct-Transition Andlysis of Photoemission from Palladium J. F. Jancak, D. E. Eastman, and A. R. Williams IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598 The energy distribution of optically excited electrons in Pd arising from direct interband transitions has been calculated assuming constant momentum matrix elements. Principal features of new photoemission data (d-band structure with four peaks at 0.15, 1.2, 2.2, and 3.5 eV below the Fermi level and a d-band width of ~3.8 eV) are successfully explained by these calculations. The data can be analyzed with comparable success using the nondirect-transition model, but only by assuming a free- electron density of unoccupied states, which is shown to be unjustified for Pd. In addition to the photoemission spectra and the density of states, the imaginary part of the dielectric constant is com- puted and compared with experiment. Key words: Copper; dielectric constant; direct interband transitions; electronic density of states; interband transition; Korringa-Kohn-Rostoker (KKR); “muffin-tin” potential; palladi- um (Pd); photoemission; plasmon; secondary emission; silver (Ag). 1. Introduction Subsequent to the interpretation of the optical prop- erties of Cu and Ag by Ehrenreich and Phillip [1] in terms of direct, or k-conserving, interband transitions a controversy has arisen over the applicability of the direct transition model to photoemission data. Although there had been, no reflectance data which contradicted the model, the failure of early attempts to account for the characteristic behavior of photoemission data (the linear shift of structure with photon energy) led Spicer and others [2–4] to postulate an alternative model based on nondirect (k not conserved) transitions. While photoemission data for several transition metals have been successfully fit using this model [5], it has proven very difficult to account theoretically for the postulated nondirect transitions. We show, via a first-principles calculation of the opti- cally excited electron spectra in Pa, that the direct- transition model is capable of explaining the dominant features (viz, structure in the energy distribution, and the frequency dependence of the quantum yield) of new photoemission results for Pd. Similar results obtained by Smith and Spicer for Cu [6] (using a histogram technique employed earlier by Brust [7]) and recent data for Au [8], suggest that the optical excitations in- volved in photoemission from metals are due to direct transitions, at least in the transition and noble metals. The nondirect transition model shows comparable success in fitting the experimental data, but only if a smooth unoccupied density of states is postulated; the latter assumption is inconsistent with the calculated density of states in palladium (see fig. 1). Both analyses O.5 O.4 | O3H O. |- | | |→ l | | | | 5 |O (a) eV ABOVE EF O = E | | | | | –5 – 4 -3 –2 - O =EF (b) ev BELOWEE FIGURE 1. Density of states (both spins) of Pa (a) above, and (b) below the Fermi level [N(E)= 2.28 states/e/|atom]. 181 indicate a d-band width of ~3.8 eV for Pd, with four peaks in the density of ~0.15, 1.2, 2.2, and 3.5 eV below the Fermi level. The quantities considered in this paper are the densi- ty of states N(E), the optical absorption e2(a)), and the photoemission energy distribution. The density of states is given by o°e, (a)) = X. nin' where fm is the Fermi function, En(k) the energy at k in band n, and ha) the photon energy. We shall assume that the momentum matrix element |(nk|p|n'k)|* is a constant independent of n, n', and k. d°kf, (1-fº)|(nk|p|n'k)|*6(Enº (k) – En (k) – ho), (2) D(E, o) =X ſ d°kf, (1–f,) (nk | p |n'k(|*6(E – En (k))6 (E-ho-En(k)), (3) nm' where E is the energy of the excited electron. In this equation, the first 6-function ensures that only final states of energy E are considered, while the second 6- function expresses conservation of energy for the ex- citation process; the Fermi functions guarantee that the initial state |nk) is filled and the final state |n'k) is empty. Note that D(E,00) is a density function for the op- tical absorption: o'e (o) = | D(E, ode (4) Ef The second and third steps in the photoemission process have been analyzed using the assumptions nn' D' (E, o] - ſ D(E", *| X. ſ d°kf, (1–f,) Ef 6(E – En (k))6(E – ha) – En (k))T(k, E, o] (5) The first factor in eq (5) normalizes D'(E,00) to a single absorbed photon and the hot-electron transport factor (nk | p |n'k)|* *- o–HE N(E, o] =1.0-Tºp (e. o) +ſ fi E Here |-(## )" N(E)=#|x| dº(E-E,(k)) _ ] dSn | 473 X |… | V.En and, except for a constant factor, Our direct-transition photoemission analysis as- sumes a three-step process [2] consisting of (1) excita- tion via direct interband transitions; (2) hot-electron transport to the surface; and (3) escape into vacuum. The first step is described by the energy distribution D(E,00) of optically excited electrons arising from direct interband transitions: described by Berglund and Spicer [2], which include isotropic scattering of the hot electrons, and a free-elec- tron surface escape probability (momentum escape cone). According to this model the transport and escape processes factorize into a hot-electron transport factor T(k,E,00) (which describes the k-dependence of the group velocity) and an energy-dependent surface escape probability. Integration over the k-dependence of the excitation and transport processes leads to the energy distribution D'(E,00) of the fraction of optically excited electrons per absorbed photon which reach the crystal surface: T is defined in eq (12). The emission intensity N(E.0) including once-scattered electrons, is then given by See (E", E) D'(E", olde) (6) (E > (b) (7) 182 is the surface escape probability for free electrons (q) is the work function), D'(E,0) is the lifetime-broadened D', and See (E",E) is the probability distribution for secondary internal electrons of energy E produced by a primary electron at energy E" (eq. (38) in [2] and eq (17) in [9]). The energy distributions D'(E,00) are con- volved with lifetime-broadening Lorentzians to account for many-electron interactions. The first and second terms in eq (6) correspond respectively to the emission of primary electrons and secondary electrons arising from a single pair-creation process [2]. The quantity N(E,00) is the photoemission energy distribution to be compared with experiment. Our method of computing D and D' is described in section 2; the remaining steps in the analysis are described in section 3. Our results are described in sec- tion 3 and conclusions are summarized in section 4. 2. Computational Details The energy distribution D(E,00) of optically excited electrons is given in eq (3). Each of the 6-functions in this equation defines an energy surface, and the integral can thus be rewritten as a line integral along the curve of intersection of these two surfaces: |(nk|p|n'k)|* V.Enºx V, Enº | D(E, o] =X. dl nn' (8) (E), (k) = E > EF, E), (k) = E – ha) s EP) Our method of computing this integral is an exten- sion of the Gilat-Raubenheimer [10] procedure of replacing an integral over a single 6-function (as in the density of states) by a sum over small cubes in each of which an energy surface like En(k)= E is approximated by a plane: E = En (k0) + (k - ko) o V.E, (ko) 2 where ko is the cube center. The density of states at energy E, for example, becomes the sum over cubes of the areas of the plane within each cube divided by |V}. En (k0) |. Given accurate values of the energies and gradients at each cube center, it is found that 10,100 cubes in 1/48 of the foc Brillouin zone are necessary to give the density of states over a range of energies with an rms error of less than 1 percent, although the relative error could be larger for a particular energy where the surfaces are small or highly curved. (It should be noted that the problem of loss of relative ac- curacy of volume integrals which are numerically small because the contributing subvolume is small is not unique to the Gilat-Raubenheimer procedure or our generalization of it. It can be avoided only by holding the number of mesh points roughly constant in the con- tributing volume, whatever its size. Thus, for example, histogram techniques also have this problem.) Our generalization of this procedure to integrals con- taining two 6-functions, such as that in eq (3), consists of approximating both energy surfaces in a given cube by planes and computing the integral along the straight line of their intersection. The integral in eq (3) thus becomes the sum over cubes of the length of the line of intersection in each cube weighted by V.E.XV.E." -" evaluated at the cube center. This procedure depends on having a formula for the length of an arbitrary line within a cube, which is ob- tained as follows: let the line be given by k = d -H v t, (d : v = 0) (9) with respect to the cube center (v is a unit vector along the line, and d is the shortest vector from the cube center to the line). If the cube edge is 2b, a portion of the line will lie within the cube if and only if there is a range of values of t for which the three inequalities – b s di + vit s b, i = 1, 2, 3 (10) are simultaneously satisfied. Each inequality defines a lower limit li and an upper limit ui on t. If (11) a portion of the line lies in the cube if, and only if, ti > tº , and the length of this portion is simply ti – tº. Thus eqs (10) and (11), along with the geometry required to find d and v in terms of the equations for the planes, furnish a method of computing the integral in eq (3). The method replaces a smooth curve by a number of straight-line segments, and the cubes must be chosen small enough to ensure that each curve of intersection is replaced by enough segments. For cubes of a given size, the error is largest where the curvature is largest; just as in the density of states, this will usually occur for energies where a contribution from a particular band pair has just begun to appear or is about to disappear, i.e., for those cases where the length of the entire curve of intersection becomes comparable to the cube size. The largest relative errors produced by the method will therefore occur where the contribution to D(E,00) is small. This is illustrated in figure 2, which compares the exact D(E,00) to the computed D(E,00) for the test case (k in units of 27T/a) El (k) = 2krky -- k3; E2(k) =2kºk, -- kº k = 1-k a = 1.5 Ry, EF-0.7 Ry. t = min (ui, u, us), tº - max (li, lº, la), 183 f |2|6 -- \ : 8 |- | \ \. 2 \, \ \ |.8 e-or O L.------.” | |.5 |.6 |.7 ENERGY (ry) FIGURE 2, Test of integration procedure for saddle-point singularity. Solid curve is analytic result: points are computed using the algorithm described in the text. This pair of bands has no physical significance; it simply forms a convenient test. For this case, for which D(E,00) can be worked out analytically, a saddle-point (logarithmic) [11] singularity occurs at E = 1.6875 Ry. The only appreciable errors in the computations occur for the lower energies, where the curve of intersection passes through just a few cubes. In practice, it appears that 10,100 cubes in 1/48 of the foc Brillouin zone (48 cubes along T — X) are adequate to give the line in- tegral within an error of about 5% which is sufficiently accurate for our purposes. The energies and gradients required by the method were obtained by first computing the energies of 3345 points in 1/48 of the Brillouin zone using the 4d"—configuration Hartree-Fock-Slater potential [12] for atomic Pa, and the nonrelativistic KKR method of band calculation (a summary of computation times is given in table 1). The energies and gradients on the finer mesh were then obtained by 27-point Lagrangian interpolation [13]. The large number of KKR points is required to get good interpolated energies and gradients for palladium (the rms interpolation error in energy is ~5 × 10−4 Ry, the maximum interpolation TABLE 1. CPU times on the IBM System/360 Model 91 1. 3345 KKR energies for 9 bands........................ ......30 min 2. Interpolation to 10,100 energies and gradients............ 30 S 3. Density of states, 20 V energy range in steps of 0.068 V (9 bands)........................................................ 30 s 4. Photoemission EDC's, 12 V energy range in steps of 0.05 V (9 bands)............................................ 3 s/value of a) 5. Construction of first 12 difference bands (from tape cre- ated in step 2)................................................. 20 S 6. Evaluation of e2(a)); 8 V range in steps of 0.136 V (12 PALLADIUM (f) E 2 I, > || ſº - <ſ 0. H. CD Orº <ſ - H. ~~ 3 / s' / . § /)^ (d) EXPERIMENT (YU AND SPICER) /2& (b) DIRECT TRANSITIONS (c) NONDIRECT TRANSITIONS | | | | |-|--|--|-- O || 2 3 5 & 7 8 9 |O || |2 haſ (eV) FIGURE 3. Comparison of wºes (a) for direct transitions (b) and nondirect transitions (c) with experiment (a)|16). The ordinate is given in arbitrary units for both calculated curves. error in energy is ~5 × 10−8 Ry, and the rms interpola- tion error in the gradient is ~1 × 10−8 Ry/(2Tſa)). The maximum error in the interpolated energies increases rapidly as the number of KKR points is decreased (for example, the energies interpolated from 3345 KKR points differ by as much as 0.05 Ry from those interpo- lated from 89 KKR points in palladium). The density of states of palladium, computed using existing Gilat-Raubenheimer programs [14], is shown in figure 1. The most surprising feature of this density of states is the rather large amount of structure for energies about EP, particularly the pronounced dip at 8.3 eV above Ey. Thus, the density of excited one-elec- tron states in Pol is by no means free-electron-like. We have also computed e2(a)), the imaginary part of the dielectric constant, assuming direct transitions and constant matrix elements of momentum. This quantity, which is shown in figure 3, was obtained by forming the difference in energies and gradients for various band pairs, and then computing the density of states of the difference bands. Because interpolation errors are mag- nified in taking differences, this calculation is less ac- curate than the ordinary density of states. To construct theoretical EDC’s for photoemission, we need not only the energy distribution of optically excited electrons, but also the fraction of these per ab- sorbed photons which reach the crystal surface. D'(E,00), as defined in eq (5), includes the hot-electron transport factor [2] - T(k, E, a y = (12). h-'o (a) T(E) |V}.E, {1 + h-ſo (a) T(E) |V.E." |}- which takes into account the photon absorption depth and the mean free path of the excited electron. The 184 (arbitrary units) \ w - - A | | | – 4 – 3 – 2 |NITIAL ENERGY E – aſ (eV) FIGURE 4. D(E, o] (broken curves) and D'(E, a (solid curves) for Pa, plotted versus initial energy E-ha). Solid and broken curves are on different scales. Quantities o T' (a)), T (E), and h\7F'En(k) are, respec- tively, the optical absorption depth, and the lifetime and group velocity of the excited electron. Under the frequently-used assumption [2,4,5] that En(k) is free- electron-like, V.En becomes proportional to E!”; T is then independent of k, and may be taken out of the integral. The density of states given in figure 1 shows, however, that this assumption is untenable for Pd. The factor |V} En, in Tenters the integrand in eq (5) almost linearly (for Pol, T ~ 0.1), and this tends to reduce the effect on D'(E,00) of structure in the unoccu- pied density of states [15] (which involves |V}.E., "). Figure 4, which shows both D(E,00) and D'(E,00) for two values of a), gives some indication of the effects of the ransport factor. Note that D' (E,d) is the “bare” dis- tribution; to compare with experiment, the curves must e convolved with Lorentzians to take account of many- lectron lifetime broadening, they must be multiplied y an energy-dependent surface escape function, and he distribution of secondary electrons must be in- luded, as in eq (6). These calculations, and further TABLE 2. k-space locations and variances of contribu- tions to photoemission energy distribution. Last column is the integrated contribution of the band to the photoemission [see eq (4)]. Band k (Units of 27t/a) or (Units Contribution of 27|a) to ex(a)) ha) = 7.7 eV 1... . . . . . . . . . . (0.66, 0.39, 0.20) 0.18 22.2] 2...... . . . . . . . (0.70, 0.37, 0.14) 0.18 17.16 3...... . . . . . . . (0.78, 0.40, 0.13) 0.17 15.50 4..... . . . . . . . . (0.82, 0.42, 0.14) 0.17 15.52 5. . . . . . . . . . . . . (0.85, 0.48, 0.12) 0.17 6,02 6. . . . . . . . . . . . . 0.0 ha) = 9.9 eV 1...... . . . . . . . (0.81. 0.35, 0.11) 0.20 10.12 2...... . . . . . . . (0.84, 0.37, 0.12) (). 19 10.24 3....... . . . . . . (0.85, 0.48, 0.09) 0.15 5.19 4...... . . . . . . . (0.73, 0.50, 0.21) 0.24 8.31 5... . . . . . . . . . . (0.70, 0.44, 0.19) 0.18 11.74. 6... . . . . . . . . . 0.0 ha) = 11.6 eV 1... . . . . . . . . . . (0.89, 0.40, 0.11) (). 16 4.72 2....... . . . . . . (0.74, 0.46, 0.25) 0.21 11.80 3....... . . . . . . (0.75, 0.43, 0.18) 0.22 12.2] 4. . . . . . . . . . . . . (0.76, 0.39, 0.15) 0.21 10.66 5...... . . . . . . . (0.63, 0.40, 0.15) 0.16 5.32 6....... . . . . . . (0.39, 0.36, 0.33) 0.02 0.05 details on the choice of T(E) are described below. The most important feature of the energy distribu- tions in figure 4 is that they show structure which (1) shifts approximately linearly with photon energy and (2) reflects the d-band density of states. The first of these explains the central feature of photoemission data which provided the original motivation for the non- direct transition model [2-4]. This behavior occurs in Pa because the contributing transitions are restricted to a relatively small region in k-space near the point (2T/a)(3/4,1/2,1/4) which is midway along the line Q from L to W on the (111) zone face (this was determined by computing the average of the cube center locations, weighted by the contribution to D(E,00). Details for several values of a) are given in table 2). The standard deviation, -0.2(2Tſa), is a measure of the localization. In this region each d-band is almost flat, leading to linearly shifting structure via the second delta function in eqs (3) and (5). The excited band En (k), on the other 185 — – T- | T | PALLADIUM (3) (3) (2 * 3H THEORY--> INITIAL ENERGY (eV) FIGURE 5. Experimental energy distributions (solid curves) and theoretical energy distributions (for direct transitions) for Pd, plotted versus initial energy. All curves are normalized to the measured and calculated quantum yields, respectively. IO-l PALLADIUM $=5.55eV # IO-2H- H. O T ſh- N. O LLl —l Lll C IO-3– —l Lll / (q) EXPERIMENT >- f (b) DIRECT TRANSITIONS = (c) NONDIRECT TRANSITIONS Ž s C; IO-4– IO-5 | | | –– | 5 6 7 8 9 |O || |2 fia (eV) FIGURE 6. Quantum yield (per absorbed photon) for Pd. Curves (a), (b) and (c) respectively give the measured results and calculated results for direct and nondirect transitions (see text). hand, changes by ~3 eV in this region, thereby prevent- ing severe modulation of the structure (with a) by the other delta function. While this localization in k-space explains the linear shift of structure with a), it throws doubt upon any direct relationship between photoemission data and the d-band density of states. The localization implies that, at best, photoemission measures the d-band density of states in a small fraction of the Brillouin zone; the similarity in structure between this partial density of states and the total density of states in Po appears to be coincidental. The localization also justifies our neglect of the k dependence of |(nk|p|n'k)|*. Thus, the inclu- sion of matrix elements should affect only the relative amplitudes of the peaks in figure 4 and not their posi- tion in energy or their movement with a). 3. Experimental Results and Comparison with Theory Photoemission spectroscopy measurements were made in the energy range 5.5 × ha) < 11.6 eV on evaporated polycrystalline Pd films (see 7. app.). Nor- malized energy distribution curves (EDC’s) are shown in figure 5 (solid lines). The experimental quantum yield Y(a)) (number of emitted electrons per absorbed photon) is shown in figure 6 (solid line). The work function (b = 5.55 + 0.1 eV was determined from the usual Fowler plot of (Y(a)))/2 vs ha) as ha) → (b. The experimental EDC’s show 4 peaks (labeled (1), (2), (3) and (4)) at ~ 0.15, 1.2, 2.2, and 3.5 eV below Ef; this structure is stationary in initial energy, with the exception that the first two peaks merge for hoo 2 11 eV [16,17]. Before describing analyses of the data, we note that the leading edges of the EDC’s are sharper ( ~ 0.2 to 0.25 eV from 10 to 90% points) than the theoretical ener gy uncertainty T-(E) determined by normalizing to the quantum yield. Our conclusion is that the broadenin is one-sided below the Einstein cutoff (ha) above Ef due to overall energy conservation for the photoelectri process. There is no obvious mechanism for impartin extra energy (of order T-1 = 0.5 eV) in addition to hot the photoelectron during the excitation and transpor process. Thus the observed broadening of the leadin edge is due to thermal smearing ( ~ 0.1 eV at 300 K) the electron spectrometer resolution, work function in homogeneity, etc. Direct transition analysis. Theoretical EDC’s base upon direct interband transitions (secs. 1 and 2) ar compared with experiment in figure 5. The theoretica EDC’s show 4 peaks at ~ 0.1, 1.2, 2.2, and 4 eV belo Ef, with the second (-1.2 eV) peak disappearing fo had ~ 1.1 eV. The theoretical EDC’s show good agree 186 ment with experiment, especially in the energy loca- tions of observed structure. Identifying the lowest theoretical EDC structure (at —4 eV) with the lowest observed structure (at –3.5 eV) indicates that our cal- culated d-bands are a=0.5 eV too wide. This difference is well within the uncertainties in the band energies due to uncertainties in the muffin-tin potential. A central feature of the theoretical EDC’s is that structure remains essentially stationary in initial-state energy. The theoretical EDC’s in figure 5 were computed using eq (6) as follows. The required quantities include: the band density of states N(E) in figure 1 for the occu- pied and unoccupied bands within homar of Ep, the op- tical absorption coefficient oſſa) [16], the energy de- pendent lifetime T (E), the work function (q) = 5.55 eV), and the band energies En(k) and energy gradients W.En(k) which enter the computation of D(E,00) and D'(E,00) (see sec. 2). The lifetime T (E) has been computed in the ran- dom – k approximation (with constant Coulomb matrix elements) [9] and depends only on energy. Evaluation of the energy-dependence of T(E), using the band densi- ty of states for Pol (fig. 1) and normalizing T(E) to the measured quantum yield (total number of emitted elec- trons per absorbed photon) at ha) = 10 eV, yields T(E) =28 (E–E) -1.15 eV-1 for 5 - E - 12 eV, e.g. the lifetime is T = 1.7 eV-1 at 10 eV. The transport factor T(k,E,00) is of order 0.1 for Pol; i.e., the average value of the mean free path l (Eny, k) = T va is estimated to decrease from ~25 A to ~10 Å in the 5 to 12 eV range [18]. Lifetime effects on the energy distributions have been treated in an ad hoc way by convolving the calcu- lated sharp distributions with an energy-dependent Lorentzian lifetime function. Here we have used the above-mentioned T (E) determined from the quantum yield, i.e., D'(E,0) is D'(E,d) convolved over E with a Lorentzian of halfwidth hiſt(a)). This broadening (with 0.4 eV - T-1 < 0.6 eV for energies from 7 to 12 eV) results in theoretical EDC’s with resolution comparable to that of the experimental EDC's (changing this broadening does not affect the energy positions of dominant structure). We have folded back the smeared energy distributions (D") about EP-F ho so as to con- serve the number of excited states while maintaining overall energy conservation [9]. (This procedure produces no new structure in the final EDC’s). In summary, all quantities involved in the theoretical emission spectra (the energy bands, density of states, See(E, E), va, etc.) are specified in terms of a small number of quantities: the one-electron muffin-tin potential, the measured work function (b, the measured optical absorption coefficient oſſo) [16] and the lifetime normalization constant. Therefore, within the model, structure in the EDC’s is directly related to the one- electron potential. In the theoretical EDC’s shown in figure 5, secondary emission contributes less than 25% of the total emission and does not lead to any observable peaks; thus peak locations and, to a large extent, amplitudes of the theoretical EDC’s are not sensitive to our treatment of secondary emission. Comparison of the theoretical EDC structure with the d-band density of states (fig. 1) shows correlation, i.e., peaks in the EDC’s tend to occur near energies corresponding to peaks in the den- sity of states. We have also studied the sensitivity of the theoretical EDC’s by altering the muffin-tin potential so as to nar- row the d-bands by ~0.5 eV. The principal effect of this change was to shift the locations of the dominant EDC peaks into better agreement with experiment; however, amplitude agreement was not improved. The relative peak amplitudes of the theoretical EDC’s in figure 5 show more variation with photon energy than is ob- served. This behavior might be due to the assumption of constant matrix elements in the present analysis. The quantum yield for Ó 3 ha) < 11.6 eV has been evaluated for the direct transition model (dashed line in fig. 6) and shows very good agreement with experiment. This agreement is a measure of the adequacy of the ap- proximations of a free-electron escape probability and energy-dependent lifetime, both of which significantly influence the quantum yield. The dielectric constant e2(a)) has been calculated as- suming direct transitions with constant matrix ele- ments. Comparison with experiment (solid curve in fig. 3) shows some correlation (e.g., the structure at ~4 eV and minimum near 8 eV); however, overall agreement is mediocre. The increase in absorption observed ex- perimentally for ha) > 8 eV is likely to be due in part to plasmon excitations (the imaginary part of e-' shows a peak at ~7.5 eV [16]. Inclusion of momentum matrix elements is necessary for a more detailed comparison with the experimental e2(a)), especially for ha) < 4 eV. Nondirect transition analyses. An analysis of the ex- perimental data has been made assuming nondirect transitions (with constant matrix elements) for two cases. First, a best fit to the experimental EDC’s was made using the empirical occupied optical density of states [5] and the unoccupied conduction band density of states shown in figure 7. This ODS describes the energy positions and amplitudes of observed structure (except for the merging of the upper two peaks for ho > 1.1 eV). 187 PALLADIUM ODS > ãº- s > S. / (ſ) / H / š / bo / / / / a ME / / / ...” / _-- ~~ V | | |-- | ~ 1 | | | -6 -5 -4 -3 -2 - O=EF 2 3 4 5 ENERGY (eV) FIGURE 7. The optical density of states (ODS) for Pd determined from a best fit to the experimental EDC’s. The nondirect transition analysis closely follows the work of Berglund and Spicer [2]; the emission N(E,d) is approximately given by eq (6) with the energy dis- tribution D'(E,00) given by pe (E - ho)pc (E) ſ Er-i-ha, Ef with T(E, o] = o(a))l (E) (1+ ol)- (E)-(E) Wº Here pe and pc are the occupied and unoccupied optical densities of states (fig. 7). The energy dependence of the inelastic mean free path l (E) = Tv, with v, & VE, was determined in the random-k approximation [9] using the ODS in figure 7. The mean free path was set equal to 15 A at 8 eV in order to give a best fit to the quantum yield Y(a)). The resulting quantum yield (broken line in fig. 6) shows very good agreement with experiment. The dielectric constant e2(a)) has been calculated using the nondirect transition model with the ODS of figure 7 (see curve (c) in fig. 3) and shows much less structure than is observed experimentally. As with the direct transition analysis, the nondirect transition anal- ysis (with constant matrix elements) fails to explain the observed increase in e2(a)) for ha) > 8 eV; this increase is likely to be due in part to plasmon excitations. A nondirect transition analysis was also made using the theoretical band density of states (fig. 1) for both the occupied and unoccupied bands [19]. It is logical to describe both the unoccupied states and occupied states by band states if either are to be so-described. The resulting EDC’s also exhibited stationary structure similar to that of the direct transition analysis in figure (13) T(E, o] ki,(E, (0) F pe (E - ho)po (E) dB PALLADIUM ha) = 10.7 eV ) __NONDIRECT TRANSITIONS TRANSiTIONS 16 º |- (BAND p(E)) 2 S 2 (b) Cl- s— DIRECT > TRANSITIONS S. § 0 —l LL] 2 C (ſ) QQ > LL O (q) EXPERIMENT O O | | | | | | –6 –5 – 4 -3 –2 —|| O=EF |N|T|AL ENERGY (eV) FIGURE 8. Comparison of the experimental EDC (a) at ha) = 10.7 eV with calculated curves for (b) direct transitions, (c) nondirect transitions using the band density of states (fig. 2), and (d) non- direct transitions using the ODS of figure 7. 4, but gave poorer agreement with experiment because of the strong structure in the unoccupied density of states at = 8.3 eV above Eſ. Nondirect-transition (and direct transition) analyses are compared with experi- ment in figure 8; curve (c) clearly illustrates the effect of the conduction band structure at ~8 eV in N(E) (i.e., sharp dip at ~ – 2.5 eV). This result raises an interesting point concerning nondirect transitions — any reasonable muffin-tin poten- tial for Pd would appear to lead to strong conduction band structure, which should be observable in non- direct transitions between band states are dominant. This is not observed in practice. Of course it is possible (but unlikely) that the escape process can suppress the effect of such structure. (Direct calculation has shown that inclusion of the transport factor using the k-depen- dent band group velocity diminishes only slightly the ef- fect of structure in the unoccupied density of states on the nondirect-transition EDC’s.) Thus, an optimal fit with the nondirect transition model requires the addi- tional assumption (unjustified for Pa) of a smooth unoc- cupied density of states. 188 4. Conclusions The photoemission energy distributions of palladium calculated using the direct-transition model (constant matrix elements) and the nondirect-transition model (with a free-electron unoccupied density of states), show about equal agreement with experiment. While we are therefore unable to offer conclusive proof of either model we have added two new elements to the controversy. First, the original motivation for the non- direct transition model has been removed by showing that the direct transition model does in fact predict structure which shifts linearly with photon energy over the limited range (p → ha) < 11.6 eV [6]. Second, we have shown that the success of the nondirect transition model depends on the unjustifiable assumption of a free-electron unoccupied density of states which is at best inconsistent with the identification of the optical density of states with the band density of states. Nonetheless, we are struck, particularly in view of these developments, by the apparent success of the nondirect transition model for many metals [5]. We are hopeful that experimental data over a broader range of photon energies will unequivocally establish one or the other model [8]. The discrepancy between theoretical and experimen- tale2(a)) curves above 8 eV can possibly be attributed in part to the observed plasma edge at 7.5 eV [16]. The remaining discrepancies between theoretical and ex- perimental ex(a)) are most likely due to our assumption of constant momentum matrix elements. Presumably, the plasmons will also affect the photoemission spectra. We have not taken them into account in our direct- transition photoemission analysis. 5. Acknowledgments The authors gratefully acknowledge the technical assistance of J. Donelon with the measurements. 6. References [1] Ehrenreich, H., and Philipp, H. R., Phys. Rev. 128, 1622 (1962). [2] Berglund, C. N., and Spicer, W. E., Phys. Rev. 136, A1030 (1964). : [3] Spicer, W. E., Phys. Rev. 154, 385 (1967). [4] Seib, D. H., and Spicer, W. E., Phys. Rev. Letters 22, 711 (1969). [5] Eastman, D. E., J. Appl. Phys. 40, 1387 (1969). [6] Smith, N. V., and Spicer, W. E., Optics Comm. 1, 157 (1969). We were motivated to perform the present calculations by an early communication of their work, for which we are indebted. [7] Brust, D., Phys. Rev. 139, A489 (1965). [8] Photoemission data for Au (D. E. Eastman) at 16.8 and 21.2 eV shows structure which contradicts the nondirect transition model. It remains to be seen if direct transitions can explain these Au data. [9] Kane, E. O., Phys. Rev. 159,624 (1967). [10] Gilat, G., and Raubenheimer, L. P., Phys. Rev. 144, 390 (1966). [ll] Kane, E. O., Phys. Rev. 175, 1039 (1968). [12] Dimmock, J. O., and Freeman, A. J., private communication. [13] Steffenson, J. F., Interpolation, Chelsea Publishing Co., New York, 1950. [14] Janak, J. F., Phys. Letters 28A, 570 (1969). [15] Kane, E. O., Proc. Int. Conf. Semiconductor Physics, J. Phys. Soc. Japan Suppl. 21, 37 (1966). [16] Yu, A. Y. C., and Spicer, W. E., Phys. Rev. 169,497 (1968). [17] Previous photoemission studies of Pa [16] have shown that the first two peaks below EP for low energies (ha) < 9 eV), but did not clearly resolve structure at higher energies. [18] We have ignored the k-dependence of the mean free path of once-scattered electrons in eq (6). This has no significant ef- fect on our results for Pd since secondary emission is esti- mated to be <25% for ha) < 11.6 eV and its energy distribution is smoother than that of the primaries. [19] The occupied and unoccupied band densities of states were convolved with lifetime functions =T-1 (E). [20] Blodgett, A. J., Jr., and Spicer, W. E., Phys. Rev. 146, 390 (1966). 7. Appendix. Experimental Conditions Photoemission EDC’s and quantum yields were mea- sured in the 5.5 to 11.6 eV range using experimental techniques similar to those described by Spicer and coworkers [20]. An ac synchronous detection system with a 2–1/2 inch diameter gold-plated spherical mesh collector was used (the mesh has 70 lines/inch and is ~ 85% transparent). An ac capacitance bridge was used to cancel stray capacitance to <.05 pf; with a 10°0 sensing resistor (Keithley Model 601 electrometer) and 1 second integrating time constant, the input noise was typically ~ 1 × 10−14 amperes. The quantum yield was measured using a calibrated Cs.Sh photodiode (with a LiF window) which is traceable to Prof. W. E. Spicer’s laboratory calibration. Palladium films (~ 1 – 2 × 108 Å thick) were prepared by evaporation onto smooth Cr-plated 5/8 inch diameter quartz discs using an electron-beam gun in an ion-pumped ultra-high-vacuum system. The pres- sure (mainly hydrogen) rose from a base of ~7 × 10-11 torr to ~ 1 × 10−8 torr during evaporation (at ~2 Aſsee) and then rapidly fell to < 4 × 10−10 torr within 10 minutes after evaporation. 189 Discussion on “Direct-Transition Andlysis of Photoemission from Palladium” by J. F. Janak, D. E. Eastman, and A. R. Williams (IBM Thomas J. Watson Research Center) W. E. Spicer (Stanford Univ.); Historically, when we saw peaks moving with hu until they disappeared, we interpreted this structure in terms of direct transitions. The examination of modulation is very important in choosing between direct and nondirect models. Secondly, I guess I would have to comment that it seems to me the agreement between your calculated transitions and the experimental data does not seem to be quite as strong as the abstract indicates. Would you comment on this? J. F. Janak (IBM): Well, as I said, I think the agree- ment on the slide we showed is within about 20-40%. We never claimed in the abstract that the agreement was perfect. But inasmuch as we have assumed that the matrix elements are constant, and we have heard today how much they can vary, I think that 30%, assuming constant momentum matrix elements, is really pretty good. J. T. Waber (Northwestern Univ.); I would like to rein- force what you have said about the need of very large number of points in k space to achieve accuracy in determining the Fermi energy. A more detailed state- ment of the reliability of density of states curves is ap- pended as a short post-deadline paper [1]. The graphs shown there are for a pure parabolic band, E = k”, with k evaluated at points on a regular grid in the Brillouin zone. The QUAD scheme of Mueller et al. [2] was used to obtain additional points in the zone. The first graph is for 5000 points and the relative deviation from the parabola is 9.8%. It is necessary to go to 100,000 points to achieve a relative precision of 3.6%. M. S. Dresselhaus (MIT): Since you deal with a range of k values wouldn’t it be possible to include the k dependence of the momentum matrix elements without too much trouble? I am referring here to the method whereby the momentum matrix elements are computed from derivatives of the Hamiltonian. J. F. Janak (IBM): Well, we have been trying. That is the next step in the analysis. I should point out that since the region is so small that the k-dependence is not really the important thing, the main thing is the band-to-band variation. In fact, the transitions occur in a region with a standard deviation or radius essentially 0.2 (in units of 27/a) around a point halfway between L and W on the (111) zone face. That actually covers about 20% of the zone, I think, when you use all the symmetry. But 20% of the total volume of a sphere is a pretty small shell on the outside. In that kind of a shell we would want the k-dependence of the matrix ele- In entS. [1] Kennard, E. B., Koskimaki, D., Waber, J. T., and Mueller, F. M., these Proceedings, p. 795. [2] Mueller, F. M., these Proceedings, p. 17. 190 Photoemission Determination of the Energy Distribution of the Joint Density of States in Copper* N. V. Smith Bell Telephone Laboratories, Murray Hill, New Jersey 07974 Measurements have been made of the photoemission properties of cesiated copper, with improved sample preparation over the previous work by Berglund and Spicer. The energy distribution curves (EDCs) of photoemitted electrons show structure in the region associated with the copper d bands which was not seen in the previous data. The behavior in the photon energy range 6 to 8 eV is particularly in- teresting in that some of these new peaks in the EDCs are observed to move, appear and disappear in a manner characteristic of direct transitions. Parallel calculations have been performed of the energy distribution of the joint density of states (EDJDOS) similar to those reported recently by Smith and Spicer. The band structure used was the in- terpolation scheme of Hodges, Ehrenreich, and Lang fitted to the APW calculation of Burdick. In a con- stant matrix element approximation, the EDJDOS represents the energy distribution of photoexcited electrons. This was converted to an energy distribution of photoemitted electrons by introducing ap- propriate threshold and escape factors. The overall agreement with experiment is good. In particular, some of the peaks in the theoretical EDCs are predicted to disappear and reappear on varying the photon energy, and there are strong similarities with the changes observed experimentally. It is found, therefore, that the optical transitions from parts of the copper d bands can be identified as direct. Theoretical calculations based on the EDJDOS work quite well for these and other transitions. Photoemission provides a very sensitive tool for verifying and even determining the EDJDOS. Burdick’s bands for copper appear to be essentially correct over a wide range of energies including the whole of the d-band region, although minor modifications of a few tenths of an eV would improve agreement. It is found that the most persistent peaks in the calculated EDJDOS tend to coincide with the peaks in the calculated true density of states. This indicates that when a phenomenological “optical density of states” can be obtained, it may well be a good approximation to the true density of states even if transi- tions are direct. Key words: Aluminum-insulator-palladium (Al-Pa); augmented plane wave method (APW); cesium; copper; direct transitions; electronic density of states; joint density of states; non- direct transitions; optical density of states; optical properties; photoemission. 1. Introduction One of the major landmarks in the photoemission in- vestigation of metals was the observation of the d bands in Cu and Ag by Berglund and Spicer [1]. Their work has stimulated much of the subsequent effort in this area and it is now fair to say that photoemission is one of our most powerful tools for probing band structure. , The measurements by Berglund and Spicer were per- formed on samples whose surfaces were covered with *The experimental part of this work was performed while the author was on assignment at Stanford University from Bell Telephone Laboratories. The facilities used at Stanford are supported in part by the Advanced Research Projects Agency through the center for Materi- als Research at Stanford University and by the National Science Foundation. a thin layer of cesium in order to lower the work func- tion. More recent work by Krolikowski and Spicer [2] on clean Cu has revealed more structure in the energy distribution curves (EDCs) of photoemitted electrons than was seen on cesiated Cu. However, the work func- tion of clean Cu is almost 3 eV greater, so that this structure could be observed only over a more limited energy range. So far, the photoemission data on Cu and other noble and transition metals [3,4] has been interpreted in terms of a predominantly nondirect model. The im- portance of the direct (i.e., k-conserving) nature of the optical transitions from the Cud bands did not manifest itself in any noticeable way, although the direct nature of the transitions in the vicinity of L. -> L1 was clearly 191 recognizable. Energy conservation appeared to be the main factor in determining the behavior of the d band structure in the EDCs. In fact, it was possible to unfold the EDCs and extract an “optical density of states” which bore a strong resemblance to the calculated band structure density of states [2]. The author has attempted to assess the merits of a purely direct transition model by performing band cal- culations of the energy distribution of the joint density of states (EDJDOS). A comparison of these calculations with the photoemission data on clean Cu has been re- ported by Smith and Spicer [5]. In performing the cal- culations, it became clear that the most interesting behavior characteristic of direct transitions would oc- cur, if at all, at energies below the vacuum level for clean Cu. It was decided, therefore, to reinvestigate the photoemission properties of cesiated Cu. Since vacuum and sample preparation techniques had improved over the intervening years, there was hope that it might be possible to resolve more structure in the EDCs than had been seen by Berglund and Spicer. A selection of the results of these experiments is presented below, and it will be seen that this hope was indeed realized. 2. Photoemission and Direct Transitions In the conventional theory of optical absorption by solids, transitions are allowed only between states which lie at the same point in k-space in the reduced zone; i.e., the transitions must be direct. If ei(k) and eſ(k) denote the energies in an initial band i and a final band f, optical transitions at photon energy ha) are restricted to the surface in k-space given by Qſì (k) = eſ (k) – e i (k) – ha) = 0. (1) The total number of direct transitions at this photon energy is represented by the well-known joint density of states given by JA (ha)) = (27)-3 X. f d°kö(QF) (k)). (2) i, f The prime on the integral denotes that the integration is to be performed only over those portions of k-space for which ei < EF - ef where EP is the Fermi energy. In photoemission experiments, we are interested in not just the total number of transitions, but also the energies at which the excited electrons emerge from the metal. A more relevant quantity for our purposes is what we call the energy distribution of the joint density of states (EDJDOS) defined by (3) 2 (e, ho) = (27)- > f d°kö (Qf;(k))6(e–eſ (k)). i, f The additional 6 function picks out those transitions whose initial energy equals e. In a constant matrix ele- ment approximation, Ø(e.ha) represents the energy dis- tribution of photoexcited electrons referred to initial states. Strictly speaking, we should weight each transi- tion with the square of an appropriate momentum matrix element. This refinement has not been at- tempted so far, and for the remainder of the paper we will remain within the constant matrix element approxi- mation. This has the advantage of making the calcula- tion of the EDCs extremely straightforward, since 3(e.ho) is a property solely of the e,k-dispersion CUITV6 S. The EDJDOS of Cu has been evaluated numerically from eq (3). The band structure used was the interpola- tion scheme of Hodges, Ehrenreich, and Lang [6] with its parameters fitted to the APW calculation by Bur- dick [7]. A computer was programmed to sample the energy eigenvalues at more than 10° points in the primi- tive 1/48 of the zone, deduce the permitted transitions, and to keep running scores of these transitions catalogued according to photon energy and initial ener- gy. Calculations of this kind were first performed by Brust on silicon [8]. The EDJDOS at a given ha) obtained in this way still represents the energy distribution of photoexcited electrons. This was converted to an energy distribution of photoemitted electrons (i.e., an EDC) by multiplying by an appropriate threshold function [1]. To take ac- count of electron loss by inelastic scattering, this threshold function contains a factor of /(1 + O(6), where o is the optical absorption coefficient and 6 is the elec- tron mean free path. The details of the threshold func- tion do not affect the qualitative features of the results, such as the shape of the EDCs. They do, however, determine the total photoelectric yield and therefore the normalization of the absolute magnitude of the EDCs. In these calculations, o was taken to be constant at 7.2 × 10−5 cm-' over the whole frequency range. The energy dependence of the mean free path & was as- sumed to be given by C(E – EF)-” va, where E is the energy at which the electron emerges from the metal (i.e., final state energy), and v, is a free electron group velocity for an electron of this energy inside the metal. C is a constant determined by insisting that & should be 22 Å for E = E + 8.6 eV; this value was found by Krolikowski [2] in his nondirect analysis. The imagina- ry part of the dielectric constant was calculated from the joint density of states making the same matrix ele- ment approximation. This enabled the EDCs to be nor- malized to the yield per absorbed photon. Finally, some broadening was introduced into the curves by convolv- Q 192 /~\ \ P \ PRESENT WORK N (hw- |O.2 eV) \ –––––BS (IO.4 eV) —-—--KS (10.2 eV) / \ / \ / \-—s | | / | | | | –8 —7 –6 –5 –4 –3 –2 – | O E-ha, +eq (eV) Photoelectron energy distribution curves for Cu referred to initial state energy. The upper two curves were taken on cesiated copper; the full curve is the present work (ha) = 10.2 eV) and the dashed curve is from Berglund and Spicer (ha) = 10.4 eV). The lower curve was taken on clean Cu by Krolikowski and Spicer (ha) = 10.2 eV). FIGURE 1. ing them with a Lorentzian function whose width at half maximum was 0.3 eV. The results of these calculations will be compared with experiment in section 4 below. 3. Experimental Details The Cu sample was prepared by evaporation in a stainless steel vacuum system for which the base pres- sures were on the 10-11 torr scale. During the evapora- tion of the Cu, the pressure rose to 5 × 10−" torr but quickly dropped to 1.5 × 10−19 torr after the evapora- tion. It was found that photoemission measurements performed on this clean Cu sample reproduced the previous results of Krolikowski and Spicer [2]. The sample was then cesiated. The cesium was contained in a glass ampule which had been broken open in the vacuum chamber at an earlier stage [9]. The cesiation was performed by gently heating the ampule and then exposing the Cu sample to the cesium beam for a few minutes. The EDCs were measured by the conventional a.c.-modulated-retarding potential technique [10]. The Cu samples used by Berglund and Spicer were prepared in glass vacuum systems operating at base pressures of around 40-8 torr. Their cesiation procedure was also rather different, but it is believed fia) = 7.1 eV PRESENT WORK ~ ---- *s –5 –4 –3 –2 — | O E-fia) + eq (eV) Photoelectron energy distribution curves for Cu at ha) = 7.1 eV referred to initial state energy. The upper two curves were taken on cesiated copper; the full curve is the present work and the dashed curve is from Berglund and Spicer. The lower curve was taken on clean Cu by Krolikowski and Spicer. FIGURE 2, that the differences in the data are due primarily to dif- ferences in the overall vacuum conditions under which the experiments were performed. In figure 1 we compare the EDC obtained at ha) = 10.2 eV in the present work with that obtained at 10.2 eV by Krolikowski on clean Cu and that obtained at 10.4 eV by Berglund and Spicer. The EDCs have been plotted against E — ha) + ed), where E is the electron kinetic energy in vacuum, and eq) is the work function. This choice of scale refers the photoelectrons to their initial states and places the zero of energy at the Fermi level. It is seen that the structure associated with the Cu d bands is much more blurred in the data of Berglund and Spicer. Also, the piece of structure labelled P observed earlier at about 7 eV below the Fermi level is absent in the present data. One possible explanation for this structure proposed by Berglund and Spicer was that it was due to a low energy peak in the density of states unanticipated by band calcula- tions. The same structure finds its way into Krolikowski’s more refined optical density of states [2]. The present work, however, indicates that its ex- istence is questionable. A parallel may be drawn here with a similar situation in nickel. Early work on Ni [3] indicated the existence of a low energy peak in the den- sity of states. However, when samples were prepared under better vacuum conditions by Eastman [4], it was found that this peak was much reduced. The effects of Cesium, or any surface contaminants for that matter, are still very imperfectly understood. 417–156 O - 71 – 14 193 Ideally, the only effect of cesium on the surface should be to lower the work function. If this were the case in practice, we would expect a detailed correspon- dence between the pieces of structure seen in the EDCs on cesiated Cu and those seen in clean Cu. It can be seen in figure 1 that the present results satisfy this requirement better than the data of Berglund and Spicer. This observation holds true at other photon energies. We will therefore proceed on the assumption that the present measurements are more representative of the properties of pure Cu. The corresponding curves for ha) = 7.1 eV are shown in figure 2. In all the curves, a peak due to electrons from the uppermost d band can be clearly seen at about –2.3 eV. The energies below this peak were inaccessi- ble in the clean Cu experiments, but are quite accessi- ble in the cesiated experiments due to the lower work functions. In the range —2.6 to — 4.0 eV, there is a marked discrepancy between the earlier data and the present data. Berglund and Spicer observed a single large piece of structure in this range which changed lit- tle with photon energy. The present work reveals two pieces of structure in this range and the profile is found to change on varying the photon energy. Indeed, this is the most interesting feature of the present work from Cu(+Cs) | Oxld" O.8 O.6 O.4 EXPERIMENT O.2 | | | | | | – 6 —5 – 4 – 3 – 2 – | O E-fia) + eq, (eV) FIGURE 3. Experimental EDC’s from cesiated Cu between ho- 6.5 and 8.2 eV. The curves are referred to initial state energy, and the zero of energy is placed at the Fermi level. the point of view of the direct versus nondirect in- terpretation, and it will be explored in more detail in the next section. The origin of the differences between the present results and those of Berglund and Spicer is not clear, although we have associated it with vacuum con- ditions during sample preparation and cesiation. Another indication that the surfaces are physically dif- ferent is that the work function in the present experi- ments was 1.75 eV which is 0.2 eV higher than that of Berglund and Spicer. 4. Comparison of Theory and Experiment We confine ourselves here to a very limited selection of the experimental data. Roughly speaking, the EDCs at high photon energies (ha) > 9 eV) are in fair accord with Krolikowski and Spicer’s data on clean Cu. At low photon energies (ha) < 5 eV) the new data agrees well with that of Berglund and Spicer. It is in the photon energy range 6.5 eV s ha) s 8.2 eV that the new and most noteworthy information has been obtained and so we will concentrate on this region. The EDCs for photon energies in the range 6.5 to 8.2 eV are shown in figure 3. As before, we plot the EDC against the quantity E — ha) + eq, which places the zero of energy at the Fermi level. Let us focus attention on the behavior of the d-band structure in the energy range –2.6 to — 4.0 eV indicated by the vertical lines. At ha) = 6.5 eV, we have two pieces of structure in this range. At ho) = 8.2 eV, we once again have a doublet, but for photon energies in between there is a continual change in the profile of the EDC. As the photon energy is in- creased, we find that the peak on the left fades away. While this is happening, the peak on the right expands by moving its low energy edge to lower energies, until at ha) = 7.8 eV there is one broad piece of structure filling the whole range. On increasing the photon ener- gy further, this broad peak splits into a doublet. Such behavior is difficult to explain on a nondirect model, and is more characteristic of direct transitions. The theoretical EDCs calculated according to the prescription outlined in section 2 are shown in figure 4 for the same photon energies. Let us once again focus attention on the behavior in the energy range –2.6 to –4.0 eV, indicated as before by the two vertical lines. At hoo = 6.5 eV, there are two peaks within this region. On increasing ha), the left-hand peak fades away and then returns. At ha) = 7.8 eV we have a single broad piece of structure filling the whole range which then splits into a doublet on going to ha) = 8.2 eV. The similarity between these trends and those shown by the experimental data in figure 3 is very striking, and would 194 Cu (+CS) |Oxidº THEORY O. 8 O.O. O24. O. 6 Æhaj = 8.2 eV º º |ſ OH- OH- OH —l | | | | –6 –5 – 4 –3 –2 - O E-fia) + eq, (eV) FIGURE 4. Theoretically calculated EDCs from cesiated Cu between ha) = 6.5 and 8.2 eV. The curves are referred to initial state energy, and the zero of energy is placed at the Fermi level. seem to support the direct transition interpretation. Theory and experiment are shown together for three photon energies in figure 5. The experimental EDCs all show a large contribution at the low energy end. This is due to electrons which have suffered an inelastic scat- tering but are still sufficiently energetic to escape from the metal. These have not been included in the theory which considers only those electrons which escape without scattering. It is seen that the calculations based on the EDJDOS are quite successful in predicting the energy location of structure in the EDCs. No great reliance can be attached to the relative peak heights in the theoretical curves in view of the crudity of the con- stant matrix element approximation; but even so, the agreement is encouraging. The peaks in the theoretical curves are much more pronounced than in experiment in spite of the 0.3 eV broadening we have introduced. Note, incidentally, that theory and experiment are plotted on the same absolute scale, all curves having been normalized to the yield per absorbed photon. Let us very briefly consider the behavior at other photon energies. Figure 6 shows a comparison of the theoretical and experimental EDCs at 10.2 eV. These are typical of the high photon energy behavior. The relative number of slow scattered electrons is higher than at the lower photon energies shown in figure 3. |x|O’ O i i –6 –5 – 4 –3 – 2 - O E-ha, +ed (eV) FIGURE 5. Comparison of the theoretical and experimental EDCs from cesiated Cu for three representative photon energies. The full curves are experimental; the dashed curves are calculated theoretically as- suming direct transitions. 1.2 x 16°- fiuj = 10.2 eV E-tiw 4 eq (eV) FIGURE 6. Comparison of the theoretical EDC (dashed curve) and the experimental EDC (full curve) on cesiated Cu at ha) = 10.2 eV. 195 1.2|− xIOT” –3 –2 —| O E-fla + ed (eV) FIGURE 7. Comparison of the theoretical EDC (dashed curve) and the experimental EDC (full curve) on cesiated Cu at ha) = 4.9 eV. The structure at the high energy end is due to direct transitions in the vicinity of l.2, — 1.1. The structure predicted in the theoretical EDC agrees well for the lower d bands, but not so well for the upper- most d bands. The uppermost d-band peak in the ex- perimental curve is a composite of two peaks labelled 1 and 2 which merge at about this photon energy. The two uppermost pieces of structure in the theoretical curve are again labelled 1 and 2. It is seen that the theory places peak 2 a couple of tenths of an eV too low. A similar discrepancy can be discerned at ha) = 8.2 eV in figure 5. It may be necessary to make some empirical adjustment to Burdick's bands to remove this discre- pancy. However, it should also be borne in mind that the interpolation scheme is likely to go astray for final states so far removed from the Fermi energy. The typical behavior at lower photon energies is represented by the ha) = 4.9 eV curve shown in figure 7. The d-band-to-conduction-band transitions are now distinct from the conduction-band-to-conduction-band transitions. The latter occur in the vicinity of L2 – Li and give rise to the rectangular shaped contribution at the high energy end of the EDC. At even lower photon energies, the EDCs are very similar to those of Berglund and Spicer. In particular, it is found that at photon energies below the L2 -> Li threshold, there is still a large contribution to the EDCs all the way up to the high energy cut-off determined by the Fermi ener- gy. Our calculations confirm that there are no direct transitions in bulk Cu which could account for these photoelectrons. By definition, these transitions must be categorized as nondirect, and their origin would be worthy of further study. 5. Conclusions Concerning the Density of States It has been found in new photoemission experiments that some of the optical transitions from the Cud bands can be identified as direct. Theoretical calculations of the EDCs based on direct transitions work reasonably well for these and other transitions. If a direct model is more appropriate model, then photoemission EDCs measure the energy distribution of the joint density of states rather than the density of states itself. (The relative heights of peaks in the EDC may, of course, differ from those in the EDJDOS because of optical matrix element variations which have been ignored here.) To determine the density of states in such circumstances, the procedure must be to find a band structure whose EDJDOS is consistent with the photoemission EDCs; the density of states is then simply calculated from the band structure. The availa- bility of high speed interpolation schemes is therefore of great importance here. The currently available interpolation schemes [5,11] are very versatile in that the band structures of many noble and transition metals can be simulated by adjust- ments of only a dozen or so disposable parameters. The photoemission EDCs are rich in structure, so that to ob- tain the kind of agreement shown in figures 3–7 im- poses very strong constraints on the permitted band structures. This holds out the exciting prospect that it may be possible, armed only with an interpolation scheme and the photoemission data, to arrive at an al- most purely experimental determination of the E, k curves for any arbitrary d-band metal! How similar is the EDJDOS to the density of states? Figure 8(b) shows the calculated EDCs for cesiated Cu based on direct transitions at three widely spaced photon energies. Figure 8(a) shows the histogram of the density of occupied states calculated from the same band structure. Certain similarities are evident. Prominent peaks in the EDC quite often coincide in energy location with peaks in the density of states. This is merely a consequence of the fact that in order to have initial states for optical transitions one must first of all have states. In other words, it is possible for the density of states to impose itself on the EDJDOS and, in certain cases (such as flat bands) to become the dominant con- sideration. Noting that the structure in the experimen- tal EDCs is more blurred than in theory, only the stron- gest and most persistent peaks will survive. This may go some way towards explaining the success of the non- direct approach, and the similarity of the empirically derived optical densities of states to the band structure densities of states. than a nondirect 196 (d) DENSITY OF STATES | | | | | | (b) EDC /* | l fia) | | -----65 eV ſ\ | L: | - | -- - - tº | \ ...~. f \ ~ l | j-V} \ | / V \\ g /~\ / \ w / \-Z \ / / \s s / | | | === – 6 –5 –4 –3 –2 – O |N|T|AL STATE ENERGY (eV) FIGURE 8, Density of filled states for Cu is compared with the theoretical EDCs for ha) = 6.5, 8.2 and 10.2 eV calculated on the basis of direct transitions. The zero of energy corresponds to the Fermi level. 6. Acknowledgments I would like to thank Professor W. E. Spicer for plac- ing the facilities of his laboratory at my disposal and for his interest. Useful conversations with Dr. L. F. Mattheiss and Dr. C. N. Berglund are also gratefully acknowledged. 7. References [1] Berglund, C. N., and Spicer, W. E., Phys. Rev. 136, A1030, A 1044 (1964). [2] Krolikowski, W. F., and Spicer, W. E., Phys. Rev., in press. [3] Blodgett, A. J., Jr., and Spicer, W. E., Phys. Rev. 146, 390 (1966), and 158, 514 (1967); Yu, A. Y.C., and Spicer, W. E., Phys. Rev. 167,674 (1968). [4] Eastman, D. E., J. Appl. Phys. 40, 1387 (1969). [5] Smith, N. V., and Spicer, W. E., Optics Communications 1, 157 (1969). [6] Hodges, L., Ehrenreich, H., and Lang, N. D., Phys. Rev. 152, 505 (1966). [7] Burdick, G. A., Phys. Rev. 129, 138 (1963). [8] Brust, D., Phys. Rev. 139, A489 (1965). A more sophisticated method of performing this kind of calculation is outlined by J. F. Janak, D. E. Eastman, A. R. Williams, these Proceedings, p. 181. [9] Further details of this equipment are given by N. W. Smith and W. E. Spicer, Phys. Rev., in press. [10] Spicer, W. E., and Berglund, C. N., Rev. Sci. Instr. 35, 1665 (1964); and Eden, R. C. (to be published). [ll] Mueller, F. M., Phys. Rev. 153,659 (1967). 197 The Band Structure of Tungsten as Determined by Ultraviolet Photoelectric Spectroscopy C. R. Zeisse” Naval Electronics Laboratory Center, San Diego, California 92.152 The technique of photoelectron spectroscopy has been used to probe the band structure of tung- sten in the energy region where the 5d bands are most prominent. The work function of the clean sam- ple, a 25 micron thick polycrystalline foil, was found to be 4.36+ 0.02 eV, and the yield rose by three or- ders of magnitude from 5.0 to 11.3 eV without showing prominent structure of any other sort. The elec- tron energy spectra, on the other hand, contain two pieces of reliable structure which are found to in- crease in energy at the same rate as the exciting photon energy. A simple analysis of the data gives evidence that the density of d states in tungsten consists of a shoulder just below the Fermi level, a peak located about 1.5 eV below the shoulder, and a broad peak which extends at least 7 eV below the Fermi level. Key words: Carbon contamination; electronic density of states; photoemission; tungsten (W); UV photoemission; work function. 1. Introduction It is expected that the 5al electrons, often cited as the cause for the high resistivity of transition metals, will produce the most prominent structure in the density of states in tungsten. Each of the two tungsten band struc- ture calculations, WI and W2, which have been done by Mattheiss [1] show three peaks below the Fermi sur- face. Experimentally, the density of states for tungsten is well known near the Fermi level, but not at other ener- gies, and the object of this work is to present experi- mental photoemission data confirming the existence of this structure in the density of states below the Fermi energy. 2. Apparatus and Procedure Particular attention has been paid to cleanliness and carbon contamination, known to have caused difficul- ties in photoemission, ion neutralization, and electron diffraction work on transition metals [2]. The standard ac technique of Spicer and Berglund [3] has been used with a spherical, gold plated collector and a Lif win- dow. The incident light intensity was measured outside the window with a sodium salicylate wavelength con- vertor and photomultiplier tube. The maximum photon *Mailing address: Code S350.2, NELC, San Diego, California 92.152. energy used was 11.2 eV, and the total resolution in the worst possible case of highest photon energy and highest retarding potential was 1.5 eV. The samples were 25 micron thick rolled tungsten foils [4], initially cleaned ultrasonically in detergent and methyl alcohol and then processed in a continuously pumped ultrahigh vacuum in the following manner: The sample was clamped between heating bars and flashed to high tem- peratures by passing a high ac current through it for about 10 seconds. The pressure rose as the gases were desorbed from its surface, but returned to the base pressure of 1-2 × 10-11 torr within 1/2 minute. The max- imum pressure during the flash depended linearly on the time spent by the cold sample between flashes and was typically 3 × 10−19 torr for each intervening minute. Assuming a sticking coefficient of unity, this figure is just what would be expected from complete vaporiza- tion of all gases which could collect on the sample sur- face during the intervening time interval at the base pressure. The flashing temperature was 2100 K, chosen because the use of higher temperatures up to 2500 K produced no change in any of the energy spectra. An energy spectrum could be started within I minute of the flash and was usually finished within 4 minutes. Carbon was removed by heating for at least one diffu- sion time constant (11.4 hours at 2200 K for a 25 micron thick sample) in an atmosphere of 5 × 10−7 torr oxygen [5]. 199 Unfortunately, a measurement could never be re- peated exactly because of small but systematic changes in both the energy spectra and yield with the time from flash. These changes were on the order of a 1% in- crease in the area of a spectrum in 4 minutes and as much as a 50% decrease within 15 minutes in the yield near threshold. After several days in the vacuum the energy spectra showed increasing numbers of electrons at all kinetic energies, a factor of 2 increase in area being commonly observed. Upon flashing again the en- tire process could be repeated. The increase in area after long exposures is similar to that observed by Waclawski et al. [6] during yield measurements from a polycrystalline tungsten ribbon exposed to various gases, but the behavior immediately after the flash is more difficult to explain as a contamination effect since only about 1/2% of a monolayer can collect in 4 minutes at 1 × 10-11 torr. Furthermore, the process did not de- pend on total pressure or light exposure in any syste- matic way, so that it seems most likely that some change is occurring in the sample due to the recent flash. In short, the phenomenon is complicated and not understood but was taken into account wherever neces- sary by extrapolating the data back to the time of the flash. 3. Data and Discussion The relative yield is shown in figure 1, uncorrected for reflectivity but corrected for the transmission of the LiF window. A Fowler plot, used to determine the work function, is shown in the inset. The sodium salicylate was at most 2 weeks old for this sample, but was 10,000 I I I | H | I | I | I | I I 1000 H. e º º g 100 H. © & - º --I-I-I-I-I-I-I-I-I-I-T- $ o % = 4.36 + 0.02 eV º o’ 10 F- © S. 2’ G 28 e —l - …” - I 1.0 .” º > eſ e’ e e’ •’ º 2^ 0 I H––––––––– 4.50 5.0 5.50 o PHOTON ENERGY (eV) 1 ——— l |→ | 1–1 I | I | l 4 5 6 7 8 9 10 11 12 PHOTON ENERGY (eV) FIGURE 1. The relative yield of Sample II, uncorrected for reflectivity. The insert shows a Fowler plot of the data near threshold. The inclusion of data from Sample I would increase the scatter by a factor of 2. several months old for Sample I, and since it was ex- posed to the diffusion oil environment of the monochro- mator its response probably is only flat to within 25% [7], which is considered to be the accuracy of the mea- surement. At any rate, there is no fine structure to the curve at this precision, but the continued rise in yield at high energies is interesting. The yield of chromium, for example, rises by a factor of about 2.5 in the energy | I | | | ; | | I I T — — — BEFORE CARBON REMOVAL AFTER CARBON REMOVAL hu = 10.20 eV SAMPLE || -4 –3 -2 - 1 O 1 ENERGY (eV) FIGURE 2: A comparison of the electron energy spectra for the two samples before and after carbon removal. The curves are unnormalized and have been shifted in energy to make the structure coincide. 200 : E - hv (eV) FIGURE 3. Normalized energy spectra for Sample II, referred to initial energy states and labeled with the value of the exciting photon energy in eV. The Fermi energy occurs at about -5 eV on this plot, but its exact location is obscured by the high energy tail caused by scattered light. range from 8 to 11 eV [8], whereas the yield for tung- sten, two elements below chromium in that column of the periodic table, rises by a factor of 10 in this same energy region. The following argument implies an ap- preciably strong density of states in tungsten 11 eV below the vacuum level: According to Spicer [9], the yield far from threshold is proportional to A/(A+B), where A is a factor due to transitions from filled states to empty states above the vacuum level and B is a fac- tor due to transitions from filled states to empty states between the Fermi and vacuum levels. An immediate consequence of this relation is that the yield is constant for all energies where B remains zero. Neglecting processes such as the creation of two electrons from a single photon or energetic electron (i.e., pair production and scattering effects), B will be zero for photon ener- gies larger than the separation between the bottom of the d band and the vacuum level. Taking the bottom of the d band as the energy of the point H12 in the Mattheiss calculation and using the experimentally determined value of the work function, this energy turns out to be 10.5 eV for W1 and 12.1 eV for W2. Since there is no indication of a plateau in the yield up to 11.3 eV, and since the s electron contribution to the density of states is negligibly small compared to the d electron contribution, the yield results indicate a d bandwidth at least as great as 11.3 eV, tending to favor the W2 calculation. Figure 2 shows the energy spectra from the two tung- sten foils, both before and after carbon removal. The shoulder and high energy peak are considered to be re- liable structures, whereas the small hump at 1.5 eV was on the order of the noise in the trace and was not present in Sample I before carbon removal. The lowest energy peak in the first sample is attributed to scatter- ing, since its amplitude increased but its position remained constant in energy as hu was increased. This interpretation is supported by the fact that the shoulder smeared out in Sample I for energies larger than 10.5 eV, whereas it was still present at 11.2 eV in Sample II. The rounding of the high and low cut-offs due to scat- tered light is particularly conspicuous at zero energy in Sample II after carbon removal. The energy spectra for Sample II are shown in figure 3. These curves have been normalized by dividing each by the incident light intensity and the transmission of the LiF window. For each curve the retarding potential E has been shifted down by hy in order to emphasize structure which “moves with hu.” It can be seen that the shoulder and high energy peak, labeled A and B in figure 3, line up nicely after this procedure, a charac- teristic feature of transitions from d states which have 201 been observed in other photoemission studies [10]. As mentioned above, the existence of the small peak labeled C is dubious experimentally. Examination of other spectra in the vicinity of 10.2 eV shows that it comes out of the threshold at 10.1 eV and disappears into the noise at 10.4 eV. Although this behavior is characteristic of a direct (i.e., k-conserving) transition [11], a search of the calculated band structures failed to reveal any direct transition to which this peak could be attributed. Photoemission studies in which the structure moves with hu have customarily been interpreted on the basis of a nondirect (i.e., non k-conserving) model, but a recent calculation in copper assuming direct transitions and using the full interpolated band structure has also been successful in predicting much of the data experi- mentally found in the d band [12]. This tends to blur the distinction between direct and nondirect models of the photoemission process, although the ability of both models to successfully predict the structure in the case of copper may merely be due to the flat E versus k behavior of the d states in that metal. In any case, a detailed analysis has not been made here, the major point being that the shoulder and high energy peak are somehow related to the density of d states in tungsten. The main point in favor of this interpretation is the coincidence of structure when plotted in terms of initial energy. In figure 4 the 10.20 eV spectrum is compared with the W1 density of states below the Fermi level. The main difference in the calculation of W1 and W2 is that the exchange potential in W1 is 30% less than in W2. This results in the W2 density of states having a slightly larger separation between the two high energy peaks (about 1.7 eV) and a broader peak at low energy (a full width at half-maximum of about 1.7 eV). The energy spectra provide a slightly better fit to the Wi calculation, although the yield data favor the deeper reach of the d band indicated by the W2 calculation. Unfortunately, the energy spectra do not give a good in- dication of the total d bandwidth because the threshold function cuts off the low energy portion of the spectrum even at 11.2 eV. 4. Conclusion Photoemission spectroscopy of two tungsten foils gives evidence of a shoulder just below the Fermi level and a peak about 1.5 eV below the shoulder. A broad peak occurs at lower energies. The evidence for the highest energy structure comes primarily from the elec- tron energy spectra, which display these structures | | | | | | -8 -6 -4 -2 0 ENERGY (eV) FIGURE 4. A comparison of the 10.20 eV energy spectrum from Sample II and the W density of states calculated by Mattheiss. The zero of energy is taken at the Fermi level, and the position of the experimental curve has been shifted to give the best agreement with the calculated structure. moving with the exciting photon energy and thereby due to initial states in the d band. The evidence for the broad low energy peak comes from the continued rapid increase of the tungsten yield between 8 and 11.3 eV. Comparison with the band structure calculation of Mattheiss provides no convincing reason for choosing his potential Wi over W2, but does confirm the ex- istence of three peaks in the density of states below the Fermi level. 5. Acknowledgments The author acknowledges fruitful discussions with Dan Pierce, Neville V. Smith, and William E. Spicer, and the expert help of George Van Vleck in managing the ultrahigh vacuum system. 6. References [1] Mattheiss, L. F., Phys. Rev. 139, A1893 (1965). [2] For example, see D. E. Eastman and W. F. Krowlikowski, Phys. Rev. Letters 21, 623 (1968); D. W. Vance, Phys. Rev. 164, 372 (1967); and P. J. Estrup and J. Anderson, J. Chem. Phys. 46,567 (1967). [3] Spicer, W. E., and Berglund, C. N., Rev. Sci. Instr. 35, 1665 (1964). [4] Sample I was obtained from Materials Research Corporation and was stated to have less than 250 ppm total impurities in- cluding 20 ppm carbon. Sample II was obtained from the Rembar Company, Inc. and was listed as 99.99% pure. 202 [5] Becker, J. A., Becker, E. J., and Brandes, R. G., J. Appl. Phys. Burns, J., and Tuzzolino, A. J., J. Opt. Soc. Am. 54, 1381 32, 411 (1961). One annoying aspect of oxygen processing was (1964). that the long times spent at high temperature caused the foils [8] Lapeyre, G. J., and Kress, K. A., Phys. Rev. 166,589 (1968). to become very bumpy, scattering light onto the collector and [9] Spicer, W. E., Phys. Rev. 112, 114 (1958), and Spicer, W. E., causing a reverse photocurrent which contributed unwanted Phys. Chem. Solids 22, 365 (1961). high and low energy tails to the spectra. [10] Spicer, W. E., Phys. Rev. 154,385 (1967). [6] Waclawski, B. J., Hughey, L. R., and Madden, R. P., private [ll] Berglund, C. N., and Spicer, W. E., Phys. Rev. 136, A1030 communication. (1964). [7] Samson, J. A. R., J. Opt. Soc. Am. 54, 6 (1964), and Allison, R., [12] Smith, N. V., and Spicer, W. E., private communication. 203 Photoemission Studies of Scandium, Titanium, and Zirconium D. E. Ecustmon IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598 Photoemission spectroscopy studies of the hexagonal metals Sc., Ti and Zr in the 4 to 11.6 eV range have resolved d-band structure and have determined occupied d-band widths (at 1/2 maximum) of 1.6, 2.0 and 2.3 eV respectively. Resolved structure for all three metals correlates with structure in energy band density of states; however, the observed band widths for Ti and Zr are much narrower than previ- ously calculated band widths. The relation of the data to the controversy concerning the nature of opti- cal excitations in transition and noble metals (direct vs nondirect transitions) is discussed. Key words: Copper; direct versus nondirect transitions; electronic density of states; gold (Au); nondirect transitions; optical density-of-states; photoemission; scandium (Sc); silver (Ag); titanium (Ti): zirconium (Zr). 1. Introduction Photoemission spectroscopy (PES) studies have been made on Sc, Ti and Zr in the 4 to 11.6 eV range. Energy distribution curves (EDCs), quantum yields, and work- functions are reported. The data show narrow occupied d-bands with structure that correlates with structure in energy band densities of states. For example, the EDCs of Sc show a peak at the Fermi energy Ep, while the EDCs of Ti and Zr show a low intensity shoulder at Ep; this behavior is consistent with specific heat measure- ments and APW band calculations, both of which show a large density of states at EF for Sc and much smaller densities of states for Ti and Zr. The experimental data are summarized via the photoemission optical density of states (ODS), which has been determined using the nondirect transition model (with crystal momentum k not conserved in the excitation process) [1,2]. In this case the ODS is a den- sity of states function (weighted by transition probabili- ties) which describes the energy locations and am- plitudes of observed structure, and which is expected to correspond to the occupied band density of states. Currently, it is not clear that optical excitations in transition metals are due to nondirect transitions; in fact, there is evidence that direct (k-conserving) inter- band [3] can account for observed photoemission structure. If the excitations are due to direct transitions, the ODS describes the energy loca- tions and amplitudes of structure in the measured transitions EDCs. In this case, the ODS only coincidently reflects structure in the total band density of states since only a small fraction of the occupied electron states are excited. Recent PES evidence for direct transitions is discussed in section 3. 2. Experimental Results Photoemission measurements were performed on evaporated polycrystalline films of Sc, Ti and Zr. Ex- perimental techniques are described elsewhere [4]. Scandium. Normalized energy distributions (EDCs) are shown in figure I and the quantum yield (electrons emitted per incident photon [5]) is shown in figure 2. The workfunction was determined as b = 3.5 + 0.15 SCANDiUM to % = 3.5eV O 3 3 H. C # 2 S &T O / Lil A – / lil hy = |O.2eV--l [i] | 9.2—- E | F- | / / / / / / O | " -8 –7 –6 –5 -4 –3 –2 -] O= E INITIAL ENERGY (eV) FIGURE 1. Energy distributions (EDC’s) for Sc. The EDC's are plotted versus the initial energy E = E-hy--ó, with the Fermi level Ef set equal to zero. - 205 Sci + =35+0.15eV o:2| Ti 4-4.33 to lev Zriº -405:0.1eV IO-3 O-4– - IO-5 - loº-º-º-º-; 8 9 |O || hv (eV) FIGURE 2: The quantum yield (per incident photon) for Sc., Ti and Zr. eV. For Sc., maximum emission occurs from energy states within -2 eV of Ep for all photon energies; these states are identified as the occupied d states. A sta- tionary peak near Ep and a shoulder at ~ – 0.9 eV are observed in all EDCs. The low quantum yield (Y = 10−9 at 10 eV) is charac- teristic of the typically short hot electron mean free paths ( ~ 10 A at 10 eV above EF) in transition metals [6]. An analysis of secondary electron emission due to electron-electron scattering [16] for Sc indicate that this mechanism accounts for most of the observed slow electron emission (with energies less than –2 eV in fig. 1); our analysis assumed the usual optical absorption depth of ~ 100-150 A. A fraction of the slow electron emission is 2 eV above the kinetic energy threshold which is observed for hy = 10 eV (at ~ – 5 eV for hy= 10.2 eV in fig. 1) is believed to be due to surface impuri- ties (similar comments apply for Ti and Zr). The ODS for the occupied d bands of Sc is compared with a theoretical band density of states in figure 3. The ODS was determined using the nondirect transition model with a smooth unoccupied conduction band [7]. The band density of states shown for Sc in figure 3 is the density of states calculated for Y by Loucks [8], which has been scaled in energy by 0.85 to account for the - 15% narrower d bands in Sc [9]. This approxima- tion is expected to adequately represent the band width and major structure for Sc [9]. Agreement between the ODS and band density of states is excellent. The band widths agree, and the observed peak at Er and shoulder at – 0.9 eV agree with the principal structure in the band density of states. Titanium. Energy distributions (EDCs) are shown in figure 4 and the quantum yield is shown in figure 2. The workfunction was determined as d = 4.33 + 0.1 eV using the usual Fowler plot. Emission from d states within -2 eV of EF dominates the EDCs for all photon energies. A low amplitude shoulder is observed at Ep., with two peaks at ~ – 0.7 and — 1.2 eV. Transition probability effects are observed for Ti which cannot be simply fitted using an ODS with constant matrix ele- ments: the structure at —0.7 eV, which is a shoulder for low energies (hu = 8 eV), increases in relative am- plitude until it equals the amplitude of the – 1.2 eV peak at hu = 10 eV, and then decreases (and washes out) at still higher energies. This observed behavior contributes to the increasing evidence against the non- direct transition model (see sec. 3). As with Sc, most of the slow electron emission (ener- gies < – 2 eV in fig. 4) is attributed to secondary elec- trons created by inelastic electron-electron scattering. The slow electron peak at = — 5 eV which increases for hy - 10 eV is believed to be due in part to surface impurities. An ODS is shown for Ti in figure 5 which describes the energy locations of observed structure, and the am- plitudes for hu = 10 eV [10]. The existence of a low amplitude shoulder at Er and 2 peaks at lower energies (at ~ – 0.7 eV and — 1.2 eV) correlates with the major structure in the band density of states (fig. 5), however, the ODS band width (2 eV) is significantly narrower than the theoretical band width (2.8 eV) [11]. - SCAND|UM 2H THEORY (FLEMING 8, LOUCKS) ODS –7 -6 -5 –4 –3 –2 - | ENERGY (eV) 'FIGURE 3. ODS and theoretical density of states for Sc. 206 TITANIUM % = 4.33 eV º 2 9 C I Cl- ^. 3. |O- O LL! —l Lil * hv = ||.2eV LL. > |O.2 O.5H 9.2—- | ~86 /− 7.7 \ O | | || | | | | | –8 -7 -6 -5 -4 -3 -2 - |NITIAL ENERGY (eV) FIGURE 4. Energy distributions (EDC’s) for Ti. The ODS's of Ti and Sc show a rigid band relation; if the Fermi level of the ODS for Ti is lowered by ~0.7 eV to account for 3 rather than 4 valence electrons, the resulting ODS closely resembles that determined for Sc. Zirconium. Energy distributions (EDCs) are shown in figure 6 and the quantum yield is shown in figure 2. The measured workfunction is q = 4.05 + 0.1 eV. The EDCs show structure and amplitude effects which are very similar to those described for Ti. An ODS for Zr is shown in figure 7 which summarizes the observed d- state structure and occupied band width (~ 2.3 eV). The band density of states for Zr calculated by Loucks [12] using the APW method is also shown in figure 7. The observed band width ( ~ 2.3 eV) is narrower than the calculated band width (~ 3 eV). TITANIUM 2.OH- |.OH- –6 –5 –4 –3 –2 - | O=EF ENERGY (eV) FIGURE 5. ODS and theoretical density of states for Ti. 3| ZIRCONUM $ = 4.05eV rº, Q < z 2F- O H O # ^ hy = ||.2eV ź S. O LL] —l * || g 2 ol— |→ \ –8 -7 -6 -5 - 4 -3 –2 - |NITIAL ENERGY (eV) FIGURE 6. Energy distributions (EDC’s) for Zr. O=EF 3. Discussion Photoemission energy distributions for Sc., Ti and Zr show d-state structure (which is stationary in initial- state energy) which indicates narrow (~ 1.5-2.3 eV) oc- cupied d bands. Observed structure tends to correlate with structure in theoretical densities of states. How- ever, amplitude effects are observed for Ti and Zr which cannot be simply described using the nondirect transition model [1]. This result adds to recent developments which cast doubt upon nondirect transi- tions in transition and noble metals. Two other papers (for Cu [13] and Pa [14]) presented at this conference show that the observation of stationary d-band structure (which has been as- sumed to imply nondirect transitions [1,2]) is con- 2 Z|RCONIUM (1) ODS (2) THEORY (LOUCKS) > O H. sº > Sº C | H LL —l LL J Q- | | | -6 –5 –4 –3 –2 - O=EF ENERGY (eV) FIGURE 7. ODS and theoretical density of states for Zr. 207 sistent with direct interband transitions. This stationa- ry structure results from the relative flatness of the d- bands ( ~ 3 eV) compared to the broad unoccupied bands ( ~ 40 eV), coupled with the limited range of photon energies usually accessible (d. 3 hp < 11.6 eV). Further evidence which casts doubt upon nondirect transitions is given by recent high resolution photoemission measurements [15] on Au, Ag and Cu at 16.8 eV and 21.2 eV. These data show structure which is quite different from that observed at energies below LiF window cutoff (11.6 eV) and cannot be ex- plained using the nondirect transition model [15]. It remains to be seen if direct transitions can explain this high energy data. 4. Acknowledgment The technical assistance of J. Donelon is gratefully acknowledged. 5. References [1] Berglund, C. N., and Spicer, W. E., Phys. Rev. 136, A1030 (1964). [2] Spicer, W. E., Phys. Rev. 154,385 (1967). [3] Ehrenreich, H., and Philipp, H. R., Phys, Rev. 128, 1622 (1962). [4] See Janak, J. F., Eastman, D. E., and Williams, A. R., Pro- ceedings of this conference (appendix). Sample chamber pressures rose to ~ 2 to 5 × 10−" torr during evaporation (at ~ 2-5 Aſsec) and then rapidly fell to < 2 × 10−" torr within 3 minutes after evaporation. All three metals are very reactive and require careful surface preparation. [5] The yield Y(a) is expressed in electrons per incident photon (rather than per absorbed photon) since the reflectance has not been measured. [6] Eastman, D. E., to be published. [7] The estimated accuracy in the determination of E, is +0.1 eV. [8] Loucks, T. L., Phys. Rev. 144, 504 (1964). [9] Fleming, G. S., and Loucks, T. L., Phys. Rev. 173, 685 (1968). [10] This ODS doesn't represent the observed amplitudes for hys 9 eV and thus is not a unique ODS. [11] The band density of states shown for Ti is the density of states calculated for Zr by Loucks [12] with the d band narrowed by ~ 10% to agree with the band width calculated for Ti by L. M. Mattheiss (Phys. Rev. 134, 192 (1964)). [12] Loucks, T. L., Phys. Rev. 159, 544 (1967). [13] Smith, N. V., these Proceedings, p. 191. [14] Janak, J. F., Eastman, D. E., and Williams, A. R., these Pro- ceedings, p. 181. [15] Eastman, D. E., to be published. 208 Photoemission and Reflectance Studies of the Electronic Structure of Molybdenum * K. A. Kress** and G. J. Lapeyre Department of Physics, Montana State University, Bozeman, Montand 59715 Normalized energy distributions of photoemitted electrons for 4.3 < hu < 11.8 eV (threshold is 4.3 eV) and near normal reflectance for 0.5 - hv < 11.8 eV are measured for molybdenum films prepared with ultra high vacuum. The nondirect transition model with constant matrix elements is found to be consistent with the photoemission data. The above model, in conjunction with calculations for the emis- sion of scattered electrons, is used to obtain the optical density of states (ODS) for the occupied states. Three peaks due to d-electrons are observed at E – E = –0.5, – 1.6, and —3.9 eV where EP is the Fermi energy. No structure is observed for E – E - 4.3 eV. The imaginary part of the dielectric con- stant, e.g., is obtained by Kramers-Kronig analysis, and the occupied ODS are used to obtain the ODS for 0 < E – E - 4.3 eV. The latter analysis is done by writing the finite difference approximation for the integral expression of e2 and solving for the empty ODS. The ODS is compared to the band calcula- tions of Mattheiss where a molybdenum density of states is obtained by scaling his tungsten (W) results. Both the measured and calculated occupied densities of states have three peaks and both empty states have one dominant peak. The calculations predict a low density of states for – 1 < E – EF - 0 eV which is not observed in the data. The absorption coefficient has a minimum at 11.3 eV which correlates with a dip in the quantum yield. The energy distributions of the photoemitted electrons show small structural changes above the spectral range of the peak in the energy loss function at 10.8 eV. The rela- tion of these data to the explanations based on the electron density of states for the anomalous isotopic mass dependence of the superconducting transition temperature is discussed. Key words: Dielectric constant; electronic density of states; molybdenum (Mo); optical density of states; optical properties; photoemission; reflectivity; tungsten (W); zinc (Zn). 1. Introduction Photoemission and optical measurements were used to study the electronic structure of the 4d metal molyb- denum. The photoemission data were found to be con- sistent with the nondirect model with one possible ex- ception. Because of the essentially nondirect character of the photoemission data, an optical density of states (ODS) graph was constructed. The results are com- pared with band calculation of molybdenum's 5al counterpart tungsten and its 3d counterpart chromium. 2. Experimental Procedures The photoemission and optical data were taken from vapor deposited films of Mo. The measurements were made with the base pressure of the ion-pumped- vacuum system in the 10-19 torr range and the films were vapor deposited with an electron beam evaporator in the 10-9 torr range. The photoemission and optical data were obtained by techniques reported elsewhere [1,2] with the exception of the quantum yield measurement. The spectral yield per incident photon was obtained directly by electronic division of the simultaneous measurement of the photoelectron current and the light flux incident on the photocathode. The yield per absorbed photon was ob- tained from the above data and the reflectance data. The signal proportional to the incident light flux was ob- tained by using sodium salicylate inside the high vacuum chamber to detect a small fraction of the in- cident radiation which passed through a small hole in the photocathode. The sodium salicylate was coated on the inside of a glass window and its fluorescence was measured by an external photomultiplier tube. The procedure produced a measurement of the quantum yield of gold that compared well with that measured by others [3,4]. In addition these studies showed that *Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR contract/grant number AF-AFOSR-838-65 and 68-1450. Based on a thesis submitted by Kenneth A. Kress to Montana State University in partial fulfillment of the requirements of the Ph.D. degree. **Supported in part by the National Aeronautics and Space Administration Traineeship. 417–156 O - 71 - 15 209 sodium salicylate had stable fluorescent properties in ultra high vacuum for several weeks if it was main- tained at room temperature. This latter observation is in agreement with other observations [5]. The above technique does not depend on the optical transmission of the LiF window used on the experimental vacuum chamber. Since the window transmission was found to change with time, this technique gave improved reproducibility of the measured quantum yield. 3. Optical Studies The results of the optical studies that are used in the photoemission analysis are reported here. Complete discussion of the optical studies are reported elsewhere |3.6]. The reflectance of Mo films at near normal in- cidence is presented in figure 1. The reflectance of Mo falls rapidly as the photon energy increases in the near infrared spectral region indicating the onset of nu- merous optical excitations at small energies. The reflectance obtained in this study is smaller in mag- nitude for 2 eV is hu is 6 eV than that reported by earli- er investigators but is consistent in structure [7]. The magnitude is consistent, however, with the reported reflectance of Mo’s 5d counterpart tungsten [7]. The optical constants were obtained by Kramers- Kronig analysis. Detailed discussion of the analysis is given elsewhere [3,6]. The reflectance was measured between 0.5 eV s h v s 11.0 eV under ultra high vacuum conditions. The spectral range of the reflectance used in the analysis was extended by several techniques. The reflectance for hy - 0.5 eV was obtained from the Hagen-Ruben formula. It was ex- tended to 14 eV by measurements on polished samples and to 23 eV by data obtained from the literature [7]. The magnitude of the latter data was adjusted to fit the 5.O |.O \ | \ -— 6. MO oëH \ 2/5 \ — 4.O \ \ -—ez O.6 H \ \ –3.O (l) \ \ £ \ \ CVI C * (u) 3 O.4|+ \ — R \ 2^ — 2.O do N / o \ Cº. N present measurement at 14 eV and is represented by the dotted line in figure 1. In the Kramers-Kronig analy- sis the reflectance was extrapolated by an inverse power function for hu - 23.0 eV. The imaginary part of the dielectric constant eg of Mo is shown in figure 1. Shoulders are observed in e2 centered at 1.8 eV, 3.8 eV, and 7.4 eV and are con- sistent with structure observed in the photoemission data. The optical functions (h v)"e2(hu) and hun(hy) which are useful in studying the photoemission data are shown in figure 2. A peak in the energy loss function, Im(1/e), is observed at 10.8 eV and a relative minimum in the absorption coefficient, Oſhv), is observed at 11.3 eV. 4. Photoemission Medsurements The quantum yield of Mo is shown in figure 3 along with the plot of the square root of the quantum yield used to determine the value of 4.3 eV for the photoelec- tric work function d [2.8]. The value obtained for the work function agrees with those reported in the litera- ture [9]. This value for d, was reproducible from all films deposited at sl × 10−8 torr with a base pressure of 6 × 10−1" torr achieved a few minutes after the deposition. The work function of films deposited at higher pressure varied from 4.4 to 4.7 eV. For films prepared at the lowest pressure, 7 × 10−9 torr, the value of q increased a few tenths of an eV in eight hours with little change in the EDC’s except for a decrease in width. A significant drop in the yield is centered at ap- proximately 11.0 eV. This drop is interpreted as an opti- cal effect. The maximum value of the quantum yield is less than 2% indicating that few of the excited electrons escape the metal even for the highest photon energies. A representative set of the energy distribution curves (EDC’s) is shown in figure 4. The EDC’s are plotted as O.2 H. `s = 1.0 N tº ſº e g is | | | | O o: 5 |O 15 #00 hv (eV) FIGURE 1. The reflectance and imaginary part of the dielectric constant of molybdenum. The dotted portion represents the shape of the high energy reflectance reported in the literature (ref. [6]). 2O O * º, > > J2 Sp * |OO 2 C. jºu * CN * *> *…* 5. ** – I-1–1–1–1–1–1–1– 2 4 6 8 |O || 2 || 4 ||6 |8 20 22 hv (eV) FIGURE 2: The optical functions hun(hv) and (hu)*eg(hv) of molybdenum. 210 Te |- O * 2 a 162 Mo ~5 Q1) º- -ch *— |- O (ſ) .C. º- C N. Ç O X- h– * C) (ſ) Q) — + 3, 10°H 5 º- > TE – 5 do .*- >– º- + º E -j mº S. *E C; l | ë 10-4 4. OeV 4.5eV 5,OeV | | | 6.O 8,O |O.O | 2.O hv (eV) FIGURE 3. Quantum yield of molybdenum. a function of the kinetic energy, E – d. and normalized to the quantum yield. The energy E is referred to the Fermi energy. The data from all other films studied were essentially identical with those shown in figure 4. Even the EDC's from film deposited at higher pres- sures differed only by a small attenuation of the popula- tion of high kinetic energy electrons and a small in- crease in the work function. Inspection of the EDC’s in figure 4 reveals no structure with fixed kinetic energy and all structures move to higher kinetic energy as the photon energy is increased. As shown in figure 5, the structures in the EDC's have fixed positions when plotted as functions of (E – hy). Three structures are observed over the entire spec- tral range studied. The energies of the structures obey the equal increment rule, Estructure = A(h v) and are due to initial state structure at —0.5, – 1.6, and —3.9 eV. The peak labeled (1) has a marked change in relative amplitude as the photon energy is increased from 10.0 to 11.0 eV. The analysis of this property is complicated by the existence of a peak in the energy loss function (10.8 eV) and a relative minimum in the absorption coefficient (11.3 eV) in this same spectral region. A definite shoulder appears at E — hu = —5.0 eV on the low kinetic energy edge of the EDC’s for hu > 10.0 eV (see figs. 4 and 5). This shoulder persists and moves toward higher energy with increasing photon energy and is taken to indicate the bottom of the 4d bands of Mo. 5. Optical Density of States Andlysis The characteristics of the structure in the EDC’s which are described in the last section indicate that the data are consistent with the nondirect transition model 8x IO-3– 2T->s 2^ ~ MO -> º © & \ 4. / %/ >~l --~~TWTTS \ . . . g 6x1O-3 / . * . . . *\\ * . 2 — / Z * e! & \\ • ||,8 C. / \ N & G º N II.O © C ... // \ 'N' & S / Z 3 / .% \ IOO \ © # 4x1O-3H- /, // 9 O \ \ Q e-se /.../ \ \ Q E /// 8.O \ \ * Q SA // / \ \ * - -- // 7O \ \ Q Sº 2x|O 31– // W \ 2 / 6.O \ \ Q - d % \ \ //~ hv = 5.0 eV \ \ | | | | | | \ . \ O | 2 3 4 5 6 7 E - Ø (eV) FIGURE 4. Normalized energy distribution curves of molybdenum versus kinetic energy. C gy 211 3 2 | MO 72 S-s hv = 110 eV /~ 2/ \ / N__/ \ º | \ 'E loº /\ > / 2/ | S | 2^ 5 | 99 / \ 5 | / \ ~ / / * / 8.0) º / / s / / U / / 7. O 2 / / / / 6.O /N | / / \ | / / \ | / / \ | / / \ / | / / | / | | | | | | | -6 –5 - 4 –3 -2 -l O E - hv (eV) FIGURE 5. Arbitrarily normalized energy distribution curves of molybdenum versus E-hu. with the possible exception of the structure labeled (1) in the 10.0 to 11.0 eV spectral range. The optical excita- tions in the nondirect model are proportional to the product of the density of initial and final states. The energy distribution of emitted electrons in the limit that the absorption depth is much greater than the electron- electron scattering length L(E) is given by [10-12] BN)" (E- hv)N;"(E) [1+S,(E, hy)] hun (h v) N(E – d. , hy) = (1) where N;"|E)=T(E),(E)N)*(E). (2) The real part of the index of refraction, the escape func- tion and the scattering function is given by n(hu), T(E), and Si(E,hv) respectively. The optical density of initial and final states are respectively Ni"P'(E — hu) and N!!!" (E) [11,12]. Any (E) or (E - hv) functional depen- dence in the matrix elements is indistinguishable from the optical density of final and initial states respective- ly. Therefore such effects will be contained in the ODS and any other matrix elements effects are taken to be constant in eq (1) [12]. For this model the imaginary part of the dielectric constant is given by [10] hu (hy)*es (hu) = A | Nº E-hp) Nº E) dB. (3) () - Given the function Nºp' and e2(hu) one can numerically determine NPP. Writing eq (3) in the finite difference approximation and rearranging terms eq (3) becomes, Ny"[(m- 1)AE] = ( (mAE) *es (mAE) (4) AN/AE }}] – 1 -AAE S \ºtº-m-DAEW"[(L-DAE) - . L= 1 where Nº!"(0)=Nº!"(0)=N-DS at the Fermi energy. The energy scale is divided into equal increments and the increments are counted with the index m. Because N eg was not measured for small spectral values, eq (4) was not used to obtain the value of A. The constant A was adjusted to give six empty states per atom for 0 < E -< 6.0 eV. The ODS calculation proceeded in two parts [3]. First, the effect of scattered electrons, Si, was neglected in eq (1). Equation (1) was used to determine an ap- proximate initial ODS which was then divided into the set of EDC’s to obtain a set of approximate functions for Nſeſſ(E). Since the latter set was equivalent (within 5%) for all the EDC's, the initial ODS was assumed to be a reasonable approximation and provides additional evidence that the nondirect model is consistent with the data. The approximate initial density of states and measured eg were used in eq (4) to obtain an approxi- mate final density of states. With the above functions available, the second part of the analysis was to repeat the calculations including the effects of scattering, represented by the function S;(E,hv) in eq (1). Si(E,hv) was computed by the once scattered model of Berglund and Spicer [10] in the limit when the inelastic scatter- ing length is much smaller than the mean optical ab- sorption depth and when it is much smaller than the elastic scattering length. The scattering corrections were insensitive to the details of the ODS. The results of the scattering correction on a typical EDC are shown in figure 6. The ODS obtained by the above method is shown in figure 7. Because of the spectral limit imposed by the LiF window, the states for |E| < 5.0 eV cannot be stu- died in detail. These less reliable regions are indicated by dots in figure 7. Since the numerical inversion for NPP was done in 0.2 eV increments, the resolution 212 MO 6 x 10-3 || hw = | | eV ~~ / N / * / # / 2- g / \-2 § 4, 10°- / £ | S | § | Q1) *-*. / F. / -8- / | / — medsured Lil ... • - S- 2 x 10°H /. to —- corrected 2 º - • . . scattered / * / / / | | | | . | O | 2 3 4. 5 6 7 E - 4 (eV) FIGURE 6. Measured, corrected and scattered energy distribution curves of photoemitted electrons from molybdenum at hy= | 1.0 eV. could not be better than this. If, in addition, the density of initial states used in the inversion for the density of final states already contains significant broadening, the details of the density of final states will be distorted. These effects combine to lessen the significance of the fine structure displayed in the density of final states seen in figure 7. The dashed, average curve for NPP is possibly a better estimate of the density of final states even though it will not permit as accurate a recalcula- tion of the imaginary part of the dielectric constant. 6. Discussion The ODS is compared to a band calculation density of states (DS) in figure 8. The DS was obtained from Mattheiss’ calculations for W with potential “one” [13]. Since Mo and W have the same crystal structure, bec, and similar atomic electron configurations the band structures are expected to be essentially the same within an energy scale factor. The energy axis for the DS was rescaled by using Mattheiss' estimate of the relative d-band widths of Mo and W1. There are three peaks below the Fermi energy in the experimental ODS (1, 2, and 3) and in the calculated DS (a, b, and c) although the energy position and amplitude of the peaks do not agree in detail. Above the Fermi energy there is one major peak in the ODS with several minor peaks at smaller energies. The calculated DS also has one large peak and two smaller ones above the Fermi energy. As with the structures observed below the Fermi energy the energy positions and amplitudes do not agree with those in the ODS. The most serious dis- crepancy noted between the ODS and the DS is in the region near the Fermi energy ( – 1 eV S E is 0 eV). The calculated DS has a deep and wide (~ 1 eV) valley at the Fermi energy with a small number of electron states per atom. In contrast, the ODS has a peak cen- tered at E = – 0.5 eV and a larger number of states per atom at the Fermi energy. The expected essential difference between the band structure for Mo and paramagnetic Cr is an energy scale factor since they have the same atomic structure and crystal structure. In figure 9 the experimental ODS 2, OH- Mo º l E O § 16- > I.O. S 'C d2 Jº (ſ) Q O O.O | | –6 - 4 -2 E (eV) FIGURE 7. Optical density of states of molybdenum. — MO ODS 2. OH- b (left scole) —— Mo DS estimoted | c from Motthesis DS 3.0 | for WI (right scale) |* ź ź ! | 5 5 s !.O H. s (ſ) ſº (ſ) S 5 § $ Q} Top -6 E (eV) FIGURE 8. The optical density of states of molybdenum (dotted line) is compared with the density of states estimated from Matt- heiss’ tungsten (W) band structure calculation (ref [13]). 213 O6 – O.4 H : –8 -6 –4 –2 O E (eV) FIGURE 9. Comparison of the optical density of states of molybdenum (present study) and chromium obtained by Eastman (ref [14]). of Mo is compared to Eastman’s ODS for Cr [14] which is similar to that obtained by Lapeyre and Kress [15]. There are three structure points in both Mo and Cr, but the relative amplitudes are different. The energy values of the structures are compared in table 1. Mattheiss' estimate of the d-band width of Cr gives a scale factor of 0.75 with respect to Mo whereas that obtained from the data is approximately 0.6. The comparison of the ODS of Mo with the estimated DS from W and the ODS of Cr indicates that the number of structure points both above and below the Fermi energy are in reasonable agreement. The com- parison also indicates that the details such as the ener- gy position and amplitude are not in such good agree- ment. It may be that the ODS is simply not a good replica of the unperturbed ground state density of states due to systematic variation of the transition probability matrix elements with the initial state energy E – hu [14]. Variations of the matrix elements with the initial energy cannot be distinguished from the initial density of states effects in the photoemission data. Some of the difficulties in the above comparison may be due to an overly simplified model used to interpret the photoemission data. There is some theoretical evidence that the detailed predictions of the nondirect and direct transition model for processes in copper are not easily distinguished over a limited spectral range [16]. In addition to these recent calculations there is possible experimental evidence for direct transition character in the Mo photoemission data. In particular, note that the high energy peak of photoemission the EDC's in figure 5 is attenuated suddenly between hu = 10.0 eV and 11.0 eV. The sudden appearance and disappearance of structure is consistent with the direct transition model and is not consistent with the non- direct model. The above discussion indicating the possibility of direct transitions must be considered tentative as there are other possible explanations for the deterioration of the high kinetic energy edge of the EDC’s between 10.0 eV and 11.0 eV. For example, since the energy loss function has a peak at 10.8 eV, the photoexcited elec- trons with E - 10.8 eV could interact inelastically and lose 10.8 eV. With such a large energy loss, these elec- trons could not escape the metal and would be missing from the EDC’s. This phenomenon has been observed in photoemission studies on Zn [17]. The relative minimum in the quantum yield at ap- proximately 11.0 eV is interpreted as an optical con- stant effect. This interpretation is supported by the ob- servation that the amplitude of the EDC’s and con- sequently the yield, is dependent on the factor hun(hu) [cf. eq (1)] which has a relative minimum at 11.3 eV. Furthermore, the effect of the minimum in the yield is compensated by the minimum in the optical function hun(hu) in just the manner necessary to produce a self- consistent ODS without appealing to hu dependent matrix elements. A narrow peak just below the Fermi energy in the DS has been proposed to explain the quenched isotopic mass dependence in Mo’s superconducting transition temperature [18]. Although the optical studies of this particular work were not extended below 0.5 eV, the photoemission data does probe this region. Within the resolution of the photoemission experiment, ~0.1 eV, no large amplitude, narrow structure was observed near the Fermi energy. This observation is in agree- ment with a more recent treatment of the isotopic mass effect which accounts for the quenched isotope effect without any anomalous condition being imposed on the density of states [19]. TABLE 1. Energies of the structure in the optical density of states for Cr and Mo Structure Mo Cr a Crb 1................ –0.5 eV — 0.2 eV — 0.4 eV (shoulder) (shoulder) 2................ — 1.6 eV — 1.1 eV — 1.2 eV 3................ — 3.9 eV — 2.2 eV – 2.3 eV * G. J. Lapeyre and K. A. Kress, Phys. Rev. 166, 589 (1968). " D. E. Eastman, J. Appl. Phys. 40, 1371 (1968). 214 7. Summary Photoemission measurements were used to study the electronic structure of Mo. The data show the filled d- band to be approximately 5.0 eV wide with three peaks at E – E = –0.5, – 1.6, and —3.9 eV. The data show no strong electronic final-state structure for E – E - 5.0 eV. The above observations are independent of the nondirect model. The nondirect model was used to obtain an ODS for Mo. The model was used in conjunction with eº to infer an ODS above the Fermi energy. The ODS was com- pared to a band calculation and to measurements on Cr. The comparisons showed poor agreement except for the correlation in the number of peaks. 8. Acknowledgments The contributions of Mr. C. Badgley and Mr. F. Blan- kenburg to the mechanical design, electrical design, and construction of part of the apparatus used in this investigation was certainly appreciated. Discussions with Dr. A. J. M. Johnson are acknowledged. 9. References [1] Kress, K. A., and Lapeyre, G. J., Rev. Sci. Instr. 40, 74 (1969). [2] Berglund, C. N., and Spicer, W. E., Rev. Sci. Instr. 35, 1665 (1964). [3] Kress, K. A., Ph. D. Dissertation, Montana State University (1969). [4] Krolikowski, W. F., Ph. D. Dissertation, Stanford University (1967). [5] Allison, R., Burnes, J., and Tuzzolino, A. J., J. Opt. Soc. Am. 54, 1381 (1964). [6] Kress, K. A., and Lapeyre, G. J., to be published. [7] Juenker, D. W., Leblanc, L. J., and Martin, C. R. J. Opt. Soc. Am. 58, 164 (1968). [8] Fowler, R. H., Phys. Rev. 38, 45 (1931). [9] Vance, D. W., Phys. Rev. 164, 372 (1967). [10] Berglund, C. N., and Spicer, W. E., Phys. Rev. 136, Al 030 (1964). [ll] Spicer, W. E., Phys. Rev. 154, 385 (1967). [12] Schechtman, B. H., Ph. D. Dissertation, Stanford University (1968). [13] Mattheiss, L. F., Phys. Rev. 40, 1371 (1968). [14] Eastman, D.C., J. Appl. Phys. 40, 1371 (1968). [15] Lapeyre, G. J., and Kress, K. A., Phys. Rev. 166, 589 (1968). [16] Smith, N., and Spicer, W. E., to be published. [17] Mosteller, L. P., Huen, T., and Wooten, F., to be published. [18] Garland, J. W., Phys. Rev. 129, 111 (1963). [19] McMillan, W. L., Phys. Rev. 167,331 (1968). 215 Ultraviolet and X-Ray Photoemission from Europium and Barium G. Brodén, S. B. M. Hagström, P. O. Heden, and C. Norris Department of Physics, Chalmers University of Technology, Göteborg, Sweden Europium and barium are predicted to have very similar outer electronic structures with the excep- tion that europium has a partially filled 4f shell. Measurements are reported on photoemission from thin films excited with both vacuum ultraviolet and soft x-ray radiation. The results obtained using the low energy excitation indicate the similarity of the materials. Both show structure close to the leading edge in agreement with band structure calculations which indicate an increase in the density of states im- mediately below the Fermi level. Only a very small feature is observed with europium films which can be associated with the 4f electrons. On the other hand using soft x-ray excitation a large peak cor- responding to 4f states lying 2.5 eV below the Fermi level is observed. The difference in the magnitudes is attributed to the size of the matrix elements involved. Key words: Barium (Ba): 3-tungsten compounds; effects of oxidation; electronic density of states: europium (Eu); lanthanides; matrix elements; photoemission; rare-earth metals; UV and x-ray photoemission: x-ray photoemission. 1. Introduction Recent band structure calculations [1-3] of the isoelectronic metals Eu (4f76s”) and Ba (6s”) have shown that their respective conduction bands are closely re- lated. Both are characterized by a 6sp band hybridized with a 5d band which extends slightly below the Fermi surface. Eu differs from Ba in that it possesses, in com- mon with other rare-earth metals, a partially occupied 4f state. The 4f electrons are shielded by the closed 5s and 5p shells and behave in many ways as core states. This is illustrated by the chemical similarity of the rare earths. The 4fs are, however, important in connection with the complex magnetic structures of the lanthanide metals. Although band structure calculations [2.4] have been successful in correlating these structures with details of the Fermi surface, they are not able to accurately locate the energy position of the 4f state, due to the sensitivity of this energy to the exact form of the exchange. We have measured photoelectron spectra in- duced by both UV light and by soft x rays in order to in- vestigate the band structure of Eu and Ba, and in par- ticular to determine the position of the 4f level in Eu. This work is part of a wider study of the lanthanides now in progress. X-ray photoemission (XPS) and UV photoemission (UPS) are two important techniques currently em- ployed in the study of the electron density of states of solids. Their relative merits are a consequence of the very different photon energies involved, rather than competing, the two methods complement each other. With soft x rays one can probe deeper below the Fermi level and can also study core levels. This is useful in that it allows the purity of the sample to be monitored, and from a measurement of the chemical shift of the core levels, information can be gained concerning the nature of the valence states [5]. The more energetic electrons produced in ESCA have, moreover, a longer scattering length (typically 100 Å) compared to the low energy UV induced photoelectrons (typically 20 Å). Greater surface purity is thus required with UPS. On the other hand, although UPS is limited to investigating states near the Fermi level, it is at present capable of considerably better resolution. 2. Experimental Both sets of measurements were made using films evaporated from tungsten filaments. The metals were obtained with a purity of 99.5 percent (Ba) and 99.7 per- cent (Eu). Both Eu and Ba are very reactive and it was inevitable that some oxidation occurred during mount- ing. The working pressures were 7 × 10-11 torr (UPS) and 5 × 10−7 torr (XPS). This was found sufficient to 217 al 2 | | | | —l | ſ l –7 -6 – 5 – 4 - 3 -2 –1 0 Energy of initial state (eV) FIGURE 1. Electron energy distribution curves of a europium film for different photon energies. allow measurements for one to two hours without ap- preciable distortion of the spectra. The UV photoemission spectra were obtained with light from an H2 discharge lamp, dispersed by a McPherson monochromator. The monochromator was coupled to the measuring chamber via a Lif window. The photoelectrons were collected with a silver coated collector by applying a retarding field between the sam- ple and the collector and using the A.C. modulation technique [6]. With this method an electron energy resolution of approximately 0.3 eV could be achieved. For the ESCA measurements the exciting radiation was obtained from an x-ray source employing an alu- minum anode, separated from the sample by a 1.5p. thick aluminum window. The photoemitted electrons were analyzed with a spherical electrostatic analyzer with a radius of curvature of 36 cm. They were recorded with standard counting techniques. The resolution obtained was of the order of 1.5 eV of which part (0.8 eV) was due to the natural line width of the in- cident x-radiation (Al Ka) and the rest came from in- strumental broadening. 3. Results Figure 1 shows a set of electron distribution curves (EDC’s) obtained from a freshly evaporated film of Eu for different incident UV photon energies. The results are referred to the energy of the initial states by plotting them against E — ha) + (b, where E is the measured energy, ha) the photon energy, and q the work function of the collector. The zero on this scale corresponds to the Fermi level. A number of peaks, some of them rather weak are observed and for reference are labelled A to F. For all photon energies the curves are charac- terized by a relatively high number of emitted electrons near the leading edge. At the higher photon energies hao = 10.2 eV (3)x.15 Eu film exposed to oxygen : (i) New film (2) 1 min at 3x10' torr 3) 1 min at 1x10" torr (A) 1 min at 3x 10"tort –8 -7 –6 –5 –4 –3 -2 —l 0 Energy of initial state (eV) FIGURE 2, Influence on the EDC curve for europium of exposure to Oxygen. two large peaks E and F appear. F remains at a fixed kinetic energy for increasing photon energy and probably corresponds to inelastically scattered elec- trons. Electron-electron scattering is an important process for both Eu and Ba as, according to the band structure, they are electronically similar to the transi- tion metals with a high density of states near the Fermi level. To determine whether or not any features of the photoemission spectra were associated with the presence of oxygen, a freshly prepared film was ex- posed to oxygen at various pressures for a given length of time. The EDC’s obtained for ha)= 10.2 eV are shown in figure 2. Apart from the increased scattering peak which is characteristic of a contaminated surface, there is a large increase in peak E suggesting that it arises from the oxidized state. This assignment is confirmed by recent measurements on EuO [7] in which a strong peak attributed to p-states was observed at the same position (–4.6 eV). Very similar results to those of Eu have been ob- tained for Ba. EDC’s corresponding to ho-7.7 eV are shown in figure 3 for both metals. The evident similari- ty of the curves reflects the closeness of the valence band structure for the two metals. With the exception heo - 7.7 eV A Eu g D C B –6 – 5 – 4 – 3 -2 -1 0 Energy of initial state (eV) FIGURE 3. Comparison of the EDC curves for barium and europium. 218 Intensity Intensity 15 35 || 0 1s EU Af 30 H. (a) © Ç | acacaºrsº *Nº o to Š * o 25 H. 20 | 15 H l 1–Lºv I | i | | I | 6 –535 –525 –20 -10 O Energy below Fermi level (eV) FIGURE 4. Photo-electron spectrum obtained with Al Ko excitation of the Eu 4f band and 0 1s level immediately after evaporation (a) and after oxidation (b). of the region between – 1.5 and —3.2 eV the features observed in one curve can be recognized in the other. Figure 4 shows photoelectron spectra obtained from a Eu film using soft x-ray excitation. As in the previous figures the energy scale corresponds to the binding energy with the Fermi level located at zero. The upper Eu curve is dominated by a peak at – 2.5 + 0.5 eV which we believe is due to the 4f state. It is likely that the feature at — 5 to — 10 eV is associated with scatter- ing mechanisms as a similar feature was observed ad- jacent to the photoelectron peak corresponding to the 3ds/2 level. The peak near – 20 eV is due to the Eu 5p level. In order to check the contamination of the sample surface the oxygen ls line was monitored and is shown by the side of the corresponding Eu signal. The lower set of results were obtained after exposing the film briefly to the atmosphere. The intensity of the 4f line decreased and structure at lower energies appeared. A strong oxygen signal was also observed. Comparing the two sets of curves it is evident that oxygen had a little influence on the films studied immediately after evaporation, and we therefore feel that we are justified in regarding them as typical of pure Eu. XPS measure- ments of the outer electrons for bulk Eu samples have recently been reported [8]. The results which cor- respond to samples mounted in air are similar to the lower Eu curve shown in figure 4. As a further check that the peak in figure 4 cor- responds to the 4f state it is compared in figure 5 with the corresponding result for a Ba film. Clearly nothing as strong as the –2.5 eV peak in Eu is seen for Ba. The small peak at –6 eV corresponds to the oxygen 2p level. The one at — 16 eV is due to the barium 5p level. 4. Discussion From figure 1 it is seen that the structure in the Eu EDC’s remains at constant energy (on the reduced Intensity 15 H. Eu -: 13 H 11 H 9 H 42. 42. 6 H BC1 amº “ſ -º- 2 -: –20 -10 0 Energy below Fermi level (eV) FIGURE 5. Photo electron spectrum close to the Fermi level of euro- pium and barium obtained with AlKo excitation. The intensities are normalized relative to the 4d core levels. scale) for increasing photon energy. Similar behavior was found for the Ba films. This indicates that the structure is associated with features in the initial densi- ty of states. Furthermore either the initial states have low dispersion in k-space or the transitions are non- direct (i.e., transitions in which the electron momentum vector k is not conserved). Comparison with the calculated band structures of Eu and Ba suggests that the hump in the EDC’s extend- ing to — 1.5 eV is associated with the filled part of the 5d band. The broad feature between — 1.5 and –3.5 eV in the Ba EDC (fig. 3) would appear to be due to the 6sp conduction band. It is difficult to explain the feature which occurs in both EDC’s in figure 3 near-4 eV. Ac- cording to the calculations no states exist in this region. One possibility is that the peaks are associated with in- elastically scattered electrons. In this connection we note that in a recent work [9] peaks observed in alkali metals have been attributed to scattering via the crea- tion of surface plasmons. From the differences in the two EDC’s shown in figure 3 we believe that the 4f level is associated with the structure lying between — 1.5 and –3.5 eV; that is with either of the features B or C. This conclusion is supported by the occurrence at –2.5 eV of the strong x-ray induced photoemission peak in figures 4 and 5. The weakness of the 4f structure in the EDC’s of the UPS measurements, compared to the XPS measure- ments, is presumably due to the magnitude of the matrix elements involved. For 7 eV → ho < 1.1 eV the 4f level is coupled with empty 6sp states, transitions between which are not allowed by electric dipole selec- tion rules. Transitions from the occupied 5d and 6sp bands will be allowed however. This explains the similarity of the Eu and Ba EDC’s. On the other hand using soft x-ray excitation all initial states will be 219 coupled to plane wave states, approximately 1500 eV above the Fermi level. Thus in the case where there are important matrix element effects involved, soft x-ray induced photoemission spectra will reproduce more ex- actly the density of states than will measurements em- ploying UV light. Electric dipole selection rules will not, however, ex- plain the optical properties of Ba and Eu [10]. The reflection curves for Ba and Eu were found to be very similar, suggesting that the 4f electrons are not excited for ha) < 5.0 eV. The band structure calculations and the present work suggest, however, that there is a high density of d states at the Fermi level. Transitions to these states from the 4f level are allowed and would be expected to give rise to an absorption edge in Eu but not in Ba. The nonappearance of this edge is possibly due to the core-like nature of the 4f states. The matrix element between the 4f and 5d states would in con- sequence be small. The conclusion is also in agreement with recent calculations on free atoms [11]. 5. References [1] Freeman, A. J., and Dimmock, J. O., Bull. Am. Phys. Soc. 11, 216 (1966). [2] Andersen, O. K., and Loucks, T. L., Phys. Rev. 167, 551 (1968). [3] Johansen, G., Solid State Communications 7, 731 (1969). [4] Keeton, S. C., and Loucks, T. L., Phys. Rev. 168,672 (1968). [5] Fadley, C. S., Hagström, S., Klein, M., and Shirley, D. A., J. Chem. Phys. 48, 3779 (1968). [6] Spicer, W. E., and Berglund, C. N., Rev. Sci. Instr. 35, 1664 (1964). [7] Eastman, D. E., Holtzberg, F., and Methfessel, S., Phys. Rev. Letters 23, 226 (1969). [8] Nilsson, Ö., Nordberg, C. H., Bergmark, J. E., Fahlman, A., Nordling, C., and Siegbahn, K., Helvetica Physica Acta 41, 1064 (1968). [9] Smith, N. V., and Spicer, W. E., Phys. Rev. Letters 23, 769 (1969); Phys. Rev. in press. [10] Müller, W. E., Phys. Kondens. Materie 6, 243 (1967). [ll] Fano, U., Cooper, J. W., Rev. Mod. Phys. 40, 441 (1968). 220 Discussion on “Ultraviolet and X-Ray Photoemission from Europium and Barium" by G. Brodén, S. B. M. Hagström, P. O. Heden, and C. Norris (Chalmers University of Technology) S. J. Cho (National Res. Council): I have one comment about this interesting experiment on Eu. A few years ago Blodgett and Spicer (refer to my paper in this con- ference) had also obtained in gadolinium with uv photoemission studies a small peak at about 2.8 eV below the Fermi level, which is about the same position as you have obtained. I strongly feel right now that this small peak in Gd is the fBand position. Do you have any idea about this f band width in Eu? Is your f band width due to the Fj multiplets? D. E. Eastman (IBM): I have looked at gadolinium again since the earlier work several years ago by Dr. Spicer and co-workers [1] and I do not see any per- ceptible 4f-state structure down to seven volts below the Fermi level that is bigger than 1% in amplitude. In the cases of europium, Dr. Hagstrom has reported on x- ray photoemission measurements. S. J. Cho (National Res. Council): There is a very small peak over there in the early UPS works men- tioned above. I do not have any detailed information about your recent experimental works. However, I do expect that the 4f band position in Gd with respect to the Fermi level should be almost the same as those in Eu. D. E. Eastman (IBM): In uv photoemission, no 4f. state peak is observed in gadolinium within 6 or 7 volts of the Fermi level. S. J. Cho (National Res. Council): The reason for the large f band width of about 2.0 eV observed in Eu and Yb is still not clear. According to your XPS work, the f band positions for both Eu and Yb are practically the same. My own feeling is that the 4f band position in Yb could be somewhat (0.5 eV) lower than that in Eu. A. R. Mackintosh (Lab. for Electrophysics, Tech. Univ.): We have recently performed some band calcu- lations on ytterbium which may throw some light upon this question [2]. Although the position of the 4f bands is very uncertain in these calculations, due to their ex- treme sensitivity to the approximation used for exchange, we find that they are situated near the bot- tom of the conduction band, about 4 eV below the Fermi level, if Slater exchange is used. This is in reasonable agreement with the x-ray photoemission results. Spin-orbit coupling splits the bands into two sets, corresponding to j = 7/2 and 5/2, separated by about 1.5 eV. The individual bands have a width of less than 0.05 eV, because the centrifugal barrier makes the f resonance extremely narrow. Perhaps the spin-orbit splitting could be seen in a high resolution experiment. S. J. Cho (National Res. Council): I have made exten- sive studies on the fiband problem by using a parame- terized Slater pºlº exchange potential for the Eu-chal- cogenides, which are magnetic semiconductors. Not only the 4f band positions, but also the total 4f band widths are fairly sensitive to the exchange potential used. It would be a difficult problem theoretically to find the proper 4f band positions for the case of metals. In spite of such uncertainty they can explain fairly well the Fermi surface topology, because the localized 4f bands are located a few electron volts below the Fermi energy for Eu, Ga., and Yb metals, and do not influence the Fermi surfaces much. For other rare earth metals the 4f bands can cross the Fermi energy. However, it could be difficult to measure the 4f Fermi surfaces due to the heavy effective masses. In the case of the semiconductors we can adjust a reduced exchange parameter to produce the right energy gap. It turned out that the 4f (?) bands become the highest occupied bands with correct energy gaps for all the Eu-chalcoge- nides with a fixed value (3/4) of the reduced exchange parameter. In these studies the f(?) band positions are reliable, but there is no reliable experimental data available to justify the maximum f° band width of about 0.5 eV obtained from my work. W. E. Spicer (Stanford Univ.): In general, I think in looking at photoemission from something like an f state one has to take into account not only the ground state splitting but (I think Dr. Cho was thinking of this) also the various spin states that can be left after the excita- tion. I believe that in the d-band of NiO these effects can give a width of about 2 eV. I don’t know what it can give here, but I suspect it is large. I think it does have to be taken into account. J. T. Waber (Northwestern Univ.); I would like to make two observations about papers which have just 221 been discussed. The comparative plot of the atom eigenvalues for the 3d and 4d transitions series shows in figure 1 that the 4d bands should fall more rapidly with atomic numbers than the 3d bands do. This dif- ference is borne out by comparing the self-consistent band calculations of Snow on silver [3], and of Snow and Waber on copper [4]. Also, I think the relative position of the d-bands accounts for the difference in color of these two metals. In figure 2, I have compared the 3 density of states curves which have been obtained in conjunction with my colleagues at Los Alamos [5]. Titanium is compared with zirconium, vanadium with niobium and chromium with molybdenum. All six graphs are for the bec phase. One striking feature is that in contrast to the well known two peak structure of the density of states curves for the 3d transition ele- ments, the equivalent curves for the 4d transition series show three peaks. This result is consistent with East- man's experimental observations on vanadium and O.O O. H. Variation of Eigenvalues - With Group Number O.2}- O.3|- º N 0.4}- ~s *- - T- ~ 5S 5 N *- (Ag) g 0.5F As - (Cu) sesſ 5 #3 Sº, O.6- * Orº O.7 `s 3d - Hartree Fock Slater S (Cu) O.8H with full exchange n+! º for configurations (s' d") O.9}- * (Ag) |.O t i I I 4 6 7 8 9 |O | | |2 | Group Number FIGURE 1. Comparison of atomic energy levels for the 3d and 4d transition series. COMPARISON OF THE DENSITY – OF — STATES CURVES FOR B.C.C. PHASE OF 3d AND 4C TRANSITION METALS 2.O I I I I i I I I ! Lal TITANIUM (d’s) * |.2}- *- O.8– *- O.4- * O } } | H t I | } / ~ 1.6l VANADIUM (d’s) - 8– |S - - 3 I.2. sº- > X- Q1) - N 0.8– º (ſ) 92 |- - C of O.4- - 1T ſ - *=s* O | } | ~t I I } } | 2 1.61 CHROMIUM (d’s) * |.2}- * O.8|- - O.4– – O | i I f | l -2.0 -1.8 -1,6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 –0.2 O Energy (Rydbergs) I I I I I I i ZIRCONIUM | | | I I i I i i I I I I N|OBIUM * I ! l I I I T- I I I I -- l MOLYBDENUM L -1.8 -1.6 -1.4 TaTºo Los Toe T.O.T. Energy (Rydbergs) FIGURE 2, Density of states curves for the bec phases of 3d" and 4 dº transition metals. O.2 222 niobium. Since the HCP forms of titanium and zirconi- um were used experimentally in place of the bec form, which we have studied, agreement with the number of peaks in the N(E) curves is not anticipated theoreti- cally. The final point is that these six N(E) curves are not very similar. This raises question about the validity of the rigid band model for the transition metals. I would like to make a final point. It relates to the very in- teresting paper by Collings and Ho. The calculated de- pendence of N(EP) on alloy composition shown in figure 3 was obtained from a very simple model of alloying which a graduate student, David Koskimaki, is cur- rently working on. We hope to report on this model in the very near fu- ture. By oversight the curve for N(EP) for Ti-Mo alloys was not supplied to Dr. Collings. This graph is similar to the one for Ti-V-Cr alloys. The shape is similar but the discrepancy remains between the numerical experi- mental and theoretical values. [1] Optical Properties and Electronic Structure of Metals and Al- loys, F. Abeles, Editor (North-Holland Publishing Co., Amster- dam, 1965) p. 246. Stotes | O Calculated Density of States 1.6- at the Fermi Level EE for - Titanium – Molybdenum Alloys N (E-) F Ti (dº sº) eV (atom) O d 0.6F Ti(d’s) O.4- | O.2- - Q; —545–545–555—aga |.OO Ti Concentration of Molybdenum MO FIGURE 3. Calculated variation of the state density at the Fermi level for Ti-Mo alloys (Waber and Koskimaki, unpublished). [2] Johansen, G., and Mackintosh, A. R., to be published in Solid State Communications. [3] Snow, E. C., “Self-Consistent APW Bands of Silver,” Phys. Rev. [4] Snow, E. C., and Waber, J. T., Phys. Rev. 157, 570 (1967). [5] Prince, M. Y., and Waber, J. T. (in press). 223 MANY-BODY EFFECTS CHAIRMEN. L. N. Cooper J. W. Cooper What is a Quasi-Particle?" J. R. Schrieffer Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania The concept of a quasi-particle excitation in an interacting many-body system will be discussed from both the physical and the mathematical points of view. The physical origin of mass enhancement, wave function renormalization, interactions between quasi-particles, etc. will be presented. Landau's Fermi liquid theory, including the quasi-particle kinetic equation, will be reviewed. Finally, the domain of validity of the quasi-particle approximation will be discussed. Key words: Density of states; Green's function; mass enhancement; quasi-particle; super- conductors. 1. Introduction Since the early work of Drude and of Sommerfeld [1], it has been clear that an independent-particle pic- ture represents in a qualitatively correct manner the electronic properties of a metal. The electronic specific heat, the transport properties, the magnetic suscepti- bility, etc. are all roughly accounted for by elementary band theory, without recourse to explicit many-body ef- fects. Exceptions to the rule are the plasma modes ob. served in the energy loss of fast charged particles, as well as cooperative phenomena such as superconduc- tivity ferromagnetism, antiferromagnetism, etc. The success of the independent-particle approxima- tion (IPA) is particularly striking in view of the large ratio of interelectronic Coulomb energy to kinetic ener- gy experienced by electrons in metals. A measure of this ratio is the electron density parameter rs defined essentially as the mean spacing between electrons, measured in units of the Bohr radius. For rs • 1, the Coulomb interactions between electrons are weak com- pared to the kinetic energy effects, while if rs = 1 the potential effects dominate the kinetic effects. For sim- ple metals, rs is typically between two and six. Thus, one might expect qualitative changes in the properties ºAn invited paper presented at the 3d Materials Research Sym- posium, Electronic Density of States, November 3–6, 1969, Gaithers- burg, Md. * This work was supported in part by the National Science Founda- tlOn. of metals relative to the IPA since this approximation includes only the average Coulomb interaction between electrons. The “correlation energy” neglected by the IPA is of order 1 to 2 eV per electron and is by no means trivial. The qualitative reason that the IPA works so well is that typical measurements made on metals at normal temperatures (T • Tr - 10 K, where TF is the Fermi temperature) involve only the low lying excited states of the metal. There is good theoretical (and experimen- tal) evidence that these many-body states are well characterized in terms of a set of elementary excita- tions, called quasi-particles, which for the interacting system play the same role as the excited electrons (above the Fermi surface) and the excited holes (below the Fermi surface) in the IPA. As for electrons and holes, these quasi-particles are labelled by a wave vec- tor k and a spin orientation s == 1/2. It is assumed that there is a sharp Fermi surface in the actual system as T -> 0, although its shape may depend on the many- body interactions. By convention, one measures the quasi-particle energies el, relative to their (common) value on the Fermi surface so that elf – 0, where kr is a wave vector on the Fermi surface. At sufficiently low temperature, few quasi-particles are excited and therefore this dilute quasi-particle gas is nearly a “perfect” gas, in the sense that the quasi- particles rarely collide. Furthermore, at low tempera- ture only low energy quasi-particles are excited. Since their intrinsic decay rate varies as el”, they constitute long lived, weakly interacting excitations, thereby justi- 227 fying their use as the building blocks for the low lying excitation spectrum. There is no need in principle for the effective mass of the quasi-particles (q.p.) to be simply related to the free electron (or band structure) mass. For simple metals, it turns out that the q. p. mass m” is of the order of the free electron mass me, differing from it in most cases by a factor of less than two. The main source of the deviation of m” from me (aside from band structure) is the electron-phonon interaction, which in general leads to an increase of mº. Unfortunately, there is at present no truly first princi. ples proof of the above statements, i.e. the 1:1 cor- respondence of the low lying excited states of the noninteracting and interacting systems, a simple effec. tive mass spectrum, long lifetimes of the quasi-parti- cles, etc., although one has a proof that these state- ments are true to all orders in perturbation theory start- ing from the noninteracting states [2]. This lack of a rigorous foundation for the theory is not merely a mathematical nicety, since we know of many systems (e.g. the superconducting phase) which are not con- nected perturbatively with the noninteracting system; nevertheless, quasi-particles are still of use even in this case. One assumes that in normal systems (absence of cooperative effects) perturbation theory (or alternative- ly, adiabatic switching on of the interactions) gives the correct physics of the interacting system despite subtle nonanalytic effects which are likely present even in normal systems. Thus far we have discussed only the excitation spec- trum and not the many-body wave functions. Response functions (e.g. transport phenomena) require informa- tion about both quantities. The remarkable fact is that a suitably defined kinetic (or Boltzmann-like) equation for the quasi-particle distribution function gives an ac- curate account of the response of the system to long wavelength, low frequency perturbations such as elec- tric and magnetic fields. This second property of quasi- particles is the heart of why the Drude-Sommerfeld scheme works well for nonequilibrium as well as equilibrium phenomena. There is not time here to go into the details of the Landau theory of Fermi liquids, upon which the present theory of quasi-particles in metals is based. The excel- lent books of Pines and Nozieres [3] and of Nozieres [4] deal in depth with this topic. We would like, how- ever, to give a brief sketch of the theory and to make a few comments about it. 2. The Landau Picture of Fermi Liquids In the Landau picture one assumes that the low ener- gy excited states of the interacting system have ener- gies well approximated by the form l f E(6n º) F X. el,6nks –H – X. fiónk,önks. ks ksk’s' 2 (2.1) Here Önk, is a measure of the quasi-particle (q.p.) occu- pation numbers. Assuming there is a well defined Fermi surface SF at zero temperature described by the wave vectors kp, one has + 1, ks outside SF and occupied by a quasi- electron ôniº –K - 1, ks inside SF and occupied by a quasi- hole 0, otherwise. The zero-order q.p. energy ele is measured relative to the chemical potential p so that ek , = 0. One assumes that ek, and its derivatives are continuous across SF and one makes the effective mass approximation '. l hk Vks = h Velº-ji, (2.2) The approximation (2.2) often suffices, since one is usually interested in q. p. states ks in the immediate vicinity of SF (since T ~ Tr). The term involving fi represents the energy of interaction between quasi-par- ticles. This function and m” are considered to be parameters to be determined from experiment or to be roughly estimated from a more fundamental theory. Landau argued that if one views the quasi-particles as being described by wave packets whose extent is large compared to the wavelength of a q.p. at the Fermi surface (AF ~ 2.7/kF ~ 10-9 cm in metals) then one can define a distribution function Önk,(r, t)for q.p.’s which plays the role of the single particle distribution function f(r,p,t) in kinetic theory. This concept is reasonable as long as Önk,(r,t) varies slowly in space (compared to AF) and in time (compared to h/p). By the usual arguments of kinetic theory one can write down a kinetic (Landau- Boltzmann) equation for Ön. 66 º, Vik V,óñk,(r, t) – 31, við (el.) * , and I(Önks) are the external force acting on the q.p. and the collision integral respectively, while Wróñks is. defined by V,öñi,(r, t) = V,önk (r, t) –H ô(el.) > füy,önk,(r, t). k's' (2.4) 228 The term V, Önks in (2.4) describes the conventional streaming flow of q.p.’s familiar from kinetic theory. The other term, arising from the interactions, may be viewed as a dragging along of the ground state particles by the inhomogeneous distribution of quasi-particles, each q.p. dragging along its own cloud. Naively, one might guess that the particle current density J(r,t) at point r may be expressed as Xv,önk,(r,t). This is not true, but rather ks J(r, t) =X v1.6ñº (r, t) ks ->|->ſº wºoloºr, 0, 25 ks k's' The vön term represents the current of the quasi-parti- cle, while the term involving frepresents the current of the ground state particles being “dragged along” with the q.p. It is clear from the kinetic equation (2.3) that this definition is correct since the continuity equation *2+ v . J–0 (2.6) ôt is satisfied by J if we use the fact that (2.7) p=p0-FX 6n,(r, t). ks Roughly speaking fü behaves like a velocity depen- dent potential acting on particles in k and k". A change in Önks acts on the particle in k's' like a vector potential would, and induces a current even though the k" wavefunction does not change, like the Meissner effect in a superconductor. 3. Quasi-Particles in Metals *T*--_ The above picture is suitable for a system like He” which is translationally invariant in its ground state and has only the Fermion degrees of freedom. Metals are clearly different: they are invariant only under the translation group of the crystal lattice and have lattice vibrations as well as electronic degrees of freedom. How much of the Landau picture survives? The “non- interacting system” is presumably now the IPA in which the Coulomb interactions between electrons are treated in the mean field approximation. In this case the one , particle states are labelled by a wave vector k (restricted to the first Brillouin zone), a band index n and the spin s. We lump n and k together for now. There is a sharp Fermi surface at T = 0 and excited states are given by the usual electron and hole excita- tions. Since the Coulomb interaction has full transla- tion invariance, k remains a good quantum number to describe the quasi-particles, although the ground state of the actual system may be the transform of some excited state of the IPA system, due to changes in shape of the Fermi surface. Luttinger and Nozieres [2] have shown to all orders in perturbation theory that the volume of k space inside SF remains fixed, as it must for the Landau picture to make sense. The energy expres- sion (2.1) still holds but fi is in general a function hav- ing only the symmetry of the crystal, rather than full the rotational symmetry present for say He”. The effective mass expression for v, still holds except 1/m" is in general a second rank tensor having only the symmetry of the crystal. If the crystalline anisotropy of m” and f are very weak (as for Na) then the identity m*_1 + Fi - I11. 3 (3.1) relating the effective mass ratio and the spin symmetric l = 1 term F1° in a Legendre polynomial expansion of i. for a true Fermi liquid is valid. Here, if N(0) is the density of single particle states of one spin orientation at the Fermi surface, then Fi-N(0) [fit --fºl (3.2) and :=S fººp (cos kk'). (3.3) l=0 Thus, for nearly free electron metals the low tempera- ture electronic specific heat, i.e. mºlm, determines F1. Other pieces of information about fi can be extracted from other experiments such as the anomalous skin ef- fect, Azbel-Kaner cyclotron resonance, de Haas van Alphen effect, dynamic magnetic susceptibility, etc. Presumably the fi drop off rapidly with increasing lso only l = 0, 1 and perhaps 2 need be retained. For non- free electron metals, it appears that the anisotropy off is so large that unravelling this function will be quite in- volved. However, we know that the transport and the dHvA measurements have already given us a great deal of information about 1|m" in complex metals. When combined with band structure calculations these meas- urements give information on the many-body effects in these systems. As we mentioned earlier, most of the m*/m effect is due to the phonons, for which a reasonably good first principles calculation is becoming possible, in many metals. A careful comparison here would provide an important check on the approxima- tion of band theory and of the approximate methods presently used in many-body theory. 229 Another problem is that the phonons complicate the kinetic equation and the current density expression, since the phonons carry momentum and energy. The necessary generalizations of the Landau theory have been worked by Prange and Kadanoff [5], although we do not have time to discuss these questions here. 4. Green's Function Picture of Quasi-Particle An alternative way of viewing quasi-particles, which is more general than the Landau theory, is through the Green’s function scheme of many-body theory [3,4,6,7]. Suppose that an interacting system of N electrons is initially prepared to be in its exact ground state, 0, N). If cº, creates a (bare) Bloch electron in state ks, then we desire the probability distribution Ple (E) of the energy for the N-H 1 particle state p".' H 1 defined by | pººl ) = cl, |0, N). (4.1) In general, q} is not an eigenstate of the full Hamiltoni- an so that p does not have a sharply defined energy, i.e. PleſE) is not a delta function. The rules of quantum mechanics tell us that if the states |n, N + 1) are the exact energy eigenstates of the N + 1 particle problem then P.(E) => | (n, N-1 |ct.| 0, N) |*6(E – [E] r] – EX]+ u) (4.2) = S (n, N-1 | pº) |*6(E –[Eti – E]+ p.), where the Et" are the energy eigenvalues of the many-body system. Within the IPA, PisſE) is a delta function, since cº, creates a Bloch state electron, which by definition is an exact single particle eigen- state of energy ele. Thus, for the independent particle approximation (IPA) PIPA F #A(E) | 0 e is “ 0. (4.3) Clearly, according to the Pauli Principle P is zero if one tries to add a particle to an already filled state, els < 0. For the interacting system, P will be a complicated (positive) function of E in general, whose shape depends on the value of ks. The essential point is that if k is slightly above the Fermi surface, Pis (E) will consist of a narrow high peak centered about a “quasi-particle energy,” (4.4) E = €1, plus a background continuum which in general has a rather smooth behavior, as sketched in figure 1 for k just above the Fermi surface. The half-width of the peak Tks, gives the intrinsic decay rate of the quasi-par- ticle according to 1/Tk. = 2ſkºſh. Perturbation theoretic arguments show that T goes to zero as ek.” for ek, “p so that the fractional width (or the reciprocal “Q” of the particle) varies as éle, showing the quasi-particle to be a well defined excitation near the Fermi surface. To complete the story, one considers hole states defined by | dº. 1) = Cl, |0, N), (4.5) Like p".", pººl is not an eigenstate of energy for the interacting system. Thus, the probability distribution of E for the hole state, which is defined by P.(E)=S. (n, N-1 | dº.')|* 77 8(E+ [EN-1 – EN1 + p.) =>|{n, N-1 | c.|0, N)|* 71. 6(E+ [EX-1 – EN]+ p.) (4.6) is not a delta function in general. Note the change of sign of the excitation energy term in the delta functions appearing in (4.2) and (4.6). This ensures that at zero temperature the holes have negative energy and electrons have positive energy. Within the IPA, dº- is an eigenstate of H and Pºº (E) given by eks ~ 0 - () IPA – Pºº (E) | el, * 0. ô(E - e.) (4.7) For the interacting system, if ks is just below the Fermi surface, a narrow, high “quasi-hole” peak centered about ele appears in P., (E), with a continuum background again occurring as sketched in figure 2. The quasi-hole and quasi-electron energies presumably P.s (E) | | | | | | | | O Šks E FIGURE 1. Probability distribution Pºs(E)for a “quasi-particle” corresponding to a bare Bloch state ks for k > kr. 230 2ſ. | | | | | | | | *ks FIGURE 2, Probability distribution Pºs(E) for a “quasi-hole” state with k < kf. join on smoothly at the Fermi surface so that m” is con- tinuous across Sr. At nonzero temperature, one makes a statistically weighted average over initial states, rather than con- sidering only the ground state |0, N). In this case the electron and hole probability distributions overlap, in that they are both nonzero for E > 0 and for E -< 0. The overlapping corresponds to creating or destroying ther- mally excited quasi-particles, amongst other things. From the Fermi statistics of electrons, there follows the rigorous sum rule ſ [P.CE) + P. (E)]dE = 1. (4.8) One advantage of the Green’s function description is that it allows the concept of quasi-particles to be use- fully extended to systems which are not related by an adiabatic transform or by perturbation theory to the noninteracting states. For example, in a superconduc- tor, P., (E) shows at low temperature a sharp peak at the “quasi-particle” energy E = Ek, E V e.-- A. while P.(E) shows a sharp peak at E = –Els as sketched in figure 3. In addition, P and P show background continua like in the normal metal. Note however that as k approaches the Fermi surface there is an energy gap between the quasi-hole and quasi-elec- tron peaks, the gap being 2Ak., as is well known from the pairing theory. Thus a minimum energy 24k, is required to make a single-particle excitation in a super- conductor (i.e. creation of a quasi electron-hole pair). We should mention that the “background con- tinuum” mentioned above corresponds physically to the creation of more complicated excitations, such as a quasi-particle plus electron-hole pairs, phonons, plasmons, etc. Generally these extra excitations are not strongly coupled together and therefore an incoherent (smooth) continuum appears. In special cases, how- (4.9) | k > k (d) | F | | | | | | i | —P. -Eks-Are 9 AKF Fks E ^ P., (E). (b) k < k = | | /\– | | | | > "FKs TAKE O AKF Fks E FIGURE 3. Pºs(E) and PºsſE) for a superconductor showing (a) sharp peak at E = Eks in Pºs(E) and (b) peak in Pºs(E) at E=- Eks. ever, resonant scatterings states of the excitations can appear, an example being the quasi-bound state of a hole and a plasmon, as Hedin et al. [8] have discussed. There is a great deal more one should say about quasi-particles. The interested reader can follow the story further in the books mentioned above and the references contained therein. It is the present author’s view that a clearer physical picture of such questions as “drag currents,” “back flow,” “screening,” quasi- particle interactions (both forward and nonforward scattering amplitudes), particularly in real metals, deserve careful attention in the future. 5. References [1] See C. Kittel, Introduction to Solid State Physics, Chapter 10, J. Wiley & Sons, Inc., New York, and J. Ziman, Principles of the Theory of Solids, Cambridge U. Press (1964). [2] Luttinger, J. M., and Nozieres, P., Phys. Rev. 127, 1423, 1431 (1962). [3] Pines, D., and Nozieres, P., The Theory of Quantum Liquids, Vol. I, W. A. Benjamin, Inc., New York (1966). [4] Nozieres, P., Interacting Fermi Systems, W. A. Benjamin, Inc., New York (1964). [5] Prange, R. E., and Kadanoff, L. P., Phys. Rev. 134, A566 (1964). [6] Abrikosov, A. A., Gorkov, L. P., and Dzysloshinskii, I. E., Methods of Quantum Field Theory in Statistical Physics, Prentice-Hall, Inc. (1963). [7] Schrieffer, J. R., Theory of Superconductivity, W. A. Benjamin, Inc., New York (1964). [8] Hedin, L., Lundqvist, B. I., and Lundqvist, S., this Symposium. 231 Discussion on “What is a Quasi-Particle?” by J. R. Schrieffer (University of Pennsylvania) H. Ehrenreich (Harvard): I was wondering just how far above the Fermi surface one can go before things really go bad? J. R. Schrieffer (Univ. of Pennsylvania): I believe that the perturbation theory shows that, depending on the density of the particles, one can get high up into the spectrum and still have reasonably well defined quasi- particles. Calculations have been made, for example, for a free electron gas, of the damping as a function of the excitation energy. While the pair production cross- section starts up as the square of the excitation energy for a very fast particle there is a very different form for the level width, and depending upon the density, it can be that you never get into the region where the quasi- particle width is broader than the actual excitation energy. L. N. Cooper (Brown Univ.); Assuming that the ex- citations of the Fermi level are in one-to-one correspon- dence with free particle excitations, what kind of ex. perimental result could be said to contradict (or not be explainable) by the Landau theory? J. R. Schrieffer (Univ. of Pennsylvania): If the low temperature specific heat turned out not to go linearly with temperature, but rather had a form T log T, for ex- ample. L. N. Cooper (Brown Univ.); That would mean that the excitations were not in one-to-one correspondence with independent particles. Presumably it has to be something in the interaction term that goes wrong. I was just wondering what kind of a result would force you to the conclusion that you couldn’t find an interac- tion which was consistent with observations. J. R. Schrieffer (Univ. of Pennsylvania): The reason I had pulled in the T log T is that it is one deviation. Presumably, if the collision cross-section or damping rate did not vary at low energy according to e”, which is a rigorous prediction based on the Landau concepts, let’s say it varied in a more singular way, e.g., as e” log e, for example, then that would violate the Landau theory. - 232 Beyond the One-Electron Approximation: Density of States for Interacting Electrons” L. Hedin, B. l. Lundqvist, and S. Lundqvist Chalmers University of Technology, Göteborg, Sweden The concept “density of states” can be given many different meanings when we go beyond the one- electron approximation. In this survey we concentrate on the definition tied to excitation processes, where one electron is added or removed from the solid. We discuss the one-particle spectral function for conduction and core electrons in metals, how it can be approximately calculated, and how it can be re- lated to different types of experiments like x-ray photoemission, x-ray emission and absorption, photoemission and optical absorption in the ultraviolet, and the Compton effect. We also discuss the form of the exchange-correlation potential for use in band structure calculations. Key words: Density of states; interacting electrons; one-particle Green function; oscillator strengths; quasi particle density of states; x-ray emission and absorption. 1. Introduction In the one-electron approximation the concept “elec- tronic density of states” is unique and simple and is defined by the formula N. (a)=X. 6 (o-e), (1) i where ei denotes the single-particle energies. Similarly the optical density of states is given by the joint density of state for electrons and holes, thus N2(0) =X. 6(o-eh-Heſ). (2) l These functions only describe the cruder aspects of experiments such as soft x-ray emission or optical ab- sorption. They must be augmented with the proper oscillator strengths, which provide important selection rules and as well give rise to deviations from the density of state curves. When one goes beyond the one-electron approxima- tion and considers interactions not included in a self- consistent field theory the situation becomes non-trivial and one encounters a wide variety of density of states * An invited paper presented at the 3d Materials Research Symposium, Electronic Density of States, November 3-6, 1969, Gaithersburg, Md. functions. The quantity which is most easy to define is the true density of states of the fully interacting Systems N(0)=X. 6(a) – En), (3) 17. where En are the energy levels of the system. Although being a key quantity in thermodynamics it is of no use for the description for e.g. optical, x-ray or energy loss spectra. We rather need the proper extensions of eqs (1) and (2) to describe one- and two-electron properties of the system. A particularly simple case is that of parti- cles having a Lorentzian energy distribution *—º T (a) – ek)” + Tº A k(a)) = (4) The density of states is then given by N(o)=X.A.(o). (5) In cases where the line width Tk is small compared with the width of the band, the formulas (5) and (1) obviously will give similar results. The simple case just mentioned does not really carry us beyond the one-electron theory. We have in the general case to abandon a concept based on single-par- ticle levels and instead use distribution functions or 233 spectral densities of which the function Ak (a) given above is a simple and almost trivial case. Such distribu- tion functions include the proper oscillator strengths, and they refer to formally exact many-electron states, properly weighted according to the physical process considered. They are closely related to correlation functions and Green functions. An example is the density-density correlation function (p(rt)p(r't')), which correlates density fluctuations at different times and different positions. This function is of basic im- portance for describing optical experiments and energy loss spectra. Unfortunately it is very difficult to analyze its structure and we will here instead concentrate on a simpler entity, the one-particle Green function and its associated spectral function [1]. The one-particle spectral function is a generalization of the usual density of states N1(a)). It gives an asymp- totically exact description of x-ray photoemission, is connected with x-ray emission and absorption and also has some relevance for uv photoemission and absorp- tlOn. It is, however, not capable to describe edge effects in general, and does not at all account for final state or particle-hole interactions. Such effects, which in their simplest form are described by N2(a)) or generalizations therefrom provide intricate problems upon which we shall only briefly touch here. These questions will be taken up in detail in the lecture by Mahan. It has recently been noticed that the one-electron spectral function should exhibit some marked and strong structure. This structure is primarily associated with the interaction between the electron and plasmon excitations. It should be quite important in x-ray photoemission, influencing the line shape of the core electron peak and giving low energy satellites both to the core electron and conduction band structures. It is important in x-ray emission and absorption spectra showing up at the threshold, and giving rise to satellite structures. Its effect may also be noticeable in uv photoemission and optical absorption spectra. We should also mention the structure in the one-elec- tron spectrum caused by coupling to phonons. This structure is limited to a region of the order of the Debye energy around the Fermi level, but in that region it is quite pronounced, causing e.g. the well-known enhancement of the quasi particle density of states. In the next section we will discuss the connections between the spectrum and x-ray photoemission, and with x-ray emission and absorption. In section 3 we discuss the spectrum of the conduction electrons from calculations to lowest order in the one-electron the electrons and between the electrons and phonons. In section 4 we take up the core electron spectrum and discuss calculations to first order in the dynamic interaction with the valence electrons. We also survey the recent work by Langreth who obtained the exact solution to an important model problem. In section 5 we give a qualitative discussion of some experimental results for photon absorption and emission, photoemis- sion and Compton scattering. In section 6 finally we give some concluding remarks. dynamic interaction between 2. The One-Electron Density of States in an Interacting System and its Connection with X- Ray Photoemission and with X-Ray Emission and Absorption In this section we introduce some theoretical techni- calities, which however seem unavoidable in order to establish a firm connection between theory and experi- ment. The one-electron spectral weight function is defined aS A (x, x'; o)=X f(x)f(x')6(o-e). (6) In the independent-particle approximation the quan- tities f, and es are the one-electron wave functions and energies. In an interacting system the oscillator strength function f is defined as f(x) = (N-1, s||(x)|N), (7) where l;(x) is the electron field operator. It gives the probability amplitude for reaching an excited state |N-1, s), when an electron is suddenly removed from the ground state |N). The quantity es is the excitation energy es= E(N) – E(N–1, s). (8) These definitions apply for a in eq (6) smaller than the chemical potential pſ. For a larger than pſ, states with N + 1 particles are involved. In applications of the theory it is often practical to represent the spectral function as a matrix in a space spanned by some suitable complete orthonormal set of single-particle states pºk(x), thus Akkº (o) =| uš (x)A (x, x'; o).uk (x') dxdx' =X. (Na N–1, s) (N – 1, s|ak|N) 6(a) – es). (9) 234 The spectral weight function is the general distribution function describing one-electron properties, and is a generalization of the one-electron density matrix, which is obtained after an integration over the frequencies. Distributions with respect to a single variable are in the usual way obtained by summation over all other varia- bles. Thus, the one-electron density of states is defined alS N(0) =Tr A (a)) =|46. X: o) dx=X. A kº (o). (10) k. As indicated, the definition is independent of the choice of matrix basis. The spectral weight function is a key quantity in the Green function formulation of the many-electron problem. A more detailed account is given in many texts, e.g. in sections 9 and 10 of ref. [1]. We next discuss the particular cases of x-ray photoemission and soft x-ray emission, where approxi- mate reductions to the one-electron spectrum can be made. - In the x-ray photoemission experiment (XPS), an energetic photon of energy o is absorbed, exciting a photoelectron which leaves the system. If the electron has a large enough energy we can write the final state aS |V, ) = a N–1, s ), (11) where k refers to a Bloch wave function of energy e, as h°k”/(2m), describing the photoelectron. The probabili- ty of this eventis, by the Golden rule, W ~ X | (V, PV)|26(o–E,--E) f =X|(N-1, sla. X. pºo; as N)|*6(o-e, --es), (12) k,k' where P is the total momentum and pick, the momentum matrix element. For a fast electron, which is outside the region of ground state fluctuations, ak|N) = 0, and the expression for W reduces to W ~ X |(N-1, s X. p., a N)|*6(a-e, --e.) K, S k: =XXAkk (e. -o)p. p. (13) k k,k' The energy distribution of photoelectrons is hence given by I(e) - VeX Akk (e-o)p. Pºk, k,k' (14) taking ex=e. Using an average momentum matrix ele- ment and neglecting nondiagonal terms in A we obtain the simple result I(e) - Vep;N(e-o). (15) The neglect of nondiagonal terms should be a good approximation if the one-electron functions are care- fully chosen Bloch functions and core-electron func- tions. We note that if the electron is ejected from a core level and if we neglect excitations of the core electron system, the operator aſ in eq (13) must destroy a core electron (ak = ac), giving W ~ X|{N}, s|N.) p. |26(a) -e, -- es). (16) Here ||Nº) denotes the ground state of the valence elec- tron system, and ||Nº, s) are excited states in the presence of the core hole. We recognize eq (16) as the result in the sudden approximation. We next turn to soft x-ray emission. In this case we may write the initial state as [2] |Vi) = ac|N*) (17) where the star on N indicates that the valence electron system is relaxed towards the core hole. Applying the Golden rule again we have for the x-ray intensity I(0) - o X | (Vºl X. pºagara, N*) |*. f kºk' 6(a) – E + EP) = @ X. | (N, -1, s| X. peka, N.; )|*ö(o-Fee – es). If we could neglect the relaxation effects in the initial state and replace ||Nº) by ||N.), we would again have the spectral function A involved. The relaxation effects can be accounted for in an approximate way by only considering particle-hole excitations, thus |N; ) = (a+ X oftajan)|N, ). h, p (19) Insertion of this expression in eq (18) leads after some further approximations to a very simple expression I(o) – o X. |p;|*A*-(o-ec), (20) k where pºſſ involves the coefficients o and is o-depen- dent. Such an approximation is however invalid at the Fermi edge, where, as Anderson [3] has pointed out, we need an infinite number of particle-hole pairs to represent Nº). 235 The edge problem has recently been treated by sev- eral people [4–6]. We choose here to give a brief ac- count of Langreth’s version [6] of Nozieres; and de Dominicis’ (ND) treatment of the problem in order to show how the one-electron spectrum enters the problem. The basic quantity needed for the evaluation of eq (18) is the correlation function Fºk (t) = ( N*|T(a), (t) aſſac (t)a;)|Nº ). (21) To evaluate that function ND study the multiple time function Fºk (T, T'; t, t') = ( N*|T(ak (T) aft, (t') ac(t)a; (t')|N* ). (22) The simple way, in which the core state operator en- ters their model Hamiltonian, causes the equation of motion for F to close onto itself and allows a solution as a product, Fºk (T.T.';t,t')= dº (t,T';t,t') Go (t,t') (23) where G. is the core electron Green function, Gc(t, t') = (N*|T(ac(t)a; (t'))|N* ) (24) and Ó a valence electron Green function which obeys an equation with a transient potential from the core hole. The x-ray spectrum is given by a convolution of the core and valence electron spectra. Both spectra are singular at the edge as a power law and so is also the resulting convolution. The point in keeping the operator ac in eq (22) is that the time dependence of al.(t) then can be given by the same Hamiltonian as that used for N*), while when studying (Nº|T(a)(t)alº)|Nº), the time evolution of a;(t) is given by a Hamiltonian that does not include the potential from the core hole, and thus the usual many- body techniques do not apply. X-ray absorption can be described in an analogous manner by the function F, only replacing N*) by ||N). It should be mentioned that the ND treatment is based on a model Hamiltonian that does not include in- teractions between the conduction electrons. This is quite appropriate for their purposes of treating the edge problem but is not sufficient for the spectrum far away from the edge. It is not clear if their treatment can be extended to the more general case. We then have to resort to other methods, such as the approximation in eq (19) or to a frontal attack on the dielectric function itself [7]. We have in this section given a definition of the one- electron density of states and have indicated its relation to some measurable quantities. We conclude from this that a theoretical investigation of this kind of experi- ments can not avoid the calculation of the one-electron density of states of interacting electrons, but at the same time due to a variety of effects there may be es- sential modifications of the density of states as it ap- pears in the actual experiment. Explicit results from ap- proximate calculations of the spectral function A and the density of states N(a)) are discussed in sections 3 and 4, and the actual comparison with experiment is made in section 5. 3. One-Electron Spectrum of Conduction Electrons The common picture of the one-electron spectra of solids is obtained from energy band calculations, in which the Schrödinger equation of independent elec- trons in a periodic potential is solved. For the different bands one obtains the electron energy as a function of wave-vector, e(k). The main contribution to the poten- tial seen by an electron is the average electrostatic field or the Hartree field. When going beyond the one-elec- tron approximation the next step would be to include dynamical effects of the interaction as well as exchange effects. Such effects can not be described by an ordina- ry local potential. A generalized “potential,” non-local in space and time, must be introduced. Let us for simplicity study the case, where specific effects of the periodicity are of minor importance, and assume the distribution of conduction electrons to be uniform. Due to the non-locality of the generalized “potential,” its Fourier transform to momentum-energy space will show a dependence on both wave-vector k and energy e, X (k, e). The quantity X is called the self. energy of the electron. Adding the self-energy to the average potential, we have to solve the equation e = e(k) + X (k,e). (25) The self-energy includes all the interactions between the electron state considered and the system. This necessarily includes dissipative effects, which lead to the decay of the state. Consequently the self-energy must be a complex quantity, thus .. X (k,e)=XR(k,e)+ix ſ(k,e). (26) The spectral weight function defined in section 2 is re- lated to the self-energy according to the formula 1, sec- 236 tion 1 l X (k, e) A (k, e) = I 7 º'S (k, o'-(S (k, o). R (27) In the simple case that the self-energy X is independent of energy, the spectral function for fixed k will have a Lorentzian shape and Xi determines the width of the line. When X depends on energy there are no a priori restrictions on the shape of the spectrum. Most of our present knowledge about the self-energy has been based on calculations, where vertex cor- rections have been neglected, i.e. using the formulae [9–13] X (k, o] = º | e”v(k’) e-' (k', o')Go(k + k", a + ay').dk'do', (28) where Go (k, o] = (a) – e (k) + ið sign (k - kp)) T' (29) is the propagator for a non-interacting electron, and v(k) = 4Te”/k” (30) is the bare Coulomb potential. The constant kº means the Fermi momentum, and 6 is a positive infinitesimal. To make connection with another approximation, we note that the Hartree-Fock approximation corresponds to the choice e = 1 in eq (28). As it stands, eq (28) ex- presses the lowest order coupling to the density fluctua- tions of the conduction electrons, described by the wave-vector- and frequency-dependent dielectric func- tion e(k,00). This function contains information about the static screening, given by e(k,0), but also important effects of the dynamic behavior of the density fluctua- tions. The typical small-k-behavior of the spectral func- tion for this kind of excitations, –Im eT'(k,00), is in- dicated in figure 1. The clearcut classification of the ex- citations into electron-hole pairs and plasmons is characteristic for the Linearized Time dependent Har- tree (LTH) or Random Phase Approximation [14]. For small wave-lengths the electron-hole pair excitations become of increasing importance, while in the k → 0- limit the plasmon exhausts completely the sum rule for Im eT', i.e. it is the only excitation. The latter property follows from general arguments about charge conserva- tion and translation invariance [8, p. 288), and should be valid for any useful approximation for e(k,00). In the l -Im 4– 8(q,00) q Pldsmon pedk Electron-hole pairs l I A- -- (1) 2d (Upt &q” FIGURE 1. Qualitative behavior of Im eT' (q,a) for small q. LTH approximation the plasmon is undamped for wave-numbers smaller than a critical value ko, which at metallic densities is of the same order of magnitude as kr. A great part of the literature about the electron gas is devoted to the search for an improved dielectric func- tion [15–20], particularly because of the failure of the LTH formula to describe interaction effects at short distances. The last word remains to be said in this question, but it is interesting to note the relative insen- sibility of the self-energy to the choice of dielectric function [11,12,21]. An approximation, which has proved to be most useful, is to take a plasmon-like sharp absorption for all k according to the formula 2 a), e-' (k, o] = 1. --→ Gº)" – o” (k) (31) where on is the classical plasma frequency and oſk) the resonance frequency at wave number k. The frequency o(k) reduces to op when |k->0 and is proportional to |k|2 for large |k. This formula correctly represents the small and large o-limits and gives a quite reasonable in- terpolation for intermediate a values [1, 1]. With this choice one can perform the frequency integration in eq (28) and obtain v (q) no (k+ q) | 3 (27)3 ſº: e(u + q) - a)) v(q) l 20 (q) a) – e(h+ q) → 0 (q) X (k, o] =- op” Tºys ſ dºq The first term in eq (32) is a screened exchange poten- (32) tial, no(k) being the momentum distribution for inde- pendent electrons, and the second term describes the correlation hole around the electron [12,22]. Because of the plasma resonance in the dielectric function there will be a rapid variation in the real part 237 A a 1 i 1 a + 2 |ImX : Imy. : |Im X lºs wº - *{ º N - }* * sº º V-J - H - *l - 4 }* > 0 * hº (O - 4 }- CC sº- s: -- º # –2- - º º ti. l Re XI * ReX. F - Re XI F- ld -4- sº F- 4 k.k § 0.6 Kr }* 10 kri 1. FI # . 05 K, H. 10 k, H. 14 k.[ 0.6- }* F |- * * | - }* * ſ à sº | }* * I. H ū nº Z=0.56 II Z-0.63 [. I § 04: | | - * I. Z= 0.58 H # , Z-027 I . I of 0.2- I }* * - * F- |- - º § l, -- _ſ\}. º 9; 00 CS-I [1 --> y ſi º (ſ) —H-I-T-r-T—r—H-T— —r—I-I-T-I-T-I-T-I r-I-I T-r-I H-T-r T. "Tº T. " "... " ' d' ' ' ' ' ' ' '. "'b'" ; ENERGY (/, ENERGY (/, ENERGY (9/9, FIGURE 2, The self-energy X and the corresponding spectral function A for r = 5 [21]. The crossings between the Re X curves and the straight lines give the solutions to the Dyson equation. The numbers at the peaks indicate the strengths of the lines. of X at the frequency a = e(k) + (op for electrons and at a) = e(k) – op for holes. This behavior of the electron gas seems to have been first noticed and discussed by Hedin et al. [23] and has been studied in great detail by Lundqvist || 13,21]. Typical curves for X obtained with this dielectric function are shown in the upper half of figure 2. In the lower part of figure 2 we have given the results for A(k,00) calculated from eq (27). For k = kr there is only one strong peak in the spec- tral function, corresponding to the usual quasi particle. For the other k-values in the figure, we find three solu- tions of the Dyson equation a) = e(k) + XR (k, o]. (33) One of them, however, falls at a) = e(k) + (op, where the damping is very strong, and therefore this solution is ef- fectively suppressed. Of the two remaining solutions, one corresponds to the usual quasi particle, i.e. a bare electron surrounded by a cloud of virtual plasmons and electron-hole excitations. For hole states, i.e. for k < kº, a new state appears which has an energy lower than that corresponding to a hole plus a plasmon, i.e. e(k)- op. This result from a low order treatment corresponds to a coherent state of hole-plasmon pairs, and may be thought of as holes coupled to real plasmons. This cou- pled state, which has been called a plasmaron, has a large oscillator strength, and thus gives an essential contribution to the sum rule for the one-electron spec- trum. For electron states, k > kr, there is no sharp state but a broad resonance with a sharp onset at a) = e(k) + (Op. Figure 3 illustrates how the parabolic quasi particle dispersion law is accompanied by a second branch of the spectrum, the plasmaron. In figure 4 the momen- tum distribution function for the interacting electrons is given. The characteristic structure due to plasmon effects will modify the density of states. This is demonstrated schematically in figure 5. The conduction band will retain approximately its parabolic shape. At an energy op below the Fermi edge there is the onset of the second band due to the plasmaron states. This band is terminated at low energies with a rather distinct edge. For unoccupied states there is an extra contribution to the density of states starting at an energy op above the Fermi level. Figure 6 gives a survey of the density of state curves at four different values of the electron gas parameter r, (rs = 2 corresponding roughly to the elec- tron density of Al, and rs- 4 to that of Na). So far we have discussed the gross effects due to the coupling between electrons and plasmons. These are quite well represented by the plasmon-pole approxima- tion for the dielectric function in eq (31). Electron-hole pair excitations are however not included, to represent I I I I I I I T 1 -4 -4 O MOMENTUM k/k, FIGURE 3. The spectral weight function A(k,00) given by level curves indicating the value of hop.A, at rs-4 (hop = 0.435 Ry) [13]. 238 1 - - -- * - -- - - - - - - - - - *- - - - - —T *- 0 *~ Momentum k/k, FIGURE 4. The momentum distribution function n(k) at rs=2,3,4 and 5 [13]. Plasmon onset Plasmar On edge ! Hierº- Plasmon edge Fermi edge FIGURE 5. Density of states for conduction electrons. them we must turn to the LTH dielectric function or higher approximations. These excitations are responsi- ble for the broadening of the quasi particle peak and the Auger tail at the bottom of the main band in figures 5 and 6 as well as the broad spectral weight contours in figure 3. - Figure 7 shows X1(k,e(k)) calculated from eq (28) and is an indication of the high damping rate for quasi parti- cles with k greater than kr + kc, i.e., in the region where they can decay into plasmons || 10,24]. However, as il- lustrated by the typical spectral forms for the quasi par- ticle peak in figure 8, the quasi particle peak is always distinguishable from the spectral background, its width T(k) is smaller than its excitation energy E(k) – p. From eq (28) one can also draw information about the exchange and correlation “potential” for the quasi par- ticles. The inadequacy of the Hartree-Fock approxima- tion in a metal is well-known, and in band calculations the effects of exchange and correlation are commonly simulated by using a local potential, such as the Slater [25] or the Gáspár [26,27] expressions. After solving the Dyson equation one can write the resulting quasi- particle energy in the form E(k)= e(k) + V(k), (34) where V(k) can be interpreted as an effective exchange and correlation potential. In figure 9 such a potential is shown. It has a remarkably weak k-dependence for moderate wave vectors, its value lying roughly halfway between the Slater [25] and “2/3 Slater” values [26,27]. One of the shortcomings of the Hartree-Fock approx- imation is its prediction of a large bandwidth. The width of the occupied part of the main band deduced from eq (28) is practically the same as the Hartree value e(kr), a result which is in accord with the experimental findings. For large momenta (k 2 kr + kg) there is a charac- teristic k-dependence of V(k), which might influence properties of electrons involved in photoemission and gº. º Wi- l - 4 -3 –2 - | O | 2 3 ENERGY & /*r FIGURE 6. The density of states for the values of the electron gas parameter rs = 2, 3, 4, 5. The dashed curve is the result of the one- electron theory, and the vertical broken line indicates the Fermi level [13]. 75 A/k, FIGURE 7. The imaginary part of the quasi-particle self-energy Im X(k,e(k))/EF as a function of the momentum k of the electron at different metallic densities (EF = 0.921, 0.409, 0.230, and 0.147 Ry at rs = 2, 3, 4 and 5, respectively). The dashed curve is Quinn’s result at rs = 2 [10,13]. 239 80 2% ///-72 s (205H 0 05 FIGURE 8. Typical spectral forms for the quasi-particle peak of the spectral weight function in different momentum regions. The scales are different for the three curves, the energy a is measured from the Fermi level pu, and the curves are drawn for r = 4 [24]. LEED experiments. It is not clear, however, how much of this structure that may be observed due to the short- lifetime of the electrons at these energies. We want to stress again that the discussion we have given of the one-electron spectrum is based on the as- sumption that vertex corrections are small. As discussed in the next section recent work by Langreth [29] shows that vertex corrections in the core electron problem can have a quite large effect on the form of satellite structures, while their effect on the quasi parti- cle properties seems to be small. Preliminary investiga- tions by one of us (L.H.) show similar strong vertex ef- fects on the conduction band satellite. The details of the plasmaron structure should thus not be taken very seriously. The quasi particle states close to the Fermi surface are of particular interest due to their importance for thermal and transport properties [30]. To study these problems the quasi particle density of states or density of levels rather than the one-electron density of states is of importance. Because of interactions the quasi-par- ticle dispersion law is distorted, corresponding to the well-known mass enhancement at the Fermi surface. The corrections to the free-electron mass m due to the electron-electron interaction, as derived from eq (28), are small [11,12], e.g. ôme/m = −.01 for Al and Öme/m = .06 for Na, and the properties of the quasi particles close to the Fermi surface are dominated by the elec- tron-phonon interaction. The effects of the electron-phonon interaction on the quasi-particle dispersion law follow in a straightforward way using perturbation theory in the Brillouin-Wigner form, thus - 6 1 – fo cº-º-Yº... º fp "E(k)-eſpiroſk=p' | (35) In this equation gº-p is the matrix element for the elec- tron-phonon coupling and a (k—p) the phonon frequen- Cy. The qualitative effect of the phonons is to flatten out the dispersion curve in the immediate neighborhood of the Fermi surface, and this gives rise to the enhance- ment of the density of states (fig. 10), or, equivalently, of the thermal effective mass and one obtains from eq (35) mph = m (1 + \) dOp-l. go-k |2 4T a (p − k) = No (EP) (36) where Noſºp) is the density of states without electron- phonon interaction, |p = |k = kr and the integration extends over the full solid angle. Ashcroft and Wilkins [32] first calculated the cor- rections for Na, Al and Pb using eq (36), and several similar calculations have been published over the last vſk), O I | -I P "2/3-Slater" Sldter FIGURE 9. Exchange-correlation potential for an electron gas at rs = 4 compared with the Slater and the Gaspar (2/3 Slater) approximations [28]. 240 4 ºb I I J I - 4 - 2 0 2 FIGURE 10. Density of quasi-particle levels for Na at T=0°K [31]. few years. The most accurate values of A are those deduced from tunneling data by McMillan and Rowell [33]. For example, the two methods give the values A = 0.49 and 0.38, respectively, for Al; X = 1.05 and 1.3, respectively, for Pb. It is obvious that the enhancement varies with tem- perature and that no enhancement is left at high tem- peratures, where the phonon system behaves like a fluctuating classical medium. Similarly to the case of electron-electron interaction the electron-phonon interaction gives a characteristic structure to the spectral function. Engelsberg and Schrieffer [34] pointed out that in the neighborhood of the Fermi surface the quasi particle picture will no longer apply. They calculated the spectral function for an Einstein model and for a Debye spectrum and ob- tained a spectral function with a very complex struc- ture in the region close to the Fermi surface. For sodi- um no dramatic effects occur because of the rather weak electron-phonon interaction [31]. We conclude this section by noting that inclusion of dynamical and exchange effects, in particular con- sideration of the electron-electron interaction tells us why the one-electron theory works so well in explaining many gross features of metals and what limitations it has. However, at least in a low-order treatment there are also new structures introduced by the interaction, schematically characterized as due to the resonant coupling between electrons and plasmons. In section 5 we will discuss possibilities to observe this structure. 4. One-Electron Spectrum of Core Electrons The preceding discussion has emphasized the strong effects of the electron-plasmon coupling on the spec- trum of conduction electrons. Similar strong effects occur in the spectra of core electrons. For simplicity we limit the discussion to simple metals with small cores, so that the core electrons can be physically distin- guished from the valence electrons and be well local- ized to a particular ion. The wave function of a core electron depends only weakly on the state of the outer electrons. The energy levels on the other hand are shifted by an appreciable amount compared to the cor- responding atomic levels, typically of the order of 5 to 10 eV, which is large on the scale of valence electron energies but is a small relative change in the energy of a core electron. The core shifts can be measured accu- rately by the method of x-ray photoemission spectrosco- py as well as by x-ray absorption and inelastic scatter- ing of fast electrons. The shift of the quasi particle energy of a core elec- tron comes partly from changes in the average Cou- lomb field, partly from polarization effects. The Coulomb shift is due to the different valence charge dis- tribution relative to that in a free atom. It generally results in a decrease of the binding energy. The polarization shift comes from the relaxation of the valence charge distribution around the hole created when we remove the electron. The valence electrons are drawn in towards the positive hole in the ion. This effect decreases the binding energy by half of the change in the Coulomb potential calculated at the core site, i.e. precisely the amount obtained if we calculate the self-energy of the hole using electrostatics. The shift in the core energy thus contains information about the valence electron distribution and polarizability, measured with the core electron as a probe. Theoretical calculations for simple metals (Li, Na, K, Al) are in very good agreement with the experimentally observed peaks in the XPS spectra [35]. In analogy with the strong effects of the interaction between particles and plasmons previously discussed, there is a corresponding coupling between a hole in the core and the density fluctuations of the conduction electrons. This leads to a strong structure in the core electron spectrum [36]. We assume for simplicity that we can neglect the spatial extension of the core electron wave function. Calculation of the self-energy to lowest order in the screened interaction gives the formula Sºo--ºnſ d'adovoe (a, oil | a) – e – en + i ö (37) where en is the core quasi-particle energy. Remember- ing that ev'(q, a has a strong resonance in the plasmon regime, we see that after integration over the frequen- cy, the self-energy will show a resonance behavior in the energy region e = en – op. This rapid variation will give rise to two solutions of the Dyson equation. The 417–156 O - 71 - 17 241 2 w/hwe FIGURE 11. Quasi-particle peak in the spectral function for a core electron and the associate plasmon satellite structure for different densities of the conduction electrons, measured by the electron gas parameter rs [36]. second solution, however, gives a quite broad peak in the spectral function. The results for different densities of the electron gas are illustrated in figure 11. The spec- trum is measured from the shifted quasi particle ener- gy, i.e. the zero of energy corresponds to complete relaxation of the electrons around the hole. We summarize the characteristic features of the core Spectrum a. A large polarization shift of the core quasi parti- cle level. b. The shape of the spectrum is independent of the core level considered. This results from neglecting the actual size of the core wave function. c. A pronounced satellite structure, which starts at a) = - op and has a broad peak. d. An extended tail on the low energy side of the quasi-particle peak. This tail is due to the coupling between the hole and screened elec- tron-hole pair excitations, and corresponds to states of the whole system involving two holes, one in the core and one in the conduction band plus one excited electron above the Fermi sea. e. There is an appreciable reduction of the spec- tral strength of the quasi particle. Approxi- mately half of the spectral strength cor- responds to excitations close to the quasi parti- cle state and approximately the same strength corresponds to high excitations of the conduc- tion electrons as described by the broad satel- lite structure. As discussed in section 2 the spectral function has the form (cf. eq (16)) A.(0)=X|(N;, slN,)|*6(0–e,) (38) and shows a powerlaw singularity at the Fermi edge. The first order theory correctly predicts a singularity but of a somewhat different form, namely (oln”a) '. The core electron spectrum can be obtained exactly if we use a simple model Hamiltonian H=eata + adt X. g.,(bo-H bºn) + X objba, (39) Q Q Here at is the creation operator for a core electron of energy e and bat the creation operator for a plasmon of energy on. The exact solution of this problem has been given by Langreth [29], and we now give a brief ac- count of his work, which also shows the close resem- blance of the problem to the Mössbauer or the impurity- phonon problem. The self-energy X in this model is given by the sum of all diagrams where the core electron is dressed with plasmons (fig. 12). For the true Hamiltonian including all electron-electron interactions we have the same set of diagrams, where the plasmon propagator is replaced by a screened interaction v(q) et' (q,a)). Except for the first diagram the bare Coulomb potential gives no con- tribution (the hole propagates in only one direction) and we can thus replace eT" by (eT' – 1). By choosing the dielectric function in eq (31) and the coupling gº= v(q)op"|(20), (40) the plasmon diagrams and the screened interaction dia- grams become the same. The core Green function Gc(t)=– i(T(a (t) at (0))) (41) can be written as (cf. eq (38)) Gc(t) = i (0|etht|0)e-ie/6(–t) (42) where 0) is the plasmon vacuum state and H is the Hamiltonian for the plasmon in the presence of the core > = 2 * 1 * 2° tº s tº e º 'º FIGURE 12. Diagrams for the core electron self-energy. 242 hole, H=X800-F bº)+X objb, (43) Q Q A canonical transformation shifts the zero point of the plasmon vibrations, thus e'He-4 = Ho-Ae, (44) where A =X fa(b; – ba); f, gaſon; Ho-X, objbd; Q Q Ae - X. aq.fi. (45) Q We may hence write (0 | eiht | 0) ~ (0 |e-AethoteA O)e-ièet = (0|e-40)eae)|0)e-ièet, (46) where A(t)=e "o'Ae-ºo-> f(bje"q-be-toq). (47) Q By applying the well-known formula e4+B = e4 eB e-1/2(A,B) (48) repeated times the exact solution follows, (0|eiht|0) = exp (– Aet) exp (— X'ſ.) exp (Xſerº) Q Q This gives for the spectral function 4.(o)=*Im c.(o)=#. ſ T 2T — Od –S. f. F 6 q (49) ei(o-e) (Olein [0)dt (50-e-Az)+x ſºo-e-A to | "22 Yºo-º-Aroºroot gº tº } (50) To compare the first order results (the first diagram in fig. 12) with the exact solution we have used the simple dispersion law on- ap-H q” (in units of the Fermi energy and the Fermi momentum), which allows an analytic solution. The result for the electron density of sodium metal (rs=4) is given in figure 13. Since the dielectric function in eq (31) contains no particle-hole pairs, the quasi particle peak is a 6-func- tion. A more realistic shape of the peak is indicated in the figure. The exact solution has structure also at –30p, -400p, etc., which is not shown in the figure. This structure, however, has weak edges and carries only a few percent of the oscillator strength. Comparing the results of the first order calculation and the exact solution we note: 1. The energy shift Ae is exactly the same. AcQu) Quasi- Satellites ||particle | | | | / / / "el - e. u) +--" 'up –3 -2 - O FIGURE 13. Comparison of the first order (dotted line) and the exact result (full curve) for the core spectral function from the model Hamiltonian in eq (39). In this model the quasi-particle peak is a 6- function. The dashed curve indicates a more realistic form of that peak. The results are for rs = 4. 2. The oscillator strengths in the satellites are closely the same. 3. In the exact solution the satellite has two marked peaks instead of one, and the peaks are sharper. - The large difference between the first order result and the exact solution for this model case should be a warning against taking the details of a low order calcu- lation too seriously. Knowing the importance of the higher order diagrams in this case one may ask if not other higher order diagrams, like those of the paramag- non problem, may play a role also for the core spec- trum. Finally, it should be stressed that while the satel- lite structure may be poorly accounted for in the first order theory, the position and singular nature of the quasi particle are quite well represented. 5. Qualitative Discussion of Some Experiments We shall discuss some different types of experiments utilizing the connections with the one-electron spec- trum discussed in section 2 and the results from the ap- proximate calculations reported in sections 3 and 4. We can really not put forward much more than guesses about where many-body effects may possibly occur. The difficulty is that the predictions by the one-electron approximation have seldom been worked out in enough detail to give reliable level densities and matrix ele- ments, and this knowledge is required both to evaluate the many-body effects per se as well as to find out how much of the experimental structure that is accounted for by the one-electron approximation. Also the experi- mental data are sometimes not as accurate and reliable as one would need. Thus surface conditions are often not under good enough control, background effects are 243 poorly known and disturbing secondary effects are not carefully analyzed and subtracted. The discussion will neglect the effects of final state interactions between the electron and hole. This means that the treatment is a simple extension of the usual theory for interband transitions in which the density of states for electrons and holes in band theory is replaced by the corresponding quantities including many-elec- tron interactions as illustrated in figures 6, 11, and 13. Although certainly of restricted validity it seems that the predictions of such an approach are worthwhile to summarize. 5.1. X-Ray Photoemission (XPS) This experiment has a fairly clearcut relation to the one-electron spectrum. Ideally the energy distribution of photoelectrons will be given by eq (15). However, this equation is valid only if the photoelectrons leave the solid without being scattered. Structure due to satel- lites in the density of states will thus be mixed with structure due to energy losses. The losses to volume excitations are proportional to the sample thickness, while the intensity of the satellite structure in the one-electron spectrum has a definite relation to the intensity of the quasi particle peak. Thus by varying the sample thickness one should be able to separate the two kinds of processes. Theory predicts an asymmetric form of the core- quasi particle peak [4,5,6,29,36.37] and satellite struc- ture starting at ap below the main peak. According to the Langreth model solution there should be two satel- lites also with an asymmetric line shape (cf. fig. 13), while the first order theory predicts the satellite in figure 11. There should be a satellite structure in the conduction band, as well. Even if the exact shape of this structure might differ from the results of the low- order theory discussed in section 3, the total intensity of the satellite band should be appreciable. An experimental verification of the many-body struc- ture would be of great aid for the further development of the theory. As regards the position of the core levels there seems to be a good agreement between theory and experiments but further experimental and theoreti- cal work on this problem would help to clarify how point defects polarize and distort their surroundings. 5.2. Soft X-Ray Emission We consider only the simplest possible case where we can assume complete relaxation of the Fermi gas around the hole before the emission takes place. This limits the approximate validity to the light metallic ele- ments such as Li, Be, Na, Mg, Al and K and excludes e.g. all transition metals. Through the recent work by Nozières and de Dominicis and others [4-6] the possibility of a singular structure at the Fermi edge seems to be well established. The magnitude of this Fermi edge peak and the influence of this effect on the intensity at ener- gies outside the immediate vicinity of the edge is how- ever so far unsettled, although important progress has been made [38]. As regards the main band it is clear that the presence of the core hole will give an enhance- ment of the intensity [1]. The actual magnitude of this enhancement factor and its variation over the main band is so far not well-known. To obtain the one-elec- tron density of states in the main band from experi- ments seems to require both careful calculations of dipole matrix elements and better estimates of the enhancement factor. Below the main band the edge for plasmon produc- tion is well established experimentally [1,39]. The in- tensity of the satellite structure is strongly affected by intricate cancellation mechanisms due to the presence of the core hole and the theoretical predictions are un- certain. The plasmaron edge (cf. figs. 5 and 6) has been searched for in Al, but the experiment was not conclu- sive [40]. As discussed in section 3, it is not clear whether this edge is a feature of an exact solution of the problem or just a result of the low order treatment. A clear experimental confirmation or dismissal would be of great aid for the further study of these many-body ef. fects. 5.3. X-Ray Absorption Consideration of these experiments requires a treat- ment of final state interactions but in the absence of a detailed theory for these we shall here take a simple point of view and treat the structure in the one-electron spectra as additional levels or groups of levels in a one- electron scheme. A similar discussion of plasmon ef- fects in x-ray absorption in metals was given by Ferrell [41]. Due to the presence of satellite structure in both core and conduction band spectra there should be a characteristic structure above the threshold [36]. Accurate x-ray absorption spectra for simple metals have recently [42] been obtained, which show an edge anomaly very similar to what one may expect from the Mahan exciton effect [4-6]. The fine structure of the absorption coefficient for the LII,III transition in mag- nesium is shown in figure 14 [43]. Immediately above the edge there is more detailed structure which 244 i l ſ. l l | l | l L(drbitrl 9- º 8- | 7- |- 6- }*. H 8- 5- |- 75- º l- - —I- T I T }- 80 85 90 95 (eV) density. The position of the plasmaron peak should be given better at higher densities and the discrepancy with experiment at lower densities is in no way alarm- ing. More serious is the fact that Langreth’s calculation has shown the higher order effects to be strong. The strong peak could of course also be due to many-body effects involving final-state interactions. 5.4. Photoemission in the Ultraviolet The basic mechanisms of the photoemission process are still not well understood. It may be regarded in one extreme as a pure surface effect and in another as a pure volume effect. In the latter case we have to ac- count for the very important inelastic scattering effects of the outgoing photoelectrons. If the surface effects are not too strong, and if we can sort out the inelasti- cally scattered electrons, the photoemission results in the ultraviolet should reflect structure due to the satel- lite band of the conduction electrons. There is some hint of such a structure in the recent results for cesium [46] (fig. 15) at an energy of about 4 eV below the Fermi edge. However, the structure could also be due to a loss of two surface plasmons [46]. 5 § 75 ºn go in in 1% to eV, FIGURE 14. The fine structure of the absorption coefficient for the Lii.111 transition in magnesium [43]. possibly could be due to ordinary band structure effects in the final state. At still higher energies comes a strong peak [43-45]. We will argue that this peak may be a many-body ef- fect. In table 1 we give results for the positions of the peaks measured from the threshold [43], and compare with the position of the core plasmaron peak discussed in section 3 (cf. fig. 11). There is good agreement for aluminum but not for the other metals. The important point, however, is that epeal:/op is a very smooth function of the electron density. Actually its value is closely 0.8 rs in all cases. This indicates that the peak is associated with properties of the electron gas rather than being due to oscillator strength effects. The latter would be connected with the properties of the ion core and there is then no obvious reason to ex- pect a regular variation with the conduction electron TABLE 1. Position of the strong peak in soft x-ray absorption and comparison with the location of the satellite peak (epm) in figure l l . Element Ts epeak, eV epeak/op epm/op Al...................... 2.07 24 1.6 1.7 Mg..................... 2.66 22 2.1 1.8 Na..................... 4.00 18 3.2 2.1 ha) = |O2 eV | O | | | I l I | #–4–4–4–5–4–3–3 KINETIC ENERGY (eV) FIGURE 15. Photoelectron energy distribution curves at ha) = 10.2 eV for Na, K, Rb and Cs [46]. The horizontal bars indicate the values of the surface (full curve) and volume (dashed) plasmons. 245 5.5. Optical Absorption in the Ultraviolet The description of optical properties of metals in terms of Drude (see e.g. refs 47,48) and interband con- tributions is often in qualitative agreement with experi- ment. To go beyond that description we need to know the dielectric function including final state interactions. The many-body effects may show up as changes in in- tensity and as new structures. Attempts to account for the former have been made e.g. by Mahan [49], who considered the contributions from virtually exchanged plasmons. Absorption caused by electron gas effects has been considered by Hopfield [50]. He observed that while the electron gas by itself cannot absorb radiation the effect of a weak perturbing potential from phonons or disorder is enough to provide the necessary momentum conservation for the absorption process and thus allow the plasmon resonances of the electron gas to show up. A straightforward way to extend the one-electron joint density of states expression is to make a convolu- tion of the spectral weights of occupied and unoccupied states. This has been done for semiconductors by Bar- dasis and Hone [51] who in addition considered vertex corrections. They obtained improved agreement with experiment. Calculations for metals by the convolution approximation indicate the existence of a plasmon-in- duced structure at photon energies above Eg-Hop [52], where Ea is the interband threshold energy. There are some experimental indications of structure beyond the ordinary interband absorption in this energy region. 5.6. The Compton Effect X-ray scattering from an electron gas in the regime of large momentum transfer is a direct measure of the I(p) (au) AN Free-electron Band structure (, or relation To tal Experimetht sº N | ? (94a) | +– o's ſº f FIGURE 16. The linear momentum distribution for Li [54,55]. one-dimensional momentum distribution [53]. From recent measurements of the Compton profiles of Li, Na and Al the linear momentum distribution has been derived [54]. The result for Li is shown in figure 16. In section 3 approximate values of the momentum distribution for an electron gas have been shown (fig. 4). The corresponding linear momentum distribution shows a too small reduction compared to the free-elec- tron case to reproduce the experimental results for Li (fig. 16) and Al, while for Na the electron gas curve falls almost entirely within the experimental region. As band structure effects could be expected to be more important in Li and Al than in Na, this kind of cor- rection has also been calculated using the OPW method [55]. As shown in figure 16 these effects reduce the discrepancy, even if a quantitative agree- ment has not been obtained. This kind of experiment provides a way of illuminat- ing another aspect of the distribution of electrons in metals and provides a useful way of checking theoreti- cal models including many-body interactions. 6. Concluding Remarks This paper has presented a discussion of the possible nature of many-electron effects on the density of states. It is based on a study of the one-electron spectrum in- cluding interactions and points out the existence of characteristic satellite structure in the density of states of electrons and holes in simple metals. Considering the joint density of states of electrons and holes as in the theory of interband transitions certain predictions about possible effects in x-ray and optical spectra can be made. All this material is however only qualitative and tentative. The structure in the one-particle spec- trum has been calculated in low order and considerable changes may result by including higher-order effects. Further, the convolution of the electron and hole spec- trum implies neglecting the final state interaction between the electron and the hole. The final state in- teractions may partly cancel out the structure in the electron and hole spectrum, and it leads to charac- teristic new effects such as edge singularities, and of course also gives an overall distortion of the spectrum. With all these reservations, however, the discussion points out the existence of a number of possible in- teresting effects which offer a challenge for further stu- dy. In assessing the possibility of pursuing this ap- proach to obtain quantitative theoretical results one has to consider critically the present state of the art with re- gard to ordinary band theory. Indeed, rather little has yet been done in a quantitative way to calculate spectra 246 especially with regard to oscillator strengths. Such more detailed knowledge from energy band calcula- tions also forms a necessary prerequisite for making quantitative statements about many-electron effects. [1] For a general review, see L. Hedin and S. Lundqvist, Solid [2] [3] [4] [5] [6] [7] [8] [9] 7. References State Physics 23, (F. Seitz, D: Turnbull and H. Ehrenreich, Editors (Academic Press, New York, 1969). Hedin, L., in “Soft X-Ray Spectra and the Electronic Structure of Metals and Materials,” D. 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Rev. 129, 1399 (1963). Geldart, D. J. W., and Vasko, S. H., Can. J. Phys. 44, 2137 (1966). Singwi, K. S., Tosi, M. P., Land, R. H., and Sjólander, A., Phys. Rev. 176, 589 (1968). Kleinman, L., Phys. Rev. 172, 383 (1968). Langreth, D.C., Phys. Rev. 181,753 (1969). Lundqvist, B. I., Phys. Kondens. Materie 6, 193 (1967); 6, 206 (1967). Hedin, L., Lundqvist, B. I., and Lundqvist, S., Intern. J. Quantum Chem. IS (1967) 791. Hedin, L., Lundqvist, B. I., and Lundqvist, S., Solid State Comm. 5, 237 (1967). Lundqvist, B. I., Phys. Stat. Sol. 32,273 (1969). Slater, J. C., Phys. Rev. 81,385 (1951). [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] Gáspár, R., Acta Phys. Hung. 3, 263 (1954). Kohn, W., and Sham, L. J., Phys. Rev. 140, A1133 (1965). Hedin, L., Lundqvist, B. I., and Lundqvist, S. (to be published). Langreth, D.C. (to be published). e.g., Pines, D., and Nozières, P., “The Theory of Quantum Liquids,” Vol. I, (Benjamin, New York, 1966). Grimvall, G., J. Phys. Chem. Solids 29, 1221 (1968); Phys. Kon- dens. Materie 6, 15 (1967). Ashcroft, N. W., and Wilkins, J. W., Phys. Letters 14, 285 (1965). McMillan, W. L., and Rowell, J. M., in “Superconductivity,” R. D. Parks, Editor (Dekker, Inc., New York, 1969). Engelsberg, S., and Schrieffer, J. R., Phys. Rev. 131, 993 (1963). Hedin, L., Ark. Fys. 30, 231 (1965). Lundqvist, B. I., Phys. Kondens. Materie 9,236 (1969). Doniach, S., and Sunjić, M. (to be published). Ausman, G. A., Jr., and Glick, A. (to be published). Rooke, G. A., Phys. Letters 3, 234 (1963). Cuthill, J. R., Dobbyn, R. C., McAlister, A. J., and Williams, M. L., Phys. Rev. 174, 515 (1968). Ferrell, R. A., Rev. Mod. Phys. 28, 308 (1956). Haensel, R., Keitel, G., Schreiber, P., Sonntag, B., and Kunz, C. (to be published). Haensel, R., and Kunz, C. (private communication). Fomichev, V. A., and Lukirskii, A. P., Soviet Physics–Solid State 8, 1674 (1967); Sagawa, T., Iguchi, Y., Sasanuma, M., Ejiri, A., Fujiwara, S., Yokota, M., Yamaguchi, S., Nakamura, M., Sasaki, T., and Oshio, T., J. Phys. Soc. Japan 21, 2602 (1966); Codling, K., and Madden, R. P., Phys. Rev. 167, 587 (1968). Watanabe, H., J. Appl. Phys. 3, 804 (1964); Swanson, N., and Powell, C. J., Phys. Rev. 167, 592 (1968). Smith, N. V., and Spicer, W. E., Phys. Rev. Letters 23, 769 (1969). Nettel, S. J., Phys. Rev. 150, 421 (1966); Miskovsky, N. M., and Cutler, P. H., Solid State Comm. 7, 253 (1969). Haga, E., and Aisaka, T., J. Phys. Soc. Japan 22,987 (1967). Mahan, G. D., Phys. Letters 24A, 708 (1967). Hopfield, J. J., Phys. Rev. 139, A419 (1965). Bardasis, A., and Hone, D., Phys. Rev. 153, 849 (1967); see also Brust, D., and Kane, E. O., Phys. Rev. 176,894 (1968). Lundqvist, B. I., and Lydén, C. (to be published). Platzman, P. M., and Tzoar, N., Phys. Rev. 139, A410 (1965). Phillips, W. C., and Weiss, R. J., Phys. Rev. 171, 790 (1968). Lundqvist, B. I., and Lydén, C. (to be published). 247 Discussion on “Beyond the One-Electron Approximation: Density of States for Interacting Electrons" by L. Hedin, B. Lundqvist, and S. Lundqvist (Chalmers University) W. Kohn (Univ. of California): Dr. Lundqvist unfortu- nately brought in the Slater and the 2/3 exchange cor- rection. Dr. Lundqvist had his own interpretation — he had a plot there and said that the reason why these cor- rections work is that they include things which the authors do not know they include. The corrections do not include anything except exchange. The literature shows very clearly and gives explicit formulas for how to include correlations in the locally uniform electron gas model. The question of whether you use 2/3 or 1 or an intermediate value in the exchange correction has a unique answer in every particular problem. For years now there has been a certain lack of rationalism here, although there is in every case, a clear, unique choice of how to treat exchange and how to treat correlation within the locally uniform electron gas model. All of these considerations are, of course, confined to that model and if that is not good enough, you have to go beyond it. Nevertheless, there is around an impression that there is a controversy about the exchange poten- tial. I just want to say there is no controversy. S. Lundqvist (Chalmers Univ.); I agree on the whole with this point of view. K. H. Johnson (MIT): The electron gas has very little to do with real solids. The successful implementation of energy band theory on real materials has depended rather critically on the adoption of approximate exchange potentials based on local electronic charge density (e.g., Slater, Kohn-Sham, etc.). Would it not be worthwhile to explore the possibilities of developing local approximations to the true nonlocal self-energy or mass operator, which would attempt to get at the ef- fects of electron-electron correlation on the band struc- tures and densities of states of real solids? S. Lundqvist (Chalmers Univ.); Professor Kohn and Dr. Sham have done considerable work on this problem and suggested precisely, this procedure. They start from the energy dependent and non-local self-energy and convert this into a kind of consistent local approxi- mation. We have also developed a similar approach, but in order to avoid the problems connected with large density gradients, we apply this procedure to the valence electrons only and described the ions in the or- dinary way, whereas my impressions from the papers by Kohn and Sham is that they intended to use a local density approximation for the total electron density. Maybe Professor Kohn would like to add something? W. Kohn (Univ. of California): I have very little to add to what you said. I am in general agreement with your point of view. K. H. Johnson (MIT): Yes, but I think the nature of the proof is such that it breaks down if the density changes are too severe, i.e., if the system is too in- homogeneous. Is that correct? W. Kohn (Univ. of California): Yes, that is correct, if you have extreme inhomogeneous systems, and by the way, we just completed some work on an example of such a system, namely on a metal surface. The assump- tions of the proof are a formal matter. In reality, the results hold remarkably well, even for rather rapid den- sity changes. I had a discussion with Dr. Hedin yester- day about how to treat the core electrons and we have been concerned with that and I would say, if I un- derstand you correctly, I am again in general agree- ment with Dr. Lundqvist that the core electrons are suf- ficiently different from an electron gas that they must, in effect, be handled differently. Finally, concerning in- homogeneities, I don’t know if Dr. Herman is in the au- dience now, but I suggest that perhaps he would like to add some comments. He is very much concerned with this. F. Herman (IBM): I would just like to ask Professor Lundqvist what his conclusions would be if you did the same problem and included some inhomogeneities? S. Lundqvist (Chalmers Univ.); This would in effect amount to a full self-energy calculation including Bloch electron states and all that. That has not been done. M. Harrison (Michigan State Univ.): Are there any ex- pected effects of the electron-plasmon interaction on bound states about a charged impurity center, or per- haps on scattering processes, either on level shifts or on level broadening, particularly on materials with large r,” 248 S. Lundqvist (Chalmers Univ.); Yes, we believe that this effect would occur. J. R. Schrieffer (Univ. of Pennsylvania): Back to this exchange question—if one takes the calculation that you are speaking about for the exchange and includes correlation effects which keep electrons apart then would the effective exchange be reduced towards 2/3 or even to 1/2 or 1/4? S. Lundqvist (Chalmers Univ.); The answer is that the sum of these two effects appears in the curve I was showing. If you look at the contribution of these com- bined effects to the self energy, you find that it would correspond more to a classic correlation hole which is actually more important that the screened exchange. Screening of the exchange introduces correlations, that is true, but the correlation hole around a particle is a major contribution. J. R. Schrieffer (Univ. of Pennsylvania): I am afraid I did not explain myself very carefully. There is a question of a screened U and then a correlated screened U. If you think of a low density gas, for exam- ple, you can have two different physical effects. One is to screen the effective interaction. That reduces the strength by the usual dielectric effects. With the screened interaction, one can calculate an interaction between say a pair of particles. If you go to the Born ap- proximation, that is the screened exchange. If you go beyond the Born approximation, that is scattering, and one gets a new effective interaction. The new interac- tion for the low density Fermi gas, for example, as in nickel, we know is less than U and it is limited in strength by the relative bandwidth. Therefore, the ef- fective exchange is reduced for two reasons; one is screening and the other is correlation. Would you ex- pect the correlation effect on the exchange to be ap- preciable? S. Lundqvist (Chalmers Univ.); I would not dare to guess what happens at very low densities. I have had too little experience with that problem. L. N. Cooper (Brown Univ.); Perhaps Walter Kohn would dare to guess. W. Kohn (Univ. of California): I want to point out just one little thing and not try to answer your question completely, but one thing is evident. The correlation correction has a completely different density depen- dence, and the rational way to handle it is to do it right. In the very high density limit it becomes negligible compared to the exchange. It does not go with the same power of the density, and so certainly the way not to handle it is to just put a constant in front and say it is somewhere between 2/3 and 1. 249 CHAIRMEN: R. A. Farrell C. S. Koonce RAPPoRTEUR. L. Hedin Excitonic Effects in X-Ray Transitions in Metals” G. D. McIlhan 1 Institute of Theoretical Science and Department of Physics, University of Oregon, Eugene, Oregon 94703 2 and General Electric Research and Development Center, Schenectady, New York 12301 In the study of soft x-ray transitions in solids, there has always been some hope that the results pro- vide a direct measure of the density of states. This assumes that (a) matrix element variations over the band and (b) final state interactions are small. Both of these assumptions are now known to be incorrect. To illustrate the possible strength of these effects, two approximate calculations are presented: the one electron oscillator strength of a simple bec metal as a function of energy; and the strength of the Nozieres – DeDominicis singularity at threshold, with phase shifts estimated from an assumed Yukawa interaction between conduction electrons and core hole. Key words: Density of states; exciton; many body effects; phase shifts; soft x-ray; transition probability. I. Introduction The absorption of a photon can cause an electron to change its state. The traditional viewpoint of this transi- tion assumes that if one knew the electrons initial lº and final lif state, then the oscillator strength was simply proportional to the square of the momentum matrix element. p-ſºnºpº (1.1) This simple viewpoint is now known to be incorrect. The proper picture is that the optical transition creates an electron and hole, and these two excitations interact with each other, and each separately with their environ- ment. The coulomb scattering between the electron and hole is called the exciton effect, named after the first known example of Frankel excitons in solids. The electron and hole can also emit phonons, suffer elec- tron-electron collisions, etc. The sum of all of these processes are called final-state interactions. The rate *An invited paper presented at the 3d Materials Research Symposium, Electronic Density of States, November 3-6, 1969, Gaithersburg, Md. 'Alfred P. Sloan– Research Fellow. * Research Supported by National Science Foundation Grant. of optical absorption is affected by these subsequent in- teractions of the electron and hole. Upon reviewing our present understanding of the op- tical properties of solids, one finds that some solids are well understood while others are not. We classify as un- derstood the semiconductors Si and Ge, [1] and simple metals like aluminum [2,3]. It is noteworthy that in these materials the final-state interactions are small: they are small in semiconductors because the large dielectric constant suppresses the exciton effect. Ehrenreich and his co-workers have shown that many body effects are small in aluminum. In these solids the simple one-electron picture implied in (1.1) seems to work quite well. We classify wide band gap insulators [4], and metals such as copper [5], as solids whose optical properties are not well understood. There is not yet good agree- ment between theory and experiment in these materi- als. The dominant optical transitions in these materials create hole states which are heavy, and which are sub- ject to significant final state interactions [3]. We speculate that final state interactions are important in these materials, and explain the difference between theory and experiments. In the study of x-ray transition in solids, there has al- ways been some hope that the results provide a direct 253 measure of the density of states [6]. This presumes that (a) matrix element variations over the band are small, and (b) final state interactions are small. Both of these assumptions are now known to be incorrect [7,8,9]. The importance of exciton effects in x-ray transitions was reported in an earlier reference [10]. The effects are large if the hole is highly localized, and if the con- duction band of the electron is isotropic. Because the coulomb scattering between the electron and hole oc- curs in an electron gas, the electron can only scatter into states not already occupied. This Fermi-Dirac ex- clusion, as well as the exchange interaction among the electrons, causes the problem to resemble the Kondo effect more than the Wannier-exciton case. The hole also has a large interaction with the entire electron gas—this leads to a renormalization effect which in- hibits the x-ray transition near threshold [11]. Nozieres and DeLominicis [12] have shown that the exciton and renormalization effects combine to give a threshold behavior for e2(a)) eſo)-;(zº)"0(o-o. (12) (O - (0T where or is the absorption threshold frequency, £o ~ EF is a characteristic band energy, and oi is a function of the phase shifts 61 of electrons at the Fermi energy scattering from the static potential of the hole. CO 2 a-#-2S (21+) () (13. l- 0 The term 261/7 is the exciton part which tends to make the threshold singular, while the second term in (1.3) arises from renormalization. The angular momentum l is that of the conduction electron [13]. If the x-ray hole has s-symmetry, the conduction electron must have p- symmetry (l= 1). If the hole has p-symmetry, the con- duction electron can have either s- or d-symmetry; in this case (1.2) has a separate term for each symmetry type. The x-ray spectra can only be unravelled with a knowledge of the phase shifts 61—these are the phase shifts for simple electron-hole scattering. These have been calculated for a free electron gas by Ausman and Glick [13]. They find oo - 0, and on 30 for l = 1 so that singularities only occur for the l=0 case (p-state hole). These results qualitatively agree with the experimental results. We have independently calculated the phase shifts, and we describe our result in section II. We have con- cluded that these phase shifts are qualitatively correct but for the wrong reasons. The wrong potential is used in the calculation, but it does not seem to make much difference in this case. II. Phase Shifts The present calculations have been performed by as- suming that the screened electron-hole interaction has the Yukawa form e? V(r) = -- exp (-kar) r where the Fermi-Thomas screening length is k?= 67Te?no/Ep. for an electron gas of density no and Fermi energy Ef. This is only a crude approximation to the actual poten- tial the electron feels when scattering from the hole. For example, studies of a point charge impurity in an electron gas show that its potential differs from a Yu- kawa at long range where Friedel oscillations occur, and at short range where its potential is less steep [14]. In addition, because the hole is part of an atom, there are term due to the exchange and orthogonality with the atomic wave functions [7,8]. In a pseudo-potential for- malism these latter effects contribute a short-range repulsive term to the interaction. In spite of these short- comings of the Yukawa potential, we believe that it pre- dicts phase shifts which are qualitatively correct. The reasons for this will be presented below. The phase shifts are defined in terms of the eigen- values Ek and eigen functions lik(r) of the electron in the region of the potential. V2 The wave is decomposed into spherical harmonics *(r)=X (21+1)ip (; )0(, ) (2.2) | = 0 In solving for the eigen functions in (2.1), one has the choice of specifying the boundary conditions for states which are plane wave-like outside of the potential re- gion. By choosing standing wave conditions, one is solv- ing for a reaction matrix K1(k, k") and the phase shifts are defined in terms of the diagonal k = k" component [15]. tan ô (k) =-2mkki (k, k) = – 2mk | radition()00, r) (2.3) 254 Similarly, if one chooses outgoing wave conditions, then one is solving for a T-matrix tiſk, k') whose diagonal components give the phase shift [15]. sin Öre”1–– 2mkti(k, k) ==2nkſ rºdrji (kr) V(r) (b1(k, r) (2.4) We have chosen to use the reaction-matrix formalism (2.3), mostly because it is a real function and this sim- plifies computation. The phase shifts 61(k) go to zero as k → Co. Let us now examine their behavior as k → 0. From (2.3) we have ji(kr) - (kr)', so tan 6 (k) -- *ſ rºdry(r)0 (0, r) (2.5) From Levinson's Theorem we know that ô1(0) - TMI where M is the number of bound states of (2.1) with angular momentum l. For example, if Mu-0 we get 8, - klt 1. Whereas if M = 1 we get 61– T-H ch" (where c is some constant) for small k. A typical case is shown in figure 1, where Mo- 1, and Mi- 0 for l = 1. So the s-wave phase shift comes into t with a linear slope, while all other 61- k't' at small k. So in calculating phase shifts, the first thing to de- cide is the number of bound states Mi. This is deter- mined by examining the radial part of Schrödinger's equation (2.1) which we put into dimensionless form 6° l(l-H 1) e-p | * mº ºm-º ºr mºm-mºº ºmº F 2.6 | ão." p” A. à |rd-0 (2.6) p = ker _2me” (27)1/3 x=}=(*)"VF 3 = 2meſh-k; (2.7) The parameter A determines the strength of the poten- tial. Shey and Schwartz [16] have computed the number of bound states which exist for each value of A and l. For s-waves (l=0), they find no bound states exist for Nº 1.68, one bound state for 1.68-A-6.45, two bound states for 6.45<\*14.3, etc. For p-waves (l= 1) bound states only exist for N-9.08, and d-wave bound states exist for A-21.8. In (2.7) we have written A in terms of the density parameter r; for an electron gas. Since metallic densi- ties vary between 2 < r < 6, then the range of A values in metals is 1.8 × A • 3.1. For an electron gas of metallic density, there is one bound state for S-states, and no bound states for any other value of angular momentum l. These are the predictions of the Yukawa potential. We must decide whether these are reasonable conclu- sions for the problem of interest. For an actual point charge in an electron gas, e.g., a proton, there is probably a bound state. For an impurity in a host metal, e.g., Al atoms in Mg, there is probably not a real bound state. This is because the atomic core of the impurity cuts off the attractive potential in the region where it is the strongest; for example, see Ashcroft’s remarks about the cancellation influence of the atomic cores [8]. In the x-ray problem there is certainly an atomic core. Yet there is also certainly a bound state in the potential. This bound state is the x-ray level itself. That is, if an electron scattering from the potential of the hole did not think a bound state existed in the potential, then it would have no inclination to recombine with the hole in the final emission process. Since the emission can occur, a bound state does exist which must be reflected on the phase shifts. The cancellation of the potential in atomic core is caused by the necessity of the conduction electron wave function to be orthogonal to all of the core states. In the x-ray problem, one core electron is absent so that the cancellation requirements are less stringent. The foregoing discussion shows that any phase shifts calculated from a simple Yukawa potential are only going to be qualitatively correct. Yet there is some in- terest in what this simple model predicts. Our method of calculation proceeds by solving directly the scatter- ing equation for the reaction matrix. Kı (k1, k2) = V1(k1, k2) 4m | * , , V1(k1, kg)K (ka, k2) –H T | kádka k} – kā (2.8) where the lº" component of the potential Vi is obtained from the Fourier transform V(ki-k2) of V(r) W.G. k.)=} ſ d6 sin 6 P. (cos 6) V(k1 – k2) •) € T2A, Qı((k} + k} + k})/2k1kg) (2.9) 255 when cos 0 is the angle between ki and k2, and Qi is a Legendre function. We will abbreviate the argument of the Legendre function to write it as Qı(k1,k2). Noyes’ method is used to evaluate (2.8). We find Kı (k, k) = V1(k, k)/(1 - A1(k)) (2.10) _ –4m ſº pºdp A1(k)=#Fº O p? – k” Vi(k, p)f(k, p) (2.11) fi(k, p) = V (k, p) +=*— | p'*dp'fi(k, p") TV (k, k) Jo p'*— k? – VI (p, p') V1(k, k)] [VI (p, k) V1(k, p") (2.12) One obtains fi(k,p) from (2.12), perhaps by iterating this equation. This is used in (2.11) to obtain A(k), and thus one has the reaction matrix in (2.10). This is an exact result if f(k,p) is found exactly. The Born Ap- proximation result is obtained by setting Ki(k, k)= Vi(k, k). Note that it is natural to write A1(k) as a power series in the interaction strength A=2| ksapin (2.7) A (k) => x*g,”(Elk.) in = 1 (2.13) Each successive power of A corresponds to another iteration of the equation (2.12) which determines fi(k, p). For example, the first term is 1 A-–tº–_ | * gº)(k) TQ (k. k.) | For l =0 we can evaluate the integral and express the integral as a summation. ks CC - ) l gº)(k) T2RO,(E,TE) X. TET [2p' sin (pl) * * * / 1–1 dp #0 (ºp): (2.14) – sin (2lp)] p=[1+4kºſk?]-1/2 (2.15) q2 = tan-1 (2k/ks) Q0(k. k.) =# ln (1 + 4k?/k?) This form is convenient for numerical computation. In figure 1 is shown the s-wave phase shift calculated by approximating Ao by the first term in (2.13) Q0(k, k) kap [l – Agº)(k)] Also shown in figure 1 are the phase shifts 61 and 62. We also calculated the first correction term gº")(k) to tan 60(k) = 3.0 PHASE SHIFTS rs = 4.0 k/k F FIGURE 1. The phase shifts 61 calculated by a Yukawa potential. This potential represents a coulomb interaction of unit charge, with Fermi-Thomas screening. The phase shift 62 is calculated in the Born Approximation, while 60 and 61 have corrections for multiple scattering. the p-wave phase shift, and this correction has been added into the calculation of the curve labeled 61. This only changes the Born Approximation result by 10 per- cent. The l = 2 phase shift is the Born Approximation result. The multiple scattering terms are small except for the s-wave. In the x-ray transition, the singular effects occur at the Fermi surface, so we are interested in the phase shifts evaluated at kſ. Figure 2 shows the critical ex- |.5 I I | 0.0 - | | | 0.5 2.0 3.0 40 5.0 6.0 FIGURE 2, The variation with electron density rs of the exponents oo and O. 1. These depend upon the phase shifts evaluated at the Fermi surface. Also shown is the renormalization parameter e and the Friedel sum rate result Z. The value of Z is rigorously unity, and our deviations from that value indicate the errors in the calculation. 256 ponents oo and oi, in (1.3) plotted versus electron density rs. Values of ol for l = 2 are essentially equal to – e. Also shown are the values of the Friedel sum Z and the Anderson exponent e, where Z=2 X (21+1)(3/7) l=0 e=2 S (21+1)(6/1)* l=0 The Friedel sum rule should rigorously be given by z = 1. Our deviation ~20 percent from the exact result of z = 1 provides an estimate of the error in the calcula- tion. Another estimate of the accuracy and convergence of the calculational method is obtained by seeing how well it estimates the position of bound states. As k → 0 we get lim 60(k) = tan-1 ( R - 0 2k/ks ) 1 – Ao(0) A bound state is predicted if A0-1. We find that gº)(0) = 1/2 gº)(0)=ln (4/3) - 1/4 If we just use the first term in the expansion (2.13) so Ao(0)=\/2 then the criteria Ao(0) > 1 means A-2. This is an error of 16 percent from the actual criteria A > 1.68. But using two terms gives A/2+ A*(ln 4/3–1/4)-1 or A= 1.765. So a considerable improve. ment in accuracy is obtained by including the second term in (2.13). So we conclude that the series (2.13) con- verges rapidly for the range of N values of interest. Ill. Matrix Elements The change of the matrix element with energy has been discussed before and is known to be a large effect [7,8,9]. We have calculated the change in matrix ele- ment near the first critical point in a bec solid for an x- ray transition from a 1s core level. We used a simple two band model in each 1/12 of the Brillouin Zone [17], so that the energies and eigenstates are 1/2 E- (*, +, )=|}º-º-ort * ilº(r)= Nº | 3,-Eſſºe", r + # |á, -E;|12eir (K-6 | c l * – 1/2 Nk = | (e.-a, -º-º: ; The matrix elements were calculated assuming that the core was a delta function, and no attempt was made to orthogonalize the conduction states to the core. Thus, after averaging over polarizations, the matrix elements 8.T6. (p})==#N}{k}|ák-E.--(k–G)?|3, – Ej =E2k (k–G) Vo) In figure 3 we show the density of states p(E), and also the absorption strength dºk (27)? AE-ſ: (pº-E)-(º-oº-ºo. In this figure, energy has been normalized to Eoi= h°G*/8m so that 3 = E/Eo and v= Vo/E0. The choice v = 0.2 is close to Ham's value for Li (v = 0.23). Indeed, the present calculation was done with Li in mind, since the lack of core orthogonalization should not matter here. (d) DENSITY OF STATES 0. 5 0. 4. |- 0. 2 |- O 3|- 0. - O |.0 20 10| (b) ABSORPTION STRENGTH 0.5H FIGURE 3. The density of states p (e) and absorption strength A (e) for a bec solid such as Li. The energy scale is listed in units Eo- h”G*110/8m, and v= V110/E0. The absorption strength A (e) is defined as the integral of the x-ray matrix element < p * > averaged over the energy band. The curve for v = 0.2 in figure 3(b) has the expected shape for Li [18]. Because the wave function is p-like at the lower critical point 3 = 0.8, the transition is al- lowed and the absorption strength has the same shape as the density of states in figure 3a. The density of states structure at 3 = 1.2 is washed out in A(3) because at the critical point the transition is s-like and largely 417–156 O - 71 - 18 257 t forbidden. The curve for v = –0.2 has the same density of states, but now the upper critical point is p-like and A(3) has the same structure as p(3) near 1.2. Neither the curve v = 0.2 nor v = –0.2 has any striking resem- blance to the density of states [18]. IV. Discussion Exciton effects should influence nearly all parts of the absorption spectra. The singular behavior at the ab- sorption or emission edge is just one prominent feature: Another result of final state interactions is that oscilla- tor strength is moved from one frequency range to another. Often these shifts are small and can be neglected. Yet in most cases the true magnitude of these effects are unknown, because the relevant theoretical calculations are too complicated to do realistically. The calculation of exciton effects is a formidable task which has not been performed properly. Of course Wannier excitons have been studied in detail. Some model calculations exist near critical point edges—the hyperbolic excitons [19.20,21]. But the main optical ab- sorption strength comes from states throughout the Brillouin Zone [5], and all of these states are included in the final state interactions. So before x-ray absorption and emission measure- ments can be used to provide information on the densi- ty of states, two sizable corrections need to be made. One of these is the exciton effect, and the other is the change in the matrix element with energy. V. Acknowledgments I wish to thank N. Ashcroft, L. Parrott, and M. Stoneham for informative discussions. VI. References [1] Dresselhaus, G., and Dresselhaus, M. S., Phys. Rev. 160, 649 (1967). [2] Ehrenreich, H., and Beeferman, L., (private communcation). [3] Spicer, W. E., Phys. Rev. 154,385 (1967). [4] Fong, C. Y., Saslow, W., and Cohen, M. L., Phys. Rev. 168,992 (1968); Park, K., and Stafford, R. G., Phys. Rev. Letters 22, 1426 (1969). [5] Dresselhaus, G., Sol. State Comm. 7,419 (1969); Mueller, F. M., and Phillips, J. C., Phys. Rev. 157, 600 (1967). [6] Tombouliam, D. H., Handbuch der Physik, Vol. 30, (Springer- Verlag, Berlin, 1957), 246-304. [7] Harrison, W. A., Soft X-Ray Band Spectra, edited by D. J. Fabin (Academic Press, New York, 1968), p. 227. [8] Ashcroft, N.W., ibid, p. 249. [9] Rooke, G. A., ibid, p. 3. [10] Mahan, G. D., Phys. Rev. 163,612 (1967). [11] Anderson, P. W., Phys. Rev. Letters 18, 1049 (1967). [12] Nozieres, P., and DeBominicis, C. T., Phys. Rev. 178, 1097 (1969). [13] Ausman, G. A., and Glick, A., Phys. Rev. [14] Langer, J. S., and Vosko, S. H., J. Phys. Chem. Solids 12, 196 (1960). t [15] Goldberger, M. L., and Watson, K. M., Collision Theory (J. Wiley and Sons, New York, 1964), p. 232. [16] Schey, H. M., and Schwartz, J. L., Phys. Rev. 139, Biq28 (1965). [17] Foo, E-Ni, and Hopfield, J. J., Phys. Rev. 173, 635 (1968). [18] Shaw, R. W., Smith, K., and Smith, N. V., Phys. Rev. 178,985 (1969). [19] Duke, C. B., and Segall, B., Phys. Rev. Letters 17, 19 (1966). [20] Hermanson, J., Phys. Rev. Letters 18, 170 (1967). [21] Kane, E. O., Phys. Rev. 180,852 (1969). 258 Discussion on “Excitonic Effects in X-Ray Transitions in Metals” by G. D. Mahan (University of Oregon) J. D. Dow (Princeton Univ.); I would like to make a comment about many-body effects vs one-electron ef- fects in this problem. While I believe that Professor Mahan's mechanism greatly increases our understand- ing of the optical properties of metals, I am not con- vinced that many-body effects are the sole source of anomalies in the x-ray spectra. In particular, I believe that a major portion of the broadening of the image of the lithium Fermi surface is due to gigantic phonon-in- duced core-shifts—a local lattice distortion squeezes the 2s electron, changing the 2s wave function at the nucleus, the effective nuclear charge, and the binding energy of the 1s state. This electron-phonon mechanism is similar to one proposed by Drs. Overhauser and McAlister and is consistent with the x-ray data of Li, Be, and Na (but does not explain the many-body peak in Na). It is also consistent with Knight shift data, and can explain why the x-ray data shows Li to have a broad Fermi surface image, while Na is sharp. Conclusive evidence about the relative importance of such electron-phonon and many-electron effects could be provided by differential studies of the x-ray spectra, using temperature or pressure modulation. G. D. Mahan (Univ. of Oregon): I would like to make a comment on that. I don’t mean to say that all the threshold effects are due to excitons. I think this is par- ticularly true in lithium and also in beryllium. When the X-ray emission spectra bends over in a very round way at threshold some of those effects are probably due to things not involved with the excitons, like band struc- ture effects. Unfortunately, everybody who calculates it, calculates it a different way and gets perfect agree- ment with experiment when they put in band structure effects or electron-electron effects. So I think that I agree with you in that. I think it is worthwhile to try to do the experiment that would test some of these ideas. W. Kohn (Univ. of California): Toward the end of your talk you discussed the question of the adequacy of the Yukawa approximation. It occurred to me that when you knock out an electron from one of the low-lying states, you are starting out with a closed shell and after you have knocked out the electron, that shell, for exam- ple, the L electrons, now constitutes really a degenerate level. The hole has an angular momentum associated with it and so there are several degenerate states. In other words, the situation that you now have (when you consider the interaction of the other elec- trons with this core) is quite similar to the Kondo effect situation. Is there in fact any simple potential that properly represents the interaction of the conduction electrons with the missing charge, or could the degeneracy play an important role? G. D. Mahan (Univ. of Oregon): I would guess that spin flip processes, as you suggest, are probably small. It seems to me that this exchange between a deep core and a conduction electron would be rather small. Is that a reasonable statement to make? W. Kohn (Univ. of California): I don’t see why it is. R. A. Ferrell (Univ. of Maryland): But is he referring only to spin flip? Couldn’t there be some angular, some orbital-angular momentum exchange? That could be a stronger coupling, couldn’t it? P. M. Platzman (Bell Telephone Labs.): You treat the emission and absorption problems on the same footing. I do not understand this. The transition takes place in a time T-h/E where T's h/op, the time for the electron gas to adjust. Thus, in absorption the appropriate final state, (a sort of Frank Condon argument tells us) is to be calculated in the absence of the hole. In emission the hole is there for a long time and then vanishes suddenly so that the appropriate wave functions are those in the presence of the hole. Could you amplify on this point? G. D. Mahan (Univ. of Oregon): I don’t agree. I think that in the absorption, the wave functions would adjust and in the emission that they have adjusted already. P. M. Platzman (Bell Telephone Labs.): I don’t un- derstand. It obviously takes time of the order of T ~ lſop for the electrons to adjust in the presence of a hole. The time of transition for absorption is much shorter than that. The energy involved is 50 to 100 volts. I don’t understand how the electrons readjust. It must be some kind of Frank-Condon principle where you make transitions to the states that correspond to the 259 electron gas before you make the hole, not after you make the hole. R. A. Ferrell (Univ. of Maryland): Is the implication that the kind of potential one should use is different from the usual static screened potential if one wants to talk about the effect of the core on the conduction elec- trons? G. D. Mahan (Univ. of Oregon): You’re asserting that the renormalization term shouldn’t be there? R. A. Ferrell (Univ. of Maryland): Would that mean that the Friedel sum rule should not actually be obeyed by the potential which would more realistically describe the potential acting on the conduction elec- trons? In other words, there isn’t time for static screen- ing to set in? G. D. Mahan (Univ. of Oregon): Yes. J. R. Schrieffer (Univ. of Pennsylvania): I would like to return to the question Prof. Kohn brought up for I have a closely related question here. If you are making a p-state hole, the potential is not spherically symmet- ric, and therefore you have an electron scattering off an asymmetric potential. One question is: To what extent is a phase shift analysis an appropriate language in which to discuss this scattering? Secondly, in addition to the static asymmetry of the potential there is a dynamic aspect of sharing of the hole in different mag- netic quantum states in a given atom. The third question: What is the effect of the wandering of the hole from atom to atom? G. D. Mahan (Univ. of Oregon): The last answer is easy. I don’t think the hole wanders. The first question is, what about using phase shifts? If you take a simple model where the p state is reasonably tightly bound and try to calculate what the potential is, it comes out spherically symmetric. J. R. Schrieffer (Univ. of Pennsylvania): Are you sug- gesting that there is a motional averaging to remove the asymmetry corresponding to an electron in a given magnetic substate. R. A. Ferrell (Univ. of Maryland): The question is, should you average or should you consider the potential set up by a certain core state in a certain magnetic sub- state? G. D. Mahan (Univ. of Oregon): I agree. If you don’t average, then you are going to get a definite lobe in your potential in some directions. W. Kohn (Univ. of California): As Bob said, that is very much part of the same question that I asked. Since the potential in a given magnetic state (let's talk about angular momentum rather than spin for a moment), would not be spherically symmetric, therefore when an S-like electron interacts with it, it might very well go into a d-state. There will be, let us say, an s-d coupling. Then to conserve angular momentum, that core would in general make a transition to another state. R. A. Ferrell (Univ. of Maryland): You probably agree that this is something to look at. It might turn out to be a small effect. A. J. Freeman (Northwestern Univ.); Dave Shirley spoke yesterday about some x-ray emission experi- ments that he and Chuck Fadley did at Berkeley. He did not have time to talk about some recent experi- ments in which they were able to measure the splitting of the binding energy of core s and p electrons. The splittings of the 3p binding energy appear to reflect the multiplet structure of the final possible states of the system with a hole in the 3p level. This multiplet splitting gives experimental evidence for the im- portance of taking a degeneracy effect into account, as just stated by Schrieffer and Kohn. F. Brouers (Univ. Liege): There are two problems if you calculate this exciton effect for the whole band. You can only find an exact solution at the Fermi edge; when you try to calculate the band, the terms which give this spike have too big an effect in a first order ap- proximation in the electron-hole effective potential. The problem is to see if electron-electron interactions which are missed in the Nozières and di Dominicis paper, for instance, can reproduce something which is like a free-electron band between the high and low energy features which one can presently deal with. But if you introduce electron-electron interactions, you have troubles because the theorem of linked cancella- tion is not correct in this particular case. You have terms which are divergent in a first order theory throughout the band. So one of the points is to use some tricks from nonlinear oscillation theory to sum up all these secular terms and to put them in the exponential. You then have just a shift in the energy scale which has no physical importance. The problem with this method is to see if the addition of electron-electron correlations to the contribution which gives this singular spike can reproduce something which is very like a one-electron theory over the main band. R. A. Ferrell (Univ. of Maryland): Thank you. I think it would take us too far afield to go into the details of this interesting question, but I believe it is apparent that it is in the direction of Platzman's question of mak- ing a more realistic treatment of the potential and its time dependence. 260 Vibronic Exciton Density of States in Some Molecular Crystals R. Kopelman and J. C. Laufer Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48104 Excited states of molecular crystals, which happen to be the majority of known crystals, are almost always classified as excitonic. The largest class of well studied systems are very closely described by the Frenkel model, and a majority of these systems can actually be described by a special case of the Frenkel model, one with steeply falling-off intersite (intermolecular) interactions. In this specific model the expression for the band structure depends only on the interchange symmetry of the crystal, with a small number of intersite parameters. Examples are given for some aromatic crystals, comparing band structures derived from theoretically calculated parameters, experimentally derived parameters, and completely experimentally derived band structure. Key words: Anthracene; aromatic crystals; benzene; excitonic density of states; molecular crystals; naphthalene. 1. Theory Molecular crystals are characterized by intermolecu- lar ground state energies which are much smaller than the intramolecular energies within the molecules them- selves. The intermolecular forces in these crystals are considered to be weak and of short range. Con- sequently, it is not surprising that many exciton states in molecular crystals are best described by the Frenkel model rather than the Wannier one [1]. The crystal exciton function in the Frenkel model is constructed as a linear combination of the wavefunc- tions called “one-site excitons” which are equal in number to the number of molecules per primitive unit cell. For each “one-site exciton,” the excitation for- mally resides on one sublattice only, and all the sublat- tices are of the same type if there exist symmetry operations of the crystal which operate on one of these sublattices and generate the others. These operations have been designated interchange operations and they can be formed into groups, called interchange groups [2]. Interchange groups are basic to the symmetry of crystals. For example, the unit cell group of a crystal is the direct product or union of the site group (of the molecules in the crystal) and an interchange group [2]. One-site Frenkel exciton functions (bſ, are con- structed [3] from a tentative wavefunction which is the product of a crystal function Xīq, representing a local- ized electronic, vibrational, or vibronic excitation f of a molecule located on the q" site of the n" primitive cell, and the other unexcited site functions of the crystal, X} as Nſh dº, Axiſ II X%. (1) h';* h In the above equation A is an antisymmetrizing opera- tor, N is the number of molecules in the crystal, and h is the number of sublattices which also is the order of the interchange group. From Öhn the one-site exciton functions in the Bloch representation can be formed, Nſh - pſ(k) = (NIH)-º 2. exp (ik Rºn) dº. (2) where k is the reduced wave vector, and Rna denotes the position of q" site in the nº primitive cell with respect to a common crystal origin. Rho can be written 31S Rna = r). -- 7 nq . (3) where rh is a vector from the crystal origin to the origin of the n” primitive cell and Tha is a vector from the nº primitive cell origin to its q" site. In our model the crystal Hamiltonian H is the sum of a site-adapted-molecular Hamiltonian Hº and an in- tersite Hamiltonian H': .V/h h H"- S S H'(nq) (4) n=1 q= 1 261 h h § | N N/h h H' = + T O m n r f f 2 'a'). 2n I > > (1–6 anºğan') H'(nq, n' q"). (5) – 1 | q= For the case of a single nondegenerate excited state f. the N × N Hamiltonian matrix is blocked off along the main diagonal into N/h h X h Hermitian submatrices, each submatrix being characterized by a value of k. The off-diagonal elements .2%(k) in each submatrix are due to the interactions between the h sublattices. They are the sum over the entire lattice of interaction terms between molecules on sites of type q and q', modified by phase factors. The diagonal elements 4,0s) on have k-independent terms which arise from the matrix elements in which simultaneously q=q' and n=n', and k-dependent terms which solely arise from the pairwise interactions of the molecules. The general matrix ele- ment is (pſk) |H|q'ſ,(k)) which we have designated 2%,(k). All matrix elements between states of different k vanish by symmetry. Expanding 2%,(k), we write 4. (h)-X [exp (ik (r-1) n' = 1 |exp (ik) º (rº – rh)] ſºlo.uk. (6) Separating the k-dependent terms 2%,(k) from the k- independent terms, we write .9% (k) = (&ſ-HDſ)60ſ -- Lºº (k) (7) where e-ſoºk (8) and D.ſ = ſ d'ſ HºdſºdR. (9) Dſ is a term representing a “Van der Waals-type in- teraction” between sites, and is the energy required to excite a single molecular site locally in the presence of the rest of the crystal. It shifts the exciton band energy from 3 ſ, the energy of the molecular site functions. Usually the major Van der Waals energy shift is 37- ãºſ, where 3°ſ is the free molecular excitation energy. ãºſ, the “ideal mixed crystal level,” can be derived from mixed crystal data [3]. The expression for a diagonal element Lſ,(k) written out fully is 2%(k) =&ſ-Dſ + X, eik R [26,(x{x}|H|x}x}) R – (X'X}|H"|x|x})], (10) where 6m = 1 for a singlet state and 6m = 0 for a triplet or other multiplet state. R is Raq; X, is X., where the subscript of og indicates the origin of sublattice q, and X, is Xfin, where the subscript R, indicates the site at distance Rng from the origin [4]. The off-diagonal ele- ments Lºſk) are all k-dependent and are conveniently expressed at this point as Nſh Lſ, (k) = X exp [ik (tº - T,)] exp [ik n' = 1 (r) — r,)|X ſ (b | H'(b),(b.R (1-600). (11) In general, in order to find the eigenvalues for a par- ticular k, it would be necessary to solve an h X h secular equation. The expression for the eigenvalues would be a function of all the elements of the secular determinant. Such an expression for a system where h =2 is given by Knox [4]: E(k) =#| ? (k) + 2 m (k) || + {}[..? (k) –%| m (k) | + |%|n(k) | 2}12, (12) where the two eigenvalues are Et and E-, and the two sublattices are labelled I and II. Although sublattices I and II are spacially equivalent, .2%; (k) and .%ft iſk) are not generally equal because they are k-dependent. The reduced wavevector k designates a phase factor relationship with the vectors R which has a spacial orientation that is usually not isotropic with respect to the sublattices I and II. In certain cases, such as when k = 0, 2% (k) equals Z, (k). Equation (12) can then be written: Eſ(k) · = 1% (k) = 2^{n(k). I II (13) The eigenfunctions for the two states then become simply v(loº-ºº: [ºpſ (k) + (pſ (k)], (14) The above example is for a system of two interchange equivalent sublattices. For a system having an in- terchange group G of order h, there will be h interchange equivalent sublattices. If .2%(k) is equal to *** (k) for all q', an eq (14) analog for the general case can be constructed group theoretically from the in- terchange group to give: - | h * *(k)=º X ºbſk), (15) q= 1 262 where aff’s are coefficients corresponding to the o” irreducible representation of G. The corresponding eigenvalues then become Eſe (k) =S a; Zºl,(k). (16) IQ Q=I Equations (15) and (16) can be obtained [5] for any k if certain integrals are excluded from the summation in eq (10). Consider the effect of an interchange operation A on sublattice q. Every site in q will be mapped into an interchange equivalent site in sublattice q'. In other words, A operating on q generates q'. Equation (10) for 2%,(k) will be identical to that for 2%(k) in Eſ, Dſ, and all corresponding terms within the brackets. But AR generates R' in q' and A R = R # R (17) for general R and k . R # k R' for general k (18) becomes k is a vector in the reciprocal lattice and is not operated on by A. However, for those R which are left invariant (to within a translation) by A., R is equal to R', and corresponding terms in the summations of .2% (k) and Zºº.(k) are equal. The sublattice q' may also be generated by another interchange operation B, which will fix other R's such that R = BR = R' (19) The terms in L.,(k) and L.,(k) associated with these R are also equal. By using all the interchange opera- tions in the crystal and collecting only those terms in 2,0k) and 2%,(k) which are equal, we can obtain the eqs (15) and (16) for any arbitrary k, provided that the rest of the terms are negligible. By the same procedure we can obtain the restrictions on k that lead to eqs (15) and (16) for any arbitrary R. For an interchange operation A which generates sublat- tice q' from q, there must be a corresponding symmetry operation in the reciprocal lattice which will leave some k invariant. For this k eq (10) will not change in value as A changes 2%,0k) to 2%,(k). Repeating this pro- cedure for all interchange operations, we find all k which satisfy eqs (15) and (16) for general R. All such k have the property that the group of k contains at least one interchange group. An examination of the terms which are retained in <2*(k) for general k shows that they include the nearest translationally equivalent neighbors about the origin of the sublattice. Since vibronic interactions in molecular crystals are often extremely short-range, next-nearest translationally equivalent neighbors may be dropped from the summation as they make only a negligible con- tribution to .2%(k). Equations (15) and (16) are there- fore believed to be valid expressions for many physi- cally real cases of excitions in molecular crystals. For Frenkel excitons the crystal transition dipole operator M is the sum of the dipole operators on all the crystal sites. The evaluation of the transition moment matrix element (wſ"a"(K")||M|\pſe (k)) by use of eq (15) shows it to be nonvanishing only under certain conditions. The first of these is the usual crystal selection rule, Ak = 0. As long as eq (15) is applicable the transition moment becomes k-independent [5]. An important consequence of this requirement is that every nonvanishing transition moment in a transition between two exciton bands will have the same intensi- ty; the band profile of such a transition will be depen- dent only on the density-of-states functions of the two exciton bands. The second selection rule for the non- vanishing of the transition matrix element is that the direct product o," X or X O.", where or is the interchange group representation of the electric dipole moment, must contain the totally sym- metric representation. The combined selection rules, for all k, are thus those of the interchange group. Ac- tually they turn out [5] to be those of the factor group. It follows from the above development that the band profiles of transitions between two exciton bands can be computed from the density-of-states functions of the two bands. If the low energy band is narrow (~10 cm−4) and the upper band is an order of magnitude broader, then to a good approximation the experimentally ob- served band profile for such a transition will be just the density-of-states function for the upper band [6]. This function can be calculated directly from eq (16) if the values of the excitation exchange integrals ſo, Hoºd, are known. Theoretical calculations of these integrals using molecular wavefunctions have been made, and various experimental techniques have also been used to obtain values for these integrals. The best of these values have been obtained from Davydov splittings [7], mixed crystal data [8], and exciton diffusion mea- surements [9]. 263 The crystals of naphthalene [10] and anthracene [11] both have C3, space groups with two molecules per primitive cell occupying sites of Ci symmetry. By using a C2 interchange group, eq (16) can be written as Eſ(k) =2M, cos (Kaa) +2M, cos (Kºb)+2M, cos (Kee) +2M (ac) cos (Kaa) cos (Koc) – 2M (ac) sin (Kaa) sin (Kcc) +4M in cos (Ka}a) cos (Kołb)+4M, nº cos (Koc) cos (Kºła) × cos (Kºb) F4M, n, sin (Kcc) sin (Ka}a) cos (Kołb), where the summation has been truncated after Summing out to and including the two translationally equivalent neighbors at lattice positions (101) and (101) In eq (20) Kaa = k - a, Kob = k b, Koc= k - c, (21) where a, b, and c are primitive lattice translations in the monoclinic system. Ma, Mb, Mc and M(alo) are the excitation exchange interactions between the site- adapted molecule at the origin and the translationally equivalent molecules at positions (100), (010), (001), and (101) respectively; and MI iſ and MI in are the interac- tions between the molecule at the origin and the in- terchange equivalent molecules at (; # 0) and (; ; 1) respectively. 2. Calculations We have used eq (20) to compute the exciton band profiles of the lowest *B2, states of anthracene and naphthalene. The calculations were performed by com- puter, with each band profile being composed of about 200,000 states. When these states were collected in energy increments of 1 cm−", our density-of-states func- tion was obtained. The band shape was determined solely by the values selected for the M’s of eq (20), the excitation exchange interactions. A similar equation was used to compute the band profile of the lowest *B1u electronic state of benzene. Benzene crystallizes [12] in an orthorhombic lattice of space group D3; (Poco), and has four molecules per primitive cell. We chose a D2 interchange group to construct eigen- values from eq (16). The benzene calculation involved the generation of about 800,000 states. They were collected in energy increments of 0.3 cm−" to obtain the density-of-states function, which is plotted in figure 5. The “B2, band profiles of naphthalene and anthracene, calculated from M values obtained theoretically from molecular orbital calculations by Rice and co-workers [13], are shown in figures 1 and 3, respectively. The corresponding band profiles for M values obtained from experimental data are shown in figures 2 and 4. (20) States 35,000 - 30,000 - 25,000 - 20,000 - lº,000 - 10,000 - º 5,000 - 10 5 O - 5 - l O Energy em" FIGURE 1. *B2u Exciton band of naphthalene calculated from theoretical parameters (see ref [13]). Ma -0.0, Mo - 2.4, M. =0.0, Monic) = 0.0, Min =–1.4 M, m = 0.0. The value of M12 used for the naphthalene calcula- tion in figure 2 was taken from the Davydov splitting ob- served by Hochstrasser and Clarke [14]. Hanson and Robinson [15] have shown that the mean of the “B2, Davydov components is shifted down by about 1 cm−1 from the mixed crystal *B2, energy [15]. We have at- tributed this “translational shift” [3] to the interaction between molecules at (000) and (010). Mº, thus becomes 0.5 cm-". However, the band profile would be the same if it were attributed to either Ma or Me. The anthracene calculation in figure 4 was based on the constants derived from two experiments: the Davydov splitting in 264 States 25,000- 20,000- g l;,000- e º º g l(),000- Q º º º º @ wº º g e © we & º 5 3. 000. º e º © T- T U t n IT -I- lº |O 5 O –5 —l 0 —l B Energy cm−" FIGURE 4. *B2u Exciton band of anthracene calculated from experi- mental parameters (see refs. [9 & 16]). Ma = 0.0, M,-0.0, Me = 0.0, Marc) = 0.0, M1 II = 2.5, Mi m = 0.30. the anthracene triplet as observed by Clarke and Hochstrasser [16] through the electronic Zeeman ef- fect, and the anisotropy of triplet exciton diffusion as measured by Ern [9]. Only the absolute value of the ratio of MI in to Mi II can be obtained from Ern’s work, but the band profile is not significantly affected by the choice of sign. It is interesting to note the difference in band shape between those profiles calculated from theoretical data and those calculated from experimen- & g • * * * * e ve e e e 9 e º l t; ū W —w 5 O –5 —l O —l B Energy omº States H0,000 . 35,000- 30,000- º 25,000. e 20,000. lº, , 000. l(),000 e 5,000. Energy cm-l FIGURE 2, *B2u Exciton band of naphthalene calculated from experimental parameters (see refs. [14 & 15]). Ma = 0.0, Mb = 0.5, Me = 0.0, Matc) = 0.0, M. 11 = 1.25, M it. = 0.0. States 25,000 20,000 " lº, , 000 - l(),000 - 5,000 - FIGURE 3. *B2u Exciton band of anthracene calculated from theoretical parameters (see ref [13]). Ma = 0.0, Mb = 3.6, Mc = 0.0, Marc) = 0.0, M. It = - 4.7, M. 11, =0.21. 265 States 30,000 - 25,000 - 20,000- 15,000 - 10,000- 5,000 - . . " FIGURE 5. Energy cm *B1u Exciton band of benzene calculated from experi- mental parameters (see ref [17]). Ma = 0.0, Mb = 0.0, Mc = 0.0, M. Il = 1.1, Mi III - 0.7, Mi ſv = -0.3. tal data. While the theoretical parameters account fairly well for the observed Davydov splittings and dif- fusion constants, they seem to fail the test of a more sensitive criterion — the band profile. The benzene band was computed from M values derived from Davydov splittings observed in a triplet state [17]. Here it is possible to obtain three values to be assigned to M; II 2 MI III 2 and MI IV. The individual factor group components were not classified as to sym- metry; thus a set of Mi II, MI 111, and Mi Iy can be deter- mined only to within a permutation. Fortunately, the symmetry of the density-of-states function is such that it is invariant to permutations of values of the in- terchange-equivalent M’s. There has not yet been any direct observation of any triplet band of any molecular crystal, so it is not possi- ble to conclude which of our calculated bands, if any, are accurate in shape. However, the entire "B2u exciton band of naphthalene has been observed [6]. The agreement between this band profile and the profile calculated for the 'B2u state from Davydov splittings and mixed crystal data is very good [6,8]. Such agreement is supporting evidence that eqs (15) and (16) are substantially correct for narrow molecular crystal exciton bands, and that the assumptions in the Frenkel theory which were used to derive them are valid. In fact, the agreement for the triplet bands is ex- pected to be even better than for the singlet bands, as the former are narrower and in addition cannot have even small contributions from long-range interactions (e.g. dipole-dipole) in view of the fact [7] that 6m = 0 in eq (10). 3. References [1] Knox, R. S., Theory of Excitons (Academic Press Inc., New York, 1963), Chap. II. [2] Kopelman, R., J. Chem. Phys. 47, 2631 (1967). [3] Berstein, E. R., Colson, S. D., Kopelman, R., and Robinson, G. W., J. Chem. Phys. 48, 5596 (1968). [4] Knox, R. S., ibid., p. 31. [5] Colson, S. D., Kopelman, R., and Robinson, G. W., J. Chem. Phys. 47, 27 (1967) and J. Chem. Phys. 47,5462 (1967). [6] Colson, S. D., Hanson, D. M., Kopelman, R., and Robinson, G. W., J. Chem. Phys. 48,2215 (1968). [7] Craig, D. P., and Walmsley, S. H., Excitons in Molecular Crystals, (W. A. Benjamin Inc., New York, 1968). See also references 1 and 3. [8] Hanson, D. M., Kopelman, R., and Robinson, G. W., J. Chem. Phys. 51,212 (1969) and unpublished work. [9] Levine, M., Jortner, J., and Szöke, J. Chem. Phys. 45, 1591 (1966); Ern, W., Phys. Rev. Letters 22, 343 (1969); Durocher, G., and Williams, D. F., J. Chem. Phys. 51, 1675 (1969). [10] Robertson, J. M., and White, J. G., J. Chem. Soc. 18, 358 (1947); Abrahams, S. C., Robertson, J. M., and White, J. G., Acta Cryst. 2, 238 (1949). [11] Mason, R., Acta Cryst. 17, 547 (1964). 266 [12] Cox, E. G., Rev. Mod. Phys. 30, 159 (1958); Cox, E. G., [14] Clarke, R. H., and Hochstrasser, R. M., J. Chem. Phys. 49, Cruickshank, D. W. J., and Smith, J. A. S., Proc. Roy. Soc. 3313 (1968). (London) A247, 1 (1958); Bacon, G. E., Curry, N. A., and Wil. [15] Hanson, D. M., and Robinson, G. W., J. Chem. Phys. 43,4174 son, C. A., ibid., A279, 98 (1964). (1965). [13] Jortner, J., Rice, S. A., Katz, J. L., and Choi, S. I., J. Chem. [16] Clarke, R. H., and Hochstrasser, R. M., J. Chem. Phys. 46, Phys. 42, 309 (1965); Levine, M., Jortner, J., and Szöke, A., J. 4532 (1967). Chem. Phys. 45, 1591 (1966). [17] Burland, D. M., and Castro, G., J. Chem. Phys. 50,4107 (1969). 267 Effect of the Core Hole on Soft X-Ray Emission in Metals L. Hedin and R. Sjöström Chalmers University of Technology, Göteborg, Sweden We report a simple type of calculation to estimate the enhancement factor on the intensity of soft x-ray emission in free-electron like metals due to the effect of the core hole. We consider an electron gas in the presence of a perturbing potential and calculate the x-ray intensity assuming the dipole matrix elements to be constant. The calculation is based on a simple type of trial function for the initial state of the valence electron system and the coefficients are determined from the variation principle. The calcu- lation does not give the Fermi edge singularity, which has recently aroused such a large interest, but in- stead aims at giving the gross effects for the whole spectrum. The results indicate an increase in the in- tensity by 25 to 50% at metallic densities. The enhancement factor is found to vary roughly linearly over the main band, increasing about 50% in going from the bottom of the band to the Fermi edge. Key words: Aluminum; core hole; electronic density of states; Fermi edge singularity; pseu- dopotential; sodium (Na); soft x-ray emission. 1. Calculations We consider the Hamiltonian | H=X era; a +5 X v(q)a?", aft-aakak }: kk' q -X. v(q) F(q)a; loak, (1) kg where F(q) is a form factor, taken as cos qk and as (sin qR)|q|R. The initial state is represented as [1] |N)-(at X ofo.o.)|N). (2) p0 where |N) is a Slater determinant of plane wavefunc- tions. Restricting the coefficients off to have the form o;= g(q = p) (1 - na) np. (3) the energy is minimized, neglecting some small exchange terms. The x-ray intensity is then calculated taking the dipole matrix elements as constants. We ob- tain [1] for the intensity I(a)) OCC I(o)-So-S g(4–1) = o(o-e) k (4) |g(k - k 1)-g(k-k2)|*6(a)--e-ek, -ek,). | Uln OCC OCC " ; 2 × k 1k 2 The function g(q) is singular as q-" for small q. The coefficient determines the magnitude of the polariza- tion charge [2]. The variational calculation gives a polarization charge of 20°. We can easily obtain a charge of 1 by using a subsidiary condition in the varia- tion, but the significance of this is not clear. 2. Results Calculations were made for different rs-values, with and without form factors. We find that the first term in eq (4) gives the dominant contribution in the main band while the multiple excitation term contributes to the Auger tail. The results are summarized in tables 1, 2 and 3. The total increase in intensity is proportional to the increase in charge density p(0)/po at the center of the perturbing potential. The energy difference TABLE 1. Results without form factor Ts p(0)/po o,” eoſer Mult. exc. 9% 2 3.7 0.81 0.88 5 3 4.8 0.78 1.44 4. 4. 5.9 0.76 2.03 4. 269 TABLE 2. Results with form factor F(q)=sin (qP)/(qr) Increase of Ts R" | p(0)|po o” eoſep | Mult. exc. enhance- 9% In ent factor 9% Al...... 2 2.15 | 1.25 || 0.86 || 0.59 12 40 Na..... 4 3.26 | 1.49 || 0.81 | 1.42 14 50 ! Values for R are taken from R. W. Shaw, Jr., Phys. Rev. 174, 769 (1968). TABLE 3. Results with form factor F(q) = cos qR I’s R1 p(0)/po Cy? eoſep | Mult. exc. 9% Li......... 3.26 1.06 1.4] 0.80 1.30 16 Na........ 4.00 1.67 1.01 0.8] 1.52 24 Al......... 2.07 1.12 0.90 0.86 0.66 19 Values for R are taken from N. W. Ashcroft and D. C. Langreth, Phys. Rev. 155,682 (1967). between the states |N) and ||Nº) is denoted by eo. The increase in charge density p(0)/po agrees very well with the positron annihilation rates at higher electron densi- ties. We do not give the intensity function I(a)) explicitly since the enhancement factor I(0)|Vois closely linear. Instead we give the increase in percent of the enhance- ment factor from a = 0 to a) = ep and the total enhance- ment which is proportional to p(0)/po. The column “multiple excitation” gives the integrated intensity of the last term in eq (4) in percent of the integrated inten- sity of the first term. The results we have obtained give only a very crude picture of the magnitude of the effect from the core hole. In particular we see that use of different form fac- tors give quite different results. We believe the results in table 2, to best represent the actual conditions. The reason is that the charge enhancement at r = R rather than at r = 0 should be taken as a measure of the inten- sity increase since it is at roughly that point where the pseudowavefunction should be matched to the correct Bloch function. With the Ashcroft pseudopotential, which in real space is zero for r < R, the charge density has a minimum at r = 0 while with the Shaw pseu- dopotential, which in real space is constant for r < R and joins continuously to the outer (–1/r)-part, the charge density has a maximum at r = 0 and probably does not vary much out to r = R. The results in table 2 should thus represent an overestimate of the enhance- ment effects. Another reason for believing the results to be too large is that the total screening charge, 20%, is larger than unity. Calculations with a subsidiary condi- tion to give a screening charge of unity indeed cuts down the enhancement but not however, by the full 20°- factor. Neglecting the last term in eq (4), that is, the multiple excitation term, we see that the intensity has the form I(0)=Vo Pâr, (5) where the effective matrix element is perf– 0 + £8(a)). (6) The results for 8(a) at the Fermi surface obtained with the Shaw pseudopotential are 0.21 and 0.33 for Al and Na, respectively. Even if these values are somewhat too large they indicate that the mixing in of matrix elements from states above the Fermi surface can have a significant influence if there is a strong variation in matrix elements and density of states at the Fermi surface. The contribution to the intensity from the multiple excitation term is zero at the Fermi edge, rises quadratically and reaches a maximum in the bottom of the main band. The tail intensity is down to 1/e of the maximum intensity at about 2ep below the Fermi edge. Our results essentially confirm the conclusion by Stott and March [3] that the shape of the intensity curve in general should not be very much affected by the core hole, and our results also indicate that the total intensity enhancement is quite small, contrary to the results by Glick, Longe and Bose [4]. It should be emphasized that the present calculation does not take the edge effects, just discussed by Dr. Mahan, into account. Actually if we consider the An- derson orthogonality catastrophe [5], the coefficient o in eq (2) should be equal to zero. If, however, the edge effects are restricted to a narrow energy region, as in- dicated by the experimental data, the approximations employed here should not be serious. 3. References [1] Hedin, L., Sol. State Comm. 5,451 (1967). [2] Hedin, L., in Soft X-Ray Band Spectra, D. Fabian, Editor, (Academic Press, New York, 1968). [3] Stott, M. J., March, N. H., in Soft X-Ray Band Spectra, D. Fabi- an, Editor, (Academic Press, New York, 1968). [4] Glick, A. J., Longe, P., and Bose, S. M., in Soft X-Ray Band Spectra, D. Fabian, Editor, (Academic Press, New York, 1968). [5] Anderson, P. W., Phys. Rev. Letters 18, 1049 (1967). 270 Discussion on “Effect of the Core Hole on Soft X-Ray Emission in Metals” by L. Hedin and R. Sjöström (Chalmers University, Sweden) B. Mozer (NBS): I am a little confused about the An- derson catastrophe. I agree that if you have a polarized state it might well be orthogonal to the pure state you started with before the interaction was turned on. But are you not really interested in the dipole matrix ele- ment, or interaction Hamilton matrix elements between N* and N2 L. Hedin (Chalmers Univ.); Yes, but since the states are orthogonal, then independently of the matrix ele- ments, you actually have the result, according to An- derson’s theorem, that the intensity vanishes at the edge. This is, however, not the full story as just discussed by Mahan, in the vicinity of the edge the in- tensity has a power law behavior and may show a singu- larity. B. Mozer (NBS): I think you have a problem. You have a problem that there are bound states. You get into the time dependence of the way things go. I forgot to say that the N* state is orthogonal to N. That is quite true, but you are interested in a matrix element that carries you from a case where you have no radiation in emis- sion to a final state where you have a built-up hole in the core and a gamma ray floating around. R. A. Ferrell (Univ. of Maryland): As a many-body system, suppose we have N + 1 electrons; we calculate the dipole moment for the (N+1)th electron but the con- tributions from the remaining N electrons are in terms of their overlap from their initial to final states. Ander- son pointed out that that vanishes. It is a many-body ef- fect. In other words, if one electron makes a transition you also have to consider what happens to all the others. Here you just have to calculate the overlap and that vanishes when you suddenly change the potential. I think you would agree with that, Dr. Hedin. L. Hedin (Chalmers Univ.): Yes. B. Mozer (NBS): I am surprised at that. Hartree-Fock method wave functions you would still get some overlap that is non-vanishing. R. A. Ferrell (Univ. of Maryland): It is a characteristi- cally many-body effect. Every overlap factor is finite, but the product of an infinite number of such factors, each less than unity, vanishes in the limit. One can look at it various ways. B. Mozer (NBS): Most of the Hartree-Fock wave func- tions away from the impurity are equal to the original wave functions times the phase factor. Consequently, the overlap factor is unity and the product of the in- finite number of such factor is unity times the phase factor. The problem is how you handle the region around the impurity and whether or not you have local- ized impurity states. 271 Cancellation Effects in the Emission and Absorption Spectra of Light Metals B. Bergersen Institute for Theoretical Physics, Göteborg, Sweden F. Brouers H. H. Wills Physics Laboratory, Bristol, England Key words: Electronic density of states; light metals; many-body effects; plasmon satellite; soft x-ray. 1. Summary The influence of the many-body effects and of the presence of a deep localized hole on the shape of soft x- ray spectra in light metals has been widely discussed recently. Two observed features of x-ray spectra which cannot be explained by a one electron theory have particularly been emphasized. (1) The first feature is the low energy tailing of the emission bands superposed by a weak plasmon satellite band. To obtain the correct order of magnitude, Brouers [1] and Longe and Glick [2] have shown that it is essential to take account of strong cancellations due to destructive interferences of the charge clouds surrounding the core hole and the electron performing the transition. (2) It is now well established theoretically and experi- mentally that the effect of the core hole sudden switching causes an anomalous behavior near the emis. sion and absorption Fermi edges. If effects causing the width of the core state are neglected together with many-body effects, Nozières and de Dominicis [3] have shown that there will be a power law singularity near the threshold. Independently, we have shown [4], that this result is expected already from simple positive definiteness arguments. In spite of the success of the theory to explain the low and high energy features of the band, there is not until now a complete theory satisfactory for the whole spec- trum. One difficulty in applying co entional many- 417–156 O – 71 – 19 body theory to x-ray spectra is the occurrence of dia- grams diverging throughout the parent band in the first order theory [2]. If, on the other hand, a renormalized theory is formulated to get rid of these divergences, one has no information on the relative areas of the singular and nonsingular part of the spectrum. Nozières and de Dominicis have computed an exact solution only at the Fermi edge and with a rough potential. We have obtained a number of formal and numerical results which fill a part of the gap between previous treatments and a good and complete theory for the whole band. Since a first order theory is useful for gaining insight into the structure of the theory and the physical processes which contribute to the emission, we have shown that the divergences of the first order theory can be eliminated. This can be done if certain energy shifts are extracted in a consistent fashion and if one realizes that there is not an exact cancellation of unlinked dia- grams between numerator and denominator in the in- tensity function. It has been shown [5] that this noncancellation is due to the sudden change of the potential when the electromagnetic transition occurs. In the satellite and tailing region, this divergence free theory coincides with the previous first order theory. In the main band, the effect of electron-electron correla- tions neglected in the Nozières model and the one-body effect of the core hole can be separated. Contrary to what happens in the satellite region, there is no cancel- lation effect and it is obvious from the results that 273 many-body and one-body effects cannot be treated on the same foot. The electron-electron effects give a cor- rection of the order of 10% to the one-electron Sommer- feld theory [6] but to obtain the correct contribution of the electron-core hole transient potential one has to evaluate a vertex correction which is equivalent to in- troduce ladder type diagrams as this is done in the positron annihilation theory. 2. References [1] Brouers, F., Phys. Stat. Sol. 22, 213 (1967). [2] Longe, P., and Glick, A. J., Phys. Rev. 177,526 (1969). [3] Nozières, P., and de Dominicis, C., Phys. Rev. 178, 1097 (1969). [4] Bergersen, B., and Brouers, F., J. Phys. Chem. 2,651 (1969). [5] Bergersen, B., and Brouers, F., Proc. of the Conf. on X-Ray Spectroscopy, Kiev (1969). [6] Bergersen, B., Brouers, F., and Longe, P., to be published. 274 Photoabsorption Medsurement of Li, Be, Nd, Mg, and Al in the Vicinity of K and Lu in Edges C. Kunz, * R. Haensel, G. Keitel, P. Schreiber, and B. Sonntag Deutsches Elektronen-Synchrotron, Hamburg, Germany and Physikalisches Stadtsinstitut, Il. Institut für Experimentalphysik der Universität Hamburg, Hamburg, Germany The absorption structure of five light metals has been measured in the vicinity of the onset of K shell respectively Lii.111 shell absorption. In accord with recent theoretical investigations a peaking of the cross section at the edge is observed for the Lii.111 edges of Na, Mg, and less pronounced for Al. There is structure of a different type at the K edges of Li and Be. Key words: Aluminum (Al); beryllium (Be); electronic density of states; electron synchrotron light; K spectra; light metals; lithium (Li); L spectra; magnesium (Mg); photoabsorption; sodium (Na); transmission measurements. Recent theoretical investigations [1-10] have shed new light on the old problem of the absorption and emission structure near the onset of inner shell transi- tions in simple metals. As the results are completely symmetric for emission and absorption we shall discuss here only the absorption behavior. The essential im- provement of these theories is that they include the in- fluence of the deep level hole potential on the electron states near the Fermi surface. As the charge of this hole is shielded by the conduction electrons the potential is confined to a small region around the excited atom. This implies that mainly the final states of s symmetry are influenced by this potential. As a consequence, the onset of p-electron transitions (e.g., LII,III edges) is ex- pected to show up as an infinite singularity instead of a simple discontinuity. The result is [1-10] l P. “WE. (1) In the region immediately following the onset. AE is the distance from the onset and O. a positive exponent ap- proximately equal to 1/2 [9]. (Actually a prominent peak rather than a singularity is expected due to Auger and temperature broadening.) On the other hand no sin- gularity should occur at the onset of s-electron transi- tions since for them o is expected to be negative and small. More details on the theoretical background of these anomalies are given in other papers at this con- ference. *Present address: Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742. Optical absorption at the LII,III edges of Na, Mg, Al, and at the K edges of Li, Be has been investigated for several decades. Because of the inherent experimental difficulties in the XUV range, many of these results are in disagreement with each other. As the situation has improved with the availability of electron synchrotrons as intense continuous light sources, it appeared desira- ble to investigate these materials in the light of recent theoretical interest. Our experiments were performed at the 7.5 GeV electron synchrotron DESY [11,12]. The results reported were obtained by means of transmis- sion measurements on thin evaporated films. Details of the experimental procedure will be given elsewhere. Figures 1 to 3 show the absorption coefficient of Na, Mg and Al near the L11,111 edges. The spectra of Na [13] and Mg are very similar in shape. A peaking toward the edge, as postulated by the theory, is clearly recognized for both metals. The peaking is less pronounced for Al. The spectra, as shown, were taken with the samples at 77 K. The structure is complicated, due to the spin orbit splitting of the ground state. It should be noted that the intensity ratio of Lili. Lil is not 2:1 as expected from simple statistics but 3.1:1 for Na, 2.8:1 for Mg, and 2.5:1 for Al. As present theories give no prediction about how the singularity of eq (1) should merge into the normal ab- sorption behavior for large values of AE, we have tried to fit our results to a tentative law: Ol P = Pot AFTE (2) 275 I *TI I ſ I T- — |l 4- - Nd 14|- ſº –––– emission e- | • . . . fitted curve U) | |S #13- | N - N D | N C N §2. N - :- N £ N ăul \ - #: L |\ (1) I -T- - |S 10 § 3. - Cº. .9 8– — fi. § 7- - # 7 —|lo ––––. | 307 308 309 310, 311 312 l 31 32 33 (eV) 34 FIGURE 1. Photoabsorption of Na in the neighborhood of the LII, III edge. Included is the emission curve given by Crisp and Williams [14]. The dots show the best fit of the experimental data by a function given by eq (2). for Na and Mg. The best fit is indicated in figures 1 and 2 by dots together with the level of pºo. In order to compare our results with emission data we have reflected the energy axis in these data and brought the LIII edges into coincidence for Na [14] and Mg [15]. Peaks are seen also in emission, but they are less pronounced. In the results of earlier absorption measurements on Na [16] and Mg [17] the peak at the edge can be seen, but no special attention has been attributed to it. For Al only the plate density curve of Codling and Madden [18] shows a faint peak at the edge. The absorption curves near the K edges of Li and Be are shown in figures 4 and 5. The samples were cooled I I —I- I f I |l 65- Mg #! ––– emission 5 . . . . fitted curve an U 60– - # s + 55– | o – .92 |F B o .9 • SE | $ 50- | - O | | - C | O E. | 5 45- | |lo – A - - - H 1 - O.2 H. % 3 Lu O * area 4->~~se f 3 N 2 —l }* <ſ #–02 § -9.4 F 2 - 4 — O.4 H — O.6 | | L | | | | | | | | 945 92O 925 93O 935 94O 945 ENERGY (eV) FIGURE 5. Subtraction of derivative of emission band from modulation curve. Curve 1 is original curve. Curves 2 and 3 are resultant subtractions assuming shifts of 0.010 and 0.015 eV, respectively. Location of Fermi energy and Van Hove singularities are from Burdick's calculation (ref. [11]). between 107 to 10° counts to be accumulated at each step position to assure statistical significance. Figure 3 shows the observed emission spectrum nor- malized to a peak value of 100. It is completely struc- tureless without any indication of the location of the Fermi energy which should be located somewhere in the region of 932 to 934 eV as determined by x-ray ab- sorption measurements [10]. The lack of any indica- tion of the Fermi energy is probably masked by satellite emissions or many body effects. There should only be natural lifetime and instrumental broadening at this energy as Auger processes would not contribute to the lifetime broadening. Figure 4 shows the change of the emission intensity which results when the copper specimen is stressed in tension. The modulation curve appears to be fairly representative of the derivative of the emission peak. There is also apparently additional structure present. One prominent feature, besides the small wiggles and peaks, is the abrupt change in slope that occurs at about 934 eV. For the same copper sam- ple this curve reproduces itself when rescanned, in- cluding the fine structure. Additional copper samples show different modulation curves. The gross shape remains the same and an apparent sudden change of slope at about 934 eV is evident, but there is significant modification in fine structure and intensity. This is at- tributed to the difference in crystallographic texture between specimens. If the modulation curve was the result of a rigid shift of the emission band, then the curve should exactly be the derivative of the emission band. To check this pos- sibility, the product of the derivative of the emission 283 band and a suitable constant, which would correspond to the energy shift of the band, are subtracted from the modulation curve in an attempt to produce a null curve. Figure 5 shows the attempt to carry out this nulling procedure. Curve 1 is the original curve and curves 2 and 3 are the resultant subtraction curves assuming shifts of 0.010 and 0.015 eV, respectively. It is apparent that no appropriate uniform or smoothly varying band shift can account for the total modulation curve. The energy shift which reduces most of the effect to zero is 0.010 and 0.015 eV at the bottom and top of the band, respectively. Since the sample is polycrystalline, has preferred orientation, and elastically anisotropic, the microscopic strain conditions are not known. Assuming the sample is isotropic with a modulus of 1.38 × 1011 dynes/cm2, the average deformation potential is 25 eV at the top of the band and 17 eV at the bottom. These numbers can only be considered approximate until ex- periments with single crystals are performed. The remaining portion of the modulation curve which cannot be nulled by the derivative subtraction technique must be associated with regions in the band which are extra sensitive to strain as regards altering the x-ray emission intensity. This could be at the Fermi energy and Van Hove singularities or critical points. There is some arbitrariness between the energy scale as measured by x-ray emission and the scales used in the theoretical band structure calculations. Burdick [11] has correlated his energy scale to the x-ray emis- sion scale by matching the peak in the density of states with the peak in the emission band. Burdick's calcu- lated values for the Fermi energy and Van Hove singu- larities are indicated in figure 5. The abrupt slope change at 933.8 matches the Fermi energy and much of the fine structure matches with the Van Hove singulari- 12O 68 OO k-- 4 O - 2O H. O | l | | | i 1–1 | l 835 84O 845 85O 855 86O 865 ENERGY (eV) FIGURE 6. Nickel LIII emission band. ties. Whether or not this is the correct match should be revealed by investigations of single crystals. The Lili emission spectrum of nickel is shown in figure 6. It is quite similar to the copper emission spec- trum except there is a noticeable satellite on the high energy tail. The origin of this satellite is presumably due to multiple ionizations produced by Auger transi- tions from the initial Li and Lil ionizations [12]. The peak in the emission band occurs at 14.561 Å or 851.5 eV [13]. X-ray absorption measurements [14] indicate that the Fermi energy is close to the emission peak (within 1 eV). The piezo soft x-ray effect in nickel is quite different than that observed for copper. Generally, two different types of results are seen depending upon the initial con- dition of the nickel sample. Figure 7 shows the modula- tion curve for a nickel sample which was heavily cold worked by rolling and measured using the same experi- mental conditions as previously mentioned for copper. Figure 8 shows the modulation curve when the nickel specimen is not subject to the same degree of cold work. As can be seen from the figures, there are two modulation peaks. The magnitude of the effect in- creases about 7 fold for the sample which is not heavily cold worked and the fine structure which is present on figure 6 is unresolved. Why there is such a difference between the two specimens will be discussed later. It is quite evident that the strain must be altering the matrix element for the x-ray transition since the total emitted intensity is greater when the specimen is under tension (in compression the modulation curve is nega- tive). The high energy modulation peak evidently cor- responds to the satellite emission of nickel and the low energy peak to the normal 3d-4s to 2p3/2 transition. The O.6 O. 2 }*- - O.O 2O - – O. 4 H | l | l | | | 85O 855 86O 865 ENERGY (eV) — O.6 | | - | 835 840 | 845 FIGURE 7. Modulation of emission band for heavily cold-worked nickel. 284 3.O 2,O H. O –2.O H. H –3.O f I | | l l —l- l l—l | 835 84O 845 85O 855 86O 865 ENERGY (eV) FIGURE 8. Modulation of emission band for annealed nickel. separation of these two peaks is about 6 eV and the relative intensities of the two vary from sample to sam- ple, probably dependent on the preferred crystallo- graphic orientation. First, refer to figure 7, which corresponds to the heavily cold worked sample. This sample is mostly ex- hibiting fine structure mostly associated with mechani- cal strain. The high energy modulation peak has a break at 856.9 eV and another at about 851 eV which is indicated by the arrow on the figure. This feature is also present on the high energy side of the low energy modu- lation peak. Assuming that this feature is the spin exchange splitting the measured value, from extrapola- tions of the modulation curve, is 0.6+ 0.05 eV. This agrees very well with the theoretical values of 0.4 or 0.6 eV [15]. The same kind of derivative analysis as that used for copper cannot be used for nickel because the gross shape of the curve is not representative of the derivative. However, one can attempt to null portions of the modulation curve by the derivative scheme as- suming that the high and low energy parts of the emis- sion curve shift in opposite directions with strain. By doing this, it becomes evident no simple shifts can null the modulation curve. The fine structure which can be readily seen on the modulation curve becomes enhanced as well as the abrupt changes in slope which occur on the high energy side of each modulation peak. The high energy modulation peak has its abrupt in- crease of intensity at 856.9 eV. The mechanism for the increased piezo effect for the nickel sample not subject to the heavier cold work has been discussed in detail elsewhere [16]. Apparently, rather than mechanical strain being the dominant mechanism for producing the modulation, the mag- netization of the sample rotates. This alters the spin- orbit splitting of the sub-bands and displaces the total band structure to a much greater extent than could be produced by mechanical strain, resulting in changes in x-ray emission throughout the total band. This explana- tion is tentative at this time and further investigation is required by doing solely a magnetic modulation experi- ment. 4. Conclusions A new experimental technique has been developed to probe the energy band structure of solids. Although the interpretation of the results are still in a primitive state, it is evident that this method can be a useful means for studying the electronic structure of metals and alloys. One of the notable features of these experi- ments is that we have transformed a structureless emis sion spectrum to a modulation curve which contains considerable structure and awaits theoretical terpretation. in- 5. Acknowledgments I wish to express my appreciation to H. Schreiber, E. Buehler, and D. Brasen for experimental assistance and J. H. Wernick and C. Herring for encouragement and helpful discussions. 6. References [1] Rooke, G. A., J. Phys. C: Phys. Soc. (London) Proc. I, 767 (1968). [2] Seraphin, B. O., and Hess, R. B., Phys. Rev. Letters 14, 138 (1965). [3] Batz, B., Solid State Commun. 4, 241 (1965). [4] Gerhardt, U., Phys. Letters 9, 117 (1964). [5] Zwerdling, S., Lax, B., Roth, L. M., and Button, K. J., Phys. Rev. 114, 80 (1959). [6] Macfarland, G. G., McLean, T. P., Quarrington, J. E., and Roberts, V., Phys. Rev. Letters 2, 252 (1959). Gobeli, G. W., and Kane, E. O., Phys. Rev. Letters 15, 142 (1965). [8] Willens, R. H., Schreiber, H., Buehler, E., and Brasen, D., Phys. Rev. Letters 23,413 (1969). [9] Sandström, A. E., Handbook der Physik, edited by S. Flügge (Springer-Verlag, Berlin, Germany, 1957), Vol. 30, pp. 186-187; Fisher, D. W., J. Appl. Phys. 36, 2048 (1965). Cauchois, Y., Phil. Mag. 44, 173 (1953). Burdick, G. A., Phys. Rev. 129, 138 (1963). Liefield, R. J., Soft X-Ray Band Spectra, edited by D. J. Fabian (Academic Press, New York, 1968), p. 133. Bearden, J. A., Rev. Mod. Phys. 39, 78 (1967). Cauchois, Y., and Bonnelle, C., Comptes Rendus Acad. Sci. Paris 245, 1230 (1957). Zornberg, E. I., Phys. Rev. Bl, 244 (1970). Willens, R. H., to be published. [7 | [10] [11] [12] [13] [14] [15] [16] 285 A Soft X-Ray Band Spectra and Their Relationship to the Density of States * G. A. Rooke Metallurgy Department, University of Strathclyde, Glasgow' The paper concentrates on the similarities and differences between the one-electron spectrum and the density of states; many-body effects, although important, are listed but they are not considered in detail. It is shown that the only reliable information about the density of states that can be obtained from soft x-ray spectroscopy are the energies of the Fermi surface and the van-Hove singularities, although the shape of the density of states can be derived indirectly from the energies of the van-Hove singulari- ties. It is the differences between the density of states and the one-electron spectra that may prove to be most important. These differences can give information about the symmetry and the local nature of the screening electrons. This is particularly interesting when studying alloys. The Li K, the Al L23 and the Zn La spectra are given as examples which illustrate the above argu- ments. Finally, a brief discussion on the soft x-ray spectra from the Al-Mg system show how the results may be used to study alloys. Key words: Alloys; auger transitions; density of states; many-body interactions; plasmons; singu- larities; soft x rays. l. Introduction Before commencing my discussion, I would like to state our objectives in attempting to measure the densi- ty of states: they are (a) to compare experiments that are in some way related to the density of states; (b) to derive some information about concepts of a more fundamental nature, the band structure, the effective potential, etc., and (c) to attempt to predict the properties of other metals and alloys. It is important to bear these in mind, as sometimes our objectives can be achieved more directly by not making use of the density of states. It will be remembered that when transitions involv- ing atomic core states occur, x-rays may be emitted or absorbed. The x-rays have an energy equal to the ener- gy difference between the two states involved in the transition. If one of the states lies in the valence band or the conduction band, a band spectrum is produced *An invited paper presented at the 3d Materials Research Symposium, Electronic Density, of States, November 3-6, 1969, Gaithersburg, Md. 'Present address: Ferranti Ltd., Western Road, Bracknell, Berks, U.K. (see fig. 1). Experimentally, better energy resolution is obtained from the band spectra with lower energies; these are called soft x-ray band spectra. Although this paper specifically discusses soft x-ray spectra, the con- cepts are applicable to all band spectra. For an introductory review of soft x-ray band emis- sion spectroscopy, I recommend either Skinner [1] or Tomboulian [2]. An excellent bibliography of Yakowitz and Cuthill [3] reviews the literature up to 1961. Some of the most recent work in the field is described in a conference proceedings edited by Fabian [4]. All of these omit very interesting Russian work for which no comprehensive review is known to me. 2. Many-Body Interactions The many-body interactions are treated in this paper as uninteresting complications that tend to hide the in- formation we are seeking. Each interaction is con- sidered as a perturbation on the one-electron spectrum. The interactions between individual electrons create two types of perturbations on the spectra, the excited initial and final states, which directly affect the spectra, and exchange and correlation, which affect the spectra 287 £1 Conduction Band * 2" * * - NNNNN N \\ "Valence Band º N ) N \ \ \-A ^\/* Vº- ^\}^2- Emission ^\}^- ^\/\}^/> ^\}^\}^* A/º - ^2. A/N2. Absorption ^/Ny. av Mº- ^\/\/\x y \\ Soft X-rays łłł Hard X-rays One-electron energy-level diagram showing x-ray transi- tions. FIGURE 1. through their effects on the band structure. The exchange and correlation only cause trouble when com- paring uncorrected one-electron band-structures with experimental results [5]. They are normally allowed for by theoreticians and are only important to experimen- talists when they are measuring many-body effects or when they are trying to derive the uncorrected band- StructureS. Excited states occur whenever electron transitions occur. For all radiating transitions, two single-particle excitations are involved; in emission processes both the initial and the final states are normally excited while in absorption processes the initial state is normally unexcited and the final state contains two single-parti- cle excitations. Each excitation affects the spectrum through its lifetime and through its perturbing effects on the unexcited one-electron spectrum. Three types of one-electron excitation can occur and their effects on the spectra are discussed separately. First, the hole in the core state perturbs the valence electrons, so that the intensity at the Fermi surface is reduced for K spectra (transitions to ls-states) and enhanced for L23 spectra (transition to 2p-states) [6]. The lifetime of the hole in the core state causes the spectrum to be broadened because of the uncertainty in its energy. If this lifetime is known, it is possible to correct for this broadening [7]. The hole in the core state can only affect spectra that involve x-rays. Second, a hole in the valence band can be filled by an electron in the same band, provided another valence- band electron is excited into the conduction band, thereby conserving energy; this interaction is called the Auger process. Because of this, there are both holes and excited electrons with energies close to the Fermi energy and this raises the possibility of excitons occur- ring in the metal. Also, Auger processes shorten the lifetime of the hole and this broadens the spectra con- siderably. Because many more electrons are capable of filling holes near the bottom of the band than those near the top of the band, the lifetime of a hole near the bot- tom of the band is shorter and the spectrum is broadened more near the bottom of the band than near the top. It has not yet proved possible to remove the Auger broadening from spectra, because the broaden- ing is not constant throughout the band. Auger broadening is important to any spectroscopy involving the absorption of the ultraviolet light, to x-ray emission spectroscopy and to ion-neutralization spectroscopy. Third, the scattering of electrons excited into the conduction band is important to any process involving the absorption of radiation. This scattering is very similar to the Auger process; an excited electron falls to an energy level closer to the Fermi energy, giving its energy to another electron which is excited out of the valence band and into the conduction band. The remaining holes and the scattered electrons could form excitons. For any electron emission spectroscopy the scattered electrons can contribute to the spectra and they enhance the spectra near the Fermi edge. In general, the lifetime of the excited states decreases as their energy increases, so that the resultant broadening is not constant throughout the band and is exceedingly difficult to remove. For each type of optical spectroscopy discussed at this conference, two of these three types of one-elec- tron excitations are involved and for ion-neutralization spectroscopy, four excitations occur. The first two types of excitations are involved in soft x-ray band emission spectroscopy. The perturbations of the valence band, near the Fermi edge, by the hole in the core state is possibly the cause of the drop in inten- sity near the Fermi edge in the lithium K spectrum and the cause of the small pip near the Fermi edge in the sodium L23 spectrum [6]. It probably contributes more 288 to the shape of all light metal spectra than to the heavy metal spectra, because the hole represents a larger pro- portion of the core electrons in the light metals. The core broadening prevents the detection of spectra resulting from transitions to 2s., 3s, etc. core states and it also is important in changing the shape of the spectra from heavy metals; the lifetime of the excited state is considerably reduced because of competing non-radia- tive transitions. Auger broadening is responsible for the low-energy tails in the spectra from metals and this in- troduces uncertainty into the measurement of the band- widths. For other spectroscopic techniques, these excitations are also highly important, their importance depending on both the technique used and the metal examined. Besides exciting a single electron state, it is also possible to excite the electrons collectively into plasma oscillations. When this occurs, together with the emis- sion of an x-ray photon, the photon loses enough energy to create the plasmon and a satellite spectrum is formed at lower energies [8,9]. Interactions between individual electrons and the plasmons, which are called plasmarons [10], have a measurable effect on the plasmon satellite [11] but not on the parent band. 3. The Density of States and Band Spectra In order to discuss the relationship between the den- sity of states and the spectra, it is assumed in the remainder of this paper that the effects of the many- body interactions can be corrected for. The word “spectrum” then implies “one-electron spectrum that would result if such corrections were made”; this spec- trum has the same width as the valence band, its shape is related to the density of states but it is affected by the transition probabilities. The similarities between the density of states, N(E), and the one-electron spectrum divided by the cube of I(E) 1/3 ° the radiation frequency, are seen by comparing the following equations: 2. – N(E) |a k ºf (1) 2 (2) Iſºlº ſº, ºr ||v,v,wid: where E is the energy of the state with wave vector k and the integral is taken over the surface, s, of constant energy, E. For the density of occupied states and for the emission spectrum, the integral is taken over the region 417–156 O - 71 – 20 of k space that lies inside the Fermi surface, while, for the density of unoccupied states and for the absorption spectrum, the integral is taken over the region of k space that lies outside the Fermi surface. The square of the modulus of the matrix element, ſilh, V, lifdt, is the transition probability, whose properties determine the difference between the two functions; the dipole ap- proximation has been assumed. A striking similarity between the two functions oc- curs because the integrations are terminated at the Fermi surface; this produces the Fermi edges in the functions. Unfortunately, for some heavy metals, the core-state broadening and the transition probabilities combine to make the Fermi edge almost impossible to Iſlea SUlre. The Brillouin zone effects are also common to both functions; at certain points of high symmetry on the Brillouin zone boundaries, the gradient, V, E, is zero and van-Hove singularities exist at corresponding points in the two functions. Certain van-Hove singulari- ties are of particular interest; in particular the energy of the bottom of the band can be used to estimate the bandwidth and the energy of the top and bottom of the d-band give the width of this band and its relation to the Fermi energy, both of which are of considerable theoretical interest. Because the matrix element is k-dependent, it is not possible to remove the transition probabilities from in- side the integral in eq (2). Hence, it is not possible to Write the intensity as a product of the density of states and the average transition probability and, even if the transition probabilities are fully known, it is not possi- ble to remove their effects from the spectra. Rooke [12] has shown that the k-dependence of the matrix element is large, sometimes changing it from nearly 1 to nearly 0 on the same constant energy curve, so that errors created by removing the matrix element from the integral can be serious. From the above considerations it can be seen that the detailed density of states cannot be directly derived by using soft x-ray spectroscopy; or by using any form of spectroscopy, for that matter. Sometimes a spectrum and the density of states will have similar shapes and they will certainly have the Fermi edge and some van- Hove singularities in common. If the spectrum changes under different experimental conditions, such as heat or pressure, it may be possible to assume that the changes will be entirely due to the density of states and to obtain a little more information about the density of states in this way. This technique has been used to find the effect of the Fermi-Dirac statistics on the density of states, by heating the target (1) and to find the effect of 289 alloying on the density of states. A more sophisticated version of this approach has just been developed by J. H. Willens in Bell Laboratories [13]. In another method for obtaining more information about the densi- ty of states, one derives a band structure that fits the measured van-Hove singularities and then uses this to calculate the density of states. However, as Fermi sur- face techniques are capable of making fine distinctions between band structures, they can also be used to ob- tain densities of states which are often more accurate than those obtained from soft x-ray spectroscopy. Even though it is only possible to obtain limited information about the density of states, this information is often very interesting. However, it is often even more in- teresting to examine the information that can be derived by studying the effect of the transition proba- bilities on the spectra. This information is discussed in the next section. 4. The Transition Probability and the Band Spectra The differences between the functions defined in eqs (1) and (2) are determined by the matrix element ſ ill kV, lifdt The integration is taken over all real space and V, is the component of the real-space gradient-operator in the x direction. lf is the wavefunction of the core state and, because of its high symmetry, it is a good approximation to ex- press it as lif(r, 6, b) = Ra, (r)Yı, (6) e” (3) where Rn (r) is a radical function, Yu (6) is a Legendre polynomial and n', 1' and m' are the usual quantum numbers. The radial function has appreciable mag- nitude only near the center of the atom and the state is localized. Because only one Legendre polynomial is in- volved, the state can be labeled by atomic notation; ls, 2p, 3d, etc. lik is the wavefunction of a valence state and it may be expanded as a series of spherical harmonies ill-X, Cr(k) all (r) Y. (6)” (4) l, m Normally, only the first three values of l are important. The radial terms are roughly constant throughout real space so that the state is shared by all atoms and is not localized like the core states. Substituting these expansions of the wavefunctions into the matrix elements gives jºy, lf dr & X. C1(k) × |Rºna.0rd. x ſy, ſo yº) ſº odo º ſº-wºod, (5) where f and g are functions whose form depends on the polarization. Each of these four terms contributes a distinct characteristic to the transition probability. The k-dependence of the first two terms, as shown in the last section, prevents the direct derivation of the densi- ty of states from the spectra. The third term is zero unless l= l' + 1 (6) This selection rule is particularly important for x-ray transitions, because the core state involved has a well defined symmetry described by a single spherical har- monic. If the core state is an s-state, only p-like states can make the transition to it. Similarly, only s-like and d-like states can make the transition to a p-state. This selection rule enables us to describe states in terms of partial densities of states; the density of s- states, the density of p-states and the density of d- states. K spectra (transitions to the s-state) give approx- imate estimates of the density of p-states, while L23 or M23 spectra (transitions to the 2p or 3p states respec- tively) give approximate estimates of the density of S and d states. Because these partial densities of states only approx- imate to the shape of the spectra, it is not possible to add them and to derive a meaningful density of states. However, this symmetry dependence of x-ray band spectra makes them unique in the information that they can reveal. This is particularly interesting in the case of alloys, for which this characteristic of x-ray spectra should be fully exploited to determine the nature of the screening charges. Further, it is the authors unsubstan- tiated view, that the extended fine-structure occurring in absorption spectra can also be attributed to this selection rule. Besides creating gross features in the spectra, this selection rule modifies the magnitudes of the van-Hove singularities. The symmetry of the crystal dictates that the states associated with points of high symmetry in the Brillouin zone will have high symmetry themselves. Thus, some of these states are almost entirely p-like while others are s-like or s and d like. If a state is entire- ly s-like, it will not contribute to a K spectrum and, if it 290 is associated with a van-Hove singularity, that singulari- ty will not be seen in the K spectrum. Similarly, as most states contain a mixture of symmetries, van-Hove sin- gularities associated with states that are entirely p-like will be exaggerated in the K spectra. The reverse situa- tion occurs for L23 or M23 spectra, for which p-like sin- gularities will not be observed and s and d like singu- larities will be exaggerated. The last term in eq (5) is dependent on the polariza- tion of the radiation and it is non-zero if m= m' + 1, but if the polarization is suitable it may also be non-zero if m= m'. We therefore write the second selection rule as m = m^ = + 1 or 0 (7) This rule implies that not all the transitions described in the second last paragraph are allowed. For any one type of transition, say s to p, only a fraction of transi- tions are allowed and this fraction may vary throughout the band, thereby distorting the spectrum. However, this distortion may not be serious, as the fraction may be nearly constant throughout the band. Of course, the constant fraction for d to p type transitions will be dif- ferent from that for s to p type transitions and, in fact, the fraction for these transitions can be shown to be about 2/5 of that for the s to p transitions. This means that the L23 or M23 spectra give an approximate mea- sure of the density of s-states plus 2/5 density of d- states. This is one reason why the L23 and K spectra, when added together, do not give the full density of State S. The second term in eq (5) provides the spectra with some very interesting properties. The integral effective- ly takes a weighted average of a(r), using a weighting factor of rRn(r). Because of the localized nature of Rn(r), this integral can be considered to take an average of a(r) in the region near the core state of the emitting atom. This is exceedingly interesting when studying alloys, as the spectra from each component metal will sample the wavefunction in the region of that type of atom. These local properties are of great in- terest and are only obtainable by the spectroscopies in- volving x-rays. Finally, the degree of approximation involved in as- suming that spectra are similar to the partial densities of states is discussed. Consider the case of the K spec- trum, which samples p-states. Because the wavefunc- tions are normalized, the term ſ ill.” dt may be inserted into eq (1) without changing it. This term may be written as X (21+1)* C#(k) ſac, (r) rº dr l and the density of p-states may be defined as d2 k Np (E) eſ wº C}(k) ſ a; , (r) rºdr (8) By replacing the last two terms in eq (5) by non-zero constants and restricting the summation to l = 1, eq (2) may then be written as I (E)/v4 eſ" ºr k VI, E 2 C}(k) 2 |a.0. dr (9) where the zero on the radial integral indicates that it is now restricted to the region of the core state of the emitting atom. The final terms in these equations differ, one being the average of the square of a(r) and the other being the square of a local average of a(r). For the densities of s-states and of d-states and the cor- responding spectra, the situation is complicated by the factor of 2/5 and by cross terms, but the differences arise in the same way. 5. The Spectra from Pure Metals It is shown above, that x-ray spectra sample only those wavefunctions that have specific symmetries and that lie near the core of the emitting atom. Apart from these restrictions, they tend to show some of the behavior of the density of states, particularly the Fermi edge and the van-Hove singularities. The spectra are broadened by the experimental resolution and by many- body effects and it is possible that these many-body ef. fects also change the shape of the spectra near the Fermi edge. These ideas are now illustrated by three pure metal spectra; the Li K, the Al L23 and the Zn L3 emission spectra. The Li Kemission spectrum [14] is shown in figure 2. The Auger tail is immediately obvious but the Fermi W W "M W l ſ : | \ e-AA Reproduced from Crisp r” 72O 7OO | l ſ l l FIGURE 2, Lithium K emission spectrum. and Williams [14]. 291 K, W 2. º > > 5 3 - 3. +Fermi energy Gſs N, Sº Extrapolation to zero l 1 l l I I l l | I I 7O 65 6O 55 Energy (eV) FIGURE 3. Aluminum L23 emission spectrum. Reproduced from Rooke [15]. edge at eV is not so obvious because of the fall in inten- sity towards the edge; this fall in intensity is attributed to the effect of the hole in the core state. Because of the low bandwidth, no van-Hove singularities occur below the Fermi edge. The shape of the density of states is al- most certainly close to parabolic, but, because of the lack of p-like states near the bottom of the band, the spectrum does not reflect this; its intensity is con- siderably reduced near the bottom of the band. The Al Les emission spectrum [15] is shown in figure 3. The Fermi edge and the Auger tail are particu- larly obvious and the van-Hove singularities appear as discontinuities in the top half of the band. It is seen that the p-like van-Hove singularities, X4 and L2, are much less obvious than the s-like singularities, Xi and Li. The bottom of the band rises sharply to a broad hump at about 66 eV; this reflects the parabolic rise of the densi- ty of states and the fact that the states near the bottom of the band are nearly all observed because they are mostly s-like. From 66 eV to about 70 eV, the intensity Pure metals 1– 6O 64 68 72 Energy E (ev) FIGURE 5. Magnesium L23 and Aluminum L23 emission spectra from Al-Mg alloys. Reproduced from [17]. ſ | O | | j O. 6 H 'o.4 H | | \ O ~ >- | | l — |O –5 O 5 |O Relative energy (ev) FIGURE 4. Zinc L23 emission spectrum. Reproduced from Liefeld [16]. falls but then it rises again towards the Fermi edge. This occurs because the states change from being mostly s-like at the bottom of the band to being mostly p-like near the center of the band and then to having both s and d character near the top of the band. The Zn L8 emission spectrum [16] is shown in figure 4. This spectrum has been obtained after considerable effort in allowing for the various experimental errors that occur. The tall peak at the center of the spectrum is possibly the d-band while the sharp fall in intensity at about + 7 eV may be the Fermi edge. 6. The Spectra from Alloys Soft x-ray spectra are proving to be of considerable value for studying alloys and it is for this purpose that soft x-ray spectroscopy will be developed in the next few years. An excellent review of the work done in the West has been given by Curry [17], but the Russians have also done some interesting work, particularly on alloys containing d-band metals. As an example the very interesting results from the Al-Mg system are discussed. The Mg L23 and Al L23 emission spectra from two alloys are shown together with the pure metal spectra in figure 5. At first sight, it is seen that the apparent bandwidths of the alloy spec- tra are not greatly different from those of the pure metal spectra. On alloying, the intensity of the Mg spec- trum is relatively enhanced at the top of the band while the intensity of the bottom of the Al spectrum is rela- tively enhanced. The spectra have been carefully and independently checked by Dimond [18] and it is felt that the results are not due to clustering. The following analysis produces some very interest- ing information. First, the width of the density of states must be at least as wide as the bandwidth of the Al spectrum; this fact is in contradiction with most theo- 292 ries of alloys. Second, either the states at the bottom of the band must be highly localized around the aluminum atoms or they must have mostly s-symmetry in the re- gion of the aluminum ions and mostly p-symmetry in the magnesium ions; the former alternative is the most likely. It is possible that the states are tunneling through a potential barrier in the region of the magnesi- um ions and can only contribute to the spectra when they are in the vicinity of the aluminum ions. Third, states that are localized near the aluminum ions tend to be relatively more s-like than the states at the same energy in the pure metal. For magnesium, the states near the bottom of the spectrum tend to be relatively more p-like, reflecting the fact that they are now not near the bottom of the band. This example shows how useful it can be to sample electrons locally and with selected symmetries. Another example involving d-band metals is given later in the conference by Lindsay, Watson and Fabian. 7. Acknowledgments The author wishes to acknowledge Dr. L. M. Watson, C. A. W. Marshall, and R. K. Dimond for their discus- sions and help in preparing this paper and also Dr. D. J. Fabian and the Science Research Council for a Research Associateship (at the Strathclyde). University of 8. References [1] Skinner, H. W. B., Phil. Trans. Roy. Soc. A239,95 (1940). [2] Tomboulian, D. H., Handbuch der Physik, XXX,246 (1957). [3] Yakowitz, H. and Cuthill, J. R., National Bureau of Standards (U.S.A.), Monograph 52, (1962). [4] Soft X-ray Band Spectra and the Electronic Structure of Metals and Alloys, D. J. Fabian, ed., Academic Press, London (1968). [5] Pines, D., Solid State Physics 1,367 (1955). [6] Mizuno, Y. and Ishikawa, K., J. Phys. Soc. Japan 25,627 (1968). [7] Liefeld, R. J., p. 133 in [4]. [8] Rooke, G. A., p. 3 in [4]. [9] See part 4 in [4]. [10] Hedin, L., p. 337 in [4]. [11] Cuthill, J. R., McAlister, A. J., Williams, M. L. and Dobbyn, R. C., p. 158 in [4]. [12] Rooke, G. A., J. Phys. Chem. 1,767 (1968). [13] Willens, J. H., Schreiber, H., Buehler, E., and Brasen, D., Phys. Rev. Letters 23,413 (1969). [14] Crisp, R. S. and Williams, S. E., Phil. Mag. (8)5,525 (1960). [15] Rooke, G. A., J. Phys. Chem. 1,776 (1968). [16] Liefeld, R., p. 149 in [4]. [17] Curry, C., p. 173 in [4]. [18] Dimond, R. K., Ph.D. Thesis, to be submitted to University of Western Australia, (1970). 293 SOFT X-RAY II; DISTRIBU CHAIRMEN. F. M. Mueller A. J. McAlister RAPPoRTEUR. D. J. Fabian Orbital Symmetry Contributions to Electronic Density of States of Au/Al" A. C. Switendick Sandiq Laboratories, Albuquerque, New Mexico 97115 From an augmented plane wave calculation of the valence and conduction bands of AuAl2 we have constructed density of states histograms. From further calculations of the wave functions, one can at- tribute atomic-like character of the band states, e.g., Au 5d-bands, Al3s-band, Al 3p-band. One can then partition the total density of states into atomic-like components according to the fractional atomic-like character of each state. From the total density of states an electronic specific heat coefficient of 2.81 m.J/mole K” was calcu- lated compared with the experimental value 3.03. The aluminum 3s density of states is compared with the aluminum L23 soft x-ray emission spectra. Excellent agreement with experiment is obtained for the absolute location, and location relative to the Fermi energy, of the low energy peak. About half the cal- culated peak is attributable to tails of wave functions associated with the gold d-bands. Additional struc- ture in the experimental curve is quite well reproduced in the calculation. This we take to be confirma- tion of the overall correctness of our bands. Key words: Augmented plane wave method (APW); electronic density of states; electronic specific heat; gold aluminide (AuAl2); “muffin-tin” potential; orbital density of states; soft x- ray emission. 1. Introduction Augmented plane wave (APW) energy band calcula- tions for AuAl2 [1] have been extended to nineteen in- equivalent points in 1/48% of the Brillouin zone for the energy range —0.20 to 1.15 Ry. This gives a total of 256 points in the full Brillouin zone and includes bands derived from the aluminum 3s- and 3p-states and the gold 3d-states. Two sets of bands fall in the energy range –0.2 to 0.1 Ry. The lowest one from — 0.2 to 0.1 Ry can best be described as the aluminum 3s-bonding band derived from the 3s atomic states of the two alu- minum atoms in the unit cell. The narrow set of bands from —0.05 to 0.05 Ry are the gold d-bands. These bands are filled and contain twelve of the seventeen valence electrons per unit cell and are about 0.5 Ry below the Fermi energy. From .1 Ry to slightly above the Fermi energy (0.56 Ry) lies a partially filled alu- minum 3s-antibonding band. A complex of bonding and antibonding bands formed from the aluminum 3p-levels extends from 0.3 to 1.2 Ry. The Fermi level falls in the region of aluminum 3s-antibonding and aluminum 3p- bonding and -antibonding levels. The interpretation of *This work was supported by the U.S. Atomic Energy Commission. these bands in terms of their atomic geneses is con- firmed by charge density calculations which were also made in this energy region. The APW method gives a very convenient description of the charge density, i.e., atomic-like near the nuclei and plane-wave-like in the exterior “muffin tin” region. We shall utilize this description to compare calculated values of the density of states and orbital densities with experimental values for the electronic specific heat and aluminum soft x-ray L23 emission spectra, respectively. We shall be quite explicit in describing the numerical procedures and ap- proximations which we have made, given the band eigenvalues and wave functions, leaving details of the band calculation for a more appropriate publication. 2. Fermi Energy The Fermi energy is determined by the relationship 2V N = ſ dºk I (27) 8 E(k)s EP ( ) where N is the total number of electrons, the factor W/(27) represents the density of spin states in k space, 297 and the factor 2 the fact that we can put two electrons in each spin state, one up and one down. We shall en- deavor to explicitly display this factor whenever it oc- curs. If we divide both sides by V and multiply by Q, the unit cell volume, and subtract the electrons in filled core bonds then we have =#| dºk (27)” J Ecker(R)-ep (2) where Z is the number of valence electrons in the unit cell and the integral excludes core states. The number of cases for which the integral can be done analytically are few and we will always have to resort to numerical techniques. From the definition of the integral 20 | g 20 ,. Z = ſº dºk= − lim (2T) 3 Ecº E(k)sº p (27)3 X. A°k; (3a) EC-E (k'i)cc where A*ki is the it” subvolume of the division of the Brillouin zone into n parts. For our calculation we choose Aºki to be of equal size with the volume centered on ki. This gives l Z = 2 – (3b) Pl, X 1 i Ecº-E(ki)=EF By increasing n and varying the subinterval shape we can investigate the accuracy of this approximation as shown in table 1. The asymmetry in the table between TABLE 1. Convergence of sum determining Fermi energy, EF, as a function of number of points, n, in k-space eq 3b 71. E.2 s Ef s E- z (E(ki) < Er) |z (E(k) s EP) 32 0.5442 0.5620 16.75 17.125 64 .5442 .5620 16.875 17.0625 128 .5442 .5480 16.75 17.125 256 .5480 .5620 16.97 17.02 greater than and less than or equal to reflects the fact that many ki’s (up to 48 for cubic symmetry) all have the same energy so the sum rarely comes out exactly Z as the last column indicates. The statements [2,3] that both the state No. Zn/2 and the state No. (Zn/2)—l have the same energy which is thus the Fermi energy we be- lieve to be misleading. The second and third columns represent more accurately the precision with which the Fermi energy is determined. Although the results in- dicate a value closer to .560 Ry, the results for 128 points indicate a value below .550 Ry. In what follows we shall quote values of results as X(y,z) where X is the value which seems most reasonable, y the increment gained by choosing the higher Fermi energy, and z the increment gained by choosing the lower one. The value Ä is the one one might see in less candid treatments. This presumes the correct solution of some ad hoc potential and reflects only the inaccuracy of our sam- pling and ignores any further inaccuracy due to the in- adequate physical relevance of the potential used. Thus we quote a Fermi energy of .557 (+.005, – .009) Ry. 3. Total Density of States If one replaces EP by E in eq (2) and considers Z as a function of E then the number of valence electrons states per unit cells between E and E and dE is given by 20 d 'L' - —— – dºk | dE 4. dZ (E) (27)3 dE | | C-E (R)=E | (4) or the number per unit energy range 20 d (27)3 dE EC-E ſº (R)=E z (E) = dºk (5a) Again, of course, we must resort to finite techniques and our result is 2 | z(En) *; AB X. l, (E,-ºf- E(ki) < E, tº ) (5b) If we let AE go to zero in eq (5b) then our approximate density of states becomes just a series of delta func- tions located at E(k) and is of no use to us. We there- fore choose some mesh of energy Eq= Eo-H q/AE with finite AE and hope eq (5b) represents some average density of states over this interval. If we choose AE too large then our curve will be structureless and again of no use to us. Again we look for some sort of convergent procedure, decreasing AE until the resulting curve has stability of structure yet not so small to lose this stabili- ty. When we do this, we find minor variations as a state goes from one AE to the next higher. These minor varia- tions become major ones if AE is too small and the curve reverts to regions of no states and delta-function- like square spikes. These minor variations of shifting states from one subinterval to another, similarly de- 298 pend on E, where we start our finite partition. Com- parison of averaging over various starting points as done by Snow and Waber [4] with more sophisticated calculations [5] employing the same data indicate that where the number of states is large (d-bands), credible structure develops; but where the number is small, structure develops which may or may not be believable. All of which is to say they could have chosen a smaller AE in the region of the d-bands and probably should have chosen a larger one in the region of the s-bands. Since we have about twice as many states per unit ener- gy range as Snow and Waber, and shall choose a larger AE, we shall tend to believe any structure which develops. We have then etc.) = 12 1 º'-(r. jëſ. z (£p) = m n AE > *(E,T 771. ) (E- (2p – 1)AE 2m < &, s Eo-H (2p + 1)AE *) & The integer p gives the region of interest, where E, is the arbitrary starting point, AE is the sampling width and we are averaging over m histograms starting at Eo, Eo-H AE/m, . . . , Eo-H (m – 1)AE/m. Using this procedure with Eo = — 0.281625 Ry AE + 0.03675 Ry, and m = 5 the histogram shown in figure 1 shows Z/2 for AuſAl2. The large peak near .5 eV is the d- band which has been spread out somewhat by our averaging procedure. The Fermi energy falls in a region 5. O —l —l LL > ( ) # LI] . H Z. 24 O Lll + E. > | - : ă -* # ul --4-- jui * - | Ll- . d - Z 3.OH- i | l i Cl– . f - - - - - - - - - - - * * * * H i. : Uſ) | -i- !-- - i 2 2.0— H i– O ; . . . . . . . . O : Uſ) 1.O t Lil H <ſ H. Uſ) –5.O O. 5. t 15.O ENERGY (ELECTRON-VOLTS) FIGURE 1. Density of states for AuſAlg constructed following the procedure outlined in Section III, factor of two in eq 6 has been omitted. of rising density of states. The value of the density of states at the Fermi energy which we calculate is 1.19 (+.05, - . 14) electrons/(eV-unit cell). This gives us a value for the electronic specific heat coefficient y = Ce/T of 2.81 (+.12, — .33) m.J/mole K” compared with the experimental value of 3.03. 4. Orbital Density of State and Soft X-Ray Spectra The intensity of soft x-ray emission spectra can be written as I (o) oc o'ſ |Hriſº 6(E(k)—ha) – Er(k))dºk (7) where Hf, is the transition matrix element between the initial and final state. For x-ray emission Ef is a low lying core state and its k-dependence may be neglected. If we make the dipole approximation and use the APW expansion of the wave function inside the n” sphere P lik (ru) = X. Cim l, ºn º Yº (0, b) (8) we obtain I (o) or o'ſ |PC|, Pºl rºdr|*6(E,(k) – ho)dºk (9) Specializing further to the L23 emission spectra in Au Al2 the final state is aluminum 2p and only the alu- minum s- or d-like components of the wave function contribute to eq (9). If we assume that the radial wave function Pºſr) is a slowly varying function of energy and can be written P}(r) = P(r)f(r) (10) where P^(r) is some average radial function and f'(r) is smooth and in our approximation = 1. The only k-de- pendence is now in the C's and we can write Iolo's ſco-Ri. + (C)*R}}6(E(k) – ho)dºk (11a) where Rº-ſpºopºord, (12) 299 with discrete representation (taking out a proportionali- ty factor R.) 1(c);=XI(c) + (cº-Rº/Rºl (11b) Defining o-ſ, (c) (Pºord (3) Spheres The amount of l-like charge in the aluminum spheres for the iº" state, and an aluminum l-like density of spin states NFA'(&) | 2 | - “O AJ.4 l = — — — l wº-isºxo (14) Again approximating Pºſt) by P'(r) o-cºſ (Pºoyar, (15) dropping all irrelevant factors of proportionality we get I(ſ)|& 3 Nº"(ſ) + CNº"(ſ) (16) where C = 1. Plots of the orbital densities of states are shown in figures 2 and 3. The aluminum d-contribution is small and will be disregarded. The s curve fairly well replicates the total density of states in figure 1 with the exception that the low energy peak is shifted about a volt lower and the structure on the low side is more pronounced. This low energy peak corresponds to the Hoso >- O § H ! 2 f i u T) i — | t ; : — " U ill ií Z | bl 0– Uſ) O25 || LL 2 O LL O l (ſ) Lil i ºf 5 OOO —5.O O.O 5.O 1O.O 15 O ENERGY (ELECTRON-VOLTS) FIGURE 2, Density of aluminum 3s-like-states, Nà". aluminum 3s-bonding band although about half of its magnitude is attributable to tails of the gold d-band wave functions overlapping the aluminum sites. This figure should be compared with the experimen- tal results of Williams, et al. [6] reported in these proceedings as shown in figure 4 graciously supplied by Dr. A. J. McAlister. We see that their peak =8 eV below the Fermi energy is well reproduced in our figure 2. This we take to be partial confirmation as to the loca- tion of the gold d-bands. Also the structure on the low energy side seems to be present in our calculations. Structure between the peak and the Fermi energy seems to be present in both curves although our values seem about 1 eV too high. We have calculated alu- —l —l § 0.50 >- H (D 2 | D 2. > u LL] : 2 | | | j} | | | | | ° O.25 H– 2 | O ! LL O ; : (ſ) LL] H <ſ H 90 O.OO $ –5.O O.O 5.O 1O.O 15.O ENERGY (ELECT RO N-VOLTS) FIGURE 3. Density of aluminum 3d-like states, Ná'. 1.OF-1 cr) | | O.O O.O –1O.O º E - Ee (ELECTRON-VOLTS) FIGURE 4. Relative Intensity/a)” of aluminum L2, a soft x-ray spectra (courtesy of A. J. McAlister). 300 minum 2p-core bands and find them 73.8+ 0.1 eV below the Fermi energy compared with the experimen- tal value of 73.5 + 0.5. This band is less than .02 eV wide and every state is 99.97% aluminum 2p-like. This confirms our neglect of final state energy. In table 2 we give the ratio _IC!"]+[f]" (r)] [C]"P'(r)]” [Cl2] [fºº(r)]* [Crºpſy (r)]2 g(r) (17) for the states T1 = — .194 Ry and T2' = .625 Ry, the bottom and top of the aluminum 3s-bands, respectively. Although this varies by a factor greater than three from the origin to the sphere radius, in the region of overlap with the 2p peak it varies by less than 10% (discounting the value for r=.8660 which is small because it is near a node of [P" (r)]*). To the extent we can neglect the variation of the radial function with energy and define Ps"(r) and fºſr), the ratio given in eq (17) should be equal to a constant, Q}/Qſ", which is .5114 com. pared to a variation of from 0.33 to 1.25 within the sphere. Thus we may have overestimated low energy C”s by 25% and underestimated the high energy Cº’s by 25% at most. The extra factor or r in eq (9) would tend to prejudice the integral towards larger r values, reducing these errors to about + 10%. In any case one could calculate the corrections throughout the band if more accurate results were needed. In summary we conclude that our band structure cal- culation reproduces density of states information over an energy range of 10 eV with reasonable accuracy. TABLE 2. Variation in Pºs radial wave function from bottom to top of band r (a.u.) [f]" (r)]*/[fºº' (r)]? [Popjº 0.0019 0.330] .0188 .3302 .0377 .3303 .0754 .3310 2p Peak/5. .1130 .3330 .1883 .3209 .2636 .3281 .4142 .337] 2p Peak, .5697 .3617 .8660 .1977 1. 1672 .3013 1.7697 .4469 2p Peak/10. 2.3721 1.0410 2.4549 1.2495 Sphere radius. 5. References [1] Switendick, A. C., and Narath, A., Phys. Rev. Letters 22, 1423 (1969); see figure 1 for a picture of these bands. [2] Burdick, G. A., Phys. Rev. 129, 168 (1963). [3] Bhatnager, S., Phys. Rev. 183,657 (1969). [4] Snow, E. C., and Waber, J. T., Phys. Rev. 157,570 (1969). [5] Janak, J. F., Physics Letters 28A, 570 (1969). [6] Williams, M. L., Dobbyn, R. C., Cuthill, J. R., and McAlister, A. J., these Proceedings, p. 303. 301 Discussion on “Orbital Symmetry Contributions to Electronic Density of States of AuAl2" by A. C. Switendick (Sandiq Laboratories) F. M. Mueller (Argonne National Labs.): Since you A. C. Switendick (Sandia Labs.): I think it would be have the wave functions point by point in the zone fairly easy. I think the essential variation is in how separated into s, p, and d characters, do you have any much s, p, and d character there is in the wave func- plans to calculate dipole matrix elements as a function tion, but one certainly could do it quite easily with the of position in the Brillouin zone itself?. Presumably you wave functions we have. could do this, but it would be rather difficult perhaps? 302 Soft X-Ray Emission Spectrum of Al in AuAl2 M. L. Williams, R. C. Dobbyn, J. R. Cuthill, and A. J. McAlister Institute for Materials Research, National Bureau of Standards, Washington, D.C. 20234 Recently, Switendick and Narath have reported results of a systematic calculation of the electronic band structure of the compound series AuX2(X = Al,Ga,In). We have measured the L23 soft x-ray emis- sion spectrum of Al in Au/Al2 and from it, estimated the Al L8 emission profile. We compare the latter to the distribution in energy of s-like charge at Al sites, estimated by Switendick from his band calculation. This s density is the dominant factor in a one-electron estimate of the soft x-ray emission rate. Quite good agreement is found, lending strong support to the calculations for Au/Al2. This result also supports interpretation of a recently observed low energy peak in the L23 emission spectrum of Al in Ag-Al alloys in terms of Agd and Al s-p hybridization. Key words: Electron density of states; gold aluminide (Au/Al2); gold-gallium (AuCa2); gold-indium (Auln2); intermetallic compounds CsCl structure; L spectra; silver-aluminum alloys (AgAl); soft x-ray emission; s-orbital density of states. 1. Introduction The series of intermetallic compounds, AuX2(X = Al,Ga,In), is of considerable current interest, largely because of the unusual behavior of the magnetic susceptibility and Ga Knight shift in AuCa2. Specific heat [1] and de Haas-van Alphen Fermi surface studies [2] indicate that these materials are fairly well described by the nearly free electron model, and reveal no strong differences between them. Yet while the Al and In Knight shifts are positive and essentially tem- perature independent, the Ga Knight shift displays a large and unusual temperature dependence, ranging from —0.13% at 4 K to +0.45% at 230 K [2]. The mag- netic susceptibilities of AuſAl2 and Auln2 are tempera- ture independent, but AuCa2 displays a temperature dependence [2]. Switendick and Narath [3] have recently reported a systematic calculation of the elec- tronic band structure of this compound series, and from it, suggest an interpretation of the Fermi surface and Knight shift data. A somewhat surprising feature of their results is the location of the d bands at 7 to 8 eV below the Fermi level. This is in sharp contrast to the 2 eV or so suggested by interpretation of the optical properties of these unusually colored materials via d band to Fermi level optical transitions [4]. While not giving a detailed analysis of the optical properties, Switendick and Narath suggest that the d bands are not necessary to an explanation of the optical properties, since the calculation predicts a large number of s- and p-like states both above and below the Fermi level. We report here a measurement of the L23 emission spectrum of Al in AuſAl2. From it, we estimate the Al L3 emission profile, and compare it to the s-orbital state density at Al sites (the leading term in a one-electron calculation of the L spectrum) estimated by Switendick from his band calculation, and reported elsewhere at this conference [5]. Good agreement is found, lending strong support to the validity of the AuſAl2 band calcula- tion. Analogies between the present work and the recent observation of a low energy peak in the L2.3 emission spectrum of Al in Ag-Al alloys [6] support in- terpretation of the latter in terms of Ag d and Al s-p hybridization. 2. Experimental Details Measurements were made in a previously described [7] glass grating, vacuum spectrometer using photoelectric detection. The spectrum was scanned continuously, total counts being recorded over succes- sive short time intervals. Successive runs were summed to enhance signal to noise ratio. The relative counting error in the raw data, (N)-11”, where N is the accumulated count per channel, ranged from 1.8 to 0.8% This degree of statistical assurance was achieved at the expense of instrumental resolution, which we estimate to be 0.35 eV at the Al emission edge. Mea- 303 surements were made at an average pressure of 7 × 10−8 Torr, at approximately 500 °C. Electron beam ex- citation was used, at an energy of 2.5 keV. The sample was a polycrystalline rod, lightly machined and washed in acetone and absolute alcohol before mounting in the instrument. It was prepared from 99.999% pure Au and Al starting materials, by first adding Au in stoichiomet- ric proportion to Al induction melted in an alumina boat under vacuum, then drawing the final melt into a graph- ite lined quartz tube. Optical metallographic exam- ination of, the sample'showed that toward the center, the last part to freeze in this preparation technique, traces of free Al (much less than 1%) occurred at the grain boundaries. None was observed at the surface. Such Al contamination as might occur at the surface should not noticeably affect the results. A practical check of this is possible. The pure Al profile differs strongly from that of Al in the compound, and any sig- nificant distortion from this source would have been evident. A further complicating factor is the Au O2.3 spectrum which overlaps the Al L2.3. Supplementary measure- ments on Au, not reported here, show the O2.3 band to be very weak in the pure metal. If we make the reasona- ble assumption [8] that the Au O23 and N67 bands in the alloy have the same relative intensity as in the pure metal (Au N67 does not overlap Al L2,3), then we can as- sign a maximum peak intensity to Au O2.3 in the alloy of no more than 2% of the Al L2.3 maximum. Since an oil diffusion pump was used to evacuate the instrument, the fourth order of the C K band might also be expected to distort the observed spectrum. However, scans of the first and second order C K bands show them to be quite weak, and past experience indicates that C K lies very near or beyond the fourth order cutoff of our instru- ment. In view of the weakness of these distorting fac- tors, we make no attempt to correct for them, and in treating the data, ignore their presence. 3. Comparison with Calculated s-Density The upper curve of figure l is our corrected experi- mental estimate of the L8 emission profile of Al in AuſAl2. The following procedure was used to construct it. First, a curve was drawn through the raw data in a manner consistent with the standard counting error. The background continuum was estimated according to a prescription given elsewhere [7], and subtracted off. The spectrum was then corrected for the presence of the L2 band, using pure Al values of the L9/L3 intensity ratio (0.21) and spin orbit splitting (0.4 eV) determined from inspection of the pure Al emission edge in second order. Next, a first order correction was applied for the changing energy resolution of our spec- trometer [7]. Finally, the spectrum was divided by the cube of the photon energy to reduce it as far as possible to a representation of the s-orbital density. The ordinate of the plot is arbitrary. In the one-electron approximation, the soft x-ray L3 emission rate per unit energy from the ith component of a compound may be written R(ho) & (E(R)-E.)* > |< 2p || 7 || > |6(ho–E(k) +E.) k, n The matrix element is evaluated at the site of the ith- type ion from normalized band wave functions |k), of energy E(k). 2p.) is the core wave function and Ee its energy. ha) = E(k) — Ec is the photon energy. n is a band index. The sum extends over the Brillouin zone and all bands. Within the framework of an augmented plane wave (APW) calculation, one can estimate for a given state |k) the fraction of the total charge within a unit cell which resides in the plane wave region, wow, and in the various orbital components within APW spheres of given ion type, wi'. These orbital weights can be in- troduced into a sum over states to yield orbital state densities, ni'(E). If the inner level wave function is well localized in the APW spheres (this is the usual case), then it is a straightforward, although tedious, task to show that the L3 emission rate may be written R(no (E-Boºne). Mºſen (b)] where A; M2p, = ſ dr r" R,(r)R}(r, E) () 304 SMEARED Al Ns (E) Al Ns (E) E - EP (ev) FIGURE 1. The lower curve is the s-orbital state density at Al sites in AuſAl2 estimated by Switendick [4]. The middle curve illustrates, in a rough approximation, the effects of spectrometer, inner level lifetime, and final state lifetime smearing on the calculated curve. The upper curve is the measured Al L8 emission profile from AuſAl2, corrected as described in the text. depends only on energy. The R’s are radial wave func- tions, Ri' being normalized over the APW sphere of radius Ai. For Al in Au/Al2, we anticipate negligible d contributions, on the basis of the nature of the potential within the Al spheres and symmetry considerations. The integral M2po is smoothly varying and will not af. fect structure in no"(E), the quantity estimated by Switendick [5] from his band calculation and shown as the bottom curve of figure 1. In the middle curve, we have attempted to illustrate the effects of spectrometer, inner level lifetime, and final state lifetime smearing on the spectrum by folding into Switendick’s curve a Lorentzian smearing function of energy dependent width. The width y(E) is the sum of two parts y (E) = yo-Hyſ (E) The constant yo was taken as 0.5 eV, to represent the effects of the spectrometer and the inner level. The energy dependent part varies from 0 at the Fermi level to 2.0 eV at the bottom of the band, and has been taken to be proportional to the total fraction of s-charge lying above the level in question. It is intended only as a rough approximation to the many body final state level broadening which occurs in an interacting electron gas [9]. The general agreement between the calculated S density and the measured Al L3 profile is quite striking. There is in fact, a one to one correlation of structural features, marred only by the slightly greater overall width of the measured spectrum, slight displacements of the peaks, and certain not too gross amplitude dis- crepancies. In all, the validity of the Au/Al2 band calcu- lation is strongly confirmed by this comparison. The large calculated peak at about —8.0 eV is of par- ticular interest. The band calculation [4] indicates that about half its intensity arises from a high density of states which are d-like on Au sites and strongly local- ized there, but have roughly 5% of their total charge in s-like orbitals at Al sites. The occurrence of similar sharply peaked structure at —8.3 eV in the Al L3 profile from the compound is consistent with this predicted lo- cation of the d bands. We further note that this behavior is strongly reminiscent of the distinct peak at 7.5 eV below EP recently observed by Marshall et al. [5] in the L23 emission spectrum of Al from alloys of up to 20 atomic percent of Ag in Al. It suggests that their interpretation in terms of Agd and Al s-p hybridization in the alloy is correct. 4. Acknowledgments We are most grateful to Dr. A. C. Switendick for communication of his results prior to publication. We also thank Dr. L. H. Bennett for introducing us to this problem and for many helpful discussions, Dr. R. E. Watson for instructive discussions, and Mr. D. P. Fickle for preparing the sample. 5. References [1] Rayne, J. A., Phys. Letters 7, 114 (1963). [2] Jaccarino, V., Weber, M., Wernick, J. H., and Menth, A., Phys. Rev. Letters 21, 1811 (1968). [3] Switendick, A. C., and Narath, A., Phys. Rev. Letters 22, 1423 (1969). [4] Optical measurements have been reported by S. S. Vishnubhlata and J.-P. Jan, Phil. Mag. 16, 45 (1967). Interpretation in terms of d band to Fermi level optical transitions has been suggested by J. H. Wernick, A. Menth, T. H. Geballe, G. Hull, and J. P. Maita, J. Phys. Chem. Solids 30, 1949 (1969). [5] Switendick, A. C., these Proceedings, p. 297. [6] Marshall, C. A. W., Watson, L. M., Lindsay, G. M., Rooke, G. A., and Fabian, D. J., Phys. Letters 28A, 579 (1969). [7] Cuthill, J. R., McAlister, A. J., Williams, M. L., and Watson, R. E., Phys. Rev. 164, 1006 (1967); Cuthill, J. R., Rev. Sci. Inst., 41,422 (1970). [8] Cuthill, J. R., McAlister, A. J., and Williams, M. L., J. Appl. Phys. 39,2204 (1968). [9] Landsberg, P. T., Proc. Phys. Soc. (London) A62, 806 (1949): Longe, P., and Glick, A. J., Phys. Rev. 177, 526 (1969). 417–156 O - 71 - 21 305 Soft X-Ray Emission from Alloys of Aluminum with Silver, Copper, and Zinc D. J. Fabian,” G. Mc D. Lindsay, and L. M. Watson Department of Metallurgy, University of Strathclyde, Glasgow, Scotland Measurements of the soft x-ray emission from the alloys Al-Ag, Al-Cu and Al-Zn are reported, and the effect of the d-bands of the metals Ag, Cu and Zn on the Al L2,3-emission for these alloys is ex- amined. For Al-Ag and Al-Cu alloys, sharp resonance peaks are observed in the Al L23-spectra and are attributed to transitions from states in the hybridized silver and copper d-bands to core states of the alu- minums atoms. The observations agree with the general theoretical considerations discussed by Har- rison [7] for a simple metal alloyed with a noble metal. For the Al-Zn alloys the d-bands of Zn do not contribute to the Al Lea-emission. Key words: Aluminum-copper alloys (AlCu); aluminum-zinc alloys (Al-Zn); electron density of states; silver aluminum alloys (Ag-Al); soft x-ray emission. 1. Introduction Soft x-ray emission investigations for studying elec- tronic structure have been steadily gaining the atten- tion of physicists and metallurgists interested in the theory of alloys, despite the well established problems of interpreting the spectra of pure metals. Curry [1] usefully summarizes the experimental findings for many of the alloy systems examined to date. With al- loys the problems of interpretation are further com- plicated by the absence of a fully satisfactory theory that explains the electronic behavior of a metal when it is alloyed with a second metal. Theoretical work in this field is continually develop- ing, and among the models that have been proposed for alloys are: the rigid-band model, first described by Jones in 1934 [2]; an “impurity” model that assumes restricted sharing of the valence electrons of the com- ponents, with consequent localized screening effects, developed chiefly by Friedel in 1952 [3]; and the two- band model successfully used by Varley in 1954 [4] to correlate thermodynamic heats of formation of binary alloys. The degree to which these models assume distinct valence bands for the two components of the alloy increases in the order listed. - At the one extreme, the rigid-band model assumes a common band of valence electrons, for which any in- *Present address: Visiting Professor at the Laboratory for Solid State Physics, Swiss Federal Institute of Technology, Zürich, Switzerland. coherent scattering due to the additional random “im- purity” potential, superposed on the basic periodic potential of the lattice, is negligible. Appleton [4] discusses this effect in relation to soft x-ray emission and also extends, qualitatively, the restricted-sharing approach used by Friedel to alloys with concentrations of solute that are greater than generally regarded as im- purities, and to alloys of higher valence components. At the other extreme we have the two-band model in which the valence electrons for a binary alloy are as- sumed to exist in two sets of energy states each as- sociated with the potential fields of the ions of one com- ponent; this model is a consequence of the changes that arise in Wigner-Seitz boundary conditions for the valence electrons when the pure-metal atoms are ran- domly mixed. Basically, both extremes recognize a valence band that is common to the alloy. Rooke [5] lends support to the two-band approach, pointing out that the pseudo- atom model, Ziman [6], will require local electron den- sities to remain unaffected on going from the pure metal to the alloy, provided that predominantly the electronic density for the pseudo-atom lies close to the ionic core; while Harrison [7], with a slightly different approach using pseudopotential theory, supports the view—for the simple metals—that the alloy com- ponents exhibit a common band structure. In the case of a simple metal alloyed with a noble or transition metal, characterized by a narrow d-band, 307 Harrison considers that this d-band for the transition metal will tend to retain its structure and carry over to the alloy; so that, if this is the “solute” metal, each solute atom will produce a strongly localized set of states at an energy that corresponds to the narrow d- band for the pure solute metal. We describe here measurements of the soft x-ray emission from alloys of aluminum with, respectively, the metals silver, copper and zinc, and we examine the effect of the d-bands of these solute metals on the Al L2,3-emission from the alloy. - 2. Experimental, and Specimen Preparation The spectrometer used in these investigations, and the method employed for processing the emission spec- tra, are fully described elsewhere; see Watson, Dimond and Fabian [8,9]. For the examination of alloys addi- tional difficulties arise associated with preparation and homogeneity and with precise determination of struc- ture during the excitation of soft x-ray emission. Where possible single-phase solid solutions were investigated for the measurements described here; for the Al-Ag and Al-Zn systems this necessitates quenching the alloy from the equilibrium temperature for the ap- propriate single phase. The Al-Cu alloys were mostly two-phase but were heat-treated in the same manner. The alloys were prepared from high-purity com- ponent metals by melting and casting in a reducing at- mosphere. Selected small ingots, free from imperfec- tion, were homogenized for 7 days in evacuated cap- sules at a temperature chosen according to the equilibrium phase diagram for the system and then rapidly quenched. Specimens in the form of thin slabs 10 mm × 5 mm, and ~ 1 mm thickness, were obtained from the homogenized ingots by hot or cold rolling ac- cording to the physical properties of the alloy, one sur- face was ground flat and polished, and the slab then subjected to further annealing and quenching. Struc- ture was checked metallographically before and after examination in the spectrometer. All spectra reported here were recorded using a platinum coated, 1-meter radius, blazed grating with 600 lines/mm. The alloy specimens during measure- ment were supported on, and in good thermal contact with (using conducting cement), a water-cooled stain- less steel target anode. Vacuum of ~ 1 × 107" Torr was established in the target chamber, after considerable out-gassing of the alloy, and spectra were recorded using 3 kV and 5 mA respectively for target voltage and current. For each alloy investigated, spectra were summed, in the manner previously described [8.9], until the total count obtained at a given wavelength position of the spectrum, was sufficient to reduce statistical uncertainty to <1%. The spectra are corrected for background (chiefly Bremsstrahlung) and for the intensity fall-off during a scan due to contamination of the specimen surface. The background is assumed to be linear with wavelength, and the contamination linear with time after an initial rapid fall-off which avoided by delay- ing the start of a scan. The specimen surface is scraped clean, under vacuum, between scans. 3. Results and Discussion The Al L2,3-emission spectra obtained for a series of aluminum-silver alloys, of varying silver content, are shown in figure 1. The spectra are plotted in terms of I/vº (in arbitrary units) as a function of energy (in eV). The division of the emission intensity by the fifth power of the frequency takes account firstly of vº in the rela- tIOn 1000 - # 100d. (1) 4 * , , , , , ; , , ; * * * * . . . . . º • { * * * : * 14 .,,, . '''''''', I * * * % fit tº it ! * , , § { { , , , , , , , , , , , , , ‘’’’ ‘’’’ ‘, ‘t,' ' ' ' ', ,, ...,'" . . . . . . , , , , , , " " ' " " ' ". . . * * * * * * 74 72 7O 68 66 64 62 6O 58 ENERGY (ºv) Al L2,3-emission from quenched Al-Ag alloys of com- positions 0–20 atomic percent Ag. All are single-phase fee solid solutions. The spectra, shown as I/vº in arbitrary units versus energy in eV, are compared by normalizing to equal peak maximum. Each spectrum is the sum of 20 to 50 scans. Vertical bars indicate the statistical uncertainty in count rate, calculated from the total count. FIGURE 1. 308 which converts intensity at wavelength N to intensity at frequency v, and secondly of vº in the relation P(b) < * | * > r, dº (2) which gives the transition probability for the transition from initial state 1 to final state 2. Because an accurate knowledge of the initial-state and final-state wavefunc- tions is required, and is impossible to determine, no at- tempt is made to account for the matrix elements in this correction for transition probability, and the curves obtained—I/vº vs E – are regarded as only approxi- mately but usefully related to the density of valence- band states for the alloy. With Al L23-emission these will be the valence-band states as seen by aluminum atoms in the alloy. On com- paring the emission spectra for Al-Agalloys (fig. 1) for varying silver concentration, clearly the most important feature to emerge is the peak, at ~65.5 eV, that first ap- pears at ~10 at. 9% Ag and rises sharply in intensity as the silver concentration is increased to 20 at 9%. Over the whole of the composition range investigated these alloys form single-phase solid solutions at the tempera- tures selected for annealing, and the alloys were quenched from this temperature to retain the Q-phase structure. The specimens showed, metallographically, no observable segregation before or after examination. The quenched alloys are known [10,11] to exhibit clustering of silver, but this we do not expect to observe in simple metallography. We attribute this peak, in the Al L2,3-emission from the alloys, to transitions from the lower part of the hybridized d-bands of silver to vacant aluminum core states. A preliminary comparison by Marshall et al. [12], of the emission spectrum for the 20 at. 9% Ag alloy with the band structure for silver metal (fig. 2), drew attention to the close correspondence of this peak with the energy at which the d-bands of silver are expected to hybridize with the s,p-band. This hybridization is of course common with d-bands, but for silver the hybridized states lie in a narrow band right at the bottom of the silver valence band, giving rise to a low grad kB and consequently a high density of s- and d- states. This will occur at ~ 7.2 eV (5.3 Ry, fig. 2) below the Fermi-energy and agrees well with the peak ob- served in the Al L2,3-emission from Ag-Al alloys which appears at ~ 7.5 eV below the emission edge. Our interpretation requires that the states in the upper regions of the hybridized d-bands contribute less strongly to the Al L23-emission, and this is probably so because these states will tend more to hybridize with p- states which do not contribute to L spectra. The in- terpretation further requires that the d-bands for silver will remain, for the alloy, at much the same energy at which they occur for pure silver metal. This assumption may not be unreasonable, particularly in view of the clustering or coherent precipitation of silver that is known to occur to some extent for these quenched alloys. The bandwidths observed appear to remain constant with varying silver concentration and this result lends no support to the rigid-band model for these alloys. An attempt was made to measure the Ag N2,3-emission but the intensity was too low to permit this emission band to be detected above the background. However, our results for the Al L23-emission support the conclusions reached by Harrison [7] that the d-bands of the solute metal will carry over to the alloy; we find that they con- tribute to the valence band and are “seen” by the sol- Vent at Om S. The results can also be explained, although less neatly, using the approach developed by Friedel [3]. Restricted sharing of valence electrons between com- ponent atoms will lead to local screening of the higher positive charge on the aluminum ions and cause the electron wavefunctions to increase near these ions. This will result in a relative increase in intensity of emission at the bottom of the valence band since the electrons in the higher energy states will be more in- volved in sharing with other atoms in the alloy. Our measurements of the Al L23-emission from alu- minum-copper alloys show a similar peak, first appear- ing for approximately 10 at 9% Cu and increasing sharply with additional Cu concentration. For the 20 at. % Cu alloy which is a two-phase structure of foc Cº-Al and tetragonal 6-Al2Cu, the general shape of the spec- trum, shown in figure 3, agrees well with that obtained by Curry [l] for Al2Cu; the same intense peak is ob- served at ~66 eV. The Cu M2,3-emission overlaps the Al L23, but its intensity for the 20 at 9% Cu alloy was too low to have any significant effect. The energy at which the hybridized d-bands for copper, figure 2, can be ex- pected to contribute to transitions to Alcore states does not in this case correspond as closely with the observed peak in the Al L2,3-emission as for Al-Ag. The peak oc- curs at ~6.0 eV below the emission-edge, and the bot- tom of the hybridized copper d-bands at ~5.0 eV below the Fermi energy. However, the emission in the case of this alloy will come largely from the Al2Cu phase which tends to order, and the emission from the Q-Cu phase will be affected also by the coherent precipitation of or- dered Al2Cu that occurs for this quenched alloy [13]; if the transitions that give rise to the observed peak are assumed to be transitions from d-bands of ordered 309 SILVER foc) O. 3 O., 2 | 2H * O. 1 25 * ... ." !o.o 1-T pus COPPER (fec) © - * 7 1- g § @ º re ſo ~5 *U PN > Q& & O.7 * > O. 7 § # 0.6 © g T.L ă ºr * Tz lºs - O. 3 H. O. 3 O. 2 im O. 2 O. 1 º O. 1 A r’ RL2 FIGURE 2: Band structures for the pure metals Ag, Cu, and Zn. Silver; calculations of Ballinger and Marshall [17]. Copper and Zinc, calculations of Mattheiss [18]. 310 *A*, ' ' |", W º, ,”, FF, 0 | () '. º | - h ſº N A { º, 0. A l l t 's 's o ſº 't ſº ', 0. | ſº t! & "", ** tº º '', A.V * TS T2 71 70 CQ Q 57 (.3 GS GA $3 CŞ2 51 (50 FIGURE 3. Al L2, 3-emission from quenched Al-Cu alloy containing 20 atomic percent Cu. Two-phase alloy, oft-Al (fec) plus 3-Al2Cu (tetragonal). The spectrum is the sum of 25 scans; vertical bars indicate statistical uncertainty in count rate. | ,” '', l ºn!" **. º ". 1,1,1''''''''' d ". º "Hºw" ". All | t | '', bº ! I/uſ H ſº ! 't | º º '', º '', '', '', '', ". !! ", s.V 10 * Tº Tº 71 TO Gº GE 67 CS 65 Gº GC 52 S1 G0 FIGURE 4. Al L2, 3-emission from quenched Al-Zn alloy containing 55 atomic percent Zn. Single-phase alloy, cph. Spectrum is the sum of uncertainty in count rate. Al2Cu to Al core states, we can explain the energy shift in terms of the lowering in energy of the Cu d-bands that must occur when Cu atoms form a locally ordered structure with Al atoms. This is again the d-band structure of the solute metal carrying over to the alloy and being “seen” by the sol- vent atoms. The strong intensity of the observed peak must be attributed either to a high narrow density of these d-band states or to a matrix-element effect for these transitions. 30 scans; vertical bars indicate statistical In the case of Al-Zn alloys we find no contribution to the Al L2,3-emission from the d-bands of Zn. This is en- tirely what we should expect from the band structure of zinc (fig. 2) where the d-bands do not appear because they are much lower than for Ag or Cu and are found below the 4s, so that little or no hybridization with s- states occurs. Our result for the Al L2,3-emission from the 55 at 9% Zn alloy is shown in figure 4. The band- width for these alloys remains constant and nearly equal to that for pure aluminum, indicating again that 311 the rigid-band model does not apply to these alloys. A small relative increase in intensity in the lower-energy part of the emission band does occur and is probably due to the localized screening of aluminum ions due to the restricted electron sharing discussed by Friedel, but no peak is observed such as that found for Al-Ag or Al-Cu. Clustering of Zn in the quenched alloys is also known to occur in this system [14], and clearly in itself does not affect the emission spectra. This suggests that for Al-Ag and Al-Cu, clustering cannot alone explain the features of the Al L2,3-emission. The results observed for the L2,3-emission from Al-Ag and Al-Cu alloys show marked similarities to the effects observed in the K-emission spectra for these alloys by Baun and Fischer [15], and for the K-emission from Al- Ag by Nemnonov [16]. We believe that the same in- terpretation can be applied to the K-emission results. 4. Acknowledgments This work was supported by a grant from the Science Research Council. The authors wish also to thank C. A. W. Marshall, G. A. Rooke, and Dr. Si Yuan for helpful discussion, Professor E. C. Ellwood for encouragement and support, and Professor G. Busch for his interest. 5. References [1] Curry, C., in Soft X-Ray Band Spectra and Electronic Structure [2] [3] [4] [5] [6] [7] [8] [9] [10] [ll] [12] [13] [14] [15] [16] [17] [18] of Metals and Materials, D. J. Fabian, Editor (Academic Press, London, 1968), p. 173. Jones, H., Proc. Roy. Soc. Al44, 255 (1934). Friedel, J., Phil. Mag. 43, 153 (1952). Varley, J. H. O., Phil. Mag. 45,887 (1954). Rooke, G. A., in Soft X-Ray Band Spectra and Electronic Struc- ture, D. J. Fabian, Editor (Academic Press, London, 1968), p. 185. Ziman, J. M., Proc. Phys. Soc. 91, 701 (1967). Harrison, W. A., in Soft X-Ray Band Spectra and Electronic Structure, D. J. Fabian, Editor (Academic Press, London, 1968), p. 238. Watson, L. M., Dimond, R. K., and Fabian, D. J., J. Sci. Instru. 44, 506 (1967). Watson, L. M., Dimond, R. K., and Fabian, D. J., in Soft X-Ray Band Spectra and Electronic Structure, (Academic Press, London, 1968), p. 45. Guinier, A., J. Phys. Radium 8, 124 (1942). Simerska, M., Czech. Jnl. Phys. B12, 54 (1962). Marshall, C. A. W., Watson, L. M., Lindsay, G. M., Rooke, G. A., and Fabian, D. J., Phys. Letters 28A, 579 (1969); and Fabi- an, D. J., Ellwood, E. C., Lindsay, G. M., Watson, L. M., and Marshall, C. A. W., in Proceedings X-Ray Spectra and Elec- tronic Structure of Matter, Institute of Metal Physics, Ukr. Academy of Sciences, 1969, p. 26. Turnball, D., in Impurities and Imperfections, Amer. Soc. Metals, 1955, p. 121. Rudman, P. S., and Averbach, B. L., Acta Met. 2, 576 (1954). Baun, W. L., and Fischer, D. W., International Res. Rep. (AFML-TR-66-191), Air Force Materials Laboratory, Ohio, 1966. Nemnonov, S.A., private communication, to be published. Ballinger, R. A., and Marshall, C. A. W., J. Phys. Chem. (Proc. Roy. Soc.).2, 1822 (1969). Mattheiss, L. F., Phys. Rev. 134 (4A), 970 (1964). 312 Soft X-Ray Emission Spectra of Al-Mg Alloys H. Neddermeyer Sektion Physik der Universitat München, München, Germany In recent years the interpretation of soft x-ray emission band spectra has made good progress. With a detailed knowledge of the electronic band structure, of transition probabilities, and of lifetime broadening effects, it has been possible to calculate the shape of emission band spectra of a few pure elements [1,2]. However, the situation is much more complicated in the case of alloys where the problems are far from being solved. The different shapes of emission band spectra of the components of an alloy make the applicability of the usual model to alloy spectra doubtful. As a contribution to these problems we have remeasured the soft x-ray emission band spectra of Al- Mg alloys using improved experimental techniques. The Al L2.8- and Mg L2,3-emission spectra lying in the same wavelength region can be studied in the same spectrometer. Since the spectra of the pure metals have characteristic details and the energy resolution in this wavelength region is good, shapes and changes of shape can be registered very precisely. Key words: Aluminum (Al); aluminum-magnesium alloys (Al-Mg); charging effect; electronic densi- ty of states; emission spectra; magnesium (Mg); rigid-band approximation; soft x-ray 1. Theory For electronic transitions from occupied valence states to empty core states, causing a soft x-ray emis- sion band spectrum, the basic formula for the intensity distribution of the spectrum can be written as [3] : * . 6(Er – E – ho) (l) Ic, r(a)) = A ſº |e Mc, Ico(a)) means the number of transitions of energy ha) per unit time, A is a constant, le:Mor the matrix element for the transition probability, and Er, Ee the energies of the valence and core states. The integration has to be carried out over all possible wave vectors k. Using the property of the 6-function the integration can be per- formed and we obtain dS |e Mc, ſº -a- C’s ` = A → . I 3. (a)) S 873 | VI,(E,. - Ec)| Er—Ec=ho (2) where dS represents an element of a surface in the k- space defined by the equation E, (k) – E. (k) = ha). (3) Usually one assumes Ec(k)= const for all k, thus imply- ing that the inner level is sharp, and one obtains the well known formula dS le Mc, alº Ic, r(a)) - , 87° |vkE, Er – Ec = ha) (4) In simple cases this integral can be split into a product of two factors, namely the transition probability and the density of states. - In the case of alloys the difficulty arises that we know neither the transition probability, nor the density of states nor E(k)-curves. In the case of a well defined sin- gle-phase Al-Mg alloy one would first apply the concept of the rigid-band model. This would mean that the valence electrons of the Al and Mg atoms constitute a single valence band. The validity of the rigid-band model for dilute Al-Mg alloys has indeed been proved by measurements of the Fermi surface [4]. Thus one should expect similar Al and Mg spectra. The fact that the two spectra are different [5-7] can be explained by a charging effect of Al atoms having a higher valence than the Mg atoms [8]. Charging leads to a nonu- niformity of the valence electron distribution within the crystal, i.e., the electron density near the Al atoms is higher than near the Mg atoms. 2. Experimental The Al L2.8- and Mg L2,3-emission spectra of the al- loys and of the pure metals have been obtained using a 313 concave grating spectrometer with ultrahigh vacuum conditions [9]. The x rays were excited directly by bombardment voltages of 2 kV and emission currents of 1 to 3 mA. The spectra were recorded either continu- ously with a strip-chart recorder or stepwise using digital equipment. The influence of contamination and self-absorption could be neglected. The spectra were corrected for the known reflectivity of the grating. The influence of the quantum efficiency of the multiplier photocathode seemed to be small. Besides the pure metals Al and Mg, the following Al- Mg alloys were investigated: Al;Mggs, AlloMg20, AlsoMg10, Alig Mg17, Ala Mg2, and Al-Mg. Microscopic and x-ray diffractometer studies were made to check the crystallographic order. According to the phase dia- gram [10] AlizMg17 and Ala Mg2 are single phase alloys. The dilute alloys Al;Mg35 and AlioMggo are single phases at 430 °C; at room temperature these alloys decompose slowly into binary phases. During the spec- troscopic studies this decomposition could be recog- nized slightly in the case of AlioMggo. AlsoMgro and Al2Mg are binary phases. 3. Results: Dilute Alloys The Mg L2,3-spectra of Al;Mg35 and AlioMggo agree with the Mg L2,3-spectrum of pure Mg (fig. 2) within the limits of statistical error, of less than 3%. Also the posi- tions of the emission edges agree within +0.01 eV with the emission edge of the pure metal. On the other hand, the shapes of the Al L2,3-spectra of these alloys (fig. 1), which are in agreement with measurements reported earlier [5], differ markedly from the Mg L2.3-spectra. So one has to conclude that the rigid-band model is not applicable to these alloys. The differences between 100 H. M- —Al5Mg,95 CN: ----AllOMg30 U |- 50 H -* -* Tºl--~~ -- * - - the Mg and Al spectra can hardly be explained by even a considerable influence of the unknown transition probability. As a possible explanation one would rather assume the existence of clusters and localized bound states. A further argument for this assumption is the good agreement between the Al L23-spectra of Al;Mggs and AlioMg30. 4. Nondilute Alloys The Mg L2,3-emission spectra of the nondilute alloys are shown in figure 2. In figure 3 the emission edges of these spectra are drawn in an enlarged energy scale. The Al L2,3-emission spectra of these alloys are presented in figure 4, and the emission edges on an en- larged scale in figure 5. The exact positions of the edges and peaks and the widths of the edges are listed in table 1. The spectra of Alig Mg17 and Al3Mg2 are in rough agreement with those published by Appleton and Curry [7], but some fine structure features have not been re- ported previously. The statements of Appleton and Curry concerning the general behavior of the spectra of metals on alloying are verified to a certain extent by the present measure- ments, but there are also some essential deviations. An increasing concentration of one alloy component does not always affect continuous variations of the shape of the emission bands. So the Mg L2,3-emission spectra of Al3Mg2 and Al2Mg are in good agreement with each other, and the Al L2,3-emission bands of the dilute al- loys Al;Mggs and AlioMggo also agree with the emission band of AlaoMgro. Appleton and Curry further state that the widths of the emission bands of Al-Mg alloys are equal to those of FIGURE 1. Al L2,3-emission spectra of AlgMggs, AlioMgoo, and Alaowſgro. - i l l —1–1—1—l ——1–1 | | 1– I ! —l l 1—- 60 65 70 Energie (eV) The spectra have been normalized. 314 A 100H Mg L2.3 * - —Mg * | ----Al30 Mg?0 * * * * * * * * * * Al2 Mg17 ... . . . . . . . *- ---Al3 Mg2 50– ... --- - - - - - |- 40 45 Energie (eV) FIGURE 2, Mg L2, 3-emission spectra of Mg and nondilute Al-Mg- alloys. The spectra have been normalized. -| 150 km. Al L23 - —Al 100H ——Al2Mg | --Ala Mg2 ov ----Al30 Mg?0 3. ſu H 50+ ...-- - - - | 100– ‘.S. - *.*, \ cº • \\ 3. ... ', Mg L23 Tj \ W. F. H. * \ \ — Mg - \ A ---also wº 50– \ \ ' ' ' ' Aliz Mg17 \ , \ -- Ala Mg2 - \ , \ -- Al, Mg Wºme H •. 0,07 eV --- *S. ... SS | l | | | l l ===-\º-e--l. I - 490 49.5 500 505 Energie(eV) FIGURE 3. Mg L2,3-emission edges of Mg and nondilute Al-Mg- alloys. The theoretical resolving power in this wavelength region amounts to 0.07 eV [9]. Energie (eV) FIGURE 4. Al L2, 3-emission spectra of Al and nondilute Al-Mg- alloys. The spectra of the alloys have been normalized at the broad maxima. The spectrum of pure Al has been adjusted to the low energy side. the pure elements within 0.5 eV. Our measurements show that the usual parabolic extrapolation of the low- energy side of the band is not possible since the bands always have a concave shape. Therefore it can not be concluded from the experimental data that the Al and Mg atoms have valence bands of different widths. On the other hand, if the Al L2.8- and Mg L2.3-spectra of the single phase alloys Alig Mg17 and Al3Mg2 are fitted to each other at the emission edge, one sees a similarity which is most pronounced in the high energy region (figs. 6 and 7): (1) the emission edges have the same width if the broadening by the spectral window function is taken into account;(2) the widths of the peaks agree rather well; (3) the characters and shapes of the emis- sion bands are similar; both spectra of Ala Mg2 being smooth whereas the Al as well as the Mg spectrum of AlizMg17 has a fine structure. These similarities indicate a common valence band and support the views expressed by Harrison [11]. The differences of the spectra in the low energy region seem to be connected with the fact that the Al atoms with its higher valence are screened by the nearly free valence 315. A 100 i 5 O l | _l | I l | l 720 7.5 Energie(eV) FIGURE 5. Al L2,3-emission edges of Al and nondilute Al-Mg-alloys. The theoretical resolving power in this wavelength region amounts to 0.14 eV. | 100H Alt2 Mg17 - — A L23 3. --- Mg L23 ſuſ 50H TABLE 1. Al-Mg-alloys and the pure metals Al and Mg: Positions of the L3-peaks (maxima), L3-edges (50% points on the L3-edges). Widths of the emission edges (twice the difference between 50 and 90% points on the L3-edges). Data in eV. No correction for instru- mental line width has been made. L3-peak L3-edge L3-width Mg L2, 3| Mg 49.40 + 0.04 || 49.55 + 0.03 0.14 + 0.02 Al;Mg25 49.40 + 0.04 || 49.55 + 0.03 .16 + 0.02 AlloMgoo 49.40 + 0.04 || 49.56-E 0.03 .16 + 0.02 AlsoMgro 49.41 + 0.04 || 49.58+ 0.03 .20 + 0.02 Alt2Mg17 49.44 + 0.04 || 49.67 + 0.03 .28 + 0.03 Al3Mg2 49.45 + 0.06 || 49.79 + 0.03 .34 + 0.04 Al2Mg 49.44, -i- 0.06 || 49.79 –H 0.03 .36 + 0.05 Al L2, 3 || Al 72.53 + 0.05 || 72.74 + 0.05 0.22 + 0.02 Al2Mg 72.39 + 0.08 || 72.72 + 0.05 .34 + 0.03 Al3Mg2 72.28 + 0.08 || 72.62 + 0.05 .36 + 0.04 Al 12Mg17 | 72.30 + 0.06 || 72.57+0.05 .30 + 0.04 AboMgo 72.57+ 0.05 Alio Mggo 72.59 + 0.06 Al; Mg35 72.61 + 0.06 Energie (eV) FIGURE 6. Al L2, 3 and Mg L2, 3-emission spectra of single phase alloy All 2 Mg17. The spectra have been adjusted to the emission edges. electrons. Thus in the crystal the valence electrons are not distributed uniformly. This has an influence on the matrix element occurring in eq (1). The screening part of the valence electrons mainly will make transitions to the core states of the Al atoms because of the strong overlapping effects. These electrons are absent in the corresponding Mg spectra the intensity of which is lowered at the low energy side. The differences of the emission spectra of well defined, homogeneous phases of binary alloys therefore would be a consequence of different transition proba- bilities. The knowledge of the transition probabilities is thus an essential prerequisite for statements on the density of states, and this holds especially in the case of alloys. It is clear that the emission spectra of all alloy components must also be taken into account. The situation is more complicated in the case of two- phase alloys like AlsoMgro and Al-Mg. It appears that the emission spectra of the pure phases do not super- pose corresponding to their quantity ratio. The experi- mental and theoretical investigations should therefore be restricted at first to monophase systems. 316 \ w! 3| g 50+ e O Energie (eV) FIGURE 7. Al L2, 3 and Mg L2,3-emission spectra of single phase alloy Ala Mg2. The spectra have been adjusted to the emission edges. 5. References [7] Appleton, A., and Curry, C., Phil. Mag. 12, 245 (1965). [8] Stern, E. A., Phys. Rev. 144, 545 (1966). [9] Wiech, G., Dissertation, University München (1964); Wiech, G., Z. Physik 193, 490 (1966). - [10] Eickhoff, K., and Vosskühler, H., Z. Metallkunde 44, 223 (1953); Samson, S., Acta Cryst. 19, 401 (1965). [11] Harrison, W. A., in Soft X-Ray Band Spectra and Electronic Structure, D. J. Rubin, Editor (Academic Press, London, 1968), p. 238. [1] Rooke, G. A., Thesis, University of Western Australia (1967); Rooke, G. A., J. Phys. 2, 767 and 776 (1968). [2] Klima, J., to be published. [3] Bassani, G. F., The Optical Properties of Solids, J. Tauc, Editor (Academic Press, New York and London, 1966), p. 33. [4] Ketterson, J. B., and Stark, R. W., Phys. Rev. 156, 748 (1967). [5] Gale, B., and Trotter, J., Phil. Mag. 1, 759 (1956). [6] Das Gupta, K., and Wood, E., Phil. Mag. 46, 77 (1955). 317 An L-Series X-Ray Spectroscopic Study of the Valence Bands in Iron, Cobalt, Nickel, Copper, S. Hanzely Youngstown State University, Youngstown, Ohio 44503 R. J. Liefeld New Mexico State University, University Park, New Mexico 88001 This paper presents the results of an attempt to evaluate the merits of the soft x-ray spectroscopic method by examining a group of neighboring elements possessing a variety of valence band properties. The emission lines studied were the threshold level Lo (valence → LIII shell) lines obtained from high purity, polycrystalline bulk samples under bombardment by a nearly monoenergetic (AE - 1 eV) elec- tron beam. The associated Lili absorption spectra were obtained in this work as self-absorption curves from the same anode samples. Experimental and instrumental distortions were either eliminated, minimized or explicitly corrected for. The results indicate the presence of some anomalous emissions on the high energy side of the La line in elements possessing a large density of unfilled valence levels just above the Fermi energy. The valence band emission line shape for these elements (iron, cobalt, and nickel) is found to be strongly dependent on the incident electron beam energy even for near-threshold- level excitations. Analysis of the emission and self-absorption curves demonstrates that the x-ray spec- troscopic method is capable of exposing meaningful differences among the valence band energy struc- and Zinc tures of the solids examined here. Key words: Cobalt; copper; electronic density of states; iron; nickel; satellite emissions; self ab- sorption; soft x-ray emission; x-ray spectroscopy; zinc (Zn). 1. Introduction It has been the purpose of this work to record L se- ries valence band spectra of some first transition group metals, correct these for a number of experimental and instrumental distortions, and use the results to assess the utility of the x-ray spectroscopic method for the study of valence band energy structures in these solids. 1.1. The State of the Problem X-ray spectroscopy has long been acclaimed to yield accurate and useful information about valence band densities of states in solids. An x-ray valence band emission line spectrum and the associated photon ab- sorption spectrum are said to represent, in principle, the electronic densities of states (times appropriate transition probabilities) in the filled and empty portions of the valence band, respectively. This interpretation has been questioned on theoretical grounds [1]. Furthermore, the recorded data are unavoidably distorted by effects inherent in their acquisition [2,3]. Compensation of the observed spectra for such afflic- tions has largely been neglected in the past. Finally, although the last decade has brought forth experimen- tal results and theoretical predictions that are in better general agreement [4], much discrepancy still remains between calculated band shapes and x-ray spectroscop- ic results that purport to show the densities of states in these bands. There exists, therefore, a need to acquire statistically accurate raw data, obtained under desirable operating conditions, and to remove the distortions implicitly in- volved in their accumulation. Such corrected results would facilitate their own interpretations and the evaluation of the x-ray spectroscopic method for reveal- ing significant features in the valence band charac- teristics of the source materials. 1.2. Materials The materials chosen for this study offer a unique op- portunity for accomplishing the above objectives. 319 Among them one finds a variety of crystal structures, associated band shapes and band positions, two dif- ferent states of magnetization, filled and unfilled 3d. and 4S-type valence orbitals. They are available in a homogeneous and chemically stable form of high purity and do not change their physical characteristics either in a prolonged ultrahigh vacuum environment or under the high (less than sublimation) temperature operation caused by incident electron beam bombardment. Finally, selection rule-allowed valence band transitions occur in regions of the x-ray spectrum which are ac- cessible to high resolution instruments. 1.3. Methods Of the three series of x-ray spectra which exist for these materials, the K spectrum contains no selection rule allowed transitions involving the valence electrons. It is, therefore, of no relevance here. Although the Mii.111 spectra possess the desired valence band infor- mation, the recording of the corresponding LII,III valence band spectra appears to be a better choice. For one thing, the small Mir-Mill separation, being about one tenth the corresponding LII-LIII separation of ~13 eV for Fe to ~23 eV for Zn, causes an undesirable over- lapping of the Mii and Mill bands. For another, the probability for radiative de-excitation of the initial x-ray state decreases rapidly toward the ultrasoft x-ray re- gion. Experimental factors such as vacuum conditions and clean sample surfaces become particularly impor- tant during the production and detection of the relative- ly low energy Mix rays. Considerable disagreement is apparent in previously published results on the L series valence band spectra of first transition metal elements [3,5,6]. The major source of these discrepancies can be attributed not to differences in instrumentation, but rather to the nature of the incident electron beam that was used to generate these spectra. Depending on the incident electron ener- gy, line shape distortions can be grouped into three more or less distinct categories. a. The Self-Absorption Region When the energy of the incident electrons exceeds about three or four times the LIII-state threshold ener- gy, one observes line shape changes and peak position shifts which are attributed to x-ray photon self-absorp- tion in the source material. The phenomenon has been documented for some time [7], but its effects on valence band line shapes have only recently been demonstrated [3,8,9]. b. The Satellite Region When the incident electrons possess energies in the range between the Lir-state threshold and about three times that amount, the high energy side of the diagram lines becomes distorted by the progressive develop- ment of satellite emissions from multiply ionized atoms. c. The Threshold Level Region When the incident electrons possess sub-Lil (but above-Liii) state excitation energies, multiple vacancy satellite emissions can be largely eliminated and, together with a normal electron incidence-normal x-ray take-off tube geometry, self-absorption effects become negligible. The threshold level mode of excitation is admittedly the least desirable for intensity considerations. It is evident, however, that with the expenditure of addi- tional time to accumulate statistically accurate data, this method permits the recording of valence band spectra which are free of the significant distortions mentioned above. 2. Apparatus and Techniques The equipment used to produce and detect the spec- tra discussed here consisted basically of an ultrahigh vacuum demountable x-ray tube, an externally manipu- lated two-crystal vacuum spectrometer using potassi- um acid phthalate (KAP) crystals and a flowing gas (P- 10) proportional counter. A detailed description of the instrument can be found elsewhere [10]. The x rays were generated by about a 100 mA beam of nearly monoenergetic (AE - 1 eV) electrons emitted from directly heated, thoria coated iridium filament strips and accelerated to the target anodes by the in- terelectrode potential. The accelerating potential was monitored with a volt box-potentiometer arrangement. Positioning the filament strips “edge-on” with respect to the anode allowed the x rays to pass between them, through a thin x-ray transparent Formwar window, into the spectrometer. The Formwar window, mounted in a self-supporting fashion, was used to separate the ul- trahigh x-ray tube vacuum from the moderate spec- trometer vacuum (~ 10-9 torr). The anode-filament geometry permitted the electrons to strike the target at normal incidence and the x rays to “take-off” at 90° with respect to the spectrometer. The x-ray tube vacuum was achieved by initially pumping the system with a mechanical forepump and an oil diffusion pump, both liquid nitrogen trapped. This was followed by a 3 to 5 hour bakeout period at 320 about 100 °C during which the target and the filament strips were outgassed at progressively higher tempera- tures. After isolating the system from the oil diffusion pump, the vacuum was improved to the eventual 10-9 torr operating level and maintained there by a titanium sublimation-ion pump combination. The x-ray tube components thus became immune to the effects of oil backstreaming and the resulting deposition of car- bonaceous materials on the anode surface. to introducing the high purity (99t'76) polycrystalline bulk samples into the x-ray tube, the sample surfaces were first mechanically polished and then electropolished. During actual experiments the anodes (except zinc) were generally kept red hot under about 100 watts of input power provided by the incident electron beam.The zinc anode was prepared by melting a layer of pure zinc onto a water cooled copper anode. The practice of enclosing the carefully cleaned, high purity samples in an ultrahigh vacuum environment and the subsequent operation at elevated temperatures was regarded as essential in minimizing contamination of the sample surfaces being studied. Prior 3. Correction Procedures The experimental curves used here as raw data represent intensity values accumulated at a series of regularly spaced Bragg-angle settings for preset count- ing times during which the bombarding electron energy was held constant. The spectrum constructed of such individual data points is affected first by contributions from extraneous atomic phenomena occurring simul- taneously with the production of the x rays of interest and second by changes in the response of the instru- mental components when they interact with photons of different energies. An enumeration of the relevant distortions and the procedure for their removal follows. (1) The “x rays off” or detector background was found to be a constant 25 counts per 100 seconds of counting time and was simply sub- tracted from the observed data. (2) The detector signals, after being preamplified, were stored in a multichannel analyzer. The dead time of the multichannel analyzer in the multichannel scaler mode was measured and found to be between 3 and 4 microseconds. With the counting rates afforded by the threshold level excitations used here (~ 100 cts/sec at the line peaks) dead time losses became negligible. (3) The spectrometer Bragg-angle scale was con- verted to a linear energy scale according to 417–156 O - 71 - 22 hc (2d) KAP sin {}; E = (l) where (2d)Kap = 26.6 A and where h and c have their usual significance. (4) Corrections for response variations of the KAP (5) crystals and of the proportional counter with photon energy have been discussed elsewhere [10]. Such an operation was based on data (counter pressure, active counter path length, entrance window thickness, etc.) that is probably unique to our instrumentation and, therefore, of little or no use to any other system. Removal of the continuous spectrum back- ground intensities was accomplished on the basis of a method developed by Liefeld [3]. The sensitivity of the spectrometer permitted the recording of the continuous spectrum alone (using an incident electron beam energy slightly below the Lill-state excitation threshold) with intensities at least ten times that of the “x rays off” background. The con- tinuous spectrum was then displaced with respect to the La line + continuous spectrum and subtracted from it. The amount of shift was equivalent to the difference in incident elec- tron beam energy between the two curves. (6) The true spectrum T(E) and the observed spec- trum O(E') are related by 0(e)=ſ. T(E) × S(E–E") dE (2) where S(E – E') is the instrumental smearing function and is proportional to the (1, + 1) posi- tion diffraction pattern for a two-crystal spec- trometer. Until recently [11], measurement of the dispersive position diffraction pattern has been difficult at best in the absence of adequately intense monochromatic X-ray sources. Liefeld, using the relatively narrow ls- 3p resonance absorption line of neon as a spec- tral resolution probe, estimated that for KAP crystals the (1, + 1) position spectral window is about 0.7 eV wide at half maximum and not grossly asymmetric [12]. Coupled with the result that the widths of the (1, + 1) and (1, - 1) position patterns are roughly the same [7]. the customary [13] procedure of approximat- ing the former by the nearly Lorentzian shape and width of the measured (1, - 1) position 321 rocking curve has been adopted here. The smearing function S(E – E') was formed from the product of a Lorentzian with a 0.63 eV width at half maximum and the recorded trian- gular transmission function of the spectrome- ter. Solution of eq (2), based on a method described by Schnopper [14], was carried out by electronic computers. The computer was programmed to generate, from a smoothed curve of original data points, a curve which when smeared with S(E – E") yielded the ob- served spectrum. The results converged to the extent that the change in the peak value between the last two iterations was of the order of 1%. Figure l illustrates the impact of the procedure on an iron self-absorption curve and on a threshold level iron La emission profile which were introduced into the computer as “original” data. (Except for the presence of the continuous spectrum background in the self- absorption spectra, such “original” data have previously been adjusted for the foregoing distortions (1) thru (5).) The comparison shows that removal of the spectral window smearing introduces no new structures but merely nar- rows and refines the original features. Additional distortions such as Auger electron in- duced satellite emissions and anode self-absorption were minimized or eliminated by using threshold level excitations and by carefully selecting the anode to fila- ment geometry. The remaining experimental affliction | | R O N "original" do to : –X— , — 4 — corrected for & — O – º – º – i 'ake.' '12 k ev : ſº º The effect of “spectral window” smearing on a threshold FIGURE 2, is that due to the finite width of the high energy inner state. The Lin states for these elements are judged to be Lorentzian in shape and possessing widths at half maximum of about 0.5 eV [2], although there is a marked variation among the values reported in the literature. Moreover, the correction for the inner state smearing presupposes the existence of very precise original data and accurate knowledge about the shape and width of the core state [2]. Because of the above difficulties and because such correction was not re- garded as imperative for present interpretation, none of the spectra presented here have been adjusted for the Lill-state width. 4. Results and Discussion The corrected, threshold level La valence band emis- sion line shapes for bulk iron, cobalt, nickel, copper, and zinc are presented in figures 1 through 5. The statistical precision of the original data from which these curves were derived was 1% or better at the line peak. A portion of the associated self-absorption spectra, obtained with the same anode samples, are also in- cluded in these figures. They were constructed by tak- ing point by point ratios of two emission curves, one of which was negligibly distorted by self-absorption while the other was seriously altered by it. The statistical ac- curacy of the emission data was better than 1% in re- gions of significant self-absorption curve structure. Figure 6 offers a direct comparison of the corrected, threshold level Lo emission line shapes. The curves are C O B A LT > *}= > º, (l) 5 * E – > Q} -X. _º ſ t | l | fil | l 1–1 l 766 770 77l, 778 782 786 | | | | | l | | 7 O O 702 704 7 O 6 7 O 8 7 1 O 7 12 7 l 4 Phot on Energy in ev. FIGURE 1. level iron La-line shape and on an iron self-absorption spectrum. Energy in ev. The threshold level cobalt La-line shape and a self- absorption spectrum. 322 N | C K E L i §*~ 3§ 81;6 850 854 Energy in ev. FIGURE 3. The threshold level nickel La-line shape and a self. absorption spectrum. C O P P E R i - _so§*~; I ſº | ſº | I | A | l | | | | 921; 926 928 930 932 93); 936 Energy in ev. FIGURE 4. The threshold level copper Lo-line shape and a self. absorption spectrum. matched at their estimated Fermi energies (taken as the position of the first inflection point of the relevant self-absorption curve) and their areas are proportional to the number of valence electrons possessed by each, i.e., Zn:Cu:Ni:Co:Fe= 12:11:10:9:8. 4.1. General Remarks It is well known that the recorded intensity in an x- ray valence band emission (or absorption) spectrum is proportional to the product of the density of filled (or unfilled) valence states and the probability of transition P(E) between the initial and final states. X-ray valence Z | N C >. :- > ta (1) ſº –Y (i) So E - > © —x. cr) 1–1–1–1–1–1–1–1 l | | | 1006 1010 101l, 1018 1022 1026 1030 Energy in ev. FIGURE 5. The threshold level zinc La-line shape and a self absorption spectrum. > É (1) E (#) .2 g Q1) Cz | | | -l2 —lC) -8 +l; Phot on Energy in ev. Reld five FIGURE 6. Comparison of the threshold level iron, cobalt, nickel, copper and zinc La-line shapes matched at their estimated Fermi energies. band spectra, therefore, cannot yield direct information about the distribution of valence states. Neither can they be used to determine the behavior of P(E) over the domain of the band as such information must come from theory. Inasmuch as even crude calculations of the pertinent transition probabilities are scarce, the of. tenmade first approximation that P(E) is relatively con- stant over the extent of the band will be invoked here. It is further proposed that the intense, narrow features in the emission spectrum represent contributions from valence states of 3d-type symmetry. Although both 3d and 4s electrons can (and apparently do) contribute to 323 the observed La-line intensity, the former will presumably dominate since they are more abundant. Analysis of the data shows that, except for Cu, the width of the prominent 3d band decreases as the shell gets filled in progressing from iron to zinc. In going from Ni to Cu, the orderly filling of the 3d band is inter- rupted. The configuration of valence electrons changes from 3d"4s” for Ni to 3d"4s' for Cu. The two additional electrons that are added to the Cu3d band complete it and the larger than Ni bandwidth is exhibited in the resultant Cu La emission spectrum. The broad, low in- tensity structures in the Zn and Cu Lo-line spectra sug. gest the presence of the overlapping 4s band. Theoreti- cally, the Fermi level in Zn is predicted to be in the re- gion where the 4s and 4p bands begin to overlap, but since p to p type transitions are selection rule forbid- den, our measurements should not display any 4p band contributions in the observed intensity. The density of filled valence states at Er is seen to increase gradually from Fe through and then abruptly decrease in Cu and Zn. This behavior is strongly confirmed by Slater's [15] results deduced from specific heat measurements. Table 1 summarizes the valence band structure mea- surements made from the above figures. The listed base widths include some allowance for inner state Smearing. TABLE 1. Valence band energy measurements in eV La peak to Liu Estimated base Material edge separation | width of principal feature Iron.................................... 1.2 3d: 6 4s: — Cobalt................................. 0.8 5 - Nickel................................. 0.4 3–3.5 - Copper................................ 2.2 4. - Zinc .................................... 8.4 2.5 12 The associated self-absorption curves have been in- cluded here because of their demonstrated similarity to photon absorption spectra [3] and because of the consequent parallels in their interpreta- tion. The self-absorption curves are purported here to conventional give some indication of the distribution of available, lo- calized 3d orbitals above EP, with the assumption again that transition probability variations are negligible over the region of interest. The above results are not at variance with existing knowledge regarding the valence bands of these metals. Band calculations for Cu [4,16] and Ni [17], for instance, correspond well to the values presented here. Where comparison is permitted, our emission line shapes are in remarkable agreement with the x-ray photoelectron spectroscopic results of Fadley and Shir- ley [18], although their Cu curve lacks any structure analogous to what has here been interpreted as the 4s band. In contrast to earlier photoemission spectroscop- ic studies on Cu [19] and Ni [20], the results of East- man and Krolikowski [21] show improved accord with our valence band structure measurements. Finally, and perhaps most significantly, the results demonstrate that the x-ray spectroscopic method yields information which does not merely reflect the nature of a localized high energy inner state but is capable of exposing meaningful differences in the valence band energy structures among the solids examined here. 4.2. Iron Ddid Figure 7 presents at iron La-line excitation curve ob- tained with a primary electron beam of AE -- 0.2 eV. Such contours are recorded by setting the spectrometer at the Bragg-angle position corresponding to the peak of the La line and by observing the emitted photon in- tensity at a series of incident electron energies. The ex- citation curves are used to determine the interelectrode potential in the L11-Lin region which will yield the max- imum La-line intensity. A study of their respective La- line excitation curves [10] indicates that for Cu and Zn the condition for maximum La-line intensity exists when using just-less-than Liſ-state threshold energy electrons. Figure 7 shows, however, that for Fe max- imum emission, intensity is gleaned by using just above Lill-state energy electrons to excite the iron La line. i | | | x X X 702 706 7 | O 7 || 4 Energy in ev. FIGURE 7. The iron Lo-line excitation curve. 324 When the Fe La line was recorded under such excita- tion conditions and the continuous spectrum removed, some anomalous emissions were found to distort the resultant line shape. Similar results have subsequently been observed for Co and Ni. The extraneous intensity structures possess the following properties: (1) They appear only on the high energy side of the La line. (2) Their intensities and their positions with respect to the La line are functions of the incident electron energy in contrast with conventional Auger electron induced satellites. Figure 8 compares four corrected Fe La-line shapes, plotted to the same peak intensity, which were generated with incident electrons whose energies rela- tive to the iron Fermi level are indicated by the arrows and whose values are represented by the arrows in figure 7. (3) Such structures seem to be particularly ac- centuated in the materials which are known to possess large densities of empty, localized 3d orbitals just above Ep, although the distortions become progressively less pronounced with the filling of the 3d band. Zn and Cu both have the maximum allowable number of 3d electrons, their self-absorption curves suggest the presence of a relatively low density of available states just above Ep and as a result their valence band emis- sion spectra are not expected to display such afflic- tions. (The broad and feeble features in these two metals have been interpreted here to represent con- tributions from the 4s band.) Our present interpretation of these results suggests that the anomalous intensity structures represent radia- tive transitions from resonantly excited, bound-electron orbitals. Such emissions have not been reported before; }* | R O N i ; f ſ t l ſ t ſ | { ſ f | | ſ | | l ſ O H- | –6 — 4 –2 0 Relative i I | ſ ſ ſ L EF +2 +4 + 6 +8 +10 +12 Phot on in ev. Energy FIGURE 8. Shape of the threshold level, iron La line as a function of the incident electron energies indicated by the arrows. the production of such bound-ejected-electron (BEE) excitation states, in fact, is thought to be unlikely when using electrons as the excitation source [2]. Friedel [22,23] claims that the valence electrons will respond to the presence of the inner shell vacancy by occupying discrete, bound electronic screening levels which possess the symmetry (or symmetries) of the valence electrons. His analysis shows that radiative transitions between these orbitals and the initial state vacancy can occur only if the position of the final state vacancy is at the top of the Fermi distribution, i.e., at Ep. As a result, the emission line shape should be distorted near the emission edge. (Skinner et al. [24] proposed that if such structures exist, they should be apparent in the vicinity of both emission and absorption edges. While the distortions are obvious in emission, a definite change in slope is also discernible in the lead- ing edge of the Fe, Co, and Ni self-absorption curves.) Friedel’s approach then apparently accounts for the ex- perimentally observed excess intensity on the high energy side of the La line, but it fails to explain either the incident electron energy dependence of these ex- citation satellites or the fact that they seem to appear only in the elements possessing large densities of un- filled states just above EP. There is some reason to be- lieve that two different phenomena are contributing to the line shape distortion. A closer examination of figure 8 reveals a relatively stationary structure about 2 eV above the Fermi level in all but the very-near-threshold curve (perhaps because the electron energy was insuffi- cient to excite it in this case), while the extraordinary emissions are seen to respond to changes in incident electron energy. Holliday [25] and Bonnelle [6] have also reported some structure on the high energy side of the iron La line. Neither of these observations, however, appears to be the excitation satellites discussed here as both in- vestigators employed incident electron energies well in excess of the Liſ-state threshold. Holliday reports a “double peak” near the maximum of the emission profile, but it is too far removed from the emission edge to correspond to our results. Bonnelle’s iron curve was apparently recorded under rather poor vacuum condi- tions ( ~ 10-5 torr). Considering further the fact that iron is a relatively reactive metal and that previously published results on Fe2O3 [26] match Bonnelle’s iron curve in every important detail, it appears that she was examining a somewhat oxidized iron sample. A final note of caution is perhaps in order. In spite of the careful anode preparation techniques employed here and the operation of the target at elevated tem- peratures in an ultrahigh vacuum environment, it is still 325 possible that a thin layer of oxide persisted on the anode surface. The possibility exists that the radiating source was characterized by a metal and metal oxide system, each exhibiting its own spectrum. This may be especially true for iron which is found to be very dif- ficult to separate from its oxide. It does not appear like- ly, however, that such an oxide film was accumulated during actual experimentation, because the La-line peak intensity remained within statistical limits over extended periods (typically two weeks) of observation. 5. Summary It has been the object of this investigation to extract, from corrected threshold level La-line shapes, incident electron energy independent emission contours and use these for further analysis regarding the distribution of valence states in the relevant solids. Such expectations have been thwarted in three of the elements studied here by extraordinary emissions that distort the resultant line shape and may thus prevent the observa- tion of sharp emission edges in these metals. Such results, however, clearly support the contention that the x-ray spectroscopic method is capable, when care- fully executed, of exhibiting significant differences among the valence band characteristics of the materi- als analyzed. 6. Acknowledgments The authors gratefully acknowledge the support received from the New Mexico State University Physics Department and the National Aeronautics and Space Administration during this investigation. 7. References [1] Parratt, L. G., and Jossem, E. L., Pjys. Rev. 97,916 (1955). [2] Parratt, L. G., Revs. Mod. Phys. 31, 616 (1959). [3] Liefeld, R. J., “Soft X-Ray Emission Spectra at Threshold Ex- citation” in Soft X-Ray Band Spectra and the Electronic Struc- ture of Metals and Materials, D. J. Fabian, Editor (Academic Press, New York, 1968), pp. 133-149. [4] Burdick, G. A., Phys. Rev. 129, 139 (1963). [5] Van den Berg, C. B., Ph. D. Dissertation, University of Gröningen (1957). [6] Bonnelle, C., Ann. Phys. 1, 439 (1966). [7] Compton, A. H., and Allison, S. K., X-Rays in Theory and Ex- periment (D. Van Nostrand Co., Inc., New York, 1935). [8] Hanson, H. P., and Herrera, J., Phys. Rev. 105, 1483 (1957). [9] Liefeld, R. J., and Chopra, D., Bull. Am. Phys. Soc. 9, 404 1964), [10] hº S., Ph. D. Dissertation, New Mexico State Univer- sity (1968). [ll] Bearden, J. A., Marzolf, J. G., and Thomsen, J. S., Acta Cryst. A24, 295 (1968). [12] Liefeld, R. J., App. Phys. Letters 7,276 (1965). [13] Porteus, J. O., J. Appl. Phys. 33,700 (1962). [14] Schnopper, H. W., Ph. D. Dissertation, Cornell University (1962). [15] Slater, J. C., “The Electronic Structure of Solids” in Handbuch der Physik, S. Flügge, Editor (Springer-Verlag, Berlin, 1956), Vol. 19, pp. 1-137. [16] Segall, D., Phys. Rev. 125, 109 (1962). [17] Connolly, J. W. D., Phys. Rev. 159,415 (1967). [18] Fadley, C. S., and Shirley, D. A., Phys. Rev. Letters 21, 980 (1968), & [19] Berglund, C. N., and Spicer, W. E., Phys. Rev. 136, A1044 (1964). [20] Blodgett, A. J., and Spicer, W. E., Phys. Rev. 146, 390 (1966). [21] Eastman, D. E., and Krolikowski, W. F., Phys. Rev. Letters 21, 623 (1968). [22] Friedel, J., Phil. Mag. 43, 153 (1952). [23] Friedel, J., Advances in Physics 3,446 (1954). [24] Skinner, H. W. B., Bullen, T. G., and Johnston, J. E., Phil. Mag. 45, 1070 (1954). [25] Holliday, J. E., “Soft X-Ray Emission Bands and Bonding for Transition Metals, Solutions, and Compounds” in Soft X-Ray Band Spectra and the Electronic Structure of Metals and Materials, D. J. Fabian, Editor (Academic Press, New York, 1968), pp. 101-132. [26] Gwinner, E., Z. Physik 108, 523 (1938). 326 Discussion on “An L-Series X-Ray Spectroscopic Study of the Valence Bands in Iron, Cobalt, Nickel, Copper, and Zinc" by S. Hanzely (Youngstown University) and R. J. Liefeld (Los Alamos Scientific Lab) K. J. Duff (Ford Motor Co.): For the magnetic materi- als is there any hope that the experimental results can be analyzed into separate contributions for majority spin and minority spins? S. Hanzely (Youngstown Univ.); Let me say that we have done the following. The threshold level emission spectra, say in particular for the ferromagnetic materi- als, were taken with the anodes in the paramagnetic state. In other words, the anode temperatures were somewhat above the Curie point. However, when the self-absorption spectra were constructed, one of the two curves from which such self-absorption spectra were constructed was taken with the anode in the fer- romagnetic state and it appears that there is no noticea- ble experimental evidence for any shift in the prominent structure whether the anode is in the paramagnetic or the ferromagnetic state. W. Spicer (Stanford Univ.); There was mention of or- bital states which appeared to lie above the Fermi ener- gies if I read the diagram right. I don’t understand these. Could you say something about them? S. Hanzely (Youngstown Univ.); Your question is that you would like to know more about the origin of these transitions. It appears that they are particularly prominent in materials which have a high density of va- cant states just above the Fermi level. These include iron, cobalt, and nickel in that order. The density of va- cant states above the iron Fermi level is very large. It is still large but not as large in cobalt and somewhat smaller yet in nickel. If you analyze the copper and zinc graphs carefully, in those two metals the d shells are full and it turns out that the density of vacant levels above the Fermi level is very small. Structures on the low energy side of the Fermi level in these elements have been interpreted to indicate the presence of the underlying 4s band rather than 3d band. 327 The Electronic Properties of Titanium Interstitial and Intermetallic Compounds from Soft X-Ray Spectroscopy J. E. Holliday Edgar C. Bain Laboratory for Fundamental Research, United States Steel Corporation, Research Center, Monroeville, Pennsylvania 15146 The Tilir in emission bands (3d-H 4s -> 2p transition) have been obtained from TiCo.95 and TiNa (x = 0.2 to 0.8) interstitial compounds and TiCr2, TiCo, TiNi and intermetallic compounds. Additional peaks on the low energy side of the Tilliſ band from TiC and TiNa appear to be cross transitions from the 2s and 2p bands of the nonmetal to the 2p level of titanium. Agreement was found between the soft x-ray band spectra and the band calculations of Ern and Switendick on TiC and TiN. The soft x-ray emission spectra from TiC indicated strong admixture of the titanium 3d and carbon 2p bands which is in disagreement with LCAO band calculations of Lye and Logothetis. However, the 2p band of nitrogen was shown to be below the Ti 3d band indicating a localized state and a possible transfer of electrons from titanium to nitrogen. The Tili; in bands from TiCrg, TiCo and TiNi show a progressive change with increasing elec- tronegativity difference between titanium and the combining element indicating possible ionic charac- ter to the bond. No peaks were observed on the low energy side of the Tillii bands, but a distinct splitting was observed in the peak of the Tilin band from TiNi. Key words: Electron concentration; electronic density of states; localized states; soft x ray; titani- um compounds. 1. Introduction The electronic structure of the first series transition metal interstitial compounds, especially the borides, carbides, and nitrides, are of particular interest because of their mechanical, electrical, and thermal properties. Two electronic structure models have been proposed for these materials. Utilizing LCAO (Linear Combination of Atomic Orbitals) type of band calcula- tions, Lye and Logothetis [1] have calculated energy bands for TiC and preliminary calculations on TiN. Their band calculations are semiempirical and show lo- calized states in the bands. In order to be consistent with the Madelung displacement of the energy levels and their own observed optical properties, they applied an electrostatic correction to their band calculation which resulted in the carbon 2p band being separated and lying above the titanium 3d band. As a result of this separation and from the filled portion of their density of states histogram they predicted that 1 – 1/4 electrons would be transferred from the carbon 2p band to the titanium 3d band for TiC. In the other model Ern and Switendick [2] using the APW (Augmented Plane Wave) method showed no separation in the carbon 2p and titanium 3d bands and thus no electron transfer. For TiN they showed the nitrogen 2p band at the bot- tom of the titanium 3d band. These two band pictures have generated considerable controversy over the past several years. Lye and Logothetis [1] state that the key to solving the problem is to locate the position of the carbon 2p and titanium 3d bands experimentally. Ern and Switendick stated the L spectra from TiC and TiN would be an aid in understanding the electronic struc- ture of these compounds. As a result of these com- ments the soft x-ray L emission spectra from TiC and TiN was measured to determine the relative location of the 2p and 3d bands, the degree of localization in the band and if there is some ionic character in the bond of these compounds. Localized states may also be important in the band structure of alloys as demonstrated in recent soft x-ray measurements by Curry et al. [3]. Similarly, in a paper presented at the Density of States Conference, Rooke [4] suggested localized states around the aluminum 329 atom to explain some of the soft x-ray measurements on Al-Mg alloys. Since a localized band picture and ionic character in the bond would be more pronounced in in- termetallic compounds than in alloys, the soft x-ray emission bands were measured for a series of titanium intermetallic compounds to determine if localized states and ionic character were present. The com- pounds were selected so there would be a progressive increase in the electronegativity difference between the titanium and the combining element. 2. Experimental Results The soft x-ray emission bands were measured with a grazing incidence grating spectrometer. The grating used is a 1 meter radius 3600 groove/mm with a platinum surface and a 1° blaze. The target potential was 4 kV, and the beam current was 1.5 mA. Since the spectrometer has been thoroughly described in other publications [5], the details will not be presented here. The TiC and TiNa targets were made from compressed powders of the compounds, and the titanium inter- metallic compounds were made by levitation melting of stoichiometric mixtures of the elements. The composi- tion of these intermetallic compounds, determined by chemical analysis after the formation of the compound, is shown in table 1. TABLE 1. Composition of Intermetallic Compounds (Wt. Percent of Element) Cr Fe Co Ni TiCr2 s s tº e º s p * * * * * g & e s s e e g g g º 67 s tº e º tº e º e º ºs e º & e s is tº $ tº e º 'º e g tº e º & e º e º e s tº a g º & s e s e is e Tife.......................l.............. 52.8 .............l............... TiCo......................l.............l............... 55.2 ............... Ti2Si....................................l...............l..…....... 38.0 TiNi.......................l.............l...............l.............. 55.4 TiNia....................................................….. 77.6 The Tillii.III emission band from TiC is shown in figure 1. The peak A, approximately 7.5 eV on the low energy side of the Tilliſ band (3d-H 4s -> 2p transition), is not observed for the pure metal. In addition, the Tilli/LIII intensity ratio for TiC has been reduced rela- tive to metallic titanium, and the peak of the Tillii band has shifted toward lower energy. Changes in the Tillii/LIII intensity ratio have been shown previously by both Holliday [6] and Fischer and Baun [7] to be due to changes in self absorption. 1 Prepared by Cerac, Inc., Butler, Wisconsin. The Tilii.III emission bands from TiNa, nitrides where x is varied from 0.2 to 0.8 is shown in figure 2. |OH- H. P. — 9– ; : - 8H : , - 7– ; : - :- 6H ; : **4 * • A \ 2 5F Lll i \\ - = 3|- Ti * | * - g 2------- ; – Tº- - H |H ; º - s : \ A Ol— º - 26.5% 27. 215. 28A 28.5% 29. H++++. i r—º H-H H- r-17 i H Il- | H I H H+. I-1- | r1- | H | TI H 470 460 450 440 430 ELECTRON VOLTS FIGURE 1. The Tilm.In emission bands from Ti and TiC; target voltage was 4 kV. The Peak A', on the low energy side of the Tilin band, is seen to increase in intensity relative to the Til III band with increasing x. There is also a shift in the Tilin band toward higher energy, with x (fig. 3). A slight in- crease is observed in the Tilli/Lin intensity ratio with x, but there is less of a change for a given range of x than that reported for TiO2 oxides [6]. In general, the changes in the Tillii.111 emission bands from TiNa, nitrides are the same as those observed for TiO2 oxides but are somewhat less pronounced. The Tillii.111 emission bands for the titanium inter- metallic compounds are shown in figure 4. These reveal an increase in the Tilii/LIII intensity ratio, and a slight shift of the Tillii.111 bands toward higher energy with in- creasing electronegativity difference between the Ti and combining atom. These changes are similar to that observed for the Tilii.111 band from TiNa, with increas- ing x. The Tillii.III emission band from TiNi shows a split in the Tillili peak which was not observed for the Tilin bands from the other compounds. The separation of the peaks is approximately 1.5 eV which is the same as the separation of the two peaks in the NiMII.iii bands from TiNi reported by Cuthill et al. [8]. 3. Discussion of Results As indicated in the introduction, the degree of lo- calization and the amount and direction of electron transfer in TiC and TiN will depend on the amount of separation and position of the nonmetal 2p bands rela- 330 | o ° Q, A Q? &b o o © * *, o * * o * , § 8: o Q o © *: to © L || * * *, *, *. & o°o .* & 2 * * e.” °ºogeoº" * & * t tº Ö . Ti N O 8 sº 3. * * °os, iº ...” * . A ©o °ocº, © A gº Ö o Cocoo © cooºoooººoº * * •. *. o °oooooo-ooºoo o oo coaºo-oºoºoooºººº a • *. * o o cºooºooºoºoooººoºooooº, & * °o.o.o.º. ** a * t oo? ows •. * o TiN ** * * * * *.. O. 5 o,98 a * '. *** *... sºº’ A * * *. *… ooooooº. &geococºoºººooººooºoºoºº. * A • & ©do *... © o o as º *\º e e *... TiN .* *-* . ".. ”, AAA ſº •e *AAAA O .3 5 A. - A& ...”.” ." •. *............ ***a*aaaaaaaaaaa”*A*a***** g ***********.*.*............... TiN 0 2 ....." " ' "... 0.2......…” “...***eese see e ................---------"“” * *************** --------...-. ..... gº - 1–1–1–1– H-H ri- ++++++++ H–H H H-H r–H–1–1–1– 470 465 460 455 450 445 440 435 E L E C T R O N V O L T S FIGURE 2, The Tilm.III emission bands from TiN, nitrides where x varies from 0.2 to 0.8. The target voltage was 4 kV. FIGURE 3. The shift in the Tilm band relative to pure Ti for TiN, nitrides as a function of x. tive to the Ti 3d. Of particular interest in this regard are peaks A and A' in figures 1 and 2. The fact that peak A' increases in intensity relative to the Tillii, peak with in- creasing N/Ti atom ratio indicates that the peak is as- sociated with the amount of nitrogen in TiN. A peak also appears on the low energy side of the Tilin band from TiO2 oxides, whose intensity increases with an in- crease in the O/Ti atom ratio [6]. Fischer [9] has -III . . Ti Ni • Ti C9 - TiCr2 a; ..”.” ". . .","... • ‘’ 1s quadrupole transition) emission bands of TiX compounds, where X is a 2nd period element, represents a cross transition between the 2s bands of the nonmetal and the 1s level (quadrupole transition) of titanium. From the above discussion there appears to be suffi- cient justification to call peak A' in figure 2 a cross 331 transition from the nitrogen valence band to the titani- um 2p level. Ern and Switendick [2] have compared the energy separation between the K85 band and the K/3" cross transition from (N2s-Tils) which is 11 eV [ll], with their calculated separation of 10.7 eV between the 2s and 3d-H 2p bands (fig. 5), the separation between the peak of the Tilin band (3d-H 4s -> 2p) and peak A' in figure 2 is 4.2 eV. Since the peak of the 2s band is ap- proximately 11 eV from the peak of the 3d band, then peak A' appears to be a cross transition from the nitrogen 2p band to the 2p level of titanium and any electron transfer would be from titanium to nitrogen. This indicates that there is a greater separation between the 2p and 3d bands than Ern and Switen- dick’s [2] calculations show. However, Ern and Switendick state that the discrepancy between the ini- tial and derived charges in the nitrogen sphere show that they should have assumed a greater separation in the 2p and 3d bands. Also, the experimental shifts of the K8" and K65 bands of TiN relative to TiO when compared to Ern and Switendick’s computer density of states show that TiN band picture is closer to TiO than that predicted by Ern and Switendick. The idea of elec- tron transfer and ionic character in the bond for TiN is | 165 S. | 175 / N. Aſ º ox- \ Liool,”“s 1.015H ss. § -> \ N. N \s O.875 \ N ul O.800 H. N g \ -- ~~ sº H P i — -a- O.725 H- > ---- .e.” P Fermi energy (O.680 Ryd) Fermi energy (O.636 Ryd) T *- |.O25 O.95O O. 8 6 5 H. |- | O. 9 4 O H- O. 7 9 O |- O .7 5 |- O. 6 4 O O. 4 |5E== O.425E- == Đ o,340EHF O.35O == O 2ész O.275E 7.| eV E; |0.7 eV O. O4O º o.2005 • TiC TiN - O.O.35 E--> - O.2OO Ea — O. l IOE - (a) - O. 275 (b) 2s — O. | 85 - O.35O — O.26O - O.425 2s – O.335 -o soo;' –– |→ | |→ O 2O 4 O O 2O 4 O Z (E) Z (E.) FIGURE 5. Density of states histogram for TiC and TiN calculated by Ern and Switendick using the APW method. (Reprinted by permission – V. Ern and A. C. Switendick, Phys. Rev. 137A, 1927 (1965). also shown by the increasing shift of the Tilin band toward higher energy with x in figure 3. Shifts in the kinetic energy for the nitrogen 1s and titanium 2p3/2 levels from ESCA (Electron Spectroscopy for Chemical Analysis) by Ramquist et al. [12] shows that the elec- tron transfer is from titanium to nitrogen. In the case of TiC, peak A (fig. 1) is approximately 7.5 eV from the peak of the Tillii band. Ern and Switen- dick calculate that the 2s and the 3d-H 2p bands of TiC are separated by 7.1 eV as shown in figure 5. Blochin and Shuvaev [11] show a separation in the K85 band (3d-H 4s -> 1s) and the K8" (carbon 2s – titanium 1s) of 7.0 eV. It would thus appear that peak A is a cross transition from the carbon 2s band to the titanium 2p level. Although Lye and Logothetis [1] do not give a value for the separation in the maximum of the 2s and 3d bands in TiC the maximum in the 2s band is about 2 eV below the maximum of the 3d band. Thus the soft x- ray Lemission spectra from TiC supports the band cal- culations of Ern and Switendick. It would appear that there is complete admixture of the carbon 2s and the titanium 3d bands because no peak was observed in the Till spectra from TiC corresponding to the 2p band. This would indicate equal sharing of electrons. How- ever, Ramquist et al. [13,14] have performed a number of ESCA and K x-ray measurements on the shifts of the ls level of carbon, the K813, Ka, x-ray lines, and Lili and Mill levels of titanium from TiC relative to the pure element. From these measurements they conclude that electrons are being transferred from titanium to carbon which is opposite to that predicted by Lye and Logothetis. Holliday [15] has reported shifts in the Ti Lin peak which indicates the possibility of electron transfer in TiC. The above experimental measurements on TiC show that Lye and Logothetis are incorrect in placing the 2p band of carbon higher than the 3d band of titanium. The degree of localization of the bands is somewhat un- certain. More theoretical work is required to fully un- derstand the meaning of shifts in the inner atomic levels of the atoms relative to the uncombined atom in relation to electron transfer and ionic character of the bond for TiC. In the case of TiN the experiments and the calculations of Ern and Switendick show the 2p band below the 3d band with electron transfer from titanium to nitrogen. Although Lye and Logothetis did not publish any band calculations on TiN they state that the 2p band would lie closer to the 3d than in TiC. This is not in agreement with the above measurement on the Till emission spectra from TiC which shows a wider separation in the 2p and 3d bands in TiN and none for TiC. 332 The observed progressive change in the wavelength and the intensity distribution of the Tilji, band from TiCre, TiCo and TiNi with increasing electronegativity difference between Ti and the combining element (fig. 4) suggests an increase in ionic bonding with increasing atomic number of the combining element. This in- terpretation is further substantiated by the fact that preliminary measurements of the L11 in bands from Cr, Co, and Ni do not have the same intensity distribution as the Tilin bands. This is similar to the results re- ported by Neddermeyer [16] on Al-Mg alloys where large differences were noted between the bottom of the Al and Mgliſ in bands. Neddermeyer attributed these changes to clusters and localized bound states at the bottom of the valence band which resulted in the densi- ty of states having a different distribution when in the vicinity of a given atom. Since detailed band calculations have not been car- ried out for these intermetallic compounds, it is of in- terest to compare the present results for the Tilſi, bands (fig. 4) with the 3d N (E) curve obtained by Cheng et al. [17] from specific heat measurements for bec 1st series transition metals. Cheng deals in valence elec- tron concentration rather than electron volts, and the approximate values for TiCr2, TiCo and TiNi are 5.3, 6.5, and 7, respectively. Even though TiNi is a CsCl type structure, its soft x-ray band spectra shows a dou- ble peak in the 3d band which is also predicted from the N(E) curve of Cheng et al. for an alloy with a valence electron concentration of 7. However, bec and TiCo does not have a double peak even though the W(E) curve of Cheng et al. predicts that a bec alloy with a valence electron concentration of 6.5 should have a double peak. These results appear to add further ex- perimental support to the fact that the rigid band model is a poor approximation to the density of states. How- ever, before a complete interpretation of the results on the intermetallic compounds can be made, an under- standing of the degree of oxidation of titanium in the alloy relative to uncombined titanium must be obtained. 4. Conclusion The foregoing results on the Lemission spectra from TiC and TiN support the band calculations of Ern and Switendick. Localized states and ionic character in the bond appear to be a part of the electronic structure of TiN but is somewhat uncertain in TiC. In addition the soft x-ray measurements on titanium intermetallic com- pounds have shown that the concepts of localized states, ionic character, and electronegativity appear to play a more important role in the electronic structure of metallic compounds than had been supposed previ- ously. In his summarizing comments before the Elec- tronic Density of States Conference, Ehrenreich [18] has emphasized that the idea of localized states should be given more consideration when considering the elec- tronic structure of alloys. 5. Acknowledgments The author wishes to acknowledge the help of the fol. lowing members of this laboratory: W. A. Hester for assistance with the experiments, L. Zwell for the X-ray diffraction work, and C. Sharp for the chemical analy- S1S. 6. References [1] Lye, R. G., and Logothetis, E. M., Phys. Rev. 147, 622 (1966). [2] Ern, V., and Switendick, A. C., Phys. Rev. 137A, 1927 (1965). [3] Curry, C., in Soft X-Ray Band Spectra of Metal and Materials, D. J. Fabian, Editor (Academic Press, New York and London, 1968), pp. 173-184. [4] Rooke, G. A., NBSJ. Res. 74A2,273 (1970). [5] Holliday, J. E., in The Handbook of X-Rays, E. F. Kaelbe, Edi. tor (McGraw-Hill Book Co., New York, 1968), Chapter 38, pp. 38-1 to 38-42. [6] Holliday, J. E., in Soft X-Ray Band Spectra and the Electronic Structure of Metals and Materials, D. J. Fabian, Editor (Academic Press, New York and London, 1968), pp. 101-132. [7] Fischer, D. W., and Baun, W. L., J. Appl. Phys. 39, 4757 (1968). [8] Cuthill, J. R., McAlister, A. J., and Williams, M. L., J. Appl. Phys. 39,2204 (1968). [9] Fischer, D. W., in Advances in X-Ray Analysis, B. L. Henke, J. B. Newkirk, and G. R. Mallett, Editors (Plenum Press, New York, 1970) Vol. 13, p. 168. [10] Zhurakovskii, E. A., and Vainshlein, E. E., Dokl. Akad. Nauk. U.S.S.R. 129, 1269 (1959). [English Transl: Soviet Phys. Doklady] 4, 1308 (1960). [11] Blochin, M. A., and Shuvaev, A.J., Bull. Acad. Sci. USSR Phys. Ser. 24, 429 (1962). [12] Ramquist, L., Hamrin, K., Johansson, G., Fahlman, A., and Nordling, C., J. Phys. Chem. 30, 1835 (1969). [13] Ramquist, L., Ekstig, B., Källne, E., Noreland, E., and Manne, R., J. Phys. Chem. Solids 30, 1849 (1969). [14] Ramquist, L., Jernkontorets Annaler 153, 159 (1969). [15] Holliday, J. E., J. Appl. Phys. 37,4720 (1967). [16] Neddermeyer, H., Dissertation zur Erlangung der Doktorwüde, der Ludwig-Maximilians Universität München, May 1969. [17] Cheng, C. H., Gupta, K. P., van Reuth, E. C., and Beck, P. A., Phys. Rev. 126, 2030 (1962). [18] Ehrenreich, H., NBS J. Res. 74A2,293 (1970). 333 Discussion on “The Electronic Properties of Titanium Interstitial and Intermetallic Compounds from Soft X-Ray Spectroscopy” by J. E. Holliday (U.S. Steel) F. M. Mueller (Argonne National Labs.): I have one J. E. Holliday (U.S. Steel): Even though the Til 9/L3 brief question for Dr. Holliday. I noticed in the curves intensity is greatly reduced for TiC relative to Ti, the for titanium and for titanium nitride you had an L2 Tilº band from Til is still observed. I have shown in emission spectra and this was absent in the carbide. Is other publications that the Tilg/L3 intensity ratio is there a simple explanation for this? strongly influenced by changes in self absorption. 334 Soft X-Ray Emission Spectrum and Valence-Band Structure of Silicon, and Emission-Band Studies of Germanium G. Wiech and E. Zöpf Sektion Physik der Universität München, München, Germany With a photon-counting concave grating spectrometer the L2,3-emission band of silicon and the energy range of the M23-emission band of germanium were investigated. The Si L spectrum shows new structural details. The measured intensity distribution for both the K and L-emission bands of silicon are compared with recent calculations of the K- and L-emission spectra and with the density-of-states CUIrVe. Key words: Carbon; diamond; electronic density of states; germanium; kp method; orthogonalized plane wave (OPW) method; silicon; soft x-ray emission. 1. Introduction X-ray spectroscopy has proved to be a valuable method for obtaining information about the electronic structure of solids. The x-ray emission bands resulting from valence electron transitions to excited core states allow the study of the whole energy range of the valence band. According to the selection rules, K-emission bands give information about the p electrons, L2,3-emis- sion bands about the s and d electrons in the valence band. This paper deals with the elements carbon (diamond), silicon, and germanium which are of special theoretical and practical interest. Because they have the same crystal structure (diamond type) the energy band structure of these elements is very similar. There- fore their x-ray emission bands are expected to show a similar intensity distribution. To date, theoretical work was limited to calculations of the band structure and the density-of-states. Recently the intensity distribution for the L2.8- and the K8-emission bands of silicon, and for the K-, L-, and M- emission bands of germanium have been calculated. In the case of silicon the agreement between the results of the calculated and observed bands is very good; the Si KB band was observed in our laboratory by Läuger [1], the Si L23 band was measured recently by the present authors. 2. Experimental The concave grating spectrometer used for the meas- urements of the Si L2,3 and C K band has been de- scribed elsewhere [22]. The Rowland circle radius is 1 m. A 600 grooves/mm grating with aluminum surface, and a 2400 grooves/mm grating with gold surface and blaze angles of 1°31' and 1°0' respectively were used. The slit width was 30 p. The x-ray tube was operated at a pressure of about 10−8 Torr to keep target contamina- tion by carbon low. The tube voltage was 2.5 kV, the beam current 0.6 to 3.2 mA. All measurements were performed with an open magnetic electron multiplier with a tungsten photocathode. 3. Results 3.1. Silicon The Si L2,3-emission band (and the C K-emission band of diamond) were measured stepwise. In order to obtain high accuracy about 4 × 104 pulses were taken for each point. Special care was taken of the influence of the 3rd order C K emission which is superimposed on the Si L2,3-emission band, the peak intensity of the former being less than 1% of the latter. This means that in the region of 95 eV the intensity of the Si L2,3-emis- sion band is influenced by carbon only to an extent of ().5%. 335 The measured band, expressed in terms of I(E)/vº, is presented in figure le. There are two pronounced peaks at 89.55 and 92.05 eV. The high energy part of the band shows more details than have been observed in earlier investigations [3-5]: a small dip between 94 and 95 eV, a peak at about 95.2 eV and a sudden change in the slope at 97.5 eV. The dip is about 1% of the peak intensity. The intensity obtained using the grating with 2400 grooves/mm was much less than that using the 600 grooves/mm grating, but the resulting intensity dis- tributions agree within statistical uncertainty. No cor- rections have been applied for the instrumental distor- $00r- | N(E) SOH a) | 0–F–F–F–F–F–F–F–F–F–F–3—H-5 100 - º ,-, ..” X- Si L2.3 32°ºss A \\ ---Theoyſ & ºt’ j \, # ºf & - O 3. J 50 c) 100r | - Si Kp º - ----Theory (1) : 50}. ... . . . Theory (2) r-n F — Exp OH: 1830 L $835 Tºo- Energy (eV) FIGURE 1. Valence-band structure of Silicon. (a) Density of states (Kane [9]), (b) Calculations of the Si L2, 3 emission band (Klima [6]), (c) Experimental Si L2, 3 emission band, and (d) Calculation of the Si KB emission-band (Klima [6]) and experimental curve (Läuger [1]). tion or for self absorption, because they have only little influence on the following discussion. The theoretical curves for the intensity distribution of the x-ray emission bands, shown in figures lb and 10, have been calculated by Klima [6] according to the for- mula X. (wºre |P, usalence) | 2 I(v) - | CY - E = h ly |grad, E dS, — L'COI’é Valence E=E.” Exalent". For the computation of the curves labelled (l) in figures 1 and 16 he used the energy bands as given by Cardona and Pollak [7] (kp-method). The energy bands on which the curves labelled (2) are based were recalcu- lated by Klima using the kp-OPW method. Spin-orbit splitting of the core state has been neglected, because its influence is small. The curves are corrected for broadening effects due to the lifetime of core states and valence states (Auger effect). The agreement between both the calculated curves and the measured curve (Läuger [1]) is quite good for the K8-emission band, while in the case of the L-emission band the calculated curve (1) is in better agreement with our experimental curve (fig. 1 c) than curve (2). The reason probably, is that more experimental data have been introduced into the calculations of Cardona and Pollak than in Klima’s kp-OPW calculations. A comparison of the experimen- tal K and L bands shows that both curves have charac- teristic features indicated by A, . . ., E, and in both curves these characteristics have nearly the same rela- tive energy with regard to the top of the valence band. The energy scales of the K and L bands have been cor- related by the energy of the Koºl line (1739.66 eV; Kern [8]). In the density-of-states curve (fig. la) calculated by Kane [9] the same characteristics A, . . ., E are ob- served. The energy position of these points relative to the top of the valence band depends on the particular energy band structure used for the calculation of the density-of-states. A more detailed analysis shows that due to the transi- tion probability observed emission band is considerably different from the N(E) curve, but the energy position of the characteristic features will be changed very little. As expected the K band of diamond (fig. 2) agrees qualitatively with that KB band of Si. In both cases we have electron transitions from the valence band into an S State. 336 100r tºl Diamond CK (4) < | U 50| ry |- O 260 255 270 275 250 285 750 Energy (eV) FIGURE 2, K-emission band of diamond (4th order). 3.2. Germcinium The calculated M23 bands of germanium look similar to the L23 bands of silicon. These calculated bands of Ge disagree with the experimental results of Tombou- lian [10] which are to date the only measurements of the Ge M23 bands available. To clear up this discrepan- cy the energy range of the Ge M23 band was in- vestigated. Using the crystal powder, and a thin plate of a single crystal with a target input of 2.5 kV, 3 m.A, only two relatively weak lines could be observed (of intensity 1200 and 1700 counts/min). The characteristics of these two lines are presented in table 1. In the energy range of the Ge M2,3-emission band the intensity was only about 400 counts/min higher than the background in- tensity (about 400 counts/min), and no reproducible fine structure could be observed. The Ge M23 emission is therefore of very low intensity; the same was found for the L emission by Deslattes [12]. The two weak lines observed may be interpreted as due to the intraband transitions M4.5 — M2 and M4.5 — M3 (table 1). A surprising observation is the big dif- ference between the energy splitting of M2 – M3 as tabulated by Bearden and Burr [11] (7.16 eV) and found in our measurements (3.5 eV). On the basis of these new theoretical and experimen- tal data Tomboulian’s interpretation of his results seems unlikely. 4. References TABLE 1. Characteristics of the intraband transitions M4, 5–M2 and M4, 5– N3. Tom- Bearden and bou- Present authors Burr [11] lian - [10] o Half- |Inten- A(A) | E(eV) |E(eV) | A(A) | E(eV)|width| sity (eV) M4, s—M2 | 124.94 | 99.24 ..... 129.3 95.9 | 1.7 || || M4, 5–M3 || 134.65 | 92.08 | ..... 134.2 | 92.4 || 3.] 2 M2–Ma | ......... 7.16 2.0 | ...... 3.5 417–156 O - 71 – 23 [1] Läuger, K., Thesis, University München (1968). [2] Wiech, G., Z. Physik 193,490 (1966). [3] Crisp, R. S., and Williams, S. E., Phil. Mag. 5, 1205 (1960). [4] Wiech, G., Z. Physik 207,428 (1967). [5] Ershov, O. A., and Lukirskii, A. P., Soviet Physics-Solid State 8, 1699 (1967). [6] Klima, J., preprint sent for publication. [7] Cardona, M., and Pollak, F. H., Phys. Rev. 142, 530 (1966). [8] Kern, B., Z. Physik 159, 178 (1960). [9] Kane, E. O., Phys. Rev. 146, 558 (1966). [10] Tomboulian, D. H., and Bedo, D. E., Phys. Rev. 104, 590 (1956). [ll] Bearden, J. A., and Burr, A. F., Atomic Energy Levels (U.S. Atomic Energy Commission, Oak Ridge, Tennessee, 1965). [12] Deslattes, R. D., Phys. Rev. 172,625 (1968). 337 Density of States in o. and B Brass by Positron Annihilation W. Triftshduser and A. T. Stewart Queen's University, Kingston, Ontario Positron annihilation experiments using a long slit angular correlation apparatus have been per- formed to investigate the momentum distribution of photons resulting from positron annihilating with electrons in brass. Single crystals of O. and 8 brass which had been oriented along the 100, 110 and 111 directions respectively, were used for the measurements. The counter slits subtended an angle of 0.32 mrad at the sample. Thus, keeping the samples at liquid nitrogen temperature to reduce the positron motion, a total resolution of 0.42 mrad was achieved. The results show clearly deviations from a spheri- cal Fermi surface. The observed anisotropies are found to agree very well with the theoretical predic- tions based on cross-sectional areas of the Fermi surface. Key words: Angular correlation; brass; electronic density of states; positron annihilation. 1. Introduction Positron annihilation experiments yield directly the density of states of electrons in k-space and the results are approximately independent of the electron mean free path. Thus, it can be used to examine the electron structure of alloys which, because of electron scatter- ing, cannot be easily examined by the more usual trans- port techniques. The principles of the positron annihilation technique are simple to describe. A positron from a radioactive decay is shot into a metal specimen and loses energy very rapidly, arriving at about thermal energy in a time somewhat less than the average lifetime which is of order 10-10 sec. Then the positron annihilates with one particular electron. The usual annihilation process results in the emission of two photons. These two photons have a total energy of 2hu = 2mc” and are emitted at exactly 180° to each other to conserve mo- mentum. In the laboratory a slight departure from 180° can be observed and, since the positron is thermalized, it is a direct measure of the transverse (to the photon direction) component of the electron’s momentum. Using independent particle language, the wave function product ill (r) lik(r) of the annihilating pair acts like the wave function of the center of mass and thus also of the gamma rays. The Fourier transform of this wave func- tion product is then the momentum amplitude function, the square of which is the observable distribution p(k). Of course the positron and electron are not inde- pendent. The positron always surrounds itself with a polarization cloud of negative charge and annihilates with these electrons and not with the unperturbed elec- trons. The momentum distribution of electrons in this cloud — really of one electron and the positron—has been calculated for two extreme situations: (a) the elec- tron gas, and (b) electrons in closed atomic shells. In both cases the distribution in momentum of the an- nihilation photons resembles quite closely that of the unperturbed electrons of the particular system being considered. In particular a sharp cutoff at the Fermi surface is both expected and indeed observed. Thus while the effect of the positron on the electrons in brass is not yet known, we have good reason to expect experi- mental results for momentum distributions of annihila- tion photons to resemble closely the momentum dis- tributions of the electrons in the alloy. The possibility of measuring momentum anisotropies in oriented single crystals has been first shown by Berko and Plaskett [1] for copper and aluminum. Meas- urements on other metals have been reported by other authors [2,3]. In this paper we present observations on single crystals of 8 and O brass, in order to examine how well this technique can be used in the determina- tion of the electron states in alloys and especially in dis- ordered alloys. Properties of ordered alloys, particu- larly the shape of the Fermi surface, can be obtained through de Haas-van Alphen measurements more accu- 339 rately than by the positron annihilation technique. We chose ordered 8 brass as a “test alloy” because the Fermi surface of this metal is very well known. We then used o brass to explore the possibilities of this technique. We shall discuss in this paper to what extent our results can be analyzed directly in terms of electron momentum distribution and cross-sectional areas of the Fermi surface. 2. Experimental Arrangement and Procedure Let p(k) db be the probability that a positron anni- hilates with an electron emitting two photons having total momentum between k and k+ dk in the labora- tory system. The conventional long-slit apparatus then measures the coincidence counting rate N (k2), where N(kz) = ſ ſ. p(k) dkraky (1) The integral is taken over the plane of constant k2. In the independent particle approximation p(k) is given by s) p(k)-cons. Slſwºre "... (2) where ill (r) is the positron ground state wave function, and lik (r) is the electron wave function with wave number K and band index 1. The summation is over all the occupied states. In the lowest approximation of free electrons and a zero momentum positron, W(k2) is directly proportional to the cross-sectional areas of the Fermi surface in the extended zone scheme. For a less simplified situation, the importance and influence of the positron wave function, the higher momentum com- ponents of the electron wave functions, and the core electrons has been discussed already by De Benedetti et al. [4]. We present data both in the form of momentum dis- tribution, that is N(kg), and also as the derivative of N(kg) with respect to kg. The derivative presentation of the data shows more sensitively the effects of changing Fermi surface topology and occupation probability. In the experimental arrangement we used the long- slit geometry having 30 cm long NaI detectors, with slits subtending 0.3 mrad at the specimen. The brass single crystals had been oriented by x rays and were spark cut to rectangular solids of 6 × 6 × 12 mm. After cutting, the surface of the specimens was carefully etched to remove possible distortions due to spark machining. As a positron source we used approximately 180 mCi of Coºs diffused into a very thin Cu foil. The radioactive area was sealed by an 0.005 mm thick stain- less steel window to avoid contamination. All experi- ments were performed with the specimen at 78 K in order to reduce the thermal motion of the positrons and its broadening influence on the instrumental resolution [5]. The coincidence data were taken automatically and corrected for source decay. 3. Results and Discussion The results for 3 brass are shown in figure 1. The three curves represent the angular distributions for | 100], [110] and [ll]] directions (k, was parallel to these directions) normalized to equal areas using the best visual fit through the data points. The statistical accuracy at k = 0 was about 0.65%. In order to evalu- ate the part due to conduction electrons we subtracted the higher momentum tail from the angular distribution by fitting a Gaussian function in the region from about 7 to 14 mrad. From the resultant curve, we calculated the derivative by taking the difference of adjacent meas- ured points. This is shown in figure 2 for the three crystal directions. The solid lines drawn through the points represent the momentum distributions obtained from calculations by Taylor who fitted the de Haas-van |OO F- BET A BRASS O 2 4. 6 8 | O | 2 | 4 ANGLE | N M | LL | RAD ANS FIGURE 1. Experimental angular distribution curves for [100], [110], and [111 | directions in 8 CuZn at liquid nitrogen temperature. The curves are normalized to equal areas. The statistical accuracy of the data points at the peak is 0.65 percent. 340 BETA BRASS (| OO) (||O) (|||) | 4 8 MOMENTUM IN UNITs of IoTºmc FIGURE 2, Electron momentum distributions in 8 CuZn. These data are obtained from figure 1 by differentiating as discussed in the text. The theoretical difference curves were obtained from cross-sectional areas calculated by R. Taylor [6]. The Fermi surface was confined to the first two zones. Alphen data to a six parameter description of the Fermi surface of ordered 3 brass using nonlocal pseudopoten- tials [6]. This calculated result shows the differences of adjacent cross-sectional areas of the Fermi surface. Higher momentum components are not included in this calculation. The agreement with the experimental data is remarkably good considering this approximation. The electron momentum distribution is quite different along the main cyrstallographic directions and deviates clearly from that of free electrons. The [110] necks and the [111] holes, lying at different positions for each of the three orientations are mainly responsible for the detailed structure in these momentum distributions. Under the same experimental conditions the mea- surements on single crystals of O brass were per- formed. We used a specimen with 22% Zn and 78% Cu. This alloy has a face centered cubic lattice structure like pure copper. The angular distribution for the [100], [110] and [111] directions normalized to equal areas are shown in figure 3. The statistical accuracy at the peak was again about 0.65%. The angular distribu- tion for the three orientations is different up to about the free electron momentum of 5.5 mrad. The higher momentum part of the curves is identical and was again fitted by a Gaussian distribution and subtracted. By this technique we hope to eliminate the momentum dis- tribution due to core annihilations. It should be noted that this broad component is higher for O brass than for £8 brass because of the higher copper concentration. The differentiated data are shown in figure 4 for these three orientations. As in the case of 8 brass the devia- IOOF- ALPHA BRASS | _l | | | | |− O 2 4 6 8 |O | 2 | 4 ANGLE | N MILL | RAD I ANS FIGURE 3. Experimental angular distribution curves for [100], [110], and [111] directions in O. CuZn at liquid nitrogen temperature. The curves are normalized to equal areas. The statistical accuracy of the data points at the peak is 0.65 percent. ALPHA B R ASS .* | (|OO) (|| O) (|||) # ºf à ºf | ! # * | ; : || | iſºſ || ||*|| || || || # Hiſ # | MOMENTUM IN UNITS OF IO-3 m c FIGURE 4. Electron momentum distributions in a CuZn. These data are obtained from figure 3 by differentiating as discussed in the text. The theoretical difference curves were obtained from cross-sectional areas calculated using free electron approximation and an empirical Fermi surface configuration with a [111] neck radius of (0.29 + 0.02)k, . 341 tion from a free electron momentum distribution is ob- vious. The solid line is the result of a calculation using nearly free electron theory [7] and an empirical model derived from the known Fermi surface of copper [8]. We assumed the [111] necks expand and that the total sphere expands such that the volume remains the proper volume for the number of free electrons one would anticipate in copper and zinc. We joined the necks to the sphere by a smooth curve and tested vari- ous neck diameters in our model. Using the band gap energies given by Segall [9] and Burdick [10] we ob- served that the higher momentum states are important. We then calculated the expected momentum distribu- tion taking into account the higher momentum com- ponents in an approximate way. The derivative of this calculation is shown as the full line in figure 4. The par- ticular curves we have drawn here correspond to a neck radius of 0.29 +0.02 (units of kr). For pure copper the neck radius is 0.20 in these units. This indicates an in- crease in neck size of almost 50% compared with COpper. In the case of 8 brass as well as in the o brass data, these theoretical calculations agree remarkably well with the experiment in overall shape but they exhibit differences in detail in certain regions. Even folding these calculations with the experimental resolution will not completely smooth out these differences. Possible explanations for these small discrepancies can be found in effects not considered in the approximations used, e.g. (1) anisotropy of the ground state positron wave function; (2) a more accurate consideration of the higher momentum components of the electron wave function; and (3) many-body correlation effects leading to a k-dependent annihilation probability [11,12]. 4. Conclusions In this paper, we have shown that the positron an- nihilation technique can be used to yield information about the occupation in k-space of electrons in alloys. This preliminary analysis has shown surprising sen- sitivity to small details of the topology of the Fermi sur- face and of the occupation of higher momentum com- ponents. In particular this preliminary analysis shows for this particular o brass specimen a neck size approx- imately 50% greater than that of copper. We anticipate that future calculations and experiments will yield much more detailed information. 5. References [1] Berko, S., and Plaskett, J. S., Phys. Rev. 112, 1877 (1958). [2] Stewart, A. T., Shand, J. B., Donaghy, J. J., and Kusmiss, J. H., Phys. Rev. 128, 118 (1962). [3] Lang, L. G., and Hein, H. C., Bull. Am. Phys. Soc. 2, 173 (1957). [4] de Benedetti, S., Cowan, C. E., Konnecker, W. R., and Primakoff, H., Phys. Rev. 77,205 (1950). [5] Stewart, A. T., Shand, J. B., and Kim, S. M., Proc. Phys. Soc. (London) 88, 1001:(1966). [6] Taylor, R., private communication. We are indebted to Dr. Taylor for these results in advance of publication. [7] Mott, N. F., and Jones, H., Properties of Metals and Alloys, (Clarendon Press, Oxford, 1936). [8] Roaf, D. J., Phil. Trans. Roy. Soc. (London) A255, 135 (1962). [9] Segall, B., Phys. Rev. 125, 109 (1962). [10] Burdick, G. A., Phys. Rev. 129, 138 (1963). [ll] Kahana, S., Phys. Rev. 129, 1622 (1963). [12] Carbotte, J. P., Phys. Rev. 155, 197 (1967). 342 Discussion on “Density of States in a and B Brass by Positron Annihilation” by W. Triftshöuser and A. T. Stewart (Queens University) P. Platzman (Bell Telephone Labs.): I would like to correct the impression given here that the positron an- nihilation unambiguously measures the momentum dis- tribution. It measures the momentum distribution of the electrons in the metal in the presence of the positron, not in the absence of the positron. It may be true, under certain situations, that the momentum dis- tribution is not significantly distorted by the presence of the positron but this is not obvious. Only if one as- sumes that the positron does not interact with the elec- tron gas does one get an unambiguous connection between the annihilation spectrum and the momentum distribution of the host metal. I think that in some cases you may push things too far in trying to sort out certain small details of momentum distribution of the metal and not keep in mind that the positron is really distort- ing that distribution. W. Triftshāuser (Queens Univ.): Now in one way you are right since there are certainly some effects that we are neglecting in this calculation. But as it comes out its effects are very small and therefore we have proceeded in first approximation by neglecting these effects, because it seems to be impossible at the moment to ob- tain a better calculation of the distortion of the electron momentum distribution by the positron than that of Carbotte and Kahana. This calculation shows that the distortion is surprisingly very small. Therefore, we are confident that the observed small details of the momen- tum distribution are real, which is proved also by the good agreement of the theoretical calculated cross-sec- tional areas of the Fermi surface with the experimental results. Thus it is reasonable to neglect these effects until better calculations are available. A. T. Stewart (Queens Univ.); It is true that not enough is known about the effects of the positron in anything but simple situations. However, the k-dependence of the perturbation is probably not great and probably smooth. Thus the fine structure seen in these data for brass can probably yield fairly reliable conclusions about occupation in k-space, F. S. topology, etc. Since the authors (Triftshäuser and Stewart) cannot answer this question, I (Stewart) wish to put it to the au- dience. The question is this: Why cannot the perturb- ing effect of the positron upon the electrons be handled in a way that is easy to understand and use? Since the results of these quite complicated calculations are usually simple, there should be a simpler way to look at the problem. It has long been known that in an electron gas the polarization cloud around the positron, although an order of magnitude more dense than the charge den- sity in the metal, has a momentum distribution that is much like a Sommerfeld gas. The electrons from the Fermi surface are relatively 30% more dense in the polarization cloud than they are in the rest of the metal. And this is the most perturbable of electron systems! In an atom, recent work of Drachman, and of Salvadori shows that while the “enhancement” at the positron may be a factor of four, the momentum distribution in this polarization cloud is almost the same as that of the outer electrons of the above. The need for a simpler outlook is especially impor- tant in interpreting the increasingly accurate data from positron annihilation experiments. Consider a Fermi surface touching a zone boundary as in figure 1. Along a direction like OA it is reasonable to use the usual enhancement as in figure 2. However, along the direction OB might we not find electrons less perturba- ble so that the enhancement function might look like figure 3? FIGURE 1. A Fermi surface contacting the zone face. 343 F S. Z. B. | —" ehdncement | O A FIGURE 2: The sort of momentum distribution enhancement usually assumed along k-directions intersecting the Fermi surface. | | | | ehancement | T | O B FIGURE 3. A suggested enhancement for a k-direction not intersect- ing the Fermi surface. 344 Compton Scattering from Lithium and Sodium P. Eisenberger and P. H. Schmidt Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey O7974 Key words: Compton scattering; electronic linear momentum distribution function; lithium; potas- sium; sodium. Recently Phillips and Weiss [1] have shown the ex- perimental possibility of using the Compton effect to measure the gound state electronic linear momentum distribution function. In this work will be reported ex- perimental results on single crystals of Li and Na. The results on single crystals of Li reveal for the first time the true anisotropic ground state electronic linear momentum distribution function of the conduction electrons. As expected it is in the [110] directions that the major distortion from spherical symmetry occurs. However, the band structure distortion has secondary effects due to electron-electron interactions. These ef- fects arise because the distortion of band near the [110] direction changes the ratio of the potential ener- gy to kinetic energy for those states and thus the discontinuity at the Fermi surface and the high momentum tail of the momentum distribution are af- fected. These effects are not observable by positron an- nihilation experiments because they are relatively in- sensitive to electron-electron interactions [2]. The effects of band structure distortions on the elec- tron-electron interactions are further stressed when the Li results are compared with those of Na. As expected Na is isotropic and thus the discontinuity at the Fermi surface and the high momentum tail are, except for small electron phonons effects, basically determined by the electron-electron interactions. The results for Na will be compared with the existing calculations [3] for the ground state momentum distribution function in which interactions have been in- cluded. In addition to the above completed work, experi- ments are already in progress on He, H2, and K. The He experiments will provide a good test of the accuracy of Compton scattering for measuring linear momentum distributions since the helium distribution is fairly well known. It will enable one to make more quantitative statements about Compton scattering results. The ex- periments on H2 will illustrate the power of the technique in studying molecular systems in which bonding considerations are important. Finally, the ex- periments on potassium should provide a further test of the electron gas calculations when electron-electron in- teractions are considered. electron-electron References [1] Phillips, W. C., Weiss, R. J., Phys. Rev. 171,790 (1968). [2] Donaghy, J. J., Stewart, A. T., Phys. Rev. 164, 391 (1967). [3] Goldhart, D. J. W., Houghton, A., Vosko, S. H., Canad. J. of Phys. 42, 1938 (1964). 345 Discussion on “Compton Scattering from Lithium and Sodium” by P. Eisenberger and P. H. Schmidt (Bell Telephone Laboratories) M. Dresselhaus (MIT): At one time I thought there was some controversy as to whether there was contact made by lithium’s Fermi surface with the Brillouin zone boundary. Have you concluded from your work that there definitely is no such contact? P. Eisenberger (Bell Telephone Labs.): No, as a matter of fact, one of the critical features in determin- ing whether there is contact will be that the nature of the electron core contributions or the crystal field con- tributions will vary greatly if there is in fact contact at the surface. In other words, one will get a completely different spectrum. Before I can answer the question, I have to be sure what the relative magnitudes are. If they are actually touched, then one could expect the same sort of general shape for the tail as electron-elec- tron interactions would predict. If they did not touch, then one would expect gaps. So one really has to do a calculation of all three effects before one can really sin- gle out and say anything about any one of them. By going to sodium where there definitely is no contact, we hope to be able to show what the nature of the electron- electron interaction is by itself. L. Muldawer (Temple Univ.); Is your method similar to that of Weiss [1] at Watertown? P. Eisenberger (Bell Telephone Labs.): Definitely. L. Muldawer (Temple Univ.); Have you gotten similar type results? P. Eisenberger (Bell Telephone Labs.): Weiss did not do the experiments in helium at all. He did the experi- ments in lithium and saw no single crystal effects. I think Dr. Weiss deserves great credit for pioneering this field experimentally. But I think that there is a phase shift between the experimental results and the theory as far as lithium is concerned. S. Hanzely (Youngstown Univ.); You mentioned that you were using monochromatic X rays at different wavelengths. I would like to know just how monochro- matic the x rays were and how you achieved the dif- ferent wavelengths. P. Eisenberger (Bell Telephone Labs.): By your stan- dards, they are not monochromatic. They have a spread in energy of roughly 2 to 3 eV. S. Hanzely (Youngstown Univ.); I presume then that your technique is not too sensitive to the fact that it is not monochromatic. P. Eisenberger (Bell Telephone Labs.): No, at the mo- ment I am not even limited by that width. I should have pointed out that in the Doppler shifting technique at x- ray frequencies, one has a built-in amplifying factor in the wavelength shifts one is talking about, because it is the product of the electron’s momentum and the photon’s momentum. The photon is at very high energy so you get a very spreadout spectrum. For a Fermi mo- mentum of 4 eV one is actually talking about a couple hundred eV energy shift at x-ray wavelengths. So one does not need that good a resolution as one might if one were directly measuring the Fermi surface as one does in soft x ray, for example. [1] Weiss, R. J., and Phillips, W. C., Phys. Rev. 171,790 (1968) and 176,900 (1968). 346 ION-NEUTRALIZATION; SURFACES; CRITICAL POINTS; ETC. CHAIRMEN. E. Callen R. R. Stromberg RAPPORTEURs: D. E. Aspres B. Lcux lon-Neutralization Spectroscopy” H. D. Hagstrum Bell Telephone Laboratories, Murray Hill, New Jersey O7974 The ion-neutralization spectroscopy (INS) is discussed in comparison with other spectroscopies of solids. It is shown that INS probes the local density of states of the solid at or just outside the solid sur- face. It is believed that this accounts for the clear-cut differences between INS results and those of other spectroscopies. Because of its unique specificity to the surface region INS is particularly useful in studying the surface electronic structures of atomically clean surfaces and of surfaces having ordered arrays of known atoms adsorbed upon them. In the latter case INS determines a portion of the molecu- lar orbital spectrum of surface molecules formed from the adsorbed foreign atom and surface atoms of the bulk crystal. Such spectra provide information on local bonding symmetry and structure and electri- cal charging within the surface molecule which is as yet unavailable by any other method. INS is the first attempt to base a spectroscopy of electronic states on a two-electron process. More recent work on experimental and mathematical problems which such a spectroscopy entails are also briefly mentioned in this paper. Key words: Auger processes; autoionization; density of states; ion-neutralization; transition proba- bility. 1. Introduction In general, spectroscopies of electronic states have been based on the absorption or emission of elec- tromagnetic radiation when the system under observa- tion is excited or de-excited. In absorption spectrosco- pies one can observe the absorption of the photon or ob- serve the electrons emitted when the photon is ab- sorbed as in photoelectron spectroscopy. All of these spectroscopies are based on one-electron transition processes. The ion-neutralization spectroscopy (INS), on the other hand, is the first, but not the only spec- troscopy, to be based on a two-electron process in which a band transition density function is obtained. It is like the photoelectron spectroscopies in that the spectroscopic information is obtained by measurement of the kinetic energy distribution of electrons ejected in the process. However, because INS employs a two- electron process, the kinetic energy distribution con- tains the “spectroscopic function” in folded or con- volved form, making data reduction somewhat more in- * An invited paper presented at the 3d Materials Research Symposium, Electronic Density of States, November 3-6, 1969, Gaithersburg, Md. volved than for a spectroscopy based on a one-electron process. INS is a relatively new spectroscopy of solids having its own unique set of characteristics, advantages, and limitations. It is the purpose of this paper to review these properties in comparison with other spectrosco- pies. We discuss the method and what it measures, its resolving power and operational limitations, and its unique contributions to our knowledge of electronic state densities. 2. The Nature and Method of INS When an excited and/or ionized atom is projected at a solid surface, an excited solid-atom system is formed. The ion-neutralization process upon which INS is based is one of the processes of auto-ionization by which such an excited solid-atom system de-excites it- self. Not all such processes are appropriate to INS, however. The autoionization processes can be divided into two principal classes depending upon whether un- filled electronic levels in the atom do or do not lie op- posite filled electronic levels in the solid. These are in- 349 2 He?(1s )—H- — t Het }* He' (is)+ He + + FIGURE 1. Electron energy diagram showing metal at left and two atomic wells for Het and Hett cores. V is the vacuum level, F the Fermi level and B the bottom of the filled band. Transitions l and 2 are those of the ion-neutralization process. dicated schematically in figure 1. Here we show the electronic energy level diagram of a metal to the left and two atomic wells outside. One atomic well is that of the Het (1s) core in which the levels are those of He°. The second well is that of the Hett core in which the energy levels are appropriate to Het. We see that the two wells differ in that one (Hett) has two states [Het (2s) and Het (3s)] lying in the energy range of the filled band of the metal, whereas the other (Het) has no states in this energy range. We expect that atomic levels lying in the range of al- lowed levels of the solid will become resonances or vir- tual bound states and that of these allowed levels, those lying in the range of the filled band will fill. Thus the atomic levels should control the autoionization process in some energy ranges when they can fill by tunneling. Preliminary experiments with doubly-charged Hett ions and with metastably-excited Het (2s) ions appear to bear this out. Thus if we want the autoionization process to be dominated by initial state electrons whose state density is determined by the solid or its surface there should be no atomic levels lying in the energy range of the filled band as is the case in figure 1 for Het. This is a fundamental restriction on the ion-solid systems to which INS can be applied. For Het ions the solid band should lie within the energy range from ~4.5 eV to ~22.5 eV below the vacuum level. Earlier work has shown that the effective ionization energy of He is about two eV less than its 24.5 eV free-space value [1]. The transitions (1 and 2) of the two-electron, Auger- type, ion-neutralization process are also shown in figure 1. Since (1 and (2 may vary over the entire filled band we expect the ejected electrons to have energies lying in a broad band. Experimentally the kinetic energy dis- tributions are measured by regarding potential means using apparatus we shall not describe here [2,3]. Ex- amples of recorder plots of several kinetic energy dis- tributions, X(E), are shown in figure 2. It is clear that the X distribution is sensitive to the nature of the solid and the preparation of its surface. The spectroscopic information obtained by INS resides in these distribu- tions. In order to extract it we must understand the structure of these distributions in detail. The distribution functions which we need to un- derstand the ion-neutralization process are shown for an atomically clean copper face in figure 3. Suppose we start with the simplification of constant transition probability independent of the initial energy [.. Then it x 10−3 24 He” ions 5 eV / 2O ^ Ge (111) N |AE 12 /~~ 8 \ Ni (IOO) c(2x2) Se-- T \ ~ \ al | s— O 4 8 12 - 16 E IN eV FIGURE 2, Kinetic energy distribution of electrons ejected by 5 eV Het ions from atomically clean surfaces of Ge(100), Ni(100), and Cu(100) and from Ni(100) surfaces having ordered c(2X2)0 and c(2X2)Se structures upon them. 350 FIGURE 3. Electron energy diagram showing distribution functions appropriate to copper. The arrows under the functional labels indicate the direction in which the function is plotted. Shown also is the variation of ground state position Ei'(s) in the lower right-hand quadrant of the figure. is clear that the probability of the elemental process in- volving valence band electrons initially at Çı and (2 is N((1)N(C2) where N(S) is the appropriate state density of the combined metal-atom system. If we ask the relative probability of producing excited electrons in dB at E we see that all elemental processes contribute in which the electrons are symmetrically disposed on either side of the level & which lies halfway between the level E and the ground level of the atom at — Ei'(s). Thus we must integrate over A obtaining the restricted pair distribu- tion function Fe(ſ) appropriate to the assumption of con- stant transition probability: Ç F.(9–ſ. Nºt-A,N& Ada, a Relaxation of the restriction on transition probability to obtain a general F(ſ) function requires introduction into eq (1) of a factor proportional to the square of the matrix element. Thus: { F(0° ſº Ha'NG-AMG-Ada 9. We shall sidestep questions of antisymmetrization of wave functions discussed elsewhere [4] and discuss only the one elemental matrix element: H'-ſ'ſ u; (1)uſ (1) (e”/r12)u (2) uſ, (2) dridt, (3) in which up' and up" are initial state functions in the band, up is the atomic ground state function, and ue is the function for the excited electron. In eq (3) terms have been rearranged so that functions of the variables of the same electron are brought together. We see that the matrix element may be viewed as a Coulomb interaction integral between two electron clouds of spatial extent ugu, and ueu". Since uq is limited to the general vicinity of the atom the term uguo' varies in magnitude with up'. Thus the “down” electron makes a contribution to H' which varies with energy as [u, (Ç-A)]4, the uo" function evaluated near the atom position. If the “up” electron were also restricted to the vicinity of the atom we could make a similar argument relating to the energy variation of the contribution of the up election to H' to the magnitude of [u,"((+A)].A. This requires in addition that ue' vary little and smoothly with energy as appears reasonable. Several reasons can be adduced for believing that the up electron is excited near the atom position. These are listed here without really adequate discussion: (1) Experimentally the prominence of the molecu- lar orbital peaks in the results for surface molecules indicates that the wave function magnitudes in the surface region are con- trolling. Dominance of atomic level resonances in the results for ions in which atomic levels fill also points to the dominance of wave function mag- nitude at the atom in governing the autoioniza- tion process. The difference between INS and photoelectric results for atomically clean surfaces can be un- derstood only if INS is surface dominated. Energy broadening in the X(E) distribution is reduced by a factor 10 when an ordered monolayer of O, S, or Se is formed on the sur- face of Ni(100). This must be the result of reduction of the density of states just above the Fermi level. Since this reduction can occur only in and outside the monolayer we have evidence in this result that the INS process oc- curs predominantly in this region. There appear to be many fewer inelastically scattered electrons in INS than for equivalent photon energy in photoelectric emission, again (2) (4) 351 suggesting a surface source of excited elec- tronS. (6) Theoretical considerations by Heine [5] and Wenaas and Howsmon [6] lead to the conclu- sion that the up electron is excited predomi- nantly outside and in the first layer of the solid. Large momentum transfer between the two participating electrons means a close collision. near the atom where we know the down elec- tron is concentrated. Also viewing the Auger process as photoemission by the down electron followed by photoabsorption by the up electron points to the conclusion that the up electron is most likely excited in the rapidly-decaying near field of the dipole of the down electron transi- tion. (8) If the up and down electrons made very dif- ferent contributions to Hf, we could not con- clude that F is the convolution square U*U but must be the convolution product V*W of two dissimilar factors. When V*W is inverted as though it were a convolution square it can be shown that spurious features will be introduced into U(ſ) unless V= W. These are not found. We are thus led to the general conclusion that: Hº & [u,(& – A)]aſu' (ſ-F A)]4, from which eq (2) becomes: FG) - ſuº-Alix-A) ſu (; +A)]*N(g-H A) dA. (5) eq (5) may be written as: FG)=ſ. U(§ – A)U (C + A) dA = U+U, (6) defining the transition density function U(£) which thus includes both state density and transition probability factors. We see also from eqs (5) and (6) that U(() is es- sentially the so-called local density of states in the vicinity of the atom, i.e., the actual state density weighted by the local wave function magnitude at the atom position. This wave function magnitude must, of course, include the effect of the presence of the atom itself in this vicinity. The pair distribution function F(ſ) of eq (6) becomes the distribution in energy of excited electrons, F(E), when band variable C is replaced by the outside energy variable E according to the relation: E=E. (s) –2(g-Ho). (7) This equation is obtained by equating magnitudes of the energy transitions l and 2 in figures 1 or 3. The ex- ternally observed electron energy distribution X(E) is related to F(E) by the equation: X(E)=F(E)P(E), (8) where P(E) is the probability of escape over the surface barrier and includes any other dependences on E such as variation in density of final states. The method of INS consists in reversing the above development to obtain U(ſ) from measured X(E). It proceeds in the following steps: (1) Experimental determination of two Xk(E) at ion energies K = K, and K2. Usually K = 5 eV and K2 - 10 eV. (2) Linear extrapolation of Xk, and Xk, to X0 to reduce the natural broadenings present in the Xk distributions. This is done by use of the rela- tl On: Xo (E) =XK, (E) + R[XR (E) -XR, (E)]. (9) Since it has been shown that broadening varies with ion velocity, it is possible to write Rººk, as R.K., k, = (K2/K1) 1/2 − 1. (10) ( 3 ) Division of Xoſł) by a P(E) function, reversing eq (8), to obtain F(E). This step is really not necessary since replacement of P(E) by a con- stant merely changes the intensity level of U(ſ) progressively as ( increases without disturbing the structure. However, we have usually di- vided by a parametric P(E) whose parameters are chosen so that the pieces of F(Q) obtained by Het, Net, and Art ions are essentially coin- cident. After change of variable, F(g) is inverted by a sequential deconvolution procedure. The for- mulas used are: Uo- (F1/2A() 1/2, U2 = (1/U0) (F2/2A(), (4) U2n-2 = (1/2U0) [(F/2A() — X. p= 1, n–2 U2n-2p–2 U2p), n > 2. (11) in which F and U are digitalized as Fn = F(nAſ), n = 1, m; U2n_2 = Uſ (2n-2)Aſ), n = 1, m. (5) Tests of the mathematical uniqueness of U(ſ) by variation of its origin and by comparison with F.' (C), the derivative of the fold function. These steps cannot be discussed in this paper 352 but will be discussed extensively in a forthcom- ing publication [7]. Suffice it to say that, although deconvolution is in general a difficult procedure, the sequential unfold works ex- tremely well for the general class of F(ſ) func- tions we have for which F(0) = 0, F" (0) = k, and F(ſ) does not depart drastically from F(Q) = kū. The procedure we now use is essentially that given when the INS method was first discussed [4]. How- ever, in the interim we have learned a good deal about the mathematical side of the data reduction, particu- larly the unfolding procedure. We have derived all possible digital sequential unfold formulations which invert directly or with the independent calculation of no more than the first data point Uo. We have also studied the noise characteristics and shown that the step-mid- point formulation given above in eq (11) not only is the only one which inverts directly without independent calculation of the first point but also has by far the best stability characteristics with respect to noise in the data. We have also faced up to the problems involved in the possibility that we are inverting as a convolution square (U+U) a function which is in reality a convolu- tion product (V*W) and have devised tests to determine if any spurious structure could possibly be introduced in this way. The data reduction procedures, although more complicated than for a one-electron spectroscopy, proceed smoothly on the digital computer and produce unique and correct answers. We shall discuss further some of the properties and limitations of INS in section 4. 3. Examples of INS Results We turn now to the presentation of INS results. These are in two categories: (1) results for atomically clean surfaces of the transition metals Cu and Ni [8], and (2) results for the Ni(100) surface with ordered monolayers of O, S, and Se adsorbed upon it [9]. Some unpublished results for Si and Ge will be mentioned in the discussion of item (1). In figure 4 we reproduce figure 6 of reference 8 show- ing F(Q) and U(Q) for Cu(111). Also shown in the correct relative position is the P(E) function used, indicating how flat it is over the energy range of the data. The average U() function for (100), (110), and (111) faces of Cu (fig. 15 of ref. 8) is compared in figure 5 here with the optical density of states curve (ODS) of Krolikowski and Spicer [10]. In figure 6 the U(C) curve for atomi- cally clean Ni(100) from INS is shown and compared with Eastman’s ODS curve for a nickel film obtained by photoemission [11]. t (eV) FOR F (g) 7 6 5 4 3 2 1 O | | | I I I CU (111) T He” ions == F(g), F(E) 2-1. k- 2^ P(E) - / / |- / mºm / – / U (; ) / — / / - / * / | | | | l l 14 12 1O 8 6 4. 2 O Ç (eV) FOR U (g) | | | | | ! O 2 4 6 8 1O 12 E (eV) FIGURE 4. F and Ufunctions for atomically clean Cu(111) and Het ions [fig. 6 of ref. 8]. The probability of electron escape used in the data processing is also shown. First, it is evident that the INS results show a peak in the general vicinity of the bulk d-band in both Cu and Ni. However, it is equally evident that this peak does not have the shape or width to be expected from band theory or measured by ultraviolet photoelectron spec- troscopy (UPS). A strong case can be made that the dif- ferences evident in figures 5 and 6 are due to the fact that the two spectroscopic methods are sensitive to dif- ferent things. Although the energy resolving power of INS is somewhat poorer than that of UPS, one cannot by any stretch of the imagination consider the INS U(ſ) curve as a smeared out version of the ODS curves. In reducing the Ni data of figure 6 very little digital i |O 8 6 4. 2 O [.. IN eV FIGURE 5. Comparison of the average U() function for (100), (110), and (111) faces of Cu [fig. 15 of ref. 8] compared with the optical density of states curve (ODS) of Krolikowski and Spicer (ref. 10) obtained by photoelectron spectroscopy. - 417-156 O - 71 – 24 353 gº ! e O / O / 3. | E. ºr' ſ a-vº-'vº | |O 8 6 4 2 O l, IN eV FIGURE 6. U(ſ) for Ni(100) compared with the ODS curve of Eastman (ref. 11) also obtained by photoelectron spectroscopy. The amount of smoothing used in the INS data reduction was deliberately reduced to the point of leaving in the data the noise seen in an attempt to demonstrate the best resolving power of the INS method. smoothing of the data was used in an attempt to in- crease the resolving power at the expense of letting through low-frequency noise. Some increase in resolv- ing power (about 20%) is evident when comparison with similar curves in reference 9 are made. The sharpness of the peak in U() at (= 1 eV is an indication of the INS resolving power. In view of the characteristics of INS discussed above it is believed that the U(ſ) curve is in fact the local density of states at or just outside the sur- face whereas the UPS results are characteristic of the bulk. Why the local density of states for d bands of transi- tion metals outside the surface differs from the bulk band is an interesting question in surface physics. The reduction in number of nearest neighbors as well as a probable small dilatation of the lattice at the surface could narrow the tight-binding d band and make it more like an atomic level. Tight-binding bands are particu- larly vulnerable to such modification in the surface re- gion. Unpublished work on Si and Ge appears to in- dicate that the INS results will much more closely mirror what is expected from bulk theory [12]. This is probably attributed to the fact that the s and p wave functions of the semiconductor valence bands overlap more strongly at the surface even though the surface atoms may be displaced from their “bulk posi- tions” by larger amounts than are surface atoms of the transition metals. Another interesting suggestion to ac- count for the INS results in Cu and Ni arises in the work of Pendry and Forstmann [13] who predict that on some faces of transition metal crystals a new type of surface state appears which should clearly modify the surface local density of states from the bulk density. The second category of INS experimental result to be mentioned in this paper is found for metal surfaces upon which ordered monolayers of adsorbed atoms are present. In figure 7 is reproduced the U(() functions from reference 9. Here in curve 1 is repeated the transi- tion density for atomically clean Ni(100). Curves 2, 3, 4 are for c(2X2) structures of O, S, Se, respectively, and curves 2',3', and 4' are for p(2X2) structures involving these same adsorbed atoms, respectively. We note a very interesting increase in complexity of the U. functions for the covered surfaces. These appear now Ni(1OO) 2- O A \ ſp /ºx2x} 2. "…SV | 1b2 304 c(2x2)O | / e= t 2 Sº \ _Y * | O \/ /A N N f A2.2% d º c(2x2)S | \ e” - Af (1)b2 y * Á ! M O f \ } | . | \ ! \ ". + (2x2)Se \–’ %+\ p \ N. / X- 12 4 / N --~~ ~ * N J' — |P * |c(2x2)sev * * } (1)bz (3)a, (1)b, O 14 12 1O 8 6 4 2 O Ú IN eV FIGURE 7. Transition density functions, U() for atomically clean Ni(100) (curve 1) and for the surface with c(2×2) structures of O, S, Se (curves 2, 3, 4, respectively) and with p(2×2) structures of O, S, Se (curves 2’, 3', 4', respectively). Energies labelled p, 1b1, 3a1, and 1b2 are identified in the text. 354 | E E VACUUM LEVEL O FERMI LEVEL $2 ſ\il i vº Hellº- METAL SURFACE MOLECULE He” FIGURE 8. Electron energy diagram illustrating the effect on INS of a resonance or virtual bound state of a surface molecule formed on a metal surface. The bond resonance in the surface molecule is assumed to lie at Çı (which is also the initial energy of the down electron) and to increase the magnitude of the surface wave function ill, over a broadened energy range as indicated on the right-hand side of the diagram. The increase in wave function outside the solid in this energy region is indicated by the dashed-line modification of the electronic wave function at £1. to indicate the energy spectra of electronic orbitals of electrons in the bonds of so-called “surface molecules” formed from the adsorbed atom and atoms of the sub- Strate. The electronic states to be associated with bond or- bitals in surface molecules form resonances or virtual bound states. These will evidence themselves in the transition density for reasons we attempt to make clear by figure 8. The presence of the electronic orbital at the surface will increase the wave-function magnitude in the vicinity of the surface molecule as indicated by lº in the figure. This will in turn increase the tunneling probability for band electrons into the Het well. The dashed line indicates how a band wave function in the absence of the surface molecule (full line) is increased in the presence of the surface molecule (dashed line). These wave function increases in the Het well will result in peaks in the local density of states and hence the U(ſ) transition density function observed by INS. This paper is not the place to discuss the results in figure 7 in any great detail. A preliminary discussion is to be found in the original publication [9] and an exten- sive paper is in preparation [7]. However, it is essential to an understanding of the scope of INS as a spec- troscopy of electronic states to mention briefly the prin- cipal results for these cases of chemisorption. Several energies are indicated in figure 7. These are the levels of the atomic p orbitals in free O, S, and Se, labelled p in the figure. In the figure the second, third, and fourth panels from the top refer to adsorbates O, S, and Se respectively. The lines labelled 1b1, 3a1, and 1b2 are molecular orbital energies in the free molecules H2X where X is O, S, or Se in the second, third, or fourth panels of the figure, respectively. Three types of molecular orbital spectrum are to be found among the six curves for adsorbed species in figure 7. Curves 3 and 4 are the most complex spectra having peaks near the orbitals indicated for the free H2X molecule. These have been attributed to the bridge-type bonding illustrated in figure 9(a) and (b). Relatively small negative charging of the X = S,Se end of the surface molecule is indicated by the fact that the lone-pair orbital peak near (1)b, also lies near the atomic p orbital energy as for free H2X. When the structure is changed from the c(2 × 2) [fig. 9(b) to the p(2 × 2) [fig. 9(d)] by removal of half of the adsorbate we see that the molecular orbital spectra change completely to those of curves 3 and 4 in which there is a single peak below the Nid-band peak indicat- ing a change in the local bonding structure. The only other reasonable alternative is the T-type symmetrical bonding as shown in figure 9(c) and (d) for which we ex- pect a nonbonding orbital in this energy range. Removal of the “center atom” in the c(2X2) structure removes the agent which distorts the square of Ni atoms of Car symmetry below each X atom into a rhom- bus of C2v symmetry. C2v symmetry is essential if the molecular structure is to resemble H2X. Reversion to Cao symmetry when the center atom is removed de- mands change of the molecular structure and spectrum as is indeed found. Finally, both c(2X2)O (curve 2) and p(2×2)O (curve 2') show a single peak shifted by a much larger amount toward the Fermi level from the atomic p level than is the case for either S or Se. This orbital spectrum (single peak in the available energy range) and larger negative charge (orbital energy shift) together with small work function change on adsorption can be shown to be con- sistent with a reconstructed surface in which the ad- sorbed atom is incorporated into the top layer of sub- strate atoms where relatively large charge will not result in large work function change. Although the above account of the data in figure 7 is admittedly 355 (e) FIGURE 9. Surface structures suggested (ref. 9) to account for the molecular orbital spectra of figure 7. (a) and (b) are for a bridge-type Nig)(-type structure repeating over the surface in a c(2X2) pattern to account for curves 3 and 4 of figure 7. (c) and (d) illustrate a p(2×2) structure adequate to account for curves 3' and 4" of figure 7. (e) and (f) illustrate a reconstructed c(2X2) structure to account for curve 2 of figure 7. Simple removal of the “center atom” in (f) without other change produces the p(2×2) reconstructed surface thought to account for curve 2" of figure 7. In these figures bond orbitals are indicated by the heavy arrows with conical arrowheads. sketchy, it does indicate how INS determines a portion of the molecular orbital spectrum of a surface molecule and the power such information has in elucidating sym- metry and bonding character. 4. Comparative Critique of INS A comparative critique of INS is perhaps best car- ried out by listing its characteristics and attempting to assess them as advantages or disadvantages in com- parison with other spectroscopies of solids. The other spectroscopies are the two forms of photoelectron spec- troscopy, ultraviolet photoelectron spectroscopy UPS [10,11] and x-ray photoelectron spectroscopy XPS [14]; soft x-ray spectroscopy SXS [15], and the sur- face Auger spectroscopy SAS [16]. In the first place INS is a two-electron spectroscopy as is SAS whereas UPS, XPS, and SXS are one-elec- tron spectroscopies. SAS is based on a two-electron Auger process similar to that underlying INS except that the vacant ground level in the excited system is an inner level of a surface atom rather than the ground level of the parent atom of an incoming atomic ion. The SAS process has been used extensively in the identifi- cation of surface impurities but Amelio and Scheibner [16] were the first to attempt to separate the Auger dis- tribution from the large background of secondary elec- trons and to unfold it to obtain spectroscopic informa- tion as has been done in INS. The fact that INS, like SAS, is a two-electron spec- troscopy must in itself be considered a drawback since it necessitates unfolding of the data. However, in INS the data are of such quality that unfolding now offers no significant problem. We have learned much about un- folding methods and possible errors since the last discussion of these matters in the literature [4]. A second characteristic of INS is its surface specificity and hence surface sensitivity. This means, as we have seen, that INS results can be compared with the results of bulk spectroscopies only in special cases. However, INS gives us a tool to study variation of elec- tronic band structure from bulk to surface, to study sur- face states on both metals and semiconductors, and, perhaps most importantly, to measure molecular orbital spectra of surface molecules formed in chemisorption. Some recent UPS work [17] with 21.2 eV radiation and grazing incidence has shown the possibility of detection of large molecules adsorbed on surfaces. Whether sur- face molecules of the type discussed here can be ob- served in this manner has yet to be demonstrated. The transition probability factors of INS arise from its surface specificity and the tunneling character of the electronic transitions. Four types can be listed: (1) a tunneling factor which decreases with depth in the band, (2) a symmetry factor arising from extent of the surface wave function which decreases as the character proceeds from s to p to d, etc., (3) a second tunneling factor which favors bulk states whose k vector is normal to the crystal face used, and (4) the enhancement in certain energy ranges caused by the surface resonances of adsorbed atoms. Although they are distinctive, there appears to be no particular disad- vantage associated with these transition probability fac- tors. It is the last one which makes possible the study of surface molecules and this must be listed as an ad- Vantage. The energy range which can be explored in the solid is E.;'-2'p where Ei' is the effective neutralization ener- 356 gy of the incident ion near the surface (effective ioniza- tion energy of the parent atom) and p is the work func- tion of the solid. This means that INS is the equivalent of a photoelectric process for which hy= Ei' - p. For He, Ei' - 22.5 eV and for a representative solid p - 4.5 eV. Thus Ei – p – 18 eV. To equal this range with UPS one must use the 21.2 eV He resonance radiation. XPS, SXS, and SAS, on the other hand, have essentially no energy range limitation with respect to the valence bands of solids. Like UPS, INS is limited by vacuum level cutoff making it difficult to extract data near the vacuum level because of the rapid variation of escape probability there. Energy resolving power of INS is undoubtedly somewhat less than that of UPS but as figure 6 in- dicates not greatly less. It is in all probability better than that of SXS, XPS, or SAS since each of these in- volve the relatively broad inner level of an atom at one point or other. Finally, we shall mention a series of side effects which must be considered in evaluating any spectrosco- py. There appear to be fewer inelastically scattered electrons to contend with in INS than in UPS at higher energies. SAS has a serious background problem unk- nown to INS. Plasma losses, which can be a complicat- ing interpretive factor, apparently play no role in INS results. SXS has a serious spectral superposition problem unknown to INS. The signal intensity in INS is adequate which sometimes cannot be said for SXS or SAS. INS has the possibility of variation of natural broadenings by variation of a controllable experimental parameter, namely incident ion velocity, making it possible to extrapolate out broadenings admittedly greater than those of UPS. In conclusion it is possible to state that ion- neutralization spectroscopy is a viable spectroscopy of solids having its own peculiar set of characteristics. It appears that its most important area of application at present is to the study of the molecular orbital spectra of surface molecules formed in chemisorption. Here it holds promise of extending our knowledge of surface structure beyond what low-energy electron diffraction (LEED) now can do. LEED tells us how a given adsorp- tion or bonding structure repeats itself over the surface. INS yields information about bonding symmetry, or- bital energy-levels, and electric charging within the sur- face molecular structure, which in many cases, using LEED and work function data, will permit the specifi- cation of bonding structure. Surface state and surface modifications of band structure also promise to be in- teresting fields in which INS can make a contribution. 5. References [1] Hagstrum, H. D., Phys. Rev. 96, 336 (1954). [2] Hagstrum, H. D., Rev. Sci. Instr. 24, 1122 (1953). [3] Hagstrum, H. D., Pretzer, D. D., and Takeishi, Y., Rev. Sci. Instr. 36, 1183 (1965). [4] Hagstrum, H. D., Phys. Rev. 150,495 (1966). [5] Heine, V., Phys. Rev. 151,561 (1966). [6] Wenaas, E. P., and Howsmon, A., private communication and Proc. 4th Int’l Materials Symp. on the Structure & Chemistry of Solid Surfaces, 6/19-21/68, Univ. of California, Berkeley. [7] Hagstrum, H. D., and Becker, G. E., to be published. [8] Hagstrum, H. D., and Becker, G. E., Phys. Rev. 159, 572 (1967). [9] Hagstrum, H. D., and Becker, G. E., Phys. Rev. Letters 22, 1054 (1969). [10] Krolikowski, W. F., and Spicer, W. E., Phys. Rev., to be published. [ll] Eastman, D. E., and Krolikowski, W. F., Phys. Rev. Letters 21, 623 (1968); Eastman, D. E., J. Appl. Phys. 40, 1387 (1969). [12] Hagstrum, H. D., and Becker, G. E., to be published. [13] Pendry, J., and Forstmann, F., private communication. [14] Fadley, C. S., and Shirley, D. A., Phys. Rev. Letters 21,980 (1968). [15] Cuthill, J. R., McAlister, A. J., Williams, M. L., and Watson, R. E., Phys. Rev. 164, 1006 (1967). [16] Amelio, G. F., and Scheibner, E. J., Surface Science 11, 242 (1968). Other forthcoming papers by these authors will give results for Si and discuss the experimental and mathematical methods in greater detail. [17] Bordass, W. T., and Linnett, J. W., Nature 220, 660 (1969). 357 Discussion on “lon-Neutralization Spectroscopy” by H. D. Hagstrum (Bell Telephone Laboratories) W. Plummer (NBS): I am very much concerned that you interpret the low energy structure as due to density of states when, in fact, you have assumed a constant escape probability, before and after adsorption of these gases. It is a well known fact, both experimentally [1] and theoretically [2], that adsorption will cause coverage dependent structure in the reflection which should be related to your escape probability. If you do not accept any structure, say for 4 eV above the vacuum level, then basically you have only two kinds of structure in all your curves. All of them would then have one broad hump with only the p(2×2) structure of Se having an extra peak. I understand from your 1966 paper that when you parameterized the escape proba- bility, you could not explain satisfactorily the low ener- gy escape probability for the various ions used, i.e., from 2 to 4 eV above the vacuum level. It is very impor- tant to your interpretation that some of the gases have three humps while others have only one or two. H. G. Hagstrum (Bell Telephone Labs.): Dr. Plummer is certainly correct that one should consider seriously the effect of the results of Madey and Yates for escape probability of the ion-neutralization process. If the lowest energy structure observed with Het ions were to be due to a variation of escape probability with energy, this structure would have to appear at the same energy above the vacuum level in the results for Net ions. We have looked again at our results for Net ions and find that they give the same spectrum as we observed for Het from the Fermi level down to that energy below the Fermi level to which the Net ion-neutralization energy enables us to eject electrons. From this it does not ap- pear that the structure in either of the electron distribu- tions for Het and Net ions results from the variations observed by Madey and Yates in a different system. I would suggest that the reason for this is that the elec- trons we observe in the ion-neutralization process are electrons which are in the main excited outside the solid surface and, therefore, do not interact nearly as intensely with the surface barrier as do the electrons in the experiment of Madey and Yates. I would also sug- gest that our result which shows that the lowest energy peak disappears in going from the c(2×2) Se to the p(2 × 2) Se structure is quite strong evidence that the peaks in the spectrum are of molecular orbital origin and not due to variation of the escape probability. Dr. Plummer has alluded to the discussion in my 1966 paper concern- ing the difficulty of obtaining reliable data in the range 0 to 4 eV above vacuum level. Since the 1966 paper was published, we have worked very hard to improve our data in this region. In the retarding potential configura- tion we have been using, this problem relates to the dif- ficulty in keeping the ion beam well focused at the lowest kinetic energy. We have been able to improve our situation since that time to the point where we be- lieve that the presence or absence of a peak in the range 2 to 4 eV above the vacuum level is no longer in doubt although we cannot, with that experimental con- figuration, claim to know the peak position or width with the same accuracy that we know these data for other peaks at higher energies. During the last year we have rebuilt our apparatus so as to eliminate the effect of the electron retardation upon the incoming ion beam. With this apparatus we have been able to reproduce very well the results of the earlier apparatus, thus bol- stering our confidence in the results in this energy range. [1] Madey, T. E., and Yates, J. J., Supp. Nuovo Cimento, Vol. 5, Ser. 1, p. 483. [2] Gadzuk, J. W., to be published in Surface Science. 358 Potential and Charge Density Near the Interface of a Transition Metal * E. Kennard** and J. T. Waber Northwestern University, Materials Science Department, Evanston, Illinois 60201 The early literature on methods of calculating surface energy and charge density and of dealing with potential barriers at an interface are reviewed. The three dimensional potentials and charge densities were obtained by superimposing the rele- vant atomic information which had been obtained from Dirac-Slater self-consistent field calculations on free atoms. The total charge density at each point P was found by summing the contributions from atoms located within a sphere of radius R centered at P. The local exchange potential was estimated at P by means of Slater's pºſ” method. This was included with the overlapped atomic Coulomb potentials to ob- tain the crystal potential near the surface. Key words: Absorption potential; charge density; metal-vacuum interface; platinum; surface energy. 1. Introduction The theoretical investigation of the surface energy of metals has received considerable attention in the last few years. This trend has been stimulated by a large amount of excellent experimental data which has been appearing in the literature. The first theoretical attempts to explain experimen- tal values of surface energy were made in the late 1930's; the nearly free electron theory of metals was used. This is quite a reasonable approach for the alkali metals. The results were on the whole in reasonably good agreement with the experimetal data. But, as in all such studies, the results were strongly influenced by the model chosen. It is the aim of this paper to examine some of these models and then to discuss the calcula- tion of more realistic models. In reality, all metal specimens are finite and are bounded by surfaces. However, theoretical calculations of electronic structure can be simplified by assuming. an infinite solid. When there is a regular arrangement of ion cores in a metal, calculations need only be car- ried out over one unit cell. That is, the infinite solid pos- sesses a very high degree of translation symmetry. *Research supported by the National Aeronautics and Space Agency, Washington, and by the Advanced Research Project Agency of the Department of Defense through the Materials Research Center at Northwestern University. *Submitted in partial fulfillment of the requirements for the degree of Doctor of Philos- phy, Northwestern University, Evanston, Illinois. At this conference on the electronic structure of metals and semiconductors, it is appropriate to con- sider the special quantum mechanical problems which occur when a surface intrudes into an infinite solid. There is a reduction in the symmetry; translational in- variance is lost along some directions [h k l] which is perpendicular to the free surface and in general certain planes which were mirror planes are no longer effective in “mapping” the crystal lattice onto itself. In general, Bloch's theorem will hold for translations having their components strictly parallel to the free surface, but not for translations with a component perpendicular to the surface. For a realistic model of a solid, there will be a non- vanishing probability of finding electrons in a region slightly beyond the last row of ion-cores, and the average charge densities will decrease in a more or less exponential way toward zero over a region of several interatomic distances. Surface studies are pertinent to the theme of this conference because certain experimental methods for determining N(E) curves, such as photoemission and ion-neutralization spectroscopy, involve electrons passing through or coming from the surface regions of a metal. More understanding of the effects of crystallo- graphic orientation on surface potentials would enhance our knowledge of the processes which occur at surfaces; in addition, it would increase our 359 knowledge of the bulk electronic structure of metals and semiconductors. 2. Review of Pertinent Studies To illustrate the problems associated with making realistic calculations of surface potentials and surface energy, we will review some of the early studies which utilized primarily the methods for studying the bulk density of states. Certain earlier studies of surfaces used the nearly free electron model of metals; these are related to the use of the “jellium” model. The several researchers in this field have assumed an infinite potential wall at the surface of the metal. Later, the ef. fects of a finite barrier and the penetration of that barri- er by electrons were considered. Although the “geometric” surface can be defined as the plane con- taining the last row of ion cores, those properties directly connected with the electron distribution cannot be considered to change abruptly from one side of this plane to the other. Instead, one should consider a sur- face region surrounding the last plane of ion cores and in that the properties of the finite solid may be different in this narrow region from those of the bulk. The boundary-layer region and the charge density variations in this region have been treated in recent stu- dies. Let us begin by defining the increase in energy as- sociated with “cutting” a metal and forming an inter- face. 3. Surface Energy The surface energy of a solid may be defined as the difference in energy between that of a given volume of metal contained in an infinite solid and the energy of an equivalent volume of metal removed from the metal and placed in vacuum—it is the energy increase which results from the presence of one or more surfaces. For ionic or covalent solids, one might reasonably consider surface energy in terms of broken bonds or dangling bonds. The relatively free electrons and the ion cores are the ingredients of a metal; the mutual interactions between the many electrons and the many cores can lead to an increase in energy. Thus, the surface energy must be related to a change in the local concentration and kinetic energy of electrons. The redistribution of the ion cores which apparently occurs may be another con- tributing factor near the “surface.” The attendant change in potential alters the local concentrations of electrons. If one thinks of a metal as a collection of plane waves (electrons being scattered about in a metal by the ion cores), it becomes very likely that standing waves will occur near the interface due to reflection at the interface. 4. Trecitment of the Electronic Problem Brager and Schuchowitzky [1] considered that the extra energy arose from the fact that the presence of a surface serves to define the position of the metallic electrons more exactly, i.e., it partially “localizes” them, and hence leads to an increase in their energy by the Heisenberg principle. These researchers did not clearly indicate whether the increase in surface energy would be confined to a surface region or whether it might lead to a general increase in energy of all the electrons. Because of the penetration of the electrons beyond the geometric surface, a “double layer” is formed with a net positive charge inside and a negative charge out- side. The energy of the dipole layer is also associated with surface energy. Early workers [2,3] attributed all of the surface energy to the potential energy which comes from separating the electrons from their ion cores, i.e., with the energy of the double layer. Sub- sequent calculations have shown that this contribution is negligible in comparison with the local change in kinetic energy [4,5]. 5. Nearly Free Electron Model In 1946, Brager and Schuchowitzky [1] assumed that the electrons move in a cubic box with side L and that the potential field becomes infinite at the walls of the box. The appropriate wave functions are of the form: il (r)=A sin (nkºw) sin (mk/y) sin (lkzz). (1) Each such state is represented by a point in k space, i.e., reciprocal space which lies on a grid of spacing (n"T/L, mT/L, lºſ/L). In the usual way, one finds that the density of states at the Fermi level is: dN_ſ , (L) #–4. (#) (2) where kmar is the radius of the Fermi sphere. The total kinetic energy of these states is given by: E,... = h°kºmax Q. tot T 1072 m.” (3) where () is the (cell) volume and m” is the effective electronic mass. The number of states has been over-counted in this Sommerfeld model because the particular boundary conditions require that the functions vanish on the walls or limiting planes of the octant in k space, i.e., that states where n, m, or la 0 are inadmissible. Brager 360 and Schuchowitzky obtained the number of reciprocal lattice points contained in the sphere of radius K and excluded those on the walls of the octant, using a for- mula given by Vinogradov [6], namely: 37 N=#K-4K-00& ) where O indicates the number of magnitude of the cor- rection term. This correction arises from approximating a sphere by a collection of equal volume boxes centered on a lattice of points." It leads to a total energy of _hºkinas Q 3Th” L 1. } { | 1.4 E = 107°m 32m ( ) kinax + O(L'") . T The energy of such a bounded metal is made up of a (4) (5) term dependent on L" and a term dependent on Lº plus correction term of order L**. The surface energy in (eV/cm2) (namely, the excess energy over the energy in the volume L” when divided by the surface area) becomes: - h? imax Surface T 327Tm E + O(L=0.6). (6) When L = ao, the latter term relates to surface irregu- larities and can be neglected. Sugiyama [7] proved that the same result held for any shaped metallic mass. Brager and Schuchowitzky [1] showed that the sur- face energy is primarily dependent on the electronic charge density in the bulk of the metal. Figure l is a log- log plot of the experimental values of the surface ten- sion of liquid metals against (p/A) where p is the density of the metal in gm/cmº and A is the atomic weight. Con- version of eq (6) indicates a slope of 4/3; the slope found is 1.2. Another result is that the second term, E2, in eq (5) is one-fourth of the energy of a two-dimensional Fermi gas located on the surface. The common model potentials have some form of a step at the geometric interface. When one uses an in- finite, three-dimensional potential step, as several wor- kers have done, the values of li(r)-> 0 on this boundary. Sugiyama [7] illustrated the spatial dependence p(y) of the electron gas when an infinite barrier is erected at a geometric surface. There is a surface thickness which is the region in which the electrons are excluded. As a consequence of the barrier, the charge density must in- crease in other regions. This is illustrated in figure 2. Charge oscillations are also shown. If it is assumed that the interior charge density remote from the surface is not altered by the presence of the surface and further, that kmar is not altered from its bulk value, then it can be shown [5,7] that the charge may be conserved by displacing the infinite FIGURE 1. Variation of the logarithm of the experimental values of the surface tension (O-) against the logarithm of the approximate electron density (p/A). * The slope of the line is l.2. potential barrier outward from the geometric surface, by a length of the order of a lattice parameter. How- ever, an imaginary node then occurs at y = a, a little way beyond the geometric surface. The k vector is inversely proportional to the distance between nodes of l;(r); thus k, = mT/L – 6/L where the “phase shift” 6 due to barrier penetration can be | In principle, a similar correction might be used to take account of the asphericity of the Fermi surface, but Brager and Schuchowitzky were content to use m” for that purpose. defined by: hy tan 6 = ) ) (7) 2 1, 2 \ 1/2 (2mpſh j) | P. - ſº X -> "or, ſº Q) C) Gl) E. U -E C) | | | | i 2 4. 6 8 |O Distdnce from surface FIGURE 2: Local variation of the charge density of free electron gas caused by an infinite potential wall placed at the geometric surface. pi exceeds po when the infinite step is at y = 0. 361 where q is not the work function but the inner potential observed in LEED experiments. One notes for the wave vectors, that The term a is of the order of an interatomic distance. The surface energy computed with this model contains a term which is similar to the one in eq (6). A negative k,---4–= 8 . (8) correction term O2 arises due to the reduction of each * 20,-- a) (Ly-Ha) component of k by 6/L. Thus: 4 8, - ſº [3(2–X)(x-1)*-i- (3A*–8A +8) are sin (A-9)] (9) 16mh” where the parameter A = dp/EP is greater than unity. Huang and Wyllie [4] were apparently the first researchers to use a realistic finite step at the surface. The height of the barrier (b was estimated from the Fermi energy EF and experimental values of the cohe- sive energy W. With such a finite potential barrier, the wave function li(r) is not forced to approach zero at the barrier step. Instead it undergoes a change in phase as electrons penetrate into vacuum and k has an imagina- ry component. This will be discussed below. The calcu- lations of Huang and Wyllie give better agreement with experimental data than those of Brager and Schuchowitzky. The results are compared in table 1. However, it was pointed out by Stratton [5], Su- giyama [7], and Huntington [8], that such realistic potential barriers do not necessarily conserve charge. Even a finite potential step cannot be placed at the position y = L, because such a step leads to an excess of charge in the interior. Thus, the position must be al- tered so that charge is conserved. Sugiyama [7] showed that the Brager and Schuchowitzky energy Es was in fact a poor approximation. He obtained a surface energy one fifth as large. Stratton [5] similarly used a finite step function located at y=-Ha, that is, one somewhat in front of the TABLE 1. Surface energy or due to the change in the kinetic energy of the electrons Surface energy (ergs/cm2) Metal B&S | H&W Stratton | Exp Lithium..............................] ...... 960 350 398 Sodium................................ 777 470 165 190 Potassium............................ 333 208 68 10] Copperº 3720 | 2180 | 800 || || 103 Silver.................................. 2170 || 1400 490 800 Gold................................... 2120 1490 500 580 Zine.........…. 2330 | ...... . ...... 743 Cadmium............................. 1670 | ...... . ...... 630 Mercury.............................. 1550 | ...... 34] 465 geometric surface. The Stratton charge density is similar to that of Sugiyama [7], except that the effect of penetration of a finite barrier is more explicitly in- dicated. For a finite step the relation which will con- serve charges requires that ſº [(x-1)+(2–2) are sin (VW)] Iſla X (10) (i = a- - where a. is the value for an infinite step, i.e., when p → o and A is defined to be (užſk*mar). The quantity a. becomes 37/8kmar. This idea leads to a surface energy of hºax hºax 1607tm” 1607m.” Or [(14–15A) VT-X + (8–24A-15A*) are sin (VA)] (11) agreeing with Sugiyama's calculation for q → Co. How- ever, Stratton's values of the surface energy tend to be smaller than those given by Huang and Wyllie. Stratton [5] found that the contributions to the surface energy are confined to a narrow region which contained a lat- tice boundary or edge. Some of his results are also com- pared in table 1. A similar calculation was carried out by Huntington [8]. His value of or for sodium was very close to that obtained by Stratton. Huntington [8] also calculated the surface energy of sodium using the self-consistent image barrier worked out by Bardeen [9]. Huntington's values appear to be approximately 40 percent of the experimental values. The probable cause for the substantial reduction of the surface energy can be illustrated in the following manner. In figure 3, the density of states for the bulk metal is drawn as a full curve. When the infinite barrier is located at one-half an interatomic distance beyond the last layer of atoms, those states are excluded for which one or more components of vector k are zero. The formula for the density of states remains the same. The number of disallowed states per unit energy is con- stant; because E is proportional k” and the number of points lying inside circles on the three orthogonal (Car- tesian) planes inscribed by the Fermi sphere are also proportional to k”. The excluded states which are 362 | N(E) — —A Z zºl-5 —- Er E. E f N(E) ==} T EFE; --- FIGURE 3a. Curve A is for Density of States Curve for a infinite metal assuming nearly free electrons. The excluded states on the walls of the XY, YZ, and XZ planes in the Fermi sphere are indicated by curve B. The net effect is to increase the Fermi level at 0 K to EF. FIGURE 3b. The dotted curve C is the N(E) curve obtained by ap- proximating barrier penetration by relocating the effective infinite barrier at a and thus increasing the constant of the parabola. The net density of states curve D (crosshatched) is obtained by subtracting B from C. The net effect of the increased number of occupied states is that E' – EF is smaller. shown as shaded area must be added to the top of the band thus increasing kmar. The dashed curve represents the situation with an infinite barrier. Alternatively, the electrons could be accommodated by altering the spacing of the states in k-space and keeping kmar a constant. The alteration of the spacing of allowable states in k-space is accomplished by in- creasing the distance between nodes of the wave func- tion in real space, i.e., by moving the position of the potential barrier. Due to barrier penetration the effective node lies at L'=L-Ha. The consequence of the analysis is that each new k, is reduced by (a/TL). The lowest state which is now included is somewhat lower than in the case of an infinite barrier as indicated on the lower portion of figure 3. However, the density of states is increased because L' exceeds L and hence the density of ex- cluded states remains the same, being constant for all E. In the analysis of Sugiyama, the Fermi level EF is barely shifted and the unoccupied states are accom- modated by the enhanced “width” of the N(E) curve. It is reasonable that the additional energy in the lower portion of figure 3 may be less than on the left-hand side. Huntington [8] in his paper showed that, providing the barrier was consistent at least with regard to charge conversation, i.e., that the height and position of the barrier were such as to conserve charge, then the sur- face energy varied by little more than 10 percent con- sidering a wide range of possible values of height and position. Any barrier which conserves charge gives a good approximation to the surface energy. 6. Electrostatic Energy of the Double Layer A double layer is formed near the interface of a metal because the electrons will penetrate into the “forbid- den” region if the potential is finite at the edge or boun- dary. Because of the distribution of electrons beyond the boundary, a local depletion of electrons will occur just inside the boundary. Thus, a dipole layer will be set up with the negative side outside the boundary, i.e., outside the metal. One may calculate formally the energy of this elec- trostatic double layer inside and beyond the lattice sur- face by means of the following integral: o,-- ſiſ. 000000n-º-'dºd, (2) where y is the coordinate perpendicular to the surface and p is the local charge density. Evaluation of this in- tegral requires a detailed knowledge of p. Stratton [5] used free electron wave functions and found that O-3 = UNe” (13) where v is a universal constant, 5 × 10−8, for all metals. His values for O's amount to a small increase in kinetic energy, namely, to approximately 10 to 20 percent of the major contribution to the surface energy. The small contribution to the surface energy from the double layer at the surface is in agreement with the ob- servation by Bardeen [9] that the surface barrier is due primarily to exchange and correlation forces. Ju- retschke [10], using Slater's method for the potential [11], was able to calculate the variation of the exchange potential along a line perpendicular to the surface. His results showed good agreement with those of Bardeen. 363 If one were to bring a test charge up to the vicinity of the lattice boundary, then its potential will cause an ad- ditional redistribution of electrons in the metal, i.e., it will cause local polarization. The effect would be equivalent to that of a classical image force with essen- tially an electron hole distributed over a region several Angstroms in extent. This, of course, is a further con- tribution to the apparent dipole moment at the inter- face. The photoemission of an electron causes the kind of a localized redistribution of charges which is associated with an exciton. However, the statistical approximation possibly is inadequate beyond the surface because this is a region of low density or of rapidly varying density. Loucks and Cutler [12] calculated the effect of using a screened exchange energy in the Bohm-Pines formalism. They obtained exchange potentials whose shape was very similar to that of Juretschke but whose magnitude varied with the screening parameter. These authors also included a long range correlation term which was independent of position. The exact form of the surface potential has been further investigated and applied by Davies [13] and Gadzuk [14]. At the present, it appears that these calculations of surface potential of a free electron metal are as exact as neces- sary. Gadzuk [14] remarks “. . . unfortunately there is very little experimental data which can be used to evaluate the exact shape and magnitude of the surface barrier.” 7. Metallic Interfaces Recently, a number of pertinent studies have ap- peared in the literature. Stern [15] discussed the problem of surface states in terms of scattering of nearly-free electron waves. He suggested that surface effects might extend into the metal to a depth ap- proaching 50 lattice parameters and that such a skin ef- fect should influence the optical properties of the metal since the penetration depth of electromagnetic waves in the visible range of wavelengths is comparable to this figure. Bennett and Duke [16] studied the interface formed between two metals. This problem has many formal similarities to the problem of a metal-vacuum interface and the analogy grows in accuracy when the electron density in one metal is significantly lower than that in the other. This case has been subsequently studied by many investigators [17-20]. Tunneling from the high into the low density region occurs but there are generally no surface states of either the Tamm or the Shockley type. The “localized” states which Bennett and Duke [17] found on the high density side of the in- terface in the early stages of the calculations tended to disappear during successive iterations and were not found at the stage of achieving self consistency. They observed that there are oscillations in the charge density near the interface which are similar to Friedel oscillations [2]]. Such oscillations are to be contrasted with the simpler dipole layer which occurs near a metal-vacuum interface. In a subsequent paper, Bennett and Duke [22] studied the general effect of in- cluding the spatial dependence of the total exchange and correlation terms in the potential (rather than the bulk values as in their earlier paper) was to increase the size of the depletion region in the low density region. Since these effects arise from the evanescent waves and the tunneling into the low density metallic region, similar effects are to be anticipated for the metal- vacuum case. The depletion region and the charge oscillations were stabilized by exchange-correlation forces and did not disappear on subsequent iterations. Surface states associated with a metal are similar to those one encounters with semiconductors. In the latter case, the levels typically lie in the energy gap of the semiconductors. The metallic surface states will be mixed with the Bloch states in the conduction band. The evanescent charge contributions are the electron- gas analogues of Heine’s “virtual surface states,” for metal-semiconductor interfaces [23]. Metal surface states have recently been predicted and observed ex- perimentally [24-26). Davison and his collaborators Steşlicka, Koutécky, and Cheng [27, 28], at the Univer- sity of Waterloo have studied most of the cases of sur- face and impurity states which are tractable to analytic methods. In general, either a tight binding approach or a molecular orbital approach was used. The effects of deforming a surface layer and the correlation between electrons were also incorporated. Steşlicka [28] has added the additional complication of the relativistic wave functions in the crystal. That is, the typical free electron wave function An exp (iknx) is replaced by its four component, relativistic counterpart. Additional states were found by her. The principal limitation in these researches is that a one-dimensional or linear crystal was assumed. Their concern was in proving the existence of solutions to the problems posed by the interruption of the full transla- tional symmetry of the lattice by either changing (a) the spacing between atoms, (b) the nature of certain atoms, and (c) by truncating the crystal. Beyond the formal ap- proach, they have not attempted to include interelec- tronic effects; however, because of their work, the reduction of the problem to numerical solutions and the 364 efforts to obtain self-consistency in the charge redis- tribution and surface potential will be simplified and facilitated. Sharma and Shrenk [29] recently studied analytically, the emission of electrons from a potential distribution modeled to stimulate the crystal structure of a metal. There are two further simple approaches to calculat- ing surface energy properties. We will take up these be- fore proceeding to discuss the present analysis on potentials and other surface effects. The first involves a modification of the tight-binding method used to cal- culate energy bands; the second, the Lennard-Jones potential. 8. Tight-Binding Method Cyrot-Lackmann [30] has used the tight-binding method to calculate several moments of density of States, M, = | EºN (E) dB (14) and expressed Mn in terms of the nearest neighbor resonance (overlap) integrals. For a simple non- degenerate band, the moments are easily calculated from the resonance integrals and a permutation opera- tor representing the number of ways the nearest- neighbor interactions can be counted. The surface is then readily introduced by its effect on the permuta- tions which can be made. In this way, the density of states with and without a surface can be calculated, and from this the surface energy is obtained. This model is interesting in that it is dependent on the geometry of the ions at the surface and thus enables a comparison of the surface energy of different crystallo- graphic faces to be carried out. Although absolute values of surface energy for different faces of a solid metal are difficult to obtain experimentally, it is possi- ble to obtain the ratios of surface energy of different crystallographic faces. Cyrot-Lackmann has shown her calculations to give good agreement with experimental ratios. An atomic approach to adsorption properties was used by Neustadter and Bacigalupi [31] who assumed that the binding energy of the metal adsorbate was given by a Lennard-Jones 6-12 potential VG)=–4+% (15) at a given site and then summed over the lattice. Their values of absorption energy and surface diffusion ac- tivation energy agreed very well with values obtained experimentally. These authors showed that binding was greatest at certain sites where it would be expected that the maximum numbers of substrate atoms would in- teract with the outermost electrons of the adsorbed atom (or ion). Thus, although these workers did not in- clude any electronic effects and did not investigate the actual nature of the interaction, they were able to esti- mate those properties arising from the geometric structure of the surface. It might be noted that Plummer and Rhodin [32] have presented experi- mental and analytic results for the adsorption of transi- tion metal atoms on transition metal substrates which suggest that nearest neighbor arguments are in- adequate. That is, they were unable to find a consistent set of a and b values for different crystallographic faces of the substrate. “granular” 9. Treatment of the Crystal Potential It has been convenient in the past to deal with a crystal potential and the surface barrier in a very sim- plistic way. Basically two types of model potential have been in vogue: the one-dimensional Kronig-Penney model where one replaces the atom potentials by delta and secondly, the atoms are represented by sinusoidal functions. There are evident advantages to such simple potentials; namely, (a) they lend themselves to analytic solutions, (b) they illustrate many of the features such as surface states, etc., which should also emerge from a full scale calculation, and (c) they do not involve extensive use of computers. Some surface properties, such as the average surface energy, are not so sensitive to the exact geometric structure of the surface and have been treated fairly successfully using a simple model of a surface potential as shown in figure 2. Levine [33] has made an interesting compara- tive analysis of various simple models and studied the accuracy of using a one-dimensional rather than a three-dimensional model. functions where As the greater complexities of a realistic crystal potential are incorporated into the model, the analytic solutions become less tractable. A numerical evalua- tion must be sought. Thus, a more realistic treatment of the potentials near the surface of a metal will be taken up below. Models of surface potential which do not consider the atomic structure at the surface, cannot be expected to give good information relating to effects such as surface diffusion and adsorption which are de- pendent on the arrangement of atoms at the surface. Figure 4 represents the potential along a line perpen- dicular to a metal surface. In region I we have the re- peated atomic potentials of the bulk material, and in re- gion III a zero potential corresponding to free space outside the metal. Region II is the surface region in 365 SCHEMATIC |LLUSTRATION OF THE CRYSTAL POTENTIAL NEAR A FREE M. l SURFACE P0 TENT | A ſ ºf | º º º frºm SURFACE I —GE0METRIC SURFACE REGION | REGION || REGION || FIGURE 4. A representation of surface potentials of a truncated crystal. The finite step associated with a “jellium” model is shown as a dotted line. Potential variations near the center of each atom are indicated by the full line. They are included in this schematic drawing for comparison with the model potentials previously used. The excess distance a needed to conserve charge in the interior of the metal is indicated beyond the last row of atoms. which both the potential and the electronic properties are changing from those of the bulk material to those of the free space region, i.e., are approaching zero. The calculated properties will be highly dependent on the exact model which is chosen to represent region II. The potential behavior pictured in figure 4 is, of course, very simplified. Even for the (100) surfaces of a face- centered cubic metal-like platinum, the potential sur- face resembles an egg carton. Neustadter and Bacigalu- pi had presented potential surfaces for various crystal- lographic orientations of the surface. In effect, the potential barrier shown in figure 5 will be different in shape and magnitude for each point on the surface. Added to this is the fact that LEED studies have shown that there may be a contraction or expansion of lattice spacing in a direction perpendicular to the surface, and they have also suggested that reconstruction of the sur- face layers possibly occurs. 10. Present Model The model used to calculate the potential and charge density at a free surface assumes that the lattice con- sists of free atoms brought together and that the poten- tial at any point is obtained by summing the (over- lapping) potential contributions from the surrounding lattice sites. Using this model, a free surface is represented by removing the contributions from atoms located at lattice sites which lie outside any chosen crystallographic plane. The starting data for these calculations has been the atomic charge densities obtained by Waber using a relativistic Dirac-Slater calculation [34]. The atomic 2. Öğ2. >. ÇKG. §Ilyſsºrſ. ITARTTTTTI R + 1 units. This new unit cell may be considered to be unaffected by the surface in that for all points within the unit cell, the sphere of radius R falls completely within the lattice side of the surface and all atoms con- tributing to the potential are present. Next consider some other particular point (e, f, g) which is nearer the surface, partially together with its sphere of radius R. At some such point, the sphere will lie outside the sur- face and as d becomes more positive, some lattice sites within the sphere will be empty. The potential at the point (e, f,g) will thus be increased. When one continues to increase the Y coordinate at some point (j,k,l) the sphere will be completely empty, i.e., this is a point in free space. This is the way in which the summation is carried out by the computer. The X Y Z coordinates are arranged so that the XY planes contain the origin of a unit cell. At the start |d is set equal to R + 1. A point(xyz) in the ini- tial unit cell, together with its set of vectors {l,m,n} to atomic sites, and the set of distances {r} are read. The distance rimm is multiplied by the lattice parameter and the potential contribution at that point Vinn is calcu- lated by interpolation of the values of atomic potential. The charge density pinn is also calculated. This procedure is repeated for all (lmn) values and the potential and charge density contribution are separate- ly summed, giving the Coulomb potential Wºy, at that point and the total charge density p'yº at that point. A table of values of l,m,n, r, Vimn,pinn is maintained. The exchange potential contribution is then evaluated using Slater's original free electron approximation [11], v.0)=–6(; p(0)." giving the total potential Wrye at this point. The value of d is then changed to d'-d--2, which is equivalent to moving to the next unit cell. At this position, the list of m’s is scanned for those values for which m > -- d. These represent empty lattice sites and the potential contribution Winn is subtracted from Vºys [also, pinn is subtracted from pººl. When all m values have been tested, the remaining potential Wºº (d") is added to the new exchange potential obtained from p" (d") to give Vryº (d"), the potential of the point (xyz) in the unit cell where the origin is at a distance d' from the surface. This process is repeated until d'=(R+1), i.e., we in- clude the potential outside the last row of ion cores for a distance equivalent to that inside. For unit cells which (17) have lost potential contributions due to the surface, the potential distribution no longer has full cubic symmetry and thus the potential at the point (xyz) will no longer be equal to the potential at the 48 equivalent points. Thus corresponding to each point (xyz) the potential. must be calculated separately for those equivalent points which are not produced by a rotation around Y. The procedure is thus to calculate Vrije for each value of d and then repeat this for each equivalent position before proceeding to the next (xyz) value. 11. Convergence of Potential Summation As was discussed in the method section, the radius R of the sphere which contains atoms contributing potential to any point was arbitrarily given some value and the infinite summation truncated after the same term at any point (xyz). In order to do this, we must determine whether the infinite summation converges rapidly or slowly and if so what error is to be expected by stopping the summation at any given point or R value. Let us consider a sphere of radius R (as before we as- sume the side of our cell equal to 2 units) and calculate the number of lattice points n contained within it: n(r) –º R3 (...) –# R3 for a FCC lattice (18) then dn(R) 2 dR = 27R (19) If the potential at a distance R due to a single atom is V(R), then the total potential V = | (2+R2)W(R)dR. (20) In order for this integral to be finite, the potential V(R) must decrease at a rate faster than R-4. Figure 6 shows that the atomic potential V(R) for platinum for high 4.0 IV(r)| x 10" 2,O 3O J4- 39 n (Bohr Units) FIGURE 6. Variation of atomic potential for platinum at distances far from the nucleus. 367 TABLE 2. Convergence of the summation W(xyz) for a sphere lying entirely in the lattice -- No. of atoms Potential Sphere within Potential increment % radius sphere from R = 5 5 286 — 2.26849 | . . . . . . 7 736 — 2.28695 0.815 9 1566 — 2.28879 .895 || 2796 — 2.2894] .92 values of R. It can be seen that the potential is decaying at a rate much faster than RT4 and, therefore, it may be assumed that the potential summation is bounded. In order to determine the error introduced by truncation the summation at some arbitrary value of R, the value of the potential at one point in the unit cell of platinum was calculated using different values of R. The point at which the potential was calculated was in the center of the foc unit cell where the nearest neighbor atoms con- tribute most to the potential and where the error in- troduced by series truncation would be expected to TABLE 3. Effect of Increasing Radius R on the Summation of the Potential V(di) have the highest relative value. The results of the calcu- lations are presented in table 2. The question of whether there were sufficient points in the sphere of radius R to assure convergence of the potential was investigated for a point di well beyond the surface, that is, where di > R units. The filled lattice sites will lie only in a “cap” of the sphere. The data in table 3 illustrates the effect of increasing R the number of units of ao/2. The number of atoms counted with R = 5 were 155, 74, 17, and 0 when di was 0, 2, 4, and 6 respectively. These values are underlined. The other entries in column three of table 3 are the number of ad- ditional points counted when R was increased from 5 to 8. For di = 0, the potential V(di) was multipled by 10, for di = 2, by 10*, for di = 4, by 107 and for di = 6 by 10°. It will be seen that for di < 4, the percentage changes are small. Although a large percentage change is obtained for a point at di = 6(ao/2), the magnitude of the change is trivially small. It is considered that the level of accuracy gained by using R = 5 was sufficient at least as a starting point. From the point of view of the computer calculations, the time used for calculating a potential increases with the number of neighboring atoms considered, i.e., in- creases as R*. Thus, from this standpoint very little in- crease in accuracy is gained by increasing R and a good deal of computer time is consumed. 12. Variation of the Potential Perpendicular to the Surface Figures 7, 8, and 9 show the calculated potential dis- tribution along lines perpendicular to a (010) surface of platinum. The cross section of the unit cell drawn on each figure shows the position of each line with respect to the surface atoms. The great variation of the shape Distance Potential (no from sur- || Radius |Number exchange contribu- face di R in of atoms tion) units of units of counted a0/2 a0/2 V(y) Multiplier () 5 155 | –2.805371 101 0 6 104. — 2.80537] 0 7 149 — 2.80537] 0 8 164 — 2.80537] 2 5 74 || –9.398146 104 2 6 76 — 9.398379 2 7 109 — 9.3984.19 2 8 120 — 9.398427 4. 5 17 | –2.362965 107 4. 6 36 — 2.467100 4. 7 6] — 2.491470 4. 8 88 — 2.498.100 6 5 0 0 109 6 6 () 0 6 7 25 – 1.024532 6 8 40 | – 1.323857 X SURFACE | | | | I | | -1 | ‘g | Cº. | –2 | .9 | +” C | (1) -3 & | 0– | | -4 <— Go —3. | FIGURE 7. The potential distribution along a line perpendicular to a [010| surface of platinum. The line passes through atom centers. 368 SURFACE -] Tº, Cº. –2 E --> C (1) º -3 ſ. <– C —- –4 O | | | | | l | | | | | | | | | | | | | I f | | } t l i | | I FIGURE 8. Potential along a line perpendicular to a (010) surface of platinum. The line passes near to the atom centers. of the potential along different lines is apparent. Figure 7 shows the potential on a line passing through atom centers which shows the periodic behavior expected. Figure 8 is along a line which passes close to but not through the nucleus (atom centers). Here the periodic behavior can be seen. The fact that the potential far from the atom centers is very nearly flat can also be seen very clearly. In figure 9 the potential along a [010] ray in between atomic centers is illustrated. There is a small ripple about the mean “muffin-tin” potential Vºn as each atom is “sliced.” The muffin-tin value is inde- pendent of the distance from the surface. The “step” due to charge overlap and penetration which was in- dicated in the schematic drawing of figure 4 is seen in the curves of figures 7, 8, and 9, which were redrawn from the computer generated curves. The 3-dimen- sional drawing in figure 5 was obtained by combining a number of figures such as these last three figures. SURFACE | | | | | | | | | j | | | | | | | –2 i –3 | I | | –4 POTENTIAL FOR OXYGEN AIOM ADSURBED ON PLATINUM LAST ROW OF ATOMS N. & , POTENTIAL Aſ Wººs ed;&Yº ~. ~ FIGURE 10. Three dimensional plot of the potential variation on a (001) plane through the centers of atoms. The local depression near the oxygen atom would facilitate electron tunneling. 13. Calculation for can Adsorbable Atom The previous section described a method for deter- mining the potential distribution at a perfect crystallo- graphic surface. This method can easily be adapted to determine the potential distribution around a surface defect. Once the perfect surface potential distribution has been calculated and stored, the effect of a defect such as, for example, a vacancy in the surface layer, can be determined by subtracting the potential con- tribution due to this atom (or ion) from our crystal potential distribution. The effect of a foreign ion or atom on or above the surface can similarly be deter- mined by adding the additional contributions to the ex- isting potential. The results are shown in figure 10 for an oxygen atom approaching the (010) surface of platinum. 14. Surface Vacancies and Migration Figure 11 shows equipotential contours on a (100) surface for platinum for the viewing plane cutting through the centers of the ion cores in the last plane. One atom has been removed from its position in this plane and the resulting distortion of the potential can be seen. It should be noted that the equipotential lines around the vacancy and the ions reflect the four-fold rotational symmetry of the lattice in this plane. Figure 12 shows the same surface but constructed by remov- ing two adjacent atoms removed and replacing one atom in saddle point halfway between the two vacan- cies. This is a crucial stage during vacancy migration. The results obtained for the potential distributions along lines perpendicular to a (100) surface are interest- ing in that they point out the great difference in poten- FIGURE 9. Potential along a line perpendicular to a (010) surface of platinum. Line does not pass near to the atom centers. 417–156 O - 71 - 25 369 EQUIPOTENTIALs in A PLANE ABOVE THE { loo). FACE OF PLATINUM WiTH A SINGLE VACANCY DEFECT ( —; -1.5 Ry, ---, -1.75 Ry; -----, -2.0 Ry, “... -5.0 Ry) FIGURE 11. Equipotential lines on a (010) face of platinum with a single vacancy defect. tial along different lines. In particular, along lines which do not pass close to the atom centers there is only a small oscillation in the potential. Thus the poten- tial along such a ray is similar to that employed in the free electron models of surface behavior. 15. Previous Treatments of Adsorption Potenticals The various approaches described above all had the goal of predicting the properties of a clean metal sur- face in contact with vacuum. Other theoretical work has been concerned with the investigation of the effects of a foreign ion or atom approaching the surface and being adsorbed. Bennett and Falicov [35] and Gadzuk [36] have examined the problem of an alkali atom in- teracting with the surface of a free electron metal, using perturbation techniques. The effect of the in- teraction is to shift the level of the outer electron of the alkali metal. Grimley [37] has shown that the charge density in the metal is disturbed in a long range oscillatory fashion by the presence of the adsorbate atom. The analysis he used was similar to that which leads to the Friedel oscil- lations. These three references are illustrative but do not represent the large number of approximate studies made in the past. EQUIPOTENTIALS IN A PLANE ABOVE THE HOO} FACE OF Pt. WITH A VACANCY-INTERSTITIAL-VACANCY DEFECT (—: -1.5 Ry, ---, -1.75 Ry; -----, -2O Ry; “. :-5.O Ry) FIGURE 12. Equipotential lines for a (010) face of platinum. Two neighboring atoms have been removed and one atom replaced at the saddle point between the two sites, to simulate surface migration. 16. Variation of Total Charge Density Figure 13 shows the electronic charge distribution for Pt, obtained by summation of charge contributions from neighboring atoms within a sphere of radius 5/2 a0. The density (full line) is plotted along a line in the [100] direction connecting two atoms in the lattice. For comparison, the face atom charge density is also plotted. Close to an atom site the metal charge density is very little different from the free atom charge densi- ty, the overwhelming contribution coming from the atom at the site. However, as we move away from the actual lattice sites the contribution of the neighboring atoms become more and more important. It can be seen that for distances greater than ~1 Bohr unit (B.U.) from any lattice site there is an increasing discrepancy between the crystal atom and that of the free atom den- sities. Note that the plot is made on semilogarithmic scale and the overlapped charge density at the midpoint between atom centers is 6 times as large as the free atom value. This position has an octahedral arrange- ment of atoms surrounding it. The accumulation of charge on the surface beyond the last atom is due to the Coulomb tails of the free atom. That is no redistribution of charge due to the presence of this metal-vacuum has been taken into account. 370 *ºff SURFACE ATOM f sº \9°º .2c2 & cº. T EF EF & p : P! p? P! (d) (b) FIGURE la. Density of state distributions for a magnetic impurity. The monotonic parabolic curve is a free electron metal density of states. b. Density of states for an electropositive adsorbate. The area in the cross hatched area is the amount of electron charge on the adsorbate. 375 f te $e Gj dE <--- e 4A E dj dE (b) FIGURE 2a. Model potential and total energy distribution for field emission from a metal. b. Model potential and total energy dis- tribution for resonance tunneling field emission from a metal with a narrow band adsorbate. A similar picture has been adopted for describing the chemisorption of electropositive atoms on metal sur- faces [6-12]. Obviously each theoretical approach has its own set of approximations to achieve a (hopefully) self-consistent solution to a model problem. Of particu- lar interest have been the instances when the surface impurity was either an alkali or alkaline earth atom. In these instances, the ionization potential of the atom is sufficiently small so that both the spin up and down levels lie above the Fermi energy as shown in figure lb. The occupation of the lower lying spin level n-1, (where n > n_) is small enough so that the question of the magnetic state of the impurity determined by n -n_ has not been considered at very great length. In a similar manner to the nonintegral Bohr magnetons, we see how it is possible to have a fractionally charged ion for the surface impurity. Obviously if we know the virtual sur- face impurity density of states, then the charge on the impurity, among other things, would be immediately deduced. Knowledge of the density of states would then permit theoretical calculations of the dipole moment as- sociated with the metal-impurity complex (and thus work function changes) [8] and chemisorption or bind- ing energies [8,11,12]. Duke and Alferieff have shown, through model calcu- lations, how the presence of a potential well (impurity) could give rise to altered current-voltage charac- teristics in field-emission resonance tunneling experi- ments [13]. In the language of field-emission workers, the analog of conductance versus voltage curves in junction tunneling is called the total energy distribution (TED) [14,15]. The measured quantity is the incremen- tal change in current per change in energy and is shown in figure 2a. As Duke and Alferieff pointed out, a poten- tial well can give rise to resonance effects which manifest themselves as structure in the TED as shown in figure 2b. We have presented an alternate theory and experimental evidence of resonance tunneling in which the virtual impurity level density of states is of key im- portance in our interpretation [16-18]. It turns out that the resonance aspect of the theory is equivalent to Ap- pelbaum's spin flip tunneling theory [19,20). The structure of the paper is as follows. In section 2, the main equations in the theory of surface impurities and resonance tunneling are established. It is shown how experimentally determined TED’s are related to the impurity density of states. Experimental results for Zr, Ba, Ca, and Ge on a tungsten host are presented in section 3 and the impurity density of states is thus determined. 2. Theory The Anderson Hamiltonian describing the coupling of a surface impurity single electron orbital to the con- tinuum of conduction band states is H - X. 6 kshlks + X. 6tts Il (ts + X. (Vaſcºst cast H.C.) k's S k's + Ueffman na– (1) where we have adopted the usual notation [2-4,11]. The atom-metal hopping integral Vak has been discussed in detail for particular model surface poten- tials by several people [6-12]. We write the intra- atomic Coulomb integral U → Ueſſ to note that when dealing with alkali impurities or alkaline earth impuri- ties in the single electron approximation [18], we should really be working in the low density approxima- tion of Schrieffer and Mattis [3] in which U is replaced by some Uaſſ [3]. In fact, in the present analysis of a low density impurity, since both nai, and na– are much less than unity, Ueſſ na na is neglected here relative to the other terms with the subsidiary provision that the resulting single electron impurity density of states, in- tegrated over all energies, contains only one electron. With this approximation, eq (1) can be subjected to a standard Green’s function analysis in which the virtual impurity Green’s function is |Vak|. — 1 Gaa (e) = |--|-> :* k = [e - e.g. + i Asgne]" (2) 376 with A = T X 6(e– e.)|Wall” and ea = ea + Ae where Ae is whatever"shift the real part of the impurity energy level experiences [8,18]. An impurity density of states is identified as – tº c -- * pm (e)=. Im Gn(e)=. (e– el)” + A* (3) Note that the total electronic charge on the impurity is () | 6 d. (na) =| pan (e) de = + Cot-' (*) JC T A Here eq, is measured from a zero at the Fermi level and is given by de — Vi + AE with do- metal work function and V, the first ionization potential of the isolated impu- rity atom. Typically for alkali and alkaline earth atoms el, P A so (na) < .25 and the low density approximation seems valid. We will now see how ed, and A, and con- sequently the impurity density of states, are deter- mined from resonance tunneling measurements. We have noted that the TED in field emission is the analog of a conductance versus voltage measurement in junction tunneling. The TED is given by • f –", "ſeem Jo T.J. T.; with Jo and d some constants depending upon system parameters and applied field but not energy and f(e) the Fermi function [14,15]. The idealized TED is shown in figure 2a for zero temperature. Usually .1 s d S.2 eV in field emission experiments so the width of the TED is restricted by the exponential decay to something less than 1 eV due to instrumental limitations of presently existing energy analyzers [17.22]. Since we are dealing with a tunneling problem, in the transfer Hamiltonian formulation, jo' is also proportional to the square of an appropriate tunneling matrix element = Tº-j [15.2.1]. If an impurity is on the surface, the TED is altered as shown in figure 2b. Duke and Alferieff chose to charac- terize this enhanced tunneling by a factor multiplying the original TED [13]. Thus in the presence of an impu- rity j'(e) = R(e);o'(e) hereby defining R(e) the enhance- ment factor. As with jo' (e), j'(e) is proportional to the square of a resonance tunneling matrix element Tm f.so R(e)=|Tin-ji’ſ Tº-j |*. A simple perturbation expansion allowing for the possibility of an intermediate virtual impurity state for the tunneling electron is Tin-f- T} ,-- Tm–10aaTA-f where Tm-a is the tunneling amplitude for going from the metal to the atom, Ta-f is the amplitude to go from the atom to free space and Gan is the impurity Green’s function given by eq (2). If all the amplitudes are real, then Tina Tar).” , , , Y ~! # *) (Gna (e) × G. (e)) mf Tma Taf TO mf + (Gan (e) + G. (e)). (4) (((( Since the presence of the atom effectively cuts a hole out of the tunneling barrier, the tunneling amplitude through the atom Tina Taf should be greater than the direct channel amplitude Tºf . In fact we have shown that the quantity }} | {{ i (tf & h? A. T(e)={ Taſe (e) hº k. e lºw- ( 5) () & })\f 4, 2m w with k = (2m/h?(de – e))” and w an effective well radius for the model atom potential. The function g(e)=1 if the intermediate state is an s state whereas g(e) < 1 for in- termediate d states due to the spatial contraction and thus reduced tunneling probabilities for d states degenerate with s states [18,23]. We also should note that eq (4) is identical with Appelbaum's expansion although he implicitly drops the second term on the right hand side, the direct resonance channel and con- centrates on the interference channel, the third term [19]. As he was investigating a Kondo type mechanism this was the appropriate procedure. On the other hand, for our purposes the “uninteresting” direct term is the most significant. If we neglect the unity term on the right in eq (4) and combine eqs (2-5), the enhancement factor is then R (e) = pad (e; F) Fºr 12 (e-so)T(e) | (6) A A Thus we see how an experimental determination of R can be used to obtain the virtual impurity state density of states. The situation is slightly more complicated when there is the possibility of observing excited states also. This has been discussed at some length together with methods for approximately treating two electron states [17,18]. Note that we have written paa also as a function of F, the applied electric field. The field shifts the impurity level downward such that ea is replaced by ea — eR's with s the distance of the impurity center from the surface. The field thus brings the virtual levels below the Fermi level and consequently allows these levels to be observed in a resonance tunneling experi- ment. However, all results will be extrapolated back to the zero field limit. - Finally we note that the decomposition of the enhancement factor into a product of a density of states times another function of energy, R(e)= pad (e)G(e) as we have done in eq (6), is useful for the following reason. 377 Any sharp structure in R(e) will appear through the den- sity of states factor. The factor in brackets in eq (6), G(e), is a smoothly varying monotonic structure. Thus an interpretation of the structure in an enhancement factor is almost identical with determining the impurity density of states. 3. Results It is our feeling that junction and field-emission tun- neling measurements can and should serve to comple- ment each other as each technique has strengths where the other technique is weak. In particular the ad- vantage of field-emission measurements is that one is dealing with atomically perfect surfaces. It is possible to look at emission from a patch of say 25 atoms and then add a single impurity atom and monitor the changes in the TED due to the presence of one impurity. The experimental procedure has been out- lined in great depth elsewhere [17,24]. 3. l. Zirconium The ionization potential of an isolated zirconium atom in the 5s” ground state is 6.84 eV. It is expected that such a level would shift upwards 1 to 2 eV when in- teracting with a tungsten surface [8,18]. Analyzing the experimental enhancement factors in terms of eq (6), it is found that the energy parameters in the virtual 5s” density of states are AE = 1.6 eV and A - 1.0 eV, in agreement with expectations. No excited states were observed. 3.2. Bcurium The 6s2 ground state of free barium lies 5.2 eV below the vacuum level. Triplet 6s.jd *D excited states lie between — 4.03 and — 4.1 eV and a singlet 6s5a "D lies at — 3.8 eV. With the applied field, the 6sbd states should be pulled below the tungsten Fermi level and thus be observable. We would expect the width of the 6s5d state to be about 1/10 of the 6s” state and the shift AE to be smaller [18]. An expression for the enhance- ment factor for two electron excited states, similar in principle to eq (6), has been derived [18]. A com- parison between a typical experimental R factor and the theoretical curve is shown in figure 3. The energy zero is taken with respect to the vacuum. As can be seen, structure related to excited states is both observa- ble and predictable. The spectroscopic findings for use in the density of states are: AE* = .95 eV, Ags = .75 eV and AE”= AE" = 0, A3, - A 1, =.1 eV; again in agree- ment with expectations. E (eV) (ZERO-FIELD EXTRAPOLATION) – 5.4 – 5.2 -5.0 — 4.8 – 4.6 – 4.4 –4.2 – 4.O -3.8 –3.6 –3.4 –3.2 | I I- | I I | I | I | I.T I [ I | I | I | | - |S |Eºs || |'D - of H - H = - - >— |.5 H. - Or. - - <ſ Or E [. - CD |_ Cr. – F - CD - #| || LL) à - -5.OH –5O –4.5H I |45 | - r—- r—- DISTANCE FROM SURFACE FIGURE 4. Pictorial representation of the broadening and shifting of the energy levels of Ba and Ca as they interact with the surface. The shapes and position of the virtual levels at the surface are taken from the data on the low work function planes of tungsten. | |-- | I I | | | | | T s|_ – LAYER OF GeONW 2 .6H- == ..) X- 0. *- <ſ Cr E- É .4 H <[ #5|Š .2 H. O –2.O —|.5 —I.O —O.5 O & (eV) (RELATIVE TO FERMI LEVEL) FIGURE 5. Experimental total energy distribution of a system con- sisting of approximately one monolayer of germanium deposited on tungsten. ported on resonance tunneling through Ge films from a fraction of a monolayer up to tens of layers thick [25]. A characteristic TED for an approximately monolayer film is shown in figure 5. The peak near the Fermi level is associated with pseudo-clean tungsten as the con- tinuation shown by the dashed line suggests. The in- teresting feature is the lower lying peak. We believe that this peak may be related to tunneling through the 4S4p” “level” which ultimately would become the valence “band” in amorphous or crystalline Ge. We would expect the peak height to lie somewhere between the 7.88 eV ionization potential of the isolated atom and the 4.76 eV work function or position of the top of the valence band of solid Ge which it does. If the TED in figure 5 is decomposed into an R factor, the full width at half-maximum of R is ~ 1.5 eV, a not unreasonable value. Since the theory of dense, interacting impurities on the surface is incomplete at this time, it is not possi- ble to precisely analyze these monolayer results in terms of a density of states as in eq (6) and thus this width is not to be identified with A, the single impurity width. Further studies have been made in which the layers are thicker, hopefully approaching the stage where the properties of the film can be discussed in terms of bulk “amorphous” germanium. The evolution of the Ge peak in figure 5, as the thickness increases, is not inconsistent with the interpretation given here in which we claim to see early stages of a rudimentary valence band. These studies will be reported on in more detail in the future. 4. References [1] Friedel, J., J. Phys. Radium 19, 573 (1958); Suppl. Nuovo Ci- mento VII, 287 (1958). [2] Anderson, P. W., Phys. Rev. 124, 41 (1961); in Many-Body Physics, C. DeWitt and R. Balian, Editors (Gordon and Breach Publishers, New York, 1969). [3] Schrieffer, J. R., and Mattis, D. C., Phys. Rev. 140, A1412 (1965). [4] Kjöllerström, B., Scalapino, D. J., and Schrieffer, J. R., Phys. Rev. 148, 665 (1966). [5] Koster, G. F., and Slater, J. C., Phys. Rev. 95, 1167 (1954); 96, 1208 (1954). [6] Gomer, R., and Swanson, L. W., J. Chem. Phys. 38, 1613 (1963). [7] Bennett, A. J., and Falicov, L. M., Phys. Rev. 151, 512 (1966). [8] Gadzuk, J. W., Surface Sci. 6, 133 (1967); 6, 159 (1967); Proceedings Fourth International Materials Symposium, The Structure and Chemistry of Solid Surfaces, (J. Wiley Publish- ing Co., New York, 1969); J. Phys. Chem. Solids 30, 2307 (1969). [9] Schmidt, L., and Gomer, R., J. Chem. Phys. 45, 1605 (1966). [10] Grimley, T. B., and Walker, S. M., Surface Sci. 14, 395 (1969). [11] Newns, D. M., Phys. Rev. 178, 1123 (1969). [12] Gadzuk, J. W., Hartman, K., and Rhodin, T. N. (in preparation). [13] Duke, C. B., and Alferieff, M. E., J. Chem. Phys. 46,923 (1967). [14] Young, R. D., Phys. Rev. 113, 110 (1959). [15] Gadzuk, J. W., Surface Sci. 15,466 (1969). [16] Plummer, E. W., Gadzuk, J. W., and Young, R. D., Solid State Comm. 7,487 (1969). [17] Plummer, E. W., and Young, R. D., Phys. Rev. (in press). [18] Gadzuk, J. W., Phys. Rev. (in press). [19] Appelbaum, J., Phys. Rev. Letters 17, 91 (1966); Phys. Rev. 154,633 (1967). [20] Anderson, P. W., Phys. Rev. Letters 17,95 (1966). [21] Duke, C. B., Tunneling in Solids, (Academic Press, New York, 1969). [22] Young, R. D., and Kuyatt, C. E., Rev. Sci. Instr. 39, 1477 (1968). [23] Gadzuk, J. W., Phys. Rev. 182, 416 (1969). [24] Clark, H. E., and Young, R. D., Surface Sci. 12,385 (1968). [25] Clark, H. E., and Plummer, E. W., 16th Field Emission Sym- posium, Pittsburgh, Pennsylvania, September 1969. 379 What Properties Should the Density of States Have in Order That the System Undergoes a Phase Transition? P. H. E. Meijer National Bureau of Standards, Washington, D.C. 20234 and The Catholic University of America, Washington, D.C. 2001 7 We have solved nothing that needs to be solved regarding the density of states problem in a system that undergoes a phase transition. We, however, raise the question of what conditions the density of state function of the total system (not the temperature dependent quasiparticle spectrum) should have in order that we are observing a phase transition. In general, one needs an extremely strong increase in the density — how strong can only be illustrated by using models. From these models we tried to obtain the density of states using the inverse Laplace transform. The results reveal that a vertical slope seems to be a necessary condition. There are reasons to believe that the condition is not sufficient. Key words: Density of states; Ising model; Onsager model; partition function; phase transition; Weiss model. 1. Introduction In order to obtain the physical properties of a large system, one has to calculate the partition function and this is in general preceded by the determination of the density of states. Let us take a discrete system: the density of states is a weight factor w/E) indicating how many ways an energy E can be realized. If this factor is known, the partition function is found by summation over the values of E: Z(p)=X w(E) exp (-BE) (1.1) where 8 = 1/kT [1]. Although such a summation may not be feasible analytically, this obstacle is much less of a stumbling block than the first requirement, the determination of w(E). It is of course possible to evaluate the general expres- sion for the partition function directly. One sums over the microscopic variables, such as the spin variable(s) of each individual spin and calculates the value of E for each set of values of the spin variables. Since each value of E carries a different Boltzmann factor, this in- troduces a constraint on the summation. If one tries to eliminate this restriction by introducing a delta func- tion or similar device, one arrives again at the density of states: w(E)) =X. - gº e X. ô (E (s tº º ºr Sn) – Eo) (1.2) 8 in as an intermediate step to obtain the partition function. For the sake of completeness, we give also the con- tinuous description. The partition function can be writ- ten as: Z (8) = ſ exp [-6E (p, q)|dpdq (1.3) OT a S Z (3) = ſ exp (-6E)p(E) de (1.4) Here p,q stands for p .... p3N, q1.... q8x and p(E) for the density of states. One can consider p(E) also as a Jacobian, the variables p,q are replaced by O., .... where 01 = E and all other o’s arbitrary but independ- ent. The density of states is given by CY6N, . . do.6A • p(E)-ſ/0. Q: CY 1 . . . C. V. ) do.2 - or one can introduce the accumulated density of states Q(E), also called “volume in phase space,” given by E aſp)=| dp . . . da ( 381 which leads directly to the density d() ji=p(E) The function Q(E) is, in the discrete language, the total number of states with energies less than a given value. The phase volume integral can be written with a step function OC () (Eo) = | 6(E(p q) - Eo) dpdq () and the density function in a similar way p (EO) = | 6(E(0-0)-E)dpda ( Such an artifact is useless in itself, but one may in- troduce transforms of these functions. The main question is, what properties should p(E) have so that the system undergoes a phase transition? Since it is impossible to obtain p(E) for almost any N- body system, we try to extract same information from the Z(8), which is in some cases approximately known. We observe that eq (1.4) can be considered as a Laplace transform executed on p(E). Hence we can ob- tain p(E) from Z(8) by an inverse Laplace transform p(E)=ſ. Zºeºde (1.5) This idea is not new. The use of the partition function to reconstruct the density of states was used as early as 1911 [2]. 2. Quasiparticle Description, the Three Level Model In the quasiparticle description, the individual spins are described by a density of states that depends on the temperature wiſe,8). The subscript i stands for the site, SPHT ~X = 0.5 .6 X = O. l; .5 ..! .3 .2 ... 1 O i .2 3 l! 6 7 .8 9 1 - O —-º- t FIGURE 1. Specific heat of a three level system. The temperature is measured in units AE = EA – E. : t = kT/AE. The relative location of the middle level E, is given by X = (E2 = E1)|AE. 382 or if the system is homogeneous, for “individual.” For one can try to deduce the behavior of the density of a spin 1/2 system there are only one or two values of E states. A is the amplitude or residue of the singularity for which this quantity is non-zero. In the first case, the and f(T) is a regular function. Using Weiss theory above the critical temperature, we have sº , 6 wi-2 and the system shows no spontaneous magnetiza- Crſk = 8*. ln Z we e e 68% tion; in the second case, the Weiss theory below the critical temperature, each wi- 1 and the system is or- dered. At the critical temperature, the levels move apart and this causes a peak in the specific heat. This motion of the level or levels with the temperature is a Ó R A * highly unsatisfactory description since the level struc- àK ln z=-| (Hºº-º-º: (b)a. ture should be given once and for all, independent of the temperature. Moreover, a very fine splitting between two levels does not give rise to a specific heat peak unless the level is situated at the bottom of the spectrum. This can be easily illustrated by a three level We try to determine ln Z by twice integrating the specific heat where K = 8.J., is a dimensionless inverted temperature. We know that U(K = 0) is regular, hence the singular behavior near K = 0 in the first and the second term must cancel. Around Kc we can replace the integral by scheme. If three levels are given by E1 = 0, E2 = X A E and E3 ô ln Z A A ^ =AE, (0s Xs 1), then for X = 0 one has a peak and a --ºr-- () (Hº- ()) Schottky curve, for X = 0.1 two maxima and for all other values a single Schottky-like curve. The case X = 1 does not give a peak (fig. 1). Hence the quasiparticle level structure is of no use to find out what is the typical behavior of p(E). wherefk) is K*f(K) minus the singularity at K = 0. ô ln Z 1 A K-K-r-ſºa. ln Z ok To Kg 3. Density Structure from the Accepted 1 A — rv -- “2 Theories To (a - 1) K3 |K –K, -*** +g(K) Assuming the generally accepted form of singularity Z = ex (ºn A K — Ko ~) • K [3] in Cv p(x,-i, K-Kºlº) exp [g(K)] Cº-H4++f(t) ** Compare this with eq (1.5), it leads to the inverse T. Laplace transform - i oc + iy - l_–4. - — a -i- 2 y” - p (e) –ſ.exe lº l) Ká |K – K, + g(K) +ke|ake-EU) This integral cannot be accomplished with the undeter- p(E) = ((1–E)'ſ tºp (1 + E)'ſ 25), mined function g(K), since the integrand is only known near Ko. In order to perform the line integral the value of the integrand must be known at some line or curve parallel to the imaginary K-axis. There is an additional e - & handicap. The previous integrations assumed that the A similar calculation Call be made for H # 0. first and second integrals were continuous function in For the Weiss Model we find the point Tc, which does not have to be the case. Hence (p(E))2/A = e2 in 2 (1 + VET)=(1+ \Pſ) (1– VET)-(1-> FI) this approach is not leading to much information about where E is in units We and ranges from — 1 to +1. The curve has a peak for E = 0, which becomes more pronounced for larger N. the characteristic properties of the density of states. where E1 = 1 – 2E. E is in units Nez, where z is the number of nearest neighbors and e the absolute value 4. Density of States of the Solvable Models of the negative interaction energy. E is measured from the ground state. For E=0 the degeneracy is 1, it goes The One Dimensional Ising Model. The density of up to 2N at E = 1. The slope at E = 0 is infinite and at E states is = 1 is zero, using the quantity p”/" as vertical axis (fig. 383 FIGURE 2, Effective density of states for the Weiss model. 2). If one plots p it is “infinite” everywhere. There is not such a thing as a steep increase in the density of states for E → T. For the Onsager Model. In order to obtain a solution we use the method of steepest descent. This can be done easiest in thermodynamic terms ſe-6 (6)+Bºdø = e -8.50)-be, where I is a Gaussian integral in terms of (8s – 3)*. The value of 8s is determined by E = U(8.) Hence the integral is proportional to the exp (NS/k) and p(E) = e^S/k To find p(E) one has to convert S(T) into S(E). This needs the inversion E = E(T) into T = T(E). Since the first goes like x lnx, the second cannot be found. Hence the conclusion is that although the density undergoes a very sharp (vertical) increase near Ee, the value at this point cannot be determined. It is possible that one finds superposed on the sharp increase a delta-function-like spike. 5. First Order Phase Transitions We like to consider briefly the first order transitions. Characteristic for a first order transition is the presence of another parameter X, such as the chemical potential or the external field. The density of states will depend on this parameter: p = p(E,X). The entropy un- dergoes a jump when X is varied at a temperature below the critical temperature. This discontinuity can be calculated from the Clausius Clapeyron equations: dP_AS dT AX 6. Summary In this note we raise the question as to what the con- ditions are to obtain a system with a phase transition on the density of states. A phase transition is observed by a singularity in the specific heat or by the appearance of a magnetization (or a similar form of condensation) in the absence of a field (or similar order-inducing ther- modynamic variable). Although it is easy to trace in a quasiparticle descrip- tion how such a singularity comes about, the general features are much harder to establish in terms of the proper many-body theory. An attempt is made to see what is at least required, using the general features of the scaling laws and the well known models that can be calculated. The basis of the consideration stems from the almost obvious relation between the underlying density of states on one hand and the Laplace trans- form on the other hand. 7. References [1] Tolman, R. C., The Principles of Statistical Mechanics, Oxford (1963). [2] See P. Ehrenfest, Ann. D. Phys., 36, 91 (1911). The problem was independently considered by H. Poincare, J. de Physique 2, 1 (1912). He first used a Laplace transform. He was criticized by Fowler, Proc. Roy. Soc. London A99, 462 (1921), for applying the transform to a discrete spectrum. All three papers deal with the density of states underlying the black body radiation. [3] The scaling laws are reviewed by L. P. Kaganoff, et al., Rev. Mod. Phys. 39, 395 (1967). [4] All models (1, 2 Dim. Ising, Weiss and Bethe) can be found in K. Huang, Statistical Mechanics, Wiley. [5] Glass, S.J., and Klein, M. J., Investigations on the Third Law of Thermodynamics, Case Institute of Technology, AEC Techni- cal Report No. 2 (August 1958). 384 On Deriving Density of States Information from Chemical Bond Considerations F. L. Carter U.S. Naval Research Laboratory, Washington, D.C. 20390 The chemical picture of bond formation between neighboring atoms in crystalline solids can give valuable electronic density of states information including the rough shape and relative filling of bands. In addition, from the representation of the chemical bond in momentum space one can readily predict the distortion of the Fermi surface from sphericity. This latter approach appears to provide an alternate explanation of the apparent attraction of the Fermi surface to the Brillouin zone faces. The first relationship is best demonstrated in cases where relatively unique schemes of bond for- mation can be devised. This is possible in many intermetallic compounds having high coordination by the use of orthogonal sets of bidirectional orbitals; their use leads to multicenter bonds or cycles which are approximately orthogonal. Via the Fourier transform, the series of Slater determinants (representing the multicenter bond) can be transformed into momentum p space and then related to the usual band picture. The occupation or filling of bands can be estimated from bond orders of the associated bonds and obtained from known interatomic distances by using Pauling's metallic radii. Bond hybridization is obtained from orthogonality requirements and bond angle considerations characteristic of valence bond theory. These ideas can be applied to FCC and HCP transition metals. For copper one would expect a sharp peak in the density of states corresponding to two unshared filled local d orbitals. In addition there would be a broad bonding band filled with electrons (6 per atom) containing large amounts of p character; and a half-filled s band. As one moves downward through the periodic table to iron while maintaining either FCC or HCP structures the high melting points of the elements involved indicate that the broad bonding band remains relatively unchanged (filled) while the number of electrons in the narrow d band is steadily decreased. In the rare earth cubic Laves phases of composition AB2, this relatively simple chemical approach suggests the presence of four bands. The two more important bands are: (1) a large narrow-width densi- ty of states band associated with two unshared local d orbitals per B atoms, and (2) a band which is generally more than half-filled associated with all the transition metal B–B as well as the A–B bonds. The other bands include an s band associated with the B atoms which is less than half-filled due to the transfer of electrons from the hyper-electronic B atom, and a band of unusually high d character as- sociated with the A–A bonds and probably not occupied for the lighter rare earth compounds. From the study of simple or bonds, Coulson has shown that the momentum distribution function is compressed in the direction of the internuclear axis. By using this idea in conjunction with the Fourier transform of hydrogenic atomic orbitals it is comparatively easy to show that for a CCP’ transition metal like copper the momentum distribution function has its principal projections in the (111) directions while the BCC transition metals should have projections in the (110) directions. Projections in more complicated structures can be obtained from considerations of bond hybridization and bond order. In summary, we see that the valence bond concepts of bond hybridization and bond order cou- pled with known structures and bond distances can be used to suggest band shapes, filling, and hybridization. It is apparent that the increased use of chemical concepts in the interpretation of infor- mation concerning the electronic density of states is an area of promise. Key words: Aromatic compounds; chemical bond; electronic density of states; Fermi surface; Paul- ing radii; transition metals; rare earth intermetallic compounds. *The abbreviation of CCP (cubic close packed) is preferable to the usual FCC (face centered cubic) terminology as many structures are FCC without being CCP. 417–156 O - 71 – 26 385 1. Introduction The chemical concept of bond formation between ad- jacent atoms can be used to obtain density of states in- formation such as the approximate shape and the rela- tive filling of bands for many crystalline materials. Such qualitative information should be particularly use- ful in interpreting trends of density of states in related series of compounds and in making rough predictions about compounds having new structures. Moreover, from the representation of the chemical bond in mo- mentum space, one can readily predict the distortion of the Fermi surface from sphericity. We intend to show that this representation provides an alternative ex- planation of the apparent attraction of the Fermi sur- face to the Brillouin zone faces. As indicated above, the paper is in two parts. Some of the older chemical concepts of the valence bond ap- proach are applied to a new set of hybrid orbitals in a discussion of transition metal compounds. The applica- tion of these concepts plus structural data leads to den- sity of states information via the following logic: (a) Bond distances are used to indicate the number of elec- trons involved in a given bond; (b) A modified valence bond formulation is developed for the new bidirectional orbitals that permits the association of particular sets of bonds into bands which are orthogonal in both real and momentum p space; (c) The relative filling of the bands is estimated from the bonding model and the number of electrons in the associated bonds. Examples are given for several compounds and some differences in the treatment of these compounds by the chemists and physicists are discussed. In the second part of the paper the density of states is approached from the point of view of the distribution of the bonding electron in momentum p space: Succes- sive portions of the paper deal with: (a) The effect of bond formation and orbital hybridization on the mo- mentum distribution function; (b) The qualitative similarity between the shapes of the Fermi surface and the total momentum distribution function for the outer electrons; (c) The application of a simple model predict- ing the Fermi surface for several examples; and (d) Generalization of these considerations to density of states information. The desired starting point for the applications of chemical concepts is a reasonably accurate structure determination of the compound of interest. The in- teratomic distances accurate to -0.002 A provides one with an estimate of the number of electrons of each atom involved in bond formation. In addition, the angu- lar aspects of each atom’s coordination suggests possi- ble hybridizations (i.e., LCAO) of the bonding orbitals which could give a fair amount of overlap with its near neighbors. While other types of data, such as estimates of interatomic force constants, magnetic data, conduc- tivity type, etc., are useful for refining the bonding model or choosing between models, the structural infor- mation is essential. The derivation of density of states information is best demonstrated in cases where relatively unique schemes of bond formation can be devised. This is possible in many intermetallic compounds having high coordination by the use of orthogonal sets of bidirectional orbitals. Since this approach, which we might term the bidirectional orbital approximation (BOA), makes use of s, p, and d orbital hybridization, we will generally confine our attention to transition metal compounds of high coordination. Prior to the con- sideration of particular compounds it will be useful to define some terms as well as to indicate what is in- volved in the BOA scheme of bond formation. 1.1. Definition of Terms Although the “valence” of an atom initially referred to as the “degree of its combining power,” the term “valence electrons” has widely different meanings for the physicist, the modern chemist, and those scientists preferring the purely ionic representation. To avoid confusion we will adopt the term, “valence electrons,” to indicate the number of electrons of an atom outside its largest complete rare gas shell and coin the term “covalency” or “covalent electrons” to indicate the number of electrons used by an atom in the formation of covalent bonds (i.e., shared electron bonds). The “bond order,” nij, of a bond (covalent) between atom i and atom j indicates the number of electrons con- tributed to that bond by each atom. Thus, the car- bon–carbon single bond C–C has a bond order of nij = 1.0 indicating each carbon atom contributed one elec- tron to the bond. Similarly, the carbon–carbon double bond C=C has a bond order nij = 2.0, in which each atom contributed two electrons. In the overwhelming majority of transition metal compounds the bond orders of all bonds are nj = 0.5 or less (exception Nb–Nb bonds in Nb5Sn). In general the bond order is inversely related to the interatomic distances. When the atomic distances are such as to give a bond order ni; 2, 0.05 the atoms are not considered to be bonding; in many transi- tion metal compounds this distance is about 3.8 Å. All the electrons involved in covalent bonds are considered to be paired even though the bond order is less than 0.5. 386 The covalency V, of the atom i is just the sum of the bond orders nij over all the bonding neighboring atoms j, thus: V-X. Ilij (1) j Pauling [1] has developed a semi-empirical equation for the calculation of nij from the known interatomic distance diff. It is as follows: 0.600 log nij= R (i) +R (j) - diſ (2) where R (i) and R1(j) are single bond radii for atoms i and j, respectively. These metallic radii, R1, may be cal- culated from equations given by Pauling knowing the average bond hybridization and number of valence electrons, z. For example, for the iron transition ele- mentS Ri (spº, ö) = 2.001–0.0432 – (1.627–0.1002)6 (3) where 6 is the average d character and the s to p hybridization ratio is 1 to 3. On the plus side of this em- pirical approach we note that eq (3) is suitable for calcu- lating the single bond metallic radii for = 16 elements with other very similar equations for most of the remaining elements of the periodic table. The average d character, 6, for the covalent electrons is taken by Pauling to be a smooth function of position in the periodic table and attains a value as high as 50% (6 = 0.5) for some of the transition elements (e.g., Ru, Rh, Os) [2]. Although a table of R1 has been given for the elements [1, p. 403], in their application to various compounds the author has found it necessary to use such values as a starting point in a reiterative calcula- tion where a measure of self-consistency is sought among (i) the equations, (ii) the assumed and calculated values of the covalency Vi, and (iii) in the amount of charge transferred between atoms. To the potential user it is further suggested that this approach is most suitable in discussing trends in series of related com- pounds (i.e., differences of covalencies, etc., between similar compounds is to be taken as more meaningful than the values themselves for any isolated compound). By using Pauling's metallic radii, as above, one can obtain estimates of (1) the number of electrons involved in bond formation, (2) the number of unshared electrons or localized electrons per atom, and (3) the relative oc- cupancy or hybridization of atomic orbitals for each type of electron. In order to estimate from chemical considerations the distribution of the covalent electrons among bands, a model of bond formation between atoms is required. 2. The Bidirectional Orbital Approximation In the usual valence bond treatment the angular parts of s, p, and d orbitals are superimposed to make a hybrid orbital which concentrates electron density in one major lobe toward a single neighbor. Such hybrid orbitals are of proven utility for elements of the first two rows (especially) but can give difficulties in their appli- cation to the transition elements. In those cases the bidirectional orbitals appear useful, where bidirectional orbitals are hybrid orbitals in which electron density is concentrated in two major directions. Equations between the coefficients of the nine atomic orbitals are obtained from the following restrictions: (1) All near neighbors are simultaneously bound; (2) lobes of equivalent orbitals are directed toward equivalent neighbors; and (3) all bidirectional orbitals are mutually orthogonal and orthogonal to the remaining orbitals. In structures with high coordination this approach has two advantages with respect to the usual valence bond treatment: (1) The high formal charges associated with either the ionic or single lobe valence bond treatment are avoided; and (2) the method usually results in a finite number of bonding descriptions that can be use- ful for either simple considerations or could be readily subjected to formal and extensive calculations. In con- trast, full valence bond calculations for a solid, with or without ionic contributions, are extremely difficult and have not recently been attempted to the author's knowledge. Rundle [3] used the simplest of bidirectional orbitals (the carbon p orbital) in his discussion of the transition metal carbides, and Ganzhorn [4] obtained orthogonal bidirectional orbitals of the general form as + V(a” – 1) dº for the octahedral and body-centered coordinations. By generalizing these orbitals into a C-type of py plus dry character and a G-type having s, dz , and proharacter, it is possible to discuss the bonding in a wide variety of metallic and semiconducting transition metal com- pounds (e.g., those having the Thap, CCP, HCP, NiAs, WC, and Laves phase structures [5-8]). In figure 1 the general shapes of the C and G orbitals and phase relationships between their lobes are in- dicated. In the process of maximizing the angular part of the two main lobes of a bidirectional orbital in the direction of two near neighbors one obtains, in general, three categories of orbitals. These are: (i) the bonding bidirectional orbitals of C- and G-type, suitable for the shared covalent electrons; (ii) local orbitals whose an- 387 º % & § \ º \ s | DS tº-3 Sl º b G orbital a C or bital FIGURE 1. The concentration of electron density into two major lobes is illustrated by the angular portions of the bidirectional orbitals. The included angle between the atoms to be bonded is much larger for the G-type (b) than for the (-type (a) to the left. gular lobes are either small or not directed toward near- by atoms, suitable for unshared electrons (paired or un- paired); and (iii) an orbital of primarily s character, suitable for a simple band treatment. Later we will as- sociate respectively with these orbital types: (i) a bond- ing or covalent band, (ii) a band of localized electrons, and (iii) a free-electron like band. Finally we point out that in structures of high symmetry the use of the bidirectional orbitals usually does not result in the atom of interest having the full symmetry of its atomic posi- tion in the structure. This is to be achieved at a later stage, in terms of multi-electron wave functions, by forming linear combinations which include the necessa- ry other sets of bidirectional orbitals. These sets, b%. are related to the original set di by the missing sym- metry elements corresponding to the operators S, by the eq (4). d; =S,d), (4) Bond formation between two adjacent atoms can result if the overlap integral between their orbitals is positive. This can usually be achieved by matching the phases (or signs) of the angular parts of their bonding orbitals. With bidirectional orbitals this leads very naturally to the formation of cycles which, for one elec- tron per orbital, are closely related to the cycles of early valence bond treatments. Cycles then consist of linear arrays of bidirectional orbitals on adjacent atoms, form- ing either rings or infinite chains. Figure 2 (top) sug- gests an infinite periodic cycle of C- and G-type orbitals in the cubic Laves phase of composition AB2. In figure 3 are seen cycles of four C-type orbitals in the (100) plane of a CCP structure. By a phase structure (Q) we refer to that set of cycles in a crystal such that all bonding orbitals are linked in cycles involving more than one orbital. A phase struc- ture may be considered “good” if overlap between or- bitals is high in terms of both phase matching and the directional properties of the orbital lobes. In MO terms the “good” criterion merely says that if a molecular or- bital configuration contributes significantly to the ground state, the lowest electron level, at least, should be bonding. Cycles within a phase structure are mutually orthogonal (approximately). This results from the mu- Top — A linear periodic cycle is shown for the A-B band in the Laves phase AB3. The phases of the G orbitals (A atom) and C FIGURE 2, orbitals (B atoms) are all matched to give good overlap. Bottom – In this cycle for the A-A band the use of C orbitals is symbolically illus- trated. The arrows to the right denote an anti-bonding overlap which would become increasingly prevalent with high k values in the Bloch wave functions. 388 FIGURE 3. The relative orientation of the C orbitals in a good phase structure for a (001) plane of a cubic close packed structure is shown. The C orbital with a dotted outline is orthogonal to the C orbital with a solid outline based at the same atom (black dot). Around the B position is a cycle of four C orbitals. This cycle is approximately orthogonal to all the other cycles of the phase structure. To restore the full symmetry to the lattice an equal amount of a phase structure corresponding to a reversal of A and B must be included in the n electron wave function. tual orthogonality of bidirectional orbitals located on the same atom and the fact that the nodal planes of the bidirectional orbitals usually pass near or through neighboring atoms which are not being bonded by its lobes. A phase structure Q(rn) of n electrons with coor- dinates r, can then be expressed as a product function of its cycles Ti 0(r)=II T; (5) where Ti is a Slater determinant of ni electrons involv- ing the appropriate C- and G-type spin orbitals dº; di (1) tº º tº e s = e tº e º ſº e (b1(ni) Tº = (b2(1) © tº ſº tº $ tº º & e º 'º g (b2(ni) (6) bº, (1) * & © tº tº e g º g g e dº (ni) The full symmetry of the structure may be restored by forming a wave function from a linear combination of phase structures which include the different symmetry related bidirectional orbital sets, bº, as obtained by eq (4). It is of considerable interest that for many structures the di of different v and i (but the same atom) are orthogonal. For these crystals, partial phase structures can be written in terms of bands' such that the total wave function for N electrons can be equated to a product of the phase structures for the bands y. V (ry) = TIy() (r. y) (7) where N = X.ny (8) and bands of local orbitals” are included. Alternately V(ry) can be considered as a Slater-like determinant of band phase structures Q(rn, y) for diagonal blocks and zero off-diagonal blocks. The band phase structures, however, are linear combinations of phase structures Qiy associated with the band y, and symmetry opera- tlOnS Qy(r)) =X. a (i. l/. y) Qi, y(rn) (9) where the individual phase structures Qi,y of the same band are not necessarily orthogonal. Various sym- metries about the site are possible by a suitable com- bination of the coefficients aſi, v, y). If Xi(p) is the Fourier inversion of the orbital di (r) according to eq (10) x(0)=#| 0 (or "d, (10) with p as the conjugate momentum of r then it may be shown [9] that the Fourier transform of a Slater deter- minant of li results in a Slater determinant of Xi in mo- mentum space and is obtained by simply replacing bi(r) with Xi(p). It follows that the Fourier transform of a sum of Slater determinants in real space results in a sum of Slater determinants in coordinate momentum space with the same coefficients. Accordingly we may associate with the phase structure, Qy, of the y band, a 3ny conjugate p space. In a valence bond treatment of a solid there is no ob- vious need for the phase structures, Qy to be periodic, except for reasons for organization. However, for the later discussions relating bonding to the density of * The term “bands” used in this somewhat different sense refers to group-symmetry restricted n electron functions which may be associated with several of the usual one elec- tron bands of band theory as below Qk (rn, y) =X. Am(k, y)Gm }}l Here Gm is a Slater determinant of n (or less) electrons involving the customary band wave functions (PE(r,0) with various combinations of the a bands. * For some bands of local orbitals and for a band of “free” electrons of primarily s character it is probably adequate to take Qk(r,y) the same as the customary band wave func- tions, (Pºtr,0), with y = cy. 389 states and the Fermi surface, we remark that for a Bloch representation for n electrons of the form eq (11) (11) ili (ri, • - 2 r)=e 2" u(r, º • 2 rn) the phase structures Qy, may be identified with the periodic part uk(r1, ... rh) of li. For k=0 and one elec- tron the phase structure is one associated with good bonding like figure 2 (top), while for k #0 antibonding states as in figure 2 (bottom) may play an important role. For the one-electron case a clear distinction must be made between k space and the momentum (p) space. The former space is a space of good quantum num- bers which can be considered to have the dimensions of momentum, whereas the momentum p is not a good quantum number for any nonconstant crystal potential. 3. Bonding in Some Transition Metals At this stage it is appropriate to assess our position and to indicate the forthcoming directions. Having equipped ourselves with some chemical nomenclature and concepts and a bonding model appropriate to transition metal compounds we will now apply these to some close packed metal structures, the Laves phases of composition AB2, and the NiAs structure. In doing so we will obtain coarse grain density of states information such as the number of bands (at least those partly filled), their rough shape, relative position, and percent filling. Important differences between the chemical and physical viewpoints will be discussed and a possible ap- proach to the resolution of some of these differences will be suggested. In following sections the influence of bond formation in momentum p space will be indicated and the relation between the momentum distribution function and surfaces of constant energy will be sug. gested with examples referencing the Fermi surface. In the following simple applications of the BOA pic- ture we hope to clarify some of the BOA formalism above. Among transition metal compounds the coor- dination number of 12 is one of the most common, in- cluding the icosahedral coordination and the close packed structures. Even the sixteen coordination of the A atom in the Laves phases can be separated into one of 12 B atoms and 4 A atoms. A coordination of 12 can be sought through the use of bidirectional orbitals by generating a set of orbitals (bi from one of the form eq (12) d = As + Bp. -- Cpl. -- D)). -- Edry -- Far, + Gdy- + HD, 2-y2 + Idea (12) by the use of a twofold axis in the z direction and threefold axis in the [111] direction. The six orthonor- mality conditions are: 1/2=42-1-D2 + E2 + H2 + I2 = B2+C2 + F^+G” (13) 2A2 = H2 + I? (14) O = BC + GF = CD + EF = BD+ EG (15) The requirement that equivalent atoms be bonded in an equivalent manner suggests that with respect to a 180° rotation about the x axis (bi should be either sym- metrical or antisymmetrical. The antisymmetrical case leads to two different sets of six C-type orbitals. Each set would give good bonding for an icosahedral coordination of 12. The two different icosahedra are related by a diagonal minor plane, Sa. The two C-type orbitals, eqs (16a, 16b), d), - Cyr = (p., -H dry)/ V2 (16a) dº?= Cyr = (p,--dry)/V2 (16b) are representatives of the two different sets bi and diff, which are related by the mirror plane operator Sa as in eq (17). Sadi = (b? (17) In the formation of the six C-type orbitals (of each set) the three p and the dry, drº, and due atomic orbitals are completely used. For each set (bi and q):”) the s, dz2–2, and dº orbitals are mutually orthogonal and orthogonal to the bonding C orbitals. The coordination of the twelve near neighbors in the CCP structures can be readily taken as the average of the two icosahedra just discussed, so the rehybridiza- tion of the atomic orbitals to form bonding hybrids is the same as for the icosahedron. By alternating the hybridization of the atoms in the (001) plane between the di and di" sets it is possible to form good phase structures. In figure 3, showing the xy plane of a CCP structure, it is clear that such an alternation results in a phase structure, Qc4, of the C band (covalent) in which cycles, Ti, are rings of four C-type orbitals, two of the Ö; set and two of the di" set. If the phase struc- ture, Qc2, represents the exchange of the di sets with the bi" sets, then the sum of these two-phase structures restores the full symmetry of the lattice. Other phase structures, Qc jº, composed of various arrangements among the atoms of the di and b;" sets may be expected 390 to contribute to the phase structure, QC, for the covalent band although the orbital overlap will not be as good on the average as in Qc4 and Qc2. Two additional bands, Qt, and Qs, can be constructed from the remaining atomic orbitals of the sets (bi and d;". The phase structures Q., of the L band (L for local) is clearly constructed of atomic dºº-yº and dº orbitals. These orbitals have lobes which are not directed towards any near neighbors and are accordingly suita- ble for unshared electrons, up to four electrons per atom. The phase structure, Qs, is clearly associated with the atomics orbital and for our current purpose we will identify it with a free-electron type band. In ac- cordance with our earlier discussion (see eq (7)) we note that all the Qc j are orthogonal to both Qi, and Qs. A very similar distribution of hybrid orbitals obtains if the BOA method is employed in the HCP structures [6]. If the cla ratio for the structure is ~1.63 six C-type orbitals of =50% d character are employed to bond the twelve near neighbors, leaving two local d orbitals and an s orbital to form the QL and Qs phase structures and bands. Changes in the cſa ratio result in dº character in the S-type orbitals as well as changes in the d to p ratio in the C-type orbitals. Now let us distribute valence electrons among the various bands starting with copper and working our way to manganese through the 12 coordination struc- tures, including only CCP, HCP, and icosahedral coor- dinations. Correlating the high melting point of these elements and their compounds with a high valence, let us, in an attitude of willing disregard of prior commit- ments, associate with each C orbital one electron for bonding purposes. Of the eleven valence electrons of copper, five electrons remain; four of these can be placed in the two local d orbitals, leaving one electron per atom for the S band. If one electron per bidirectional orbital (the half-bond mentioned earlier) gives rise to a filled band, then we see that this naive picture results in a diamagnetic copper having a half- filled S band and good electron mobility. The latter is related to the nondirectional character of the s orbital”. plus poor interband electron scattering associated with the existence of an energy gap in the C band at the Fermi level. This gap is between a bonding C band and a C band with appreciable antibonding character. Questions concerning the band gap, the high covalency (6 + 1), and the high percentage participation of d TABLE 1. Idealized distribution of electrons of some metals having HCP and CCP structures * Due to the non-directionality of the s orbital the S band of a CCP metal is not associated with strongly directional bonds. As indicated in the latter part of the paper the S band is then expected to have a low density of states; accordingly, the associated curvature is small and the mobility high. | Idealized distribution | Unpaired of electrons electrons State Ref C band S band L band | Ideal Obs Cu element................ 6 | 4. () () 10 Ni element................. 6 l 3 l 0.55 | 10 Co element................ 6 l 2 2 1.60 | 10 Fe: * Element, y-phase high temp............ 6 | | | 0.57 || 1 || Element precipitated from Cu, pseudo single crystal **..... 6 l I | ().78 12 Alloy, Fe in Cu........ 6 l l l l 13 Mn: * In element and alloys (icosahedral position in phases)............... 6 l 0 0 = 0 14 *We note that the magnetic behavior of the solute atoms in the dilute alloys of Mn in Cu [12] and Fe in Pd [15] are well known to be of a different type than Fe in Cu [12] and do not fit in this table. **Neutron diffraction determination. character in the C orbitals will be considered after the discussion of the NiAs structure. Applying this naive approach to the series Cu, Ni, Co, Fe, and Mn with these elements in the close packed or icosahedral coordinations leads to fair agreement (ta- ble 1) as regards to their observed magnetic properties. Excellent agreement can be obtained for Ni and Co with regard to both the number of conduction electrons in the S band and the magnetic moment by shifting =0.4 electrons from the S bands to the local d orbitals (or the QL-type bands). Good agreement is obtained for iron in its CCP forms and for Mn in the icosahedral coordinations in the or phases. Unfortunately in the or phases disorder is common among the sites and bond distances are only poorly known so that the effective covalency is uncertain. The density of states picture is then composed of three bands. A medium broad C band with a capacity of about six electrons. This covalent band is generally filled and responsible for the cohesiveness and high melting point of the element. In the case of an incom- 391 plete filling, hole character associated with low mo- bility is expected for this band. The L band has a high narrow density of states and is responsible for the mag- netic character of the metals. Finally we have the broad half-filled (or less) S band associated with high mobility and low cohesiveness. Accordingly, for these struc- tures, we see the splitting of the d orbitals into a broad hybrid band associated with bond formation and a nar- row band; the latter fills up as copper is approached. 4. The Laves Phases In the transition metal Laves phase compounds, AB2, one finds four orthogonal phase structures correspond- ing to different bands, two of which are interpenetrat- ing covalent phase structures. The B atom has a slightly distorted icosahedral coordination consisting of six A neighbors and six B neighbors. The bidirectional C orbitals as discussed above will be used for the B atom. The larger A atom has a coordination of 12 B neighbors in a reduced Friauf configuration (see fig. 4) and four A neighbors tetrahedrally distributed along the threefold axes. (For a more complete discussion of these interesting structures see [8] as well as standard texts.) Whereas the antisymmetric solution (C orbital) was sought to the eqs (13,14,15) which link the hy- bridization coefficients for the icosahedron, the sym- metric alternative is required for the Friauf polyhedron. By rotating a G-type orbital of the form eq (18) G = as + bp.r –H id-2 (18) clockwise about the x-axis, solution of eqs (13,14,15) may be obtained as a function of the angle of rotation, o, [8]. When o' = 18° the lobes of the G orbital (fig. 1b) are then well directed to form bonds with B atoms at position 1 and 1' of figure 4. A 180° rotation of this or- bital about the z-axis will permit it to bond atoms at the 2 and 2' positions while a 120° rotation about the [111] direction will give it good position with respect to the 3 and 3' atoms. Accordingly, six orbitals (G) can be obtained which are suitable for bond formation with the 12 B atoms of the reduced Friauf polyhedron. An alternate set Gº can be obtained by reflecting them through the diagonal mirror plane (Sa) at x = y. The sum of the square of the angular parts of these 12 orbitals is seen in figure 5 to reproduce the full symmetry of the site. Of the nine original atomic orbitals three remain to form the four A–A bonds. These three orbitals, of p and d character only, can be hybridized to give two C- type orbitals whose lobes project through the faces of FIGURE 4. The coordination of the 12 B atoms about the A atom is illustrated for a reduced Friauf polyhedron. Four more atoms (A) are coordinated through the centers of the four hexagons. Six G type orbitals are used by the central A atom to bond pairs of B atoms at positions like 1 and I FIGURE 5. This computer drawing illustrates the symmetry of total bonding for the G-type orbitals of the Friauf polyhedron. The viewer is looking down the threefold axis at a hexagonal face. the hexagons of B atoms toward the four tetrahedrally distributed A atoms. The remaining orbital has no large lobes. It is important to note that three orthogonal or- bitals (two C-type and one local) can be formed in three ways; in each case mutual orthogonality exists between them and the G-type orbitals (as well as their mirror images). That these C orbitals can give good bonding is suggested in figure 6. We note that in both figures 5 and 6 the direction of view is down the threefold axis of figure 4 and that the angular projections are exag- gerated somewhat to resemble the orbital distortion due to bond formation. 392 FIGURE 6. In this figure the tetrahedral bonding of the C-type orbitals is shown. These have the same angle o (22.5°) and relative orientation as the (; orbitals in figure 5. A number of transition Laves phase compounds have such short A–A distances that they are considered [16] to be in a state of compression (up to 1.1%) when compared to the A–B distances (which appear nor- mal). In terms of an average single bond metallic radius these contacts can be so short that the covalency of La in LaNiº is an impossible sixteen [17] compared to a maximum possible of four (for lanthanum with a formal charge of – 1). A solution to this dilemma may be sought in terms of the above bidirectional orbital discussion. While the G- type orbitals used in the A–B bonds have a normal d character that varies with the angle a from 42% to about 48%, the C-type orbital d character is much higher (about 75%). In terms of Pauling's metallic radii, N(E.) the effect of increasing the d character of an orbital from 45% to 75% would result in a decrease of 0.32 A in the single bond radius of a trivalent Pt transition ele- ment (as in eq (3)). This dramatic decrease, however, would be partially nullified by an increase of 0.11 A due to the absence of s character in the C-type orbital. The net change for an A–A bond, 0.44 A would reduce a ridiculous bond order of 7.2 to reasonable 0.5 or a bond order of 0.5 to one of 0.11. In other words, due to the dif- ference in hybridization of the G- and C-type orbitals, there is a good reason for not expecting the hard sphere model or an average single bond radius to be applicable for elements with the Friauf coordination. In a later sec- tion we note that the correctness of this approach can be tested by Fermiology or density of states studies. Cycles of the two orthogonal covalent phase struc- tures Q – and Q4–4 are illustrated symbolically in figure 2, top and botton, respectively. These are shown as periodic except for an anti-bond phase relation to the left in figure 2 (bottom). The phase structure Q4—b corresponds to the important A–B bonds as well as B–B bonds in the ratio suggested in figure 2 (top). Here the G-type orbital is used by the A atom. However, in the A – A bond only C orbitals of the A atom are em- ployed (fig. 2, bottom). The two remaining bands, L and S, are like those in the close packed structures and may be expressed as Slater determinants involving the drº-gº and dº orbitals (L band) and the s orbital (S band) of the B atoms only. The A atom contributes nothing to the S band because its s orbital is completely used in forming six orthogonal G orbitals. Further, its only “lo- cal” orbital a (p-d hybrid) is deeply involved in the “resonating” C orbitals of the A–A bond, and hence is not suitable for unshared electrons. f * F . s - - - - - ; * * Density of state curves are illustrated for the various bands of a hypothetical cubic rare earth Laves phase, AB2. FIGURE 7. ...The solid lines indicate bands at least partly occupied, the unfilled (AB) and (AA) bands contain appreciable antibonding character. The separation and location of the filled (F) and unfilled (E.) forbitals of the A. atoms are considered to be very sensitive to the number of occupied states. The band volumes per AB, are: AB, (AB) bands– 18 states: AA, (AA) bands–4 states. 1.(B) band –8 states, L(F + F") band– 14 states, S band–8 states. 393 Applying the above description to the rare earth com- pounds having the cubic Laves phase we obtain a den- sity of states curve associated with five bands (fig. 7). The first, previously unmentioned, would be a very sharp band associated with the highly localized f electrons of the rare earth A atom. The less sharp L band is associated with the two local d orbitals on the B atom with much the same properties as the same band in the close packed structures. The number of electrons in these orbitals (and hence the magnetic mo- ment) is dependent on the number of its valence elec- trons minus its covalence, V, and minus the electrons (up to about 1) transferred from it to the hypo-elec- tronic A atoms [1, see p. 431]. Polarization of the A — B bonds towards the more electronegative A atom may be expected to largely neutralize the resultant formal charge. In these compounds an S band is not likely to be of major importance for three reasons: (1) The s character in the A atoms will not contribute since they are totally used (A* = 1/6) in the formation of the six G- type orbitals. (2) The next nearest neighbors of the B atoms are much more distant than those, say, of the close packed structures and accordingly will not help to stabilize the S band. (3) If electron transfer takes place from the hyper-electronic element (the B atoms), then the likely source will be the B atom s electron (i.e., the S band). Correspondingly the S band should have only a fraction of an electron per B atom and should be associated with a poor mobility (compared to copper) of n type. A calculation of bond orders for the A — B and B-B bonds (n in a 1/3) suggests that the A — B band is generally more than half filled (compared to one electron per bidirectional orbital) and so may be considered as p-type conductivity of low mobility. The fifth band (A – A) — associated with just the A atoms and their tetrahedral C orbitals—is probably unoccupied for the Laves phases of the lightest rare earths but becomes increasingly important as the d character of the A atom increases with atomic number. The high d character of these orbitals is taken to be energetically unfavorable for the lighter elements which presumably use their available d character in the formation of the A–B bonds. The radius contraction for the A–A bond due to the high d character of these C-type orbitals also suggests low bond orders (hence, occupation) for this band (n-type). Finally, we note that these same considerations carry over to the hexagonal Laves phases, with due allowance for unit cell changes. 5. The NiAs Structure Among the compounds having the NiAs structure one finds semiconducting as well as semi-metallic members. Although this difference is often discussed [18] in terms of the internuclear distance between the Ni-type atoms along the co axis, the use of the BOA method suggests that the cla ratio for these compounds can have an important effect independent of the inter- nuclear distance. In addition we will note that semicon- ductivity can be associated with the half-bond (i.e., where nij = 0.5) between the Ni- and As-type atoms. In the NiAs structure both components have six near neighbors arranged in a trigonal prism (As site) or trigonal antiprism (Ni site) coordination. In addition the metal atom has two metal neighbors in the antiprism directly above and below. Both G- and C-type bidirectional orbitals can be used in a description of bond formation in these coordinations. Using the C orbitals one can obtain three orthogonal orbitals suita- ble for the formation of six half-bonds in the general twisted trigonal prismatic coordination. The first is ob- tained by rotating the C orbital about the x axis by an angle y; the other two by rotating the resultant orbital by angles of 120 and 240 degrees, respectively, about the z axis. Orthogonality relations [5] result in eq (19) 3a* sin” y = 1 (19) where a” is the p character of the orbital. This represen- tation includes the pure p-state octahedral coordina- tion, the 50% p octahedral coordination and trigonal prismatic coordinations at y = 90° and y = 37°. About 8% d character is associated with this latter value of y. For the As-site element in the prismatic position, this orbital with its high p character and low d character is probably suitable. On the basis of symmetry alone it is not possible to select a unique set of bidirectional or- bitals which are appropriate for the transition metal in its distorted octahedral site (assuming cla = 1.63). As indicated earlier, octahedral symmetry can be achieved by C orbitals of pure p character and 50% p character as well as by G orbitals of sal” hybridization. Let us as- sume for the moment that the latter hybridization is most nearly correct for a particular transition metal. Since this assumption is equivalent to assuming that the p orbitals are too high in energy for consideration we must look to a pure G-type orbital in the z direction to account for bonding between the metal atoms along the c axis. When the cſa ratio is 1.63 then the G2 orbital is pure dº, however as the cla ratio is decreased the orthonormality conditions require that s character is in- creased in such a way that overlap with the metal atoms along the z axis is increased. In figure 8 we see such an orbital with 25% s character corresponding to cſa = 1.22. (The four G orbitals are then also appropriate for 394 REAL SPACE G G Bonding 2-ºxºs.--> ŽTZTTNSºwlſº ºt. TWTNISIS *T*-ºs- É L-f TWTU) increase \ F.T. MOMENTUM SPACE zººs UTV Lº T e Sºszºzzº increase Bonding Local FIGURE 8. For a hypothetical bonding scheme in the NiAs structure the angular parts of the bonding G, orbital are shown for 25 per- cent s character with a cſa ratio of 1.22. As the cſa increases to 1.63 and beyond the s character first goes to zero and then is added with a negative sign to give a local (, orbital of 25 percent s character. The effect in momen- tum p space is also illustrated. bonding in a body centered cubic site with the body diagonal along the c-axis). Similarly, as the cla ratio is increased beyond 1.63 an orbital (local) not suitable for bonding develops. Such an orbital is labeled G in figure 8 and is appropriate for unshared electrons. It is not surprising then that for prediction of metallic character in NiAs-type compounds, attention must be directed not only to interatomic distances but also to cſa ratios, as has been mentioned [18]. For an example relating the half-bond to semiconduc- tivity we will briefly consider CrSe which has the NiAs structure. By transferring one electron from Se to Cr, Set can form six half-bonds and still satisfy the octet rule. The negative chromium ion can use three of its seven electrons in half-bond formation with Set using the octahedral C-type orbitals. The remaining four elec- trons can be localized in hybrid orbitals which consist primarily of dry-type orbitals plus a localized s-de, hybrid. This is in agreement with the use of Pauling's metallic radii which suggests that metal-metal bonding is not appreciable. Accordingly, CrSe is expected to be semiconducting (observed) with a high effective mo- ment (4.9 compared with 4.7p. obs.). From elec- tronegativity considerations we may expect the half- bonds to be polarized in such a way that the formal charges are essentially neutralized. Under compression in the c direction one predicts the appearance of electri- cal conductivity and a reduction in the effective moment. For the typical NiAs structure, in the bidirectional or- bital approximation, the density of states curve would be a composite of four bands. Of least interest would be a narrow band filled with the two s valence electrons of the As-type atom. The second is a filled band cor- responding to three electrons per atom per unit cell (or to one electron for each of three bidirectional orbitals per atom). This band is responsible primarily for the cohesive properties of the solid and would correspond to largely p but some d character of the As-type atom and a poorly definable hybridization for the unspecified metal atom. The third band, an L band, is composed of the localized orbitals on the metal atom and in general will be partly unfilled and narrow. The size of this band, as well as its filling, will be dependent on the fourth band which corresponds to the metal-metal bonds along the c axis. If the fourth band is of high energy and un- filled the third band (L) will contain the G localized states of the fourth band. On the other hand, occupa- tion of the fourth band suggests the G states are of high energy and unoccupied. Perhaps the more correct ap- proach is to consider that the third band (L) contains only three orbitals per metal atom (filled at six electrons per atom) and that the nature of the fourth band changes dramatically from a localized state to a bond- ing state depending upon a cooperative phenomena governed by the number of electrons involved, the cſa ratio and the intermetallic distance. In the bonding state the fourth band is generally unfilled and conduct- ing of n type. 6. On Some Significant Differences Between the Chemists' and Physicists' Approach to Solids The differences in approach between the chemists and the physicists are more than just semantic. One of the most important of these is the restraint shown by the physicist in hybridization of orbitals compared to the apparent wild abandon of the chemists, as exem- plified by this article. In band calculations of “hybrid- ization” by physicists the primary energy values 395 are those of the free atom (used as an approximation) and the primary mechanisms are those from spin-orbit coupling, relativistic effects, and continuity restrictions on the wave function at special points, lines, and sur- faces. However for atoms at sites of high symmetry this latter mechanism often gives no hybridization in the chemists’ The calculation scheme used is generally a first order independent particle, molecular orbital approach which yields the properties of the elec- tron near the Fermi surface in fair agreement with ex- periment. In partial contrast the chemist, in small molecule cal- culations, employs outer electron orbitals which are “quenched” of angular momentum and which contain scale parameters for total energy minimization. The in- teraction mechanism is bond formation leading to the best ground state. Although simple MO treatments give good energies for a one electron bond, the addition of a second electron to the same orbital almost totally destroys the formation of the bond, even with shifts in parameter values. This effect is due to failure of the simple MO treatments to properly account for the cor- relation of the bonding electrons and has been known for more than 35 years, since the work of James and Coolidge [19]. Of critical importance in the thinking of the chemists is the tetravalency and tetrahedral character of carbon in the millions of its organic compounds. Free carbon has a 2s22p* ground state while tetravalent carbon has a 2s2p* valence state. The associated promotion energy as calculated by Slater is 9.05 eV. On the other hand the promotion energy for iron from a free atom state (3d%3°) to a covalency of six (3dºsp?) with a hybridiza- tion of dºsp” is 4 eV [20]. Accordingly, the chemists’ query for the physicist is “If carbon employs an sp” valence state with a promo- tion energy of 9 eV to an appreciable extent in the coumpounds of carbon, one of the best understood ele- ments, is it improper to form hexacovalent iron orbitals using s, p, and d orbitals having a promotion energy of only 4 eV.” One thing is certain: for accurate ground state energies (and hence bond energies) it is extremely important to take proper account of electron correlation and nature’s result is still only very poorly approxi- mated, with or without the super-computing systems of IBM and CDC. In the simple application of BOA to copper, as in table 1, it would appear that the covalency is seven, cor- responding to one electron in the S band and six in the C band. The corresponding atomic (bonding) configura- tion we will denote as sG8. This valence is even higher than that used by Pauling (5.56) in his metallic radii Se]]. Se. table [1, p. 403] and differs considerably from that sug. gested by Brewer (2 to 3) on the basis of bond energies [21]. If the requirement that all near neighbors are simultaneously bonded is adhered to, then one must suggest that the C orbitals are not very good for simul- taneously forming twelve half-bonds in copper due to the small radial extent of the dry-type orbitals. The cor- rect explanation is probably that the configuration d'sC" is as wrong as d"s and that phase structures in- volving symmetric combinations of dēsC2 and d6sC4 should be included as main contributing terms in the wave functions. However, if the argument (BOA) must be modified for copper, the bidirectional orbital approx. imation would appear to be a reasonable starting point for the other transition elements of Groups IIIb to VIII. An oversimplified treatment of binding might lead one to expect spherical electron distributions for both carbon with its spº hybridization and for the copper d shell with the full occupancy of its d orbitals by shared and unshared electron pairs. However, the sphericity required by Unsold’s theory for atomic carbon with spº hybridization is destroyed in the tetravalent bonding state because some of the electron density shifts into the bonding regions between atoms. If d character is employed in bond formation in copper, as in the C band of the BOA picture, then some dry, drº, and dye character is involved in the unoccupied antibonding or- bitals of the C band. This then will also disturb the sphericity of the 10 d orbitals since the dra—ſº and dº, orbitals are occupied by unshared electron pairs. Now we would like to adduce a few arguments sup- porting the assumption that one may associate a filled band with the concentration of one electron per bidirectional orbital (half-bond). In MO theory, the band associated with a chain of linear atoms has its greatest density of states at an electron concentration of one electron per orbital. However, in the BOA ap- proach one generally is concerned with the interaction of chains (cycles) which involve two or three dimen- sional arrays of atoms. Accordingly, while a single par- tial phase structure Qy, associated with a band y may not have a density of states depression at one electron per bidirectional orbital, it is anticipated that the in- teraction of different partial phase structures of the same band will give such a depression. This is because cycles from different phase structures orthogonal. If this is correct then, when the metal-metal bond in an NiAs compound can be ascribed to a single phase structure with one infinite cycle of G orbitals (as in the above example), one would not expect a band depression at one electron per orbital. are n Ot 396 Considerable precedence in chemistry exists for as- sociating special stability with one electron per orbital. Depending on the number of neighbors this electron concentration can lead to fractional bond orders. Paul- ing has called attention to this [1, p. 420] and, using his metallic radii, has noted the abundance of such cases among transition metal compounds while Rundle [22] has emphasized the importance of half-bonds. We would like to point out here, that since Pauling used the elemental bond distances as reference points, a “proof” that the number of covalent electrons used by copper is 5.56 by means of his metallic radii would be highly circular. Nevertheless, his formulae, such as eq (3), are firmly related to bond distances and covalencies about which no controversy exists. Accordingly his ob- servations of fractional bond orders merit more than just passing attention. In organic chemistry, the aromatic compounds play a special role because of their unusual stability. These are planar compounds consisting of fused benzene-like rings (hexagonal) to give what looks like sections of hex- agonal bathroom tile. The p2 carbon orbitals are per- pendicular to the plane containing the O-bond structure and are occupied by one electron each. In valence bond theory the special stability of these compounds is at- tributed to the resonance (interaction) of the two Kekule structures (phase structures). These com- pounds form a continuous series starting with benzene having one ring and six p. electrons through a long sequence of compounds involving many rings (e.g., 8 rings and 30 electrons) to a graphite sheet. In graphite we will see later that the bond order associated with the p, electron is 1/3 and that the related band is described as filled with a zero band gap [23, 24]. Currently this represents the strongest argument that semiconductivi- ty can be related to the half-bond (nij = 1/2). However in practice, semiconductivity can be readily predicted, as in the C.S. – C.S, series and FeS2, etc. [5,7], by as- sociating a band gap with the half-bond. In returning to the primary difference between the chemists’ and physicists’ approach to transition metal compounds, i.e., the amount of d character involvement in bonding electrons, we will recall the suggestion that this may critically depend on the correctness with which electron correlation is handled. However, even within the band scheme a partial resolution of the dif- ference in d character use may be sought by the use of scale factors in the radial part of the wave function. It would be of particular interest to associate a different scale factor with different symmetry related d orbitals, for example, one scale factor for the dry type orbitals in CCP copper and another scale factor for the drº–y, and dº orbitals. In such a case the scale factors are parame- ters which could be adjusted for energy minimization for each value of k. If the execution of this suggestion is exceedingly difficult (as the author is aware) the task posed below for the solid state chemist is no less possi- ble. Two methods of treating the correlation of bonding electrons are currently being tested on small molecules. The first of these, the alternate molecular orbital (AMO) approach, is being actively pursued by Pauncz [25] and involves the alternation of electrons of different spin on a linear chain or ring of orbitals. Energy is minimized as a function of the Xi, which are related to the relative separations or phases between electrons of different spin. The second method is primarily due to Linnett [26, 27] and is termed the non- paired spatial orbital (N.P.S.O.) treatment. For a cycle of four orbitals, C1, C2, C3, and C4 as in figure 3, new or- bitals would be formed as below: C1 + KC, C2 + KC3, Cº-H KC1, C++ KC Symmetry must be restored to the wave function in both cases and in the N.P.S.O. method would involve inclusion of orbitals of the form KC + Co., etc. By em- ploying either the AMO or the N.P.S.O. method in con- junction with bidirectional orbitals one has an approach which contains relatively few adjustable parameters with the inclusion of electron correlation. By varying these parameters (plus scale factors) as a function of k space it may be possible to achieve a partial resolution of the differences relating to d hybridization. 7. The Chemical Bond in Momentum Space In this section we would like to consider three effects in momentum p space. The first is the effect of forma- tion of the simple chemical bond, as between ls electrons in molecular hydrogen. The second to be con- sidered is the effect of bond hybridization. In the case of strong bond formation these two effects reinforce one another although they appear to be largely indepen- dent. Finally, we note that the effects of different bonds are essentially additive in momentum p space in terms of probability density just as they are in real space. Although Podolsky and Pauling [28] had obtained the hydrogenic momentum distribution functions in 1929 and Hicks [29] had considered the effect of the momentum distribution of molecular hydrogen on Compton scattering, it remained for Coulson in 1940 to initiate a series of papers [30-34] specifically consider- ing the effect of chemical bond formation in p space. His results for the homopolar and heteropolar bond 397 were essentially the same whether he used the molecu- lar orbital (one electron) or valence bond method (two electrons). Coulson showed that contours of constant momentum density were very appreciably extended in the directions perpendicular to the internuclear axis of the bonding atoms. For the one electron momentum function he obtained eq (20) _l = cosp (ra-r) l + San X(p).Y” (p) A (p) A *(p) (20) where r and p are vector coordinates in real and mo- mentum space, respectively, ra-ri, is the internuclear distance vector, and A(p) is the Fourier transform of the atomic orbital \P aſ r) on atom a. In the bonding case (+ signs) with Waſr) (and A4(p)) spherically symmetric we see that the momentum distribution function is a max- imum when the vector momentum p is perpendicular to the internuclear direction (ra-ru). In the case of the antibonding state ( – signs, eq (20)) for molecular hydrogen the signs of the ls orbitals are different on atom a and atom b: this gives a nodal plane between the two atoms and corresponds to the exclusion of con- siderable electron density from the volume between the atoms. This phase reversal between atoms a and b now results in a maximum density when p is parallel to the internuclear axis. The situation is the same for the two electron bond. The hybridization effect is readily seen from the hydrogenic wave functions in momentum p space ob- tained by Pauling and Podolsky [28]. In the traditional valence bond method, we noted earlier that in the process of maximization of overlap, primary attention was paid to the angular parts of the orbitals. In follow- ing the same procedure we recall [28] that the Fourier transform of an atomic orbital Valm has the same angu- lar functions in terms of spherical coordinates in mo- mentum p space" as \'nın has in real space, except for aſ—i)' factor. The angular part of a hybrid orbital, Wang, based on atom a, can be maximized in the direction of atom b by varying the coefficients C; in the following equation (as- suming normalization). Wang. = C.S + Cup + Cad The corresponding momentum hybrid is given below. Xanº = Cs-iC), p-Crd Here s, p, and d refer to the angular parts of the orbitals in real space and 5.5, and d refer to the angular parts in momentum space. The results are most graphic for a bonding G-type orbital. If s character is added to a dº orbital to obtain extension in real space along the + and — z axes (fig. 8, left) then the effect in momentum space is a contraction along the + and – p- axes and extension in the pºp, plane. The effect on a localized G. type orbital is just the opposite (fig. 8, right). For bond- ing electrons and orbitals the effect of hybridization is clearly in addition to the effect of bond formation as between 1s orbitals in H2. If the eigenfunction, V, for an n electron problem is expressed as a Slater determinant of Ói, where di represent the orthogonal one electron solutions, then the one electron density function, p(r), is obtained by integrating V\V* over n – 1 electron configuration Space. }} p(r)=X diſr) by (r) (21) i – 1 We see that the result is simply the sum of the electron density probability functions of the individual occupied states di. Since the corresponding expression for V in momentum space is also a Slater determinant, we see that the total bonding electron density in momentum p space is simply the sum of the individual bonding covalent electron momentum densities. 8. That the Fermi Surface Minmics the Momentum Distribution Function of the Bonding Electrons In this section we intend to show by a simple and heuristic argument that the Fermi surface in a sense co- pies the momentum distribution function associated with the outer electrons. Although the above title emphasizes the bonding electrons, there is no reason to exclude localized electrons if these energies are near the Fermi surface. While the subsequent discussions and applications will assume the Fermi surface to be in the unreduced form this will cause no distress to those familiar with such reductions to a single Brillouin zone. Finally no special attention is paid to energy gaps at Brillouin zone faces at such gaps are more closely re- lated to crystal symmetry elements than to the effects of bond formation. The argument to be given below involves the applica- tion of the Virial theorem which states that the average kinetic energy T is related to the average potential ener- gy, V, of a stationary state according to eq (22), T= W 'Accordingly, for a ps electron we see that it is extended in the same direction (z axis) in both real and momentum space. in apparent violation of the naive application of the Heisen. berg uncertainty principle. 2 - (22) 398 where the potential V is a homogeneous function of degrees of the coordinates. Neglecting magnetic forces, the potential V for an electron is just the sum of all the coulombic potentials due to the nuclei and the other electrons; accordingly s = - 1 and V = -2T. Inserting this relation into eq (23) for the average particle energy E, we obtain E= – T. E = T-I-V =–T (23) The derivation of the Virial theorm may be found in most standard quantum mechanics texts; its derivation as indicated above is due to a method of Foch and is given with applications to small molecules by Kauz- mann [19]. The validity of eq (23), relating the average energy of a particle to the negative of its kinetic energy, will be illustrated for the hydrogenic atom. From the corresponding momentum eigenfunctions Podolsky and Pauling obtained eq (24) —r •) 2Tme”z\* p;= p = {− }] }} nh (24) for the average momentum squared of an electron of mass m having a principle quantum number n. Insert- ing this in eq (23) we obtain — , , 2 - 2~22, 4 --- - P. 27°2°e 4 m. E = — T = — 2m n” h 2 (25) which, of course, is the familiar atomic formula. Here the lowest, most stable energy state is associated with the highest average momentum squared, a typical molecular result. - For the free electron-like model, where the electron momentum is identified with the k-vector the Virial theorem as in eq (22) must be modified to include con- stant volume effects (i.e., the Virial associated with restraint of atomic nuclei); the net result for the energy Eky of the y band may be written as in eq (26). p” t Elºy =#-– Eoy = — Eoy | ? 26 2m 2m (26) Here we see that the sign associated with the momen- tum term is just reversed. The above considerations are now extended to the more general independent particle band picture. To satisfy the Bloch conditions we write eq (27) for the Y band ilky (r) = e”ury(r) (27) where uſ y(r) is that real coordinate part of the eigen- function which has a periodicity identical to that of the lattice. If UKy(p) is the Fourier transform of the uky for a single cell” in real space then the momentum eigen- function X,Y(p) corresponding to Vºy(r) is simply eq (28). X,Y(p) = Ury (p – k) (28) The average value of momentum squared of the elec- tron associated with the crystal momentum k of the y band is F-ſtep-ºp put (p-od, 29. Substituting a for p – k we obtain Pi—ſ Uky (a)) (k+ ay): (k+ ay) U. (a)) da) — k? + 2k º (or y + oft, (30) where we identify a y as the average squared-momen- tum of the electron (k, y) in a single unit cell. Thus ofty has a “molecular” character. Similarly doky is identified with the single cell average momentum corresponding to uſ...}(r), which for k = 0 and for a centrosymmetric cell, must be zero but may be expected to be non-zero other- wise. We now assert that the energy Eºy increases for k and decreases with oft, according to eq (31) below, essentially as suggested by a combination of eqs (25) and (26) above. k” oft. Ey + º- – *— Vºy-- neglected terms + constant (31) 2m 2m The potential energy term, Vry, is given by eq (32), — ` :k | %k Vºy–S uRyu; Zm in Tº. dT }}} k' m "ſm (32) where xk is the coordinate of the k electron separated from N nuclei of charge Zm by the distance rººm. For a centrosymmetric structure both oft, and Vºy are posi- tive and change in a parallel fashion. Accordingly, only the variation of Eky with a y is considered. Further, since the derivation of eqs (31) and (32) are indicated in appendix A, here we add only that m is the real mass of the electron and that the neglected terms of possible in- terest involve derivatives of doky and a y with respect to the lattice parameters. The shape of Fermi surface is contained in eq (31). For the y band it is clear that for constant energy (i.e., * A single cell is used to achieve analogy of a with p of the atomic or molecular case. X(p) is usually expressed as the Fourier series having coefficients Uky sampled at reciprocal lat- lice points. 399 Eky= Erry), the absolute value of k will be greater than k' when Jº s J. (0%) > 0. y. Accordingly, a Fermi surface for the y band closely fol- lows the variations in oft, Prior to discussing the ex- pected variations of of for the bonding electrons we will first consider the cases for deep core electrons and highly excited states. For K-shell electrons of a heavy metal we would ex- pect that the depth of the y band below the energy zero is determined essentially by eq (33), that is, eq (25) with the sum taken over equivalent atoms in the unit cell and with n referencing the principle quantum number. We may expect that (00 y = X. pſ = X. Pſi = (0, ofty is practically independent of k even though pā is (33) very large. In the highly excited electron case (0%) should be quite small, with variations of Goły with k even smaller. Accordingly, in both of these cases the simpler band arguments should apply; for example, the free electron approximation for the excited electron should be reasonably correct. -- Expectations of the variations of a)}) for bonding electrons are most simply argued from a highly anisotropic example. Consider the case of a crystal composed of linear chains of atoms bonded along the [001] direction with distances between chains cor- responding to van der Waals’ contacts. In terms of bidirectional orbitals aſ, would correspond to the “phase structure” of a single cell in which all the phases match so that overlap of the orbitals on adjacent atoms in the chain is high (e.g., see fig. 2, top). Now as k increases in the [001] direction, the value of oft, is expected to decrease corresponding to the increasing frequency of antibonding contributions (i.e., nodal planes between atoms as in fig. 2, bottom). However, as k increases in the [k, k, 0] direction the value of off, should be much less dependent on k because antibond- ing phase relations between non-bonded adjacent chains are of negligible importance. The result for a surface of constant Ey is that the surface extends further from the origin in the plane perpendicular to the bond direction than along the bond direction, i.e., [001]. This, however, is similar to the shape of the mo- mentum distribution, X(p) X*(p), for the bond forma- tion in the [001] direction and supports the suggestion that the Fermi surface mimics the momentum distribu- tion function for the bonding electrons. The generaliza- tion to include several bands of comparable energy is obvious. It remains to point out that as more and more elec- trons are added to the system there is a reversal in the trend in variations of o; with direction such that the Fermi surface becomes less spherically distorted. For example, as more electrons go into antibonding or- bitals, oº, in the [001] direction can sustain only small variations (further decreases) because it already is small. However, in the [k, k, 0] directions much larger variations will occur; with the result that the Fermi surface in the [001] direction will “catch up” with the Fermi surface in the kr ky plane as more elec- trons occupy the antibonding, higher energy, levels. In the next section a simple physical model will be described that is useful for estimating the distortion of the Fermi surface from a knowledge of the structure in real space. After a few examples the approach is generalized for a discussion of density of states func- t! On S. 9. Application with Examples By extending the above considerations to all the bands of similar energies it is possible to make qualita- tive estimates of the unreduced shapes of the Fermi surface, or other surfaces of constant energy, from a simple knowledge of the structure in real space. In making the application the primary considerations are (1) number of bonds in a given direction; (2) their bond orders, nij, and (3) the hybridizations of bonds, if known. The latter point is relevant in that the presence of nodal planes in X(p) can give reduced sensitivity of oft, with k in those planes and a similar effect on the Fermi surface of the y band. A simple and qualitative method of estimating the distortion of the Fermi surface from sphericity is to mark (e.g., with narrow tape) those great circles of a sphere" which are perpendicular to the bond directions. The effectiveness of a given circle in distorting the Fermi surface radially outward is taken to be propor- tional to the sum of the bond orders, nij, of those bonds perpendicular to the given circle. The directions of maximum distortion are taken to be at the intersections of the great circles, weighted by the number of circles involved and their associated bond orders. Of course the volume enclosed by the distorted surface must remain a constant. As the first example we will consider the BOA pic- ture of Cu in the CCP structure. Using C-type orbitals one obtains good Cu–Cu bonds in the (110) "True to his profession the author recommends a round bottom distillation flask as a trans- parent sphere. 400 - * * * * = O o –2TTC * 2 TTC FIGURE 9. The solid line gives the hypothetical elipsoidal Fermi surface of a tetragonal crystal of H; ions with unit cell edges of a0 = 2.4, C0 = 3.46A. The dashed contour lines are for constant momentum density and were calculated by Coulson |30} for the H, ion. The dotted rectangle represents the boundaries of the first Brillouin zone. - directions. The above rule of thumb gives rise to the in- tersections of three circles in the (111) directions (toward the L point) and intersections of two circles in the (100) directions (toward the X point). However, the effect of the latter two circles should be less than 2/3 the former due to the presence of nodal planes (contain- ing the [100] direction) in the C-type orbitals as- sociated with those circles. The result is in good agree- ment with the current picture of copper [35] in which the Fermi surface is strongly attracted to the large hex- agonal Brillouin zone faces and less strongly attracted the smaller square faces. By way of contrast we have shown in figure 9 that the hypothetical oblate spheroidal Fermi surface for an imaginary crystal of H2+ ions mimics the contours of constant momentum and appears primarily attracted to the smaller [100] zone face. The molecules are as- sumed to be packed with their bond directions parallel to the c axis with a van der Waals’ separation of 24 A between nuclei of different molecules. For the alkali metals with the BCC structure it is doubtful that bonding effects would be the major cause of distortions in the Fermi surface since the bond order is no larger than 1/8. However, with the WB and WIB groups of the periodic table, distortions of Fermi sur- face can be estimated even though a unique bonding picture cannot be given at this time. For these transi- tion elements (e.g., V, Cr) the most important bonds (8 nearest neighbors in the (111) directions) probably have a bond order of about 0.5. Using the above method one finds that the corresponding circles cross at the twelve (110) directions. This distortion is in agreement with the opinion that the Fermi surface is attracted to the centers of the rhombic faces of the BCC Brillouin zone. However, the effect of bond formation of an atom with its six next-nearest neighbors is to distort the Fermi surface toward the furthest point, H, from the center of the zone, that is, in the (100) directions. In going from V to Cr this distortion should show a slight increase as the additional electron is expected to par- ticipate most strongly in the next nearest neighbor bonding. To suggest this in another way, assume the bond order of the bond with the nearest neighbors remains constant at =0.5. Using Pauling's metallic sin- gle bond radius R1, the nearest neighbor distance, d, is then d—º a0 = 2R + (). 18 (34) where a0 is the cubic cell edge. However, a0 is also the next neighbor distance so that eq (35) holds, a 0–2R1 = - 0.60 log n. (35) where n is the corresponding bond order. Eliminating 2 R1 from the equations we obtain (l() ( _V3 (36) 2 )+0.18––0.00 log n. eq (36) which suggests that the observed decrease in lattice parameters with atomic number leads to an in- crease in the bonding order of the next-nearest neighbors and a corresponding greater distortion (small) in the (100) directions. An extreme example among the non-cubic structures is to be found in graphite. In this hexagonal layer struc- ture all the strong bonds are within the layers in the (1010) directions, whereas bonding between layers is very weak, i.e., van der Waals’ contacts. The bond order between carbons is 1 1/3 corresponding to whole O bonds (sp2) and 1/3 of a TT bond (using p. orbitals). All of these bonds correspond to an unusually strong distortion of the Fermi surface in the [0001] direction. This distortion is shown very clearly in the surfaces of constant energy calculated by Wallace [23]. In the un- filled zone (corresponding to the T electrons) beyond the filled O-electron zone, the above model suggests that the Fermi surface should be distorted in the (1121) directions where l is not necessarily integer. 417–156 O - 71 - 27 401 In the NiAs structures the onset of electrical conduc- tion due to bond formation along the c axis should be accompanied (using the BOA picture, fig. 8) by the out- ward movement of the Fermi surface in the basal plane to give the greatest projection of the Fermi surface in the (2130) directions. Appreciably weaker distortions occur in the (213l) and (1121') directions where l and l' are non-integers dependent on the colao ratio. In some of the cubic Laves phases we indicated earli- er that simple bond distance considerations suggested that the A – A distances corresponded to an unusual state of compression and a large bond order (nA-4 × 1). However, the BOA picture suggested that the C-type orbitals had an unusually large d character. This would make the effective single bond radius, R (A–A), so small that the A atoms were practically non-bonded. The decision between these choices could be made on the basis of Fermiological studies principally involving the projection of the Fermi surface in the (110) directions. The simple model relating bond formation to Fermi surface projection is employed. The B atoms form interconnected linear arrays of bonds in the (110) directions among themselves. As in the simpler CCP transition metals this should give a pronounced projection of the Fermi surface in the (111) directions. The A–B bonds, however, are in the (113), directions and are not expected to result in pronounced distortions due to the diffusion of the inter- sections of their great circles. The A–A bonds are in the (111) directions and would contribute to a projection in the (110) directions, where weak projections exist due to the intersection of a B– B circle with two A–B circles. A study of the ex- tent of the (110) projections as a function of the “com- pression” of the A — A bond for different compounds would make a valuable contribution to the chemical un- derstanding of these compounds. 10. Density of States, General Remarks The preceding discussion concerning the Fermi sur- face is, of course, generally applicable to any surface of constant energy in k space. On this basis a few conclud- ing remarks can be made relating the density of states function, N(Ey), to the few chemical concepts of which use has been made in this article. If |&Ey/ök, is the length of the normal derivative of the constant energy surface, S, for a single band then the related density of states may be calculated from eq (37) N(Ey) = (Vol/878) |beloſ.-ds (37) where Wol is the volume of the crystal [36]. The manner in which the length |6Ey/ök, varies may be estimated from eq (38), which is for a spherical surface in polar coordinates, where m” is an effective mass. 8Ex_k__l b2_k ôk, m 2m 6k, m* (38) The density of states is essentially then the integral of the surface area weighted by m”/k. Therefore it is ap- parent that the distortion of the surface S(Ey) from sphericity gives rise to a high density of states, N(Ey). Furthermore the bottom of the band, Eoy, is largely determined by off, according to eq (31). In the exam- ples using the BOA treatment, we also saw that particu- lar sets of chemical bonds could be associated with a particular band (or set of symmetry related bands). The use of chemical concepts would then suggest that for a high total density of states N(E), with a con- stant number of bands, one should have chemical bonds of high bond order, nij, (or strength) with relative- ly few important bond directions. It appears reasonable that the relative positions of peaks in the density of states might be correlated with bonding and hybridiza- tion through the use of the different off, . For example, in the isostructural metal acetylides CaC2, SrC2, BaC2, LaC2, etc. a strong peak may be related to the carbon triple bond in (C = C)-” radical, while the bottom (and the shape) of the partly filled conduction band may vary in a predictable manner with the metal atom. By way of contrast one would expect a comparatively flat band, neglecting local states, for a material with numerous weak bonds in a great variety of directions. Such a com- pound which was crystalline but whose unit cell con- tents had an amorphous-like structure (but repeated periodically) could be expected to have a spherical Fermi surface, not because the free electron approxi- mation was realistic but because Xofy y was not a strong function of the spherical coordinate angles bk, 6, in k space. ll. Summary By way of conclusion it will be useful to both sum- marize the current results and to attempt to forecast the benefits of further interaction between chemical concepts and those of solid state physics. For the appli- cation of the chemical approach an accurate structure determination is the essential input. If a reasonable bonding description for the compound can be devised then we have seen that in a variety of transition metal compounds the number, character, and relative filling 402 of bands can be readily estimated. Further, the distor- tion of Fermi surface from sphericity can be qualitative- ly estimated. Such information is of obvious importance in guiding the experimentalist studying the extended zone Fermi surface topology of a new structure. On the other hand we have pointed out, that in the Laves phases the aid of the physicist would be useful in refin- ing the chemical concepts. The value of the chemical approach may also be felt in the area of compounds of large unit cell size. For ex- ample, the cubic Laves phase of 24 atoms per unit cell is probably near the limit which can be accurately treated in band calculations with the larger computers. However, reasonably accurate structure determina- tions involving several hundreds of atoms are now possible. If the coordinations of the elements in such structures permit unique bonding descriptions then it may be feasible using BOA, coupled with group theory and standard k-space arguments, to develop useful band models for comparison with experimental density of states data. As an example, it may be possible to treat some of the metallurgically important sigma- and chi-phases of the iron transition metals. Band calcula- tions are not feasible for such materials, nor are they likely to be in the near future. The application of valence bond theory with the bidirectional orbital approximation to the Group IV and III-IV type compounds is less certain because cycles in the BOA sense lose some definition so that the distinc- tion between “good” or “bad” (poor overlap) phase structures diminishes. The net result is a severe orthogonality problem which is likely to be a major ob- stacle in the treatment of related amorphous materials. It is interesting to note, however, that for such elements bidirectional orbitals of the G- and C-type can be em- ployed to give tetrahedral coordination (see fig. 6). Concerning the relationship between the Fermi sur- face and momentum distribution function it is also desirable not to paint too cheering a picture. The above interpretation of off, and its variation with k is barely in its infancy and requires considerable refinement be- fore it can be applied with confidence. For example, the simple model given in this article using the sphere and marking tape to indicate distortions of the Fermi sur- face suggests that the Fermi surface of two compounds would be identical if only the unit cell sizes were the same and if the same number of bonds (X nij) were in the same directions. Since such limited information is insufficient to define a unique structure it is unlikely that the Fermi surface is as unique as the simple model Suggests. Nevertheless, it would appear that the use of these chemical concepts, as well as others, should play an in- creasingly important role in providing information to experimentalists and theorists, not only about the den- sity of states, but about the Fermi surface topology and the spectral density as well. 12. Acknowledgments The author wishes to indicate his appreciation of the continuing interest shown by Dr. L. H. Bennett of NBS in the chemical approach to solid phenomena and to thank Dr. J. Feldman of NRL for pointing out that the constant volume effect must be included in the applica- tion of the virial theorem. Thanks are due also to Mr. W. C. Sadler and Miss M. O’Hara both of NRL for tech- nical assistance. Finally, it is with abiding pleasure that the author acknowledges innumerable discussions of the relations between chemical bond and the Fermi sur- face with Dr. G. C. Carter of NBS. Appendix A The intent of appendix A is to show the correctness of eq (31), that is, that the energy Eºy of the k state in the y band can be expressed as below, Elºy = --→ *— Vºy-H neglected terms 2m 2m (31) where m is the electron mass, Vºy is a potential term, and oº, is defined by eq (30). While use will be made of eq (1A), the generalized Hellmann-Feynman Theorem [37], the kinetic energy terms for the elec- trons of the y band will be brought outside the integral &B_dH_ſ , , 0F1 ôX #–ſ * : *dr (1A) by use of eq (30) p;= k” +2k ory +oğ, (30) In eq (1A) we note that H is an explicit function of the variable A, which in our case will prove to be the unit cell edges. The Hamiltonian H is taken as a sum of the Hamil- tonians for the electrons in the y band, Hy, plus the in- terband electronic and internuclear repulsive poten- 'tials, Vyy and VR, respectively. H — H,-- Vyy + Vº (2A) J Rºk f M N Zl Zm +x. X. Rim (3A) Pºl P 1 H = H,--> X. k k' 403 In eq (3A) only the filled k states are included; here rº represents the interelectronic distance between elec- trons of different bands and Rim in the VR term cor- responds to the distance between the lih nucleus of charge z and mth nucleus of charge zm. For the y band we have eq (4A) .V. !, k Hy-> Ty-X, X → S. S. k }}} (4A) where rºm is the separation between the kth electron and the mth nucleus and Tºy is the kinetic energy term for the electron. Using eq (30) one obtains eq (5A) for T. p; k” k Goky Goſſy ky = -= - 2m 2m 77). 2m (5A) For the nuclei constrained by external forces to remain at fixed positions, the Virial theorem for the k electron can be written as eq (6A), where ðE. ða; - - 3 E ky = - Tºy - X. Cli (6A) the ai refer to the unit cell edges of the crystal. It is clear from the preceding equations (eqs (2A) through (5A) that the “external forces” at constant volume for the ky electron are primarily those due to electronic and nuclear interactions within the crystal rather than just forces external to the crystal. Of particular interest is the evaluation of the sum X ai (6k/6a) where k is expressed as eq (7A) (1) 2 º k 2 k e ajky k (li" () COky ->| m Óai i 2m m 77). as eq (13A) by making use of eq (12A). By neglecting the term involving (60.9/0a) and (6afty!0a) we obtain eq (31), the desired result. While (60.3/0a) may be negligible for a centrosymmetric structure the variation of off, with the unit cell lengths requires more careful consideration than we will give it here. We do note how- ever that its variation in the direction of strong bonds may be appreciable and sensitive to k when k is parallel to the bond directions. In considering the variation of the potential energy with respect to the ai we note that these must be zero for the intraband and interband electronic terms since the l/rij do not explicitly contain the ai. However, the 2m 2m 3 - k=27 X. s aft (7A) with Ni indicating the number of unit cells of the crystal in the ai direction and with ni and N, integers. For a triclinic unit cell [38] the reciprocal cell edges aſſº may be expressed in terms of the unit cell angles, oi, and edges as eq (8A) where S is given by eq (9A). a lisin & (8A) l (li S 3 3 1/2 s=|| 12 II cos or-X cos” al (9A) accordingly k can be written simply as 3 k = X. Ki/a; (10A) where n; sin Qi j = 27 “Hºl A K; = 27 NS (11A) The sum of interest is then eq (12A). 3 Ök Ki - Ki X (li ;--> (li a; (li k (12A) If we accept for the moment the potential energy term in X ai(0Eky/dai) as - Vºy, then eq (6A) may be written — — — | – V. 2m 6a; | Jºy kar doº .* |W (13A) - — I — , , y x * .2 2 * of ðai 2m 6a; electron-nuclear terms contain ai in the form eq (14A), 3 rºm = x + xn = x + X. aili (14A) i where x refers to position of the kth electron and the mth nucleus and the Li are integers only if the nuclei are exclusively at the lattice points. The variation of the potential terms for the y band then gives N | %k Vºy–X. uous-lº tºld: }}} ſkm ſºm (15A) Under equilibrium conditions the total potential is at a minimum with respect to the variations of ai. This per- 404 mits the sum over all the bands to be equated to the in- ternuclear repulsive potential. The constant term in eq (31) is then some appropriate fraction of this sum. [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] 13. References Pauling, L., Nature of the Chemical Bond, 3rd ed. (Cornell University Press, Ithaca, New York, 1960). Pauling, L., Proc. Roy Soc. 196A, 23 (1949). Rundle, R. E., Acta Cryst. 1, 180 (1948). Ganzhorn, K., Z. Naturforschg. 7A, 291 (1952); 8A, 330 (1953). Carter, F. L., Rare Earth Research III, L. Eyring, Editor (Gor- don and Breach, New York, 1965), p. 495. Carter, F. L., Fifth Rare Earth Research Conference, held at Iowa State University, Ames, Iowa, Aug. 1965 (preprint). Carter, F. L., Second International Conference on Solid Com- pounds of Transition Elements, held at Tech. University of Twente, Enschede, The Netherlands, June 1967. Carter, F. L., Proceedings of the Seventh Rare Earth Research Conference, held at Coronado, California, Oct. 28-30, 1968 (preprint), p. 283. Sneddon, I. N., Fourier Transforms, (McGraw-Hill, New York, 1951), lst ed., p. 368. Crangle, J., Electronic Structure and Alloy Chemistry of the Transition Elements, P. A. Beck, Editor Publishers, New York, 1963), p. 51ff. Weiss, R. J., and Tauer, K. J., Phys. Rev. 102, 1490 (1956). Abrahams, S. C., Guttman, L., and Kasper, J. S., Phys. Rev. 127, 2052 (1962). Franck, J. P., Manchester, F. D., Martin, D. L., Proc. Roy. Soc. (London) 263 A, 494 (1961). Kasper, J. S., Theory of Alloy Phases, (American Society for Metals, Cleveland, Ohio, 1956), p. 264. Clogston, A. M., Matthias, B. T., Peters, M., Williams, H. J., Corenzwit, E., and Sherwood, R. C., Phys. Rev. 125, 541 (1962). Pearson, W. B., Acta Cryst. B24, 7 (1968). (Interscience [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [3]] [32] [33] [34] [35] [36] [37] Laves. F., Theory of Alloy Phases, (American Society for Metals, Cleveland, Ohio, 1956), p. 124. Kjekshus, A., and Pearson, W. B., Progress in Solid State Chemistry 1, 160f, H. Reiss, Editor (Macmillan Co., New York, 1964). Kauzmann, W., Quantum Chemistry. (Academic Press Inc., New York, 1957). Pauling, L., Theory of Alloy Phases, (American Society for Metals, Cleveland, Ohio, 1956), p. 220. Brewer, L., Electronic Structure and Alloy Chemistry of the Transition Elements, P. A. Beck, Editor Publishers, New York, 1963), p. 221. Rundle, R. E., J.A.C.S. 69, 1327 (1947); J. Chem. Phys. 17, 671 (1949). Wallace, P. R., Phys. Rev. 71,622 (1947). Slonczewski, J. C., and Weiss, P. R., Phys. Rev. 109, 272 (1958). Pauncz, R., Alternate Molecular Orbital Method, (W. B. Saun- ders Co., Philadelphia, 1967). Hirst, R. M., and Linnett, J. W., J. Chem. Soc. 1035, 3844 (1962). Empedocles, P. B., and Linnett, J. W., Proc. Roy. Soc. (London) A282, 166 (1964); Trans. Faraday Soc. 62, 2004 (1966). Podolsky, B., and Pauling, L., Phys. Rev. 34, 109 (1929). Hicks, B., Phys. Rev. 52,436 (1937). Coulson, C. A., Proc. Cambridge Phil. Soc. 37, 55 (1941). Coulson, C. A., and Duncanson, W. E., ibid, 37, 67 (1941). Coulson, C. A., ibid, 37, 47 (1941). Duncanson, W. E., ibid, 37, 397 (1941). Duncanson, W. E., and Coulson, C. A., ibid 37, 406 (1941). Callaway, J., Energy Band Theory, (Academic Press Inc., New York, 1964), p. 184. Smith, R. A., Wave Mechanics of Crystalline Solids, (John Dickens and Company, Ltd., Northampton, Great Britain, 1963), p. 307. Frost, A. A., and Lykos, P. G., J. Chem. Phys. 25, 1299 (1956). (Interscience [38] International Tables for X-Ray Crystallography I, 13 (Kynock Press, Birmingham, England, 1952). 405 Electroreflectance Observation of Band Population Effects in InSb R. Glosser,” B. O. Seraphin,” and J. E. Fischer Michelson Laboratory, China Lake, California 93555 It is found that bias changes applied to n-InSb produce shifts in portions of the electroreflectance spectra. We attribute this to changes in the conduction band population produced as the separation between the Fermi level and the bottom of the conduction band is varied. Spectra which displays red, blue, or no shift correlates to electronic transitions starting from, ending at, or bridging the Fermi level. These observations permit a band structure identification of the shifting spectra and optical monitoring of the surface potential. Key words: Band population effects; electroreflectance; Fermi level shifts; indium antimonide (InSb); optical transitions; surface potential. It has long been known that the Fermi level in n-type InSb can easily be moved into and up the conduction band by increasing the bulk concentration of donors. This rise of the Fermi level is responsible for the blue shift of the absorption edge as was first explained by Burstein [1] and Moss [2]. In contrast to shifts produced by changing the bulk doping of n-type InSb, we have found that the position of the Fermi level can similarly be changed in the surface with respect to the conduction band by varying the surface potential with bias. Such effects have been postulated earlier [3]. This observation is made by monitoring the direction of shift of certain parts of the electroreflectance spectra which permits the classification of the observed struc- ture into three categories: spectra displaying red, blue, or no shift which correlate to transitions starting from, ending at, or bridging the Fermi level. This aids in their band structure identification. The results establish band population effects as an additional modulation mechanism in electroreflectance. It also permits opti- cal monitoring of the surface potential and possible determination of the effective mass of otherwise inac- cessible bands near K– 0. The manner in which this effect comes about is shown in figure 1. Consider a degenerate n-type crystal with a p-type surface. As it is biased positively, the bands move downward and the conduction band drops further below the Fermi level. According to this model, for transitions ending at the Fermi level, the threshold M E WºW Ey. (x) § & §§ & \ \\ \ Ev + E. (Mo) *Present address: Physics Department, University of California, Santa Barbara, Califor- nia 93.106. **Present address: Physics Institute I, Technical University of Denmark. Lyngby/Denmark. S st WTV N & NW N § \\ sº *—Hº- X Xol -K O K FIGURE 1. Schematic showing the shift of the threshold energy AE at absolute zero for an optical transition as the bands of degenerate n-type InSb are shifted with respect to the Fermi level EP by an applied bias. A valence band E, and a conduction band E, are shown as a function of depth X from the surface (left) and as a function of momentum K for an arbitrary depth Xo about I point (right). Modulation around a given bias is represented by the band edge wobble. The symbols – and + represent the bias direction for a p- and n-type surface, respectively. 407 Emax {ev) | 1.95– 1.90– | | | || Q @ * Q O O - Ll InSb, 6x10" cm^N, ,” 300 K w O us O O O 1.10 - e * O 1.05- - o I--V. FIGURE 2, Spectral shift with bias of structure associated with a transition ending at the Fermi level (lower half), as compared to reference structure associated with a transition bridging the Fermi level (upper half). Circles represent positive and squares negative extrema of the response. Arrows point in the direction of increasing peak magnitude. All data are taken in the MIS configuration. Modulation is + 300 mV. energy is shifted towards the blue. Transitions starting from the Fermi level should show a red shift for these same conditions. Those transitions not involving the Fermi level should yield electroreflectance signals which show little shift with bias and change sign upon changing the polarity of the surface potential [4]. We have observed all three of these effects in the electroreflectance spectra of InSb. The geometry and apparatus have been described earlier [5]. Samples were prepared in both the MIS [6] (metal-insulator- semiconductor) and electrolyte configurations [7]. Figure 2 brings out the contrast between a transition ending at the Fermi level and one which bridges it. The structure at 1.1 eV has previously been identified as originating from the spin-orbit split valence to conduc- tion band (ſzo to Tóc) [8]. A blue shift with bulk doping has been seen in the corresponding transition in GaAs [9]. Here we see a blue shift of about 95 MeV with ap- plied bias which should be compared to the structure at 1.9 eV where over the same range of bias practically no shift is observed. This latter spectra is well identified as originating from the A4 + A5 to A6 transition and consequently bridges the Fermi level. The blue shift at 1.1 compares very well with transmis- sion measurements where the bulk doping is varied Emax ev. © e o O 1.85 - 1.80 - I I - V. —l O +l dc FIGURE 3. Spectral shift with bias of structure associated with transitions starting from the Fermi level (upper two plots), as com- pared to reference structure associated with a transition bridging the Fermi level (lower plot). Circles represent positive and squares negative extrema of the response. Arrows point in the direction of increasing peak magnitude. All data are taken by the electrolyte tech- nique. Modulation is + 60 mV. [10] if one equates a strongly p-type surface with in- trinsic InSb (conduction band depopulated) and equates the flat band position (+0.25 V as observed by the polarity reversal of the 1.9 eV structure) with the bulk doping of 6 × 101° crim-8. A complementary red shift with bias application is found at 3.1 and 3.5 eV which we label A-A'. This is shown in figure 3. We again use the A4 + A5 to A6 transition as a comparison. In the MIS configuration, internal photoemission [11], above 2.8 eV at room tem- perature, tends to keep the surface p-type by populat- ing trapping states. In order to overcome this problem, the data was taken using the electrolyte technique [7]. Over the bias range used, we see that the A calibration structure shifts at most 20 MeV which is in marked contrast to the 80 MeV shift of the A-A' doublet. At room temperature, no other structure was seen in the 3 to 4 eV range with the electrolyte technique. Each member of the doublet exhibits only one peak at 300 K. At liquid nitrogen temperatures in the MIS structure, the A-A doublet is now complemented by weaker structures at 3.4 and 3.8 eV, labeled B-B'. The spectra 408 in H 2,O | 2.50 In Sb N Type | | 6 x 101° CM-9 2 H. | 80° K #| | |} Plane 5 X Greater •=-º- Thon Scale 3.59 | H. 3.2 O ! O __-TN | - - * - in | M | 3. – I H iss' 244 3.16 - l | | | | | | 2.O 2.5 3.O 3.5 Pillſ [IB|| || Electroreflectance spectrum of an InSb MIS-sample at liquid-nitrogen temperature. The doublets A-A and B-B are discussed in the text. FIGURE 4. are shown in figure 4. Because of the internal photoemission problem, no meaningful bias effects could be made in this temperature and energy range. With our observation of the bias produced red shift of A-A' and the weak structure at 3.4 and 3.8 eV and comparing this with recent band structure calculations for InSb we draw the following conclusions. The A-A" pair on the one hand and B-B' on the other are mated pairs in the sense of components of a spin- orbit split doublet. The peaks A and A' correlated to transitions starting from the Fermi level in the lowest conduction band and ending in higher conduction bands [12] (Tec to Tic and I go to Isc). The B-B' doublet probably ends in the same set of conduction bands but starts from the top of the valence band. Further support for the interpretation of the A-A.' structure is obtained by comparing our results and the electroreflectance spectra of Shaklee et al. [13] and Cardona et al. [9] with the thermoreflectance spectra reported by Matatagui and co-workers [14]. Electroreflectance spectra for n-type InSb always shows structure at 3.1 and 3.5 eV but weak or no struc- ture at 3.4 and 3.8 eV. Thermoreflectance in contrast, shows no structure at the former doublet but response for the latter. The spectra yielded elsewhere by the two types of modulated reflectance spectra are entirely comparable, in particular, the structural features near 2 and 4 eV. This is in line with our assignment of A-A' structure to transitions starting from the lowest conduc- tion band. Thermal modulation should only weakly af- fect a transition that starts above the band minimum at the Fermi level, in contrast to electric field modulation operating at such a transition through band population effects. As a further test of our interpretation of these results, we observed the electroreflectance spectra of a p-type sample carrying a p-type surface. As expected, the A- A' structure is practically absent except for a very small residual structure probably caused by the in- cident light and/or the electric field in the surface. Bias- ing towards the flat-band condition causes the structure to grow in agreement with the idea that conduction band population is being increased. These results and their interpretation can be com- pared with the results of Bloom and Bergstresser [15]. The observed separation of the A-A' doublet is in excel- lent agreement with their prediction of a splitting of Tzo and Tsc by 0.38 eV. The separation of A and B on the one hand and A' and B' on the other is 0.21 eV which is close to the fundamental gap of InSb at liquid nitrogen temperature as required by our assignment. Aspnes [16] has pointed out that either of the A-A' structure represents the first hope of observing an M3 critical point. The structure observed evidently stems from a superposition of the singularity at the Fermi level with that at the bottom of the conduction band. An experiment at liquid He temperature with a more highly doped sample (and an n-type surface) would be neces- sary to clearly separate the spectra due to the two sin- gularities. As for the assignment of the weak B-B' doublet, two possibilities are available. They most probably represent transitions at the T point from the top of the valence band to the higher conduction bands, but they can represent transition from the split valence band near T in the A direction to the second lowest conduc- tion band. The splitting predicted by Bloom and Berg- stresser is 0.42 eV which is sufficiently close to our ob- served value that this possibility must also be con- sidered. Further work is necessary to conclusively de- cide the assignment of B-B'. Regardless of which of the possibilities explains the exact origin of the weak doublet, we determine the separation of the conduction band at Tzo from the top of the valence to be 3.4 eV as compared to the calculated value of 3.6 eV [15]. In addition to the usefulness of band population ef- fects for band structure identification, it appears that the spectral shifts could be used to determine the effec- tive mass of the higher conduction and the split off valence band. A straightforward derivation using the 409 optical gap relation given in reference 1 yields, for ex- ample, the effective mass of the spin orbit split valence band m),(S0) = C. m., (FE) . m., (FE) . [my (FE) (1 –C) + m, (FE)]−1 where C = 6AE(FE)/6AE(SO) is the ratio of the ob- served spectral shifts for a given change in surface potential and mn(FE) and mp(FE) are the known effec- tive masses at the fundamental edge. Because the fun- damental edge was not accessible with our equipment, we could not determine mp(SO). References [1] Burstein, E., Phys. Rev. 93,632 (1954). [2] Moss, T. S., Proc. Phys. Soc. (London) B76, 775 (1954). [3] Aspnes, D. E., and Cardona, M., Bull. Am. Phys. Soc. 13, 27 (1968); Seraphin, B. O., J. Physique 29, C4-96 (1968). [4] Seraphin, B. O., Hess, R. B., and Bottka, N., J. Appl. Phys. 36, 2242 (1965). [5] Seraphin, B. O., J. Appl. Phys. 37,721 (1966). [6] Pidgeon, C. R., Groves, S. H., and Feinleib, J., Solid State Com- mun. 5, 677 (1967); Ludeke, R., and Paul. W., in II-VI Semiconducting Compounds, D. G. Thomas, Editor (W. A. [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] Benjamin Inc., New York, 1967) p. 123; Fischer, J. E., Bottka, N., and Seraphin, B. O., to be published. Shaklee, K. L., Pollak, F. H., and Cardona, M., Phys. Rev. Let- ters 15,883 (1965). Groves, S. H., Pidgeon, C. R., and Feinleib, J., Phys. Rev. Let- ters 17,643 (1966). Cardona, M., Shaklee, K. L., and Pollak, F. H., Phys. Rev. 154,696 (1967). Kosogov, O. V., and Maramzina, M. A., Sov. Phys.-Semicon- ductors 2,854 (1969). Mueller, R. K., and Jacobson, R. F., J. Appl. Phys. 35, 1524 (1964); Glosser, R., and Seraphin, B. O., Z. Naturforsch. 24a, 1320 (1969). Interconduction band transitions have previously been ob- served in transmission measurements in other materials. Zal- len and Paul found this to occur in n-Gap at the X point [Zallen, R., and Paul. W., Phys. Rev. 134, A1628 (1964)]. The recent work by Patrick and Choyke clearly shows intercon- duction band transitions in cubic n-SiC to occur also at the X point [Lyle Patrick and W. J. Choyke, Phys. Rev. (to be published)]. Other observations are discussed in these references. Shaklee, K. L., Cardona, M., and Pollak, F. H., Phys. Rev. Letters 16, 48 (1966). Matatagui, E., Thompson, A. G., and Cardona, M., Phys. Rev. 176,950 (1968). Bloom. S., and Bergstresser, T. K., Solid State Commun. 6,465 (1968). Aspnes, D. E., private communication and this conference. 410 Spin-Orbit Effects in the Electroreflectance Spectra of Semiconductors B. J. Parsons and H. Piller” Michelson Laboratory, China Lake, California 93555 Measurements have been made of the electroreflectance spectra of germanium, gallium arsenide and gallium antimonide in the range of photon energies from 0.6 to 6.7 eV. Special attention has been paid to the resolution of multiplicity within the Eo' and E1' structures. The identification of these struc- tures in terms of critical point interband transitions involving the second conduction band has an impor- tant bearing on the band structure of these materials. The data is discussed in terms of the possible identification, within these higher energy multiplets, of spin-orbit splittings associated with the valence band states at T and L. Key words: Critical points; diamond semiconductors; electroreflectance; gallium antimonide (GaSb); gallium arsenide (GaAs); germanium;'semiconductors, diamond; semicon- ductors, zinc blende; spin-orbit splittings. 1. Introduction The interpretation of electroreflectance spectra in terms of critical point interband transitions relies heavi- ly on direct comparison between experimental data and calculated band structure. The high resolution of elec- troreflectance, however, permits the direct observa- tion, in many cases of interest of spin-orbit effects, which, at low energies, are interpretable with little am- biguity. In diamond and zinc blende semiconductors, for example, the spin-orbit splitting at T and along A (A0 and A1, [1] respectively) are easily identified within the Eo and E1 structures [2] and a wealth of information is available to substantiate these assignments. At higher energies the number of critical points increases and even tentative assignments become difficult in those not infrequent cases where the complexity of the ob- served structure suggests the near degeneracy of several critical points. Transitions from the valence band to the second con- duction band are expected to exhibit a multiplet struc- ture since each band has two spin-orbit components. The degree of multiplicity observed experimentally will depend on the absolute and relative magnitudes of the spin-orbit splittings and on the selection rules. Some guidance in the assignment of certain of these higher *Present address: Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana. energy structures is, in principle therefore, to be ex- pected from the recurrence within this multiplet struc- ture, of splittings characteristic of the initial valence band states. The Ao and A1 splittings, for example, might be expected to recur within the Eo' and E1' structures, respectively, these structures having been associated in the past with transitions at T and L [2]. These expectations depend crucially on the assumption that the transitions involved occur at the same point in k-space. Thus, the Ao splitting will recur exactly within the Ed' structure only if both the Eo and Eo' transitions occur at T. In the case of the E1 and E1' transitions these almost certainly do not occur at the same point along the A direction. There is evidence to suggest, however, that the A1 splitting of the A45, and Ago bands is relatively constant over an extended region of k- space which includes the A and L critical points. This is a controversial point, but for the present we take the observed doublet nature of the E1 structure [3] as justification for this assumption. The range of photon energies beyond 4.5 eV has been investigated less extensively than the lower energy re- gion and, because of the lack of a convenient and in- tense light source in this region of the spectrum, the available data is generally inferior to that obtained at lower energies. The E1' structure observed in this re- gion in a number of semiconductors has thus received only cursory attention in the past. We have used experi- 411 mental techniques which have permitted a more detailed study of this region and have resolved the mul- tiplicity in the E1' structure in a number of semiconduc- tors. The purpose of this paper is to report data for ger- manium, gallium arsenide and gallium antimonide which extends to 6.7 eV and to discuss the results in terms of the possible identification of recurrent valence band spin-orbit effects. 2. Experimental Details Two reflectometers were used to obtain the present data. The first, based on a single pass prism monochro- mator (Perkin-Elmer model 98) has been described el- sewhere [4]. The second is based on a 0.3 meter grat- ing monochromator (McPherson model 218) and utilizes a modular electronic signal processing system (PAR). The light source used in the 4.5-6.7 eV range was a deu- terium discharge lamp fitted with a suprasil window. The sample geometry used was the so-called dry sandwich configuration [5]. Polished and etched sam- ples were coated with a thin, 200 A, film of aluminum oxide, and a thin semitransparent conducting film of nickel completed the thin film capacitor arrangement. Electric fields well in excess of 108 V/cm are readily at- tained using this technique. This sample geometry has a number of advantages over other more popular con- figurations, not the least of which for the present work is the ability to cover a wide range of photon energies using cooled samples. 3. Results and Discussion 3.1. Germcinium Germanium has been widely investigated and data on the electroreflectance spectrum for this diamond group semiconductor is well documented in the literature [2,6-8]. For reasons not fully understood at the present, the recurrence of the 300 mV Ao spin-orbit splitting within the Eo' structure has not been established. Since the Eo' structure is relatively narrow and well separated from adjacent structures this is somewhat surprising. There is evidence, however, for the near degeneracy in this region of the spectrum of several critical point con- tributions to the optical properties, some of which may well be associated with off-center critical points. The E1' structure for germanium appears as a shoulder in the reflectance spectrum [9] and weak structure in the electroreflectance spectrum has been observed [2,8] at corresponding energies. This struc- ture has generally been associated with the L3'-L3 quadruplet. In germanium the splitting of the Lé, and Lisc levels of the second conduction band is expected to be smaller than the valence band splitting at L (A1 = 200 mV.) and we expect to be able to identify the recur- rence of this 200 mV effect. The spectrum obtained at 300 K in this region of the spectrum for a 0.3 ohm-cm p- type sample is shown in figure 1. The single peak at 5.75 eV has a half-width of only 100 mV suggesting that this peak is not associated with the L. multiplet. Weak structure on the low energy side of this peak is barely resolved into a doublet with close to the correct splitting. No significant improvement is gained over this spectrum on cooling the sample to liquid nitrogen temperatures (80 K). It is noteworthy that we find no structure in the 4.65-5.1 eV region where structure has been observed in photoemission [10]. We conclude that the associated peak in the density of states is not a critical point effect. 3.2. Gallium Arsenide The lack of inversion symmetry in the III-V com- pounds causes degeneracies in the valence and conduc- tion bands at X to be lifted. The resulting multiplicity of the X transitions complicates the E2 structure for these materials and makes the identification of structure in this region more difficult. The spectrum for n-type galli- um arsenide between 3.6 and 6.7 eV is shown in figure 2. The peaks at 4.45 and 5.02 eV (300 K data) are well established [2] and have previously been labeled Eo' and E2, respectively. It is clear that considerable over- lap exists between satellites and components of these two structures. An edge between the two main peaks is resolved at 80 K. The observation of a small peak at 3.95 eV, which we interpret as an interconduction band AR x 16’4 Ge 2 I- 0.30-cm p-type 300°K | ºm o H- 2-T^\ ſº- 4 5 6 7 / El - | ºm –2 L- E 2 PHOTON ENERGY (eV) FIGURE 1. Electroreflectance spectrum for p-type germanium at 300 K in the region of the E structure. 412 GaAs n= 7 1018 cm−3 4 5 L-N a \ . A Aº. O ~~~~ x 16’4 7 | 300°K – H. Cºlºr - A E O LTN / ) /N ' 1–T | 80°K 4 5 NL-Té 7 –2 H. E6 –4 – º, PHOTON ENERGY (eV) —- FIGURE 2, Electroreflectance spectrum for n-type gallium arsenide in the region of the E. E., and E structures at 300 and 80 K. transition [11,12] is close to small structure which has been reported in the reflectance [13]. This suggests that the peak at 5.50 eV is a component of the Eo' Stru Cture. At high energies, peaks are observed at 5.70 and 6.55 eV (300 K data). The 6.55 eV peak corresponds to a shoulder in the reflectance [14] at the same energy which has been associated with the L3’ – La transitions. The observed structure is clearly in- complete in our data and confirmation of this assign- ment will await an extension of the electroreflectance spectrum to higher energies. The two peaks at 5.50 and 5.70 eV do have the correct 200 mV separation to be as- sociated with the recurring splitting, but in view of the expected multiplicity and overlap of the Eo' and E2 structures in this region this is probably accidental. 3.3. Gallium Antimonide The electroreflectance spectrum of n-type gallium antimonide at 300 K is shown in figure 3. The Ao and Al splittings (730 mV and 440 mV, respectively) are larger than in germanium or gallium arsenide and the Eo' and E1' structures are correspondingly broader and more detailed. The band structure of gallium antimonide, ac- cording to a recent calculation by Zhang and Callaway [15], yields values for the important energy level separations given in table 1 which also lists the ob- served electroreflectance peak energies. Data at the fundamental edge shows a pronounced dependence on bulk doping and the quoted energies for the Tse — Too and Tzo – Tec doublet are probably high. The spectrum for p-type material suggests that light and heavy hole transitions play an important role at low temperatures. We find no evidence for more than two peaks in the re- gion of the E structure. The single peak at 1.50 eV is al- most certainly associated with the Tzo – T6c transitions. GaSb 300°K n = 4.10'cm−3 0.5 PHOTON ENERGY eV —s- | 2 3 4 5 6 FIGURE 3. Electroreflectance for n-type gallium antimonide at 300 K. TABLE 1. Calculated energies of important interband transitions in gallium antimonide according to Zhang and Callaway [15] and the experimental peak energies observed in electroreflectance. Note that the Tzv-Tze transition is not allowed in the unperturbed crystal. Experi- Calculated mental peak Structure | Transition energy energy (eV) (eV) ------- 300 K 80 K Eo Tse—Téc 0.81 0.78 0.89 Tze-T6c 1.59 1.50 1.60 E L4, 5p–L6c 1.98 2.04, 2.13 A4, 55- A6c 2.10 L6-L6c 2.42 2.48 2.59 Age – A6c 2.60 E6 and E9 Tec – Tic 3.01 2.94. T66 – Tsc 3.45 3.11 3.20 Tso-Tze 3.82 3.31 3.34. Å79–X6c 4.26 3.38 3.4] Tso-Tsc 4.26 3.68 3.75 X68–X6c 4.29 4.03 4.13 X75–X7c 4.64 4.16 4.34 X68–X7c 4.67 4.62 4.57 T70–Tsc 5.04 4.72 E; L4, 50–L6c 5.48 4.95 5.11 L4, so-L4, 5c 5.7] 5.48 5.62 L6-L6c 5.92 5.94, 5.98 L6-L4, sc 6.15 413 (b) PHOTON ENERGY eV -º- n, 80°K oHV-4––––––––. TV NZ TV I º r I FIGURE 4. Detailed behavior of the electroreflectance of gallium antimonide in the region of the E structure: (a) for an n-type sample at 300 K, (b) for the same n-type sample at 80 K, (c) for a second n-type sample at 80 K, and (d) for a p-type sample at 80 K. The very detailed spectrum between 2.8 and 4.2 eV is shown in figure 4 (a) for an n-type sample at 300 K, (b) for the same sample at 80 K, (c) for a second n-type sample at 80 K, and (d) for a p-type sample at 80 K. The behavior in this region is considerably more complex than previously reported [2,16,17]. Peaks at 3.2 and 4.13 eV are present in all the 80 K spectra, other peaks at 2.94, 3.34, 3.41 eV and the broad peak at 3.75 eV are all dependent in some way on the bulk doping and/or the surface preparation. Further studies of the behavior of these peaks will be necessary before a satisfactory interpretation of this data can be proposed. The clear doping dependence of the lower energy members of this group indicates, however, that interconduction band transitions are important in this region. Similar behavior has been reported in indium antimonide [11,12]. Other structure in the same energy range is probably associated with critical points close to Talong the A direction and correlated to the behavior of the Aš' band. On this basis the valence band to conduc. tion band transitions at T should occur at energies close to the E2 multiplet and considerable unresolved overlap is thus expected in this region. The small peaks at 4.57 and 4.72 eV are also probable members of one or other of these two groups of structure (see fig. 5). The spectrum between 4.0 and 6.7 eV is shown for an n-type sample in figure 5. At 80 K the E1' structure is / 4 H. F2 E. 2 H. /-/Y. 300°K oHA-\7 | } ſº - - 17 -3 Trº) at room temperature. In addition, near the experimental sixth half oscillation where the light and heavy hole contributions are of opposite signs, we observe destruc- tive interference which greatly modifies the signal lineshape in that region. The unique characteristics of the resultant lineshape allow the determination of the relative magnitudes of the dipole matrix ele- ments and reduced masses for the two bands in a region of k-space somewhat removed from the T- point. The experimental results also demonstrate that neither thermal broadening nor field in- homogeneity need be a problem in electroreflectance measurements. Key words: Electro-absorption techniques; electroreflectance; Franz-Keldysh theory; germanium; oscillatory dielectric function. The predictions of the effect of an electric field on the band structure of solids show that the dielectric function becomes oscillatory as a function of the energy in the vicinity of critical points in the joint density of states of the valence-conduction bands. Therefore both the electro-absorption (EA) and the electroreflectance (ER) techniques should show a large number of oscilla- tions with a very slowly decreasing envelope [1]. Un- fortunately, in both EA and ER only four or five half oscillations have been observed for most materials stu- died [2-5]. In addition, the lineshapes that were ob- served were not in good agreement with the one-elec- tron theory for a number of reasons. In EA the discrepancy between theory and experi- ment arose because the Coulomb interaction had not been taken into account. In ER a number of serious problems were present in addition to the neglect of the Coulomb interaction. For example, in many experi- ments no effort was made to modulate from the flat band condition, although the theory was presented in those terms. Thermal broadening was incorporated in the theory at an early date and produced lineshapes which were in better qualitative agreement with experi- ment; however, it is now evident that broadening was being forced to account for other additional factors, and hence the broadening parameters so obtained were ex- cessively large. The problem of electric field variation over the penetration depth of the light was anticipated at an equally early date [5] but was not adequately treated until recently by Aspnes and Frova [6], who found that a significant field variation within one reciprocal wave vector of the surface would mix the real and imaginary parts of the dielectric function, giving rise to peculiar lineshapes. Both Frova and Aspnes [7] and Seraphin and Bottka [8] have shown good experimental verifica- tion of this point at the direct edge in Ge. The publica- tion of these results seemed to indicate that it would be very difficult to obtain easily interpretable lineshapes except at very small fields in intrinsic material. Since then, however, Koeppen and Handler [9] have shown that, although intrinsic material is best for small fields, doped material is much superior for higher fields. Figure 1 (taken from ref. 9) shows graphs of R1, the *Supported by the Advanced Projects Agency under Contract No. SD-131 and the U.S. Army Research Office (Durham) under Contract No. DA-HC04-67-C-0025. 417–156 O – 71 – 28 417 lo' F- | | | I – Usé ACCUMULATION 6 — |O H. g|S - Tol-U –|UU || 0." ſoº 4 |O lo' ELECTRIC FIELD 8. FIGURE 1. R. versus magnitude of surface electric field e for four values of impurity concentration for Ge at 300 K. Except for the up = 6 curve, only the depletion region is shown. The curves can be used for n- or p-type material. The points A, B, and C indicate the fields above which it is better to use a sample with the next higher doping level shown. logarithmic derivative of the surface electric field with respect to distance, plotted versus the magnitude of the surface electric field under equilibrium conditions for several different impurity concentrations. The smaller R1 is at any given field, the less important the problem of field nonuniformity will be. The smallest value of R for each value of the field corresponds to a different up. Thus, for a peak surface field of 2-104 V/cm, a sample of up = 6 will be optimum. Furthermore, the results so obtained should exhibit no more mixing than results ob- tained at e = 3-10° V/cm with intrinsic Ge. In com- parison, Frova and Aspnes found measurable mixing ef- fects in intrinsic Ge lineshapes for e > 104 V/cm. Using Ge samples doped in accordance with figure 1, we have obtained results in the high field limit which are in excellent agreement with simple one-electron theory. The data include both the direct edge (Tst -> Tºº) and spin orbit split (T7" -> Tri-) structures and show a number of effects not previously observed. These new results further suggest that it may become possible to determine the source in k-space of other ER signals. The standard electrolyte electroreflectance technique was used in obtaining the present data. Ex- cept for the following two modifications, the experimen- tal system was the same in most respects as that described by Hamakawa et al. [10]. First, the optical method for determining flat band described in that reference has since been improved so that it is now possible to monitor the proximity of the bands to the flat band condition every alternate half cycle while the experiment is in progress. Details of this will be presented in another paper published elsewhere. Secondly, potentiostatic control of the modulation and bias voltages is now employed for greater stability of conditions in the sample cell, and voltages quoted below were measured via a salt bridge located adjacent to the surface of the grounded Ge sample. The surface potential was square wave modulated at 310 Hz between flat band and arbitrary band bendings. The energy bands were always modulated in the depletion direction for greater field uniformity in the space charge region as suggested above. Under certain dynamic conditions it was even possible to reduce R. below the equilibrium values shown in figure 1. The sample was a 0.13 Q cm n-type wafer of Ge at room temperature (up=6.2) with a (110) reflecting face. A Polaroid Corporation type HR sheet polarizer was used to polarize the light either along the [110] or [001] axis of the reflecting surface. Several graphs of AR/R are plotted in figure 2 versus a common energy scale which extends from 0.7 to 1.2 eV. Experimental lineshapes are drawn as solid curves, and all were obtained with a peak-to-peak modulation voltage of 0.7 V from the flat band condition. Because the values of AR/R range over three decades, the data for the [110] and [001] polarizations are displayed semi-logarithmically in figures 2b and 2C, respectively. For comparison, the [110] data are also plotted versus the more customary linear scale (with one scale change) in figure 2a. Of immediate interest is the large number of half oscillations present in the Ts" -> Tº structure: the oscillations extend nearly 0.2 eV above the gap energy where they finally disappear in the rising ex- ponential tail of the spin orbit split structure. A com- parable number of half oscillations has only been ob- served in the EA spectrum of the indirect edge of Si [11], shown in figure 3. This graph serves to demon- .OO4 | .OO2 ſ\ j d:(||O) |\ O | –-> NZ ve ^2=~\ ez- —.OO2 |x2O – OO4 AR. R IO-3 - 4 |O –5 |O 16° Energy (eV) FIGURE 2, Electroreflectance spectra between ().7 and 1.2 eV for light polarized as indicated (note that the T — l’; structure exhibits no polarization dependence). ( urve (a) is plotted linearly with one scale change: curves (b) and (c) are plotted semi- logarithmically. Solid curves are experimental; the dashed line in (c) is a least-squares computer fit using one-electron theory. The rms noise magnitude during this work was approximately 2. () ". 418 | i | | | }- S|L|CON - +,O6 ELECTRO – ABSORPTION of, T = 23 o C E 404 - - 2 ID X— - - & +O2 § ſ\ H O ZºS ~~~ CO \y S-2 \Z Or <[ > - O2 H - d Tz- and Tit-> Ty- structures, respectively. Our experience with thermal broadening precludes the possibility of values signifi- cantly greater than these for this energy range at room temperature. Another striking feature of the lower energy struc- ture is the collapse of the oscillation envelope and a brief irregularity in the oscillatory nature of the lineshapes just after the 5th (3rd positive) experimental S J2 > O) § § -.6 H T. 8 —.8 f= T- –|.O H + i + Tz T- –|.2 --- | | | | O ...Ol .O2 k(atomic units) FIGURE 4. Energy band diagram of the bands contributing to the structures shown in figure 1. Arrows 1 and 2 represent equal energy transitions of 0.91 eV, the central energy of the region of destructive interference. This figure is only schematic, since it neglects changes in curvature and anisotropy of the bands, as discussed in the text. peak at approximately 0.9 eV. This effect is caused by the mutual cancellation of the light and heavy hole con- tributions to the resultant lineshapes. The period of oscillation of the electro-optic function is inversely pro- portional to the cube root of the reduced mass of the bands involved, and thus the difference in effective masses of the degenerate valence bands (see fig. 4) causes a periodic beating of contributions from equal- energy transitions, such as those indicated by arrows 1 and 2, centered at different points in k-space for the light and heavy hole bands. Since the contributions from the two bands are comparable in amplitude but opposite in sign, the resultant lineshape is a sensitive function of both the precise ratio of signal amplitudes and the ratio of reduced masses. The ratio of the reduced masses parallel to the electric field determines how far above the gap the cancellation will occur, and the ratio of oscillator strengths determines the exact lineshape in the beat region. Blossey [12] has defined a parameter F (electric field strength in units of effective Rydbergs per Bohr radius) which is a measure of the electric field strength relative to that of the Coulomb interaction. He shows that for F ~ 20, the electroreflectance spectra at the direct edge energy will not be greatly altered by the presence of the Coulomb interaction. For the data shown in figure 2, F = 174 and F = 61 for the light and heavy holes, respectively. In addition, since the region of destructive interference occurs -0.1 eV (57 to 96 ef- 419 fective mass Rydbergs) above the gap, the electron-hole pairs can certainly be thought of as nearly free parti- cles. We shall compare the observed spectra with sim- ple one-electron theory using Lorentzian-broadened electro-optic functions G(m,y) defined by Aspnes [13]. Furthermore, we shall assume that the complex Ts" -> T; structure can be represented by the algebraic sum of two G-functions, although in the immediate vicinity of the direct gap the degeneracy at k = 0 and the Cou- lomb interaction should make the fit less exact. The dashed curves in figure 2c are the result of a least-squares computer fit of the following functions to the low and high energy structures, respectively: AR A Y R tº ...) T(ho)? [BG (my, y) + G(mh, y)] (1) AR A ' f R. (T: – T. ) T(ho)? G(mso, y') (2) In the above, A and A' are amplitude factors, ha) is the photon energy, B is the ratio of light to heavy hole con- tributions, y and y' are the broadening parameters, and the m's are defined as ... — E; – ha) Ei-ha) 3 "Tº h6; (3) 29 where e is the peak effective electric field at the Gesur- face, and Ei, h0; and pºli are respectively, the effective energy gap, the electro-optic energy, and reduced mass associated with transitions from the ith hole band, i = ^ (light), h (heavy), or so (spin orbit split) to the conduc- tion band. Figure 5 shows an expanded view of the beat region for both polarizations, the solid curves representing the experimental data, the dashed lines the computer fit of eq (1). The fit between theory and experiment for both polarizations is remarkably good; the only serious dis- crepancy is the too-rapid diminution of the envelope of the experimental structure with increasing energy. We expect that even this difference would disappear if the Coulomb interaction and a slight amount of electric field inhomogeneity were taken into account. These results of simple superposition of the light and heavy hole bands are also in agreement with the calculations of Enderlein et al. [16]. Table 1 lists the values of the various parameters ob- tained from the computer fit: only pºso was assumed given; all other quantities were derived relative to it. To elaborate, the electron and hole masses for the Tºt -> Tº structure were taken to be .036 mo [14] and .077 mo [15], respectively. The fit of eq (2) then uniquely .88 .90 .92 .94 Energy (eV) FIGURE 5. Expanded view of the heat region of the Tº -> Ty structure for both polarizations. Solid curves are experimental; dashed lines are the theoretical predictions of eq (1). specified hôso, y', Eso, and the electric field e. The parameters pertaining to the Ts" -> Tº structure were subsequently derived by using the value of the field e obtained above. The reduced masses measured by this technique are average values since the effect of the electric field is to mix states in k-space, especially those within one unit of m (25-30 meV) of the point under consideration. Nonetheless, two points can be made on the basis of these data. First, ph has a magnitude greater than the electron mass at the T-point, indicating the probability that both the electron and heavy hole band have un- dergone a significant change in curvature within .025 atomic units of k = 0 (see fig. 4). Secondly, the TABLE 1. Summary of experimental parameters T; → T. Tº – T. pso - 0.0245 mi,” ” pu = 0.022 mo (.0195 mo at k=0) pun- .038 mo (.033 mo at k=0) e = 4.2 : 104 V/cm e = 4.2 : 104 V/cm h6.0 = 30.3 meV h6) = 31.5 meV h6), - 26.1 meV y' = 9 meV y = 3 meV Eso = 1.08 eV EI, n = .79 eV B110 = .71 (110 polarization) Booi = 1.16 (001 polarization) 420 anisotropy of the heavy hole band is clearly indicated by the polarization dependence of the lineshape in the beat region. Since B depends only on the ratio of the dipole matrix elements for the two transitions, which in turn can be shown to be functions of the effective masses evaluated along the polarization direction [17], the difference between B110 and Booi is an indication of the difference between corresponding components of the heavy hole mass tensor in these two directions. In fitting the data many small effects have been neglected such as the mass dependence upon the direction and magnitude of k, the energy dependence of the broaden- ing parameter and the Coulomb interaction. In fact it is quite surprising that the fit is as good as it is. In conclusion we have shown that reliable ER data can be obtained when the energy bands are modulated from flat band and the sample is doped so that the variation of the electric field with distance is not impor- tant. We anticipate that it will now be possible to separate exciton effects from one-electron behavior at the direct edge in Ge since mixing can be made negligi- ble in both high and low field regions where the respec- tive effects dominate. Furthermore, by use of these techniques at higher photon energies, it should be possible to obtain lineshapes which are sufficiently good so as to permit the unambiguous identification of critical points which gave rise to those structures. [1] [2] [3] [4] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] References Aspnes, D. E., Phys. Rev. 147,554 (1966): 153,972 (1967). Seraphin, B. O., Modulated Reflectance, (to be published in Optical Properties of Solids, F. Abeles, Editor, North Holland Publishing Co.). Seraphin, B. O., Electroreflectance (to be published in Semiconductors and Semimetals, K. Willardson and A. Beer, Editors, Vol. VI, Academic Press, New York). Cardona, M., Modulation Spectroscopy, Supplement 11 to Solid State Physics, F. Seitz, D. Turnbull and H. Ehrenreich, Edi- tors (Academic Press, New York, 1969). Cardona, M., Shaklee, K. L., and Pollack, F. H., Phys. Rev. 154,696 (1967). Aspnes, D. E., Frova, A., Sol. State Comm. 7, 155 (1969). Frova, A., and Aspnes, D. E., Phys. Rev. 182,795 (1969). Seraphin, B. O., and Bottka, N., Sol. State Comm. 7,497 (1969). Koeppen, S., and Handler, P., Phys. Rev., to be published. Hamakawa, Y., Handler, P., and Germano, F., Phys. Rev. 167, 709 (1968). Frova, A., Handler, P., Germano, F. A., and Aspnes, D. E., Phys. Rev. 145,575 (1966). Blossy, D. F., Ph. D. thesis, University of Illinois (1969). See reference 1; also, at the direct edge, AR/R = Ael - G(m,y). Zwerdling, S., Lax, B., and Roth, L. M., Phys. Rev. 108, 1402 (1957). Smith, R. A., Semiconductors (Cambridge University Press, New York, 1961), p. 350. Enderlein, R., Keiper, R., and Tausenfreund, W., Phys. Stat. Sol. 33,69 (1969). Smith, R. A., Wave Mechanics of Crystalline Solids (John Wiley and Sons Inc., New York, 1961), pp. 464-466. 421 Variations of Infrared Cyclotron Resonance and the Density of States Near the Conduction Band Edge of InSb% E. J. Johnson and D. H. Dickey Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Massachusetts O2139 The electronic density of states near the conduction band edge of InSb with and without a magnetic field is obtained from dispersion relations based upon k p interactions with nearby bands and parame- ters determined and confirmed by several intraband experiments including fundamental cyclotron resonance, spin resonance, spin-flip cyclotron resonance, phonon assisted cyclotron resonance and har- monics of cyclotron resonance. The density of states displays effects due to nearby band interactions and due to electron-phonon interaction. Key words: Cyclotron resonance; electronic density of states; electron-phonon interaction; InSb: interband transitions; Landau levels. 1. Introduction The electronic density of states can be obtained in a straightforward manner, once one has an electron ener- gy dispersion relation in which one has some con- fidence, and if one has negligible broadening of the energy levels. We have determined the experimental parameters for such a dispersion relation for states near the bottom of the conduction band of InSb [1] by obtaining experimental data that is more precise and covers a wider range of conduction band energies than previous data. Our analysis takes into account interac- tions not previously recognized and reconciles various interband optical experiments for the first time. In this paper we summarize this work and apply it in determin- ing the density of states both with and without a mag- netic field. A variety of experiments involving in- traband optical transitions confirm the dispersion relá- tion for energies to ~80 meV above the band edge. The density of states near the bottom of the conduc- tion band of InSb can be expected to depart from sim- ple parabolic band behavior, principally due to interac- tions between the conduction band and other nearby bands. These interactions for InSb have been treated theoretically by Kane [2]. The effects on the magnetic energy levels have been treated by Lax et al. [3] and by *This work was sponsored by the Department of the Air Force. Bowers and Yafet [4]. When the cyclotron energy is comparable to an LO phonon energy an additional ef- fect on the density of states occurs due to a polaron self-energy effect [5]. In InSb this effect occurs in a narrow energy range and can be easily accounted for in fitting the nonparabolic band theory to the experimen- tal results [6]. The experimental results were obtained by observing the infrared transmission of InSb samples versus mag- netic field for fixed photon energies from 4.5 to 50 meV. Absorption peaks were observed which can be identified with a variety of intraband transitions involv- ing Landau levels. The fundamental cyclotron resonance absorption occurs as a triplet, and is used to determine the band edge effective mass. A combination resonance transition is observed which involves a spin flip, in addition to a fundamental cyclotron resonance transition, and is used together with spin resonance results of Isaacson [7] to determine the band edge ef- fective g-factor. Using these band edge parameters energies for transitions involving cyclotron resonance harmonics, phonon assisted cyclotron resonance, and the variation of spin resonance energy with carrier con- centration are calculated and are found to agree well with observed transition energies. The consistency among the various intraband transitions provides con- 423 fidence in the nonparabolic band theory and the values of the band parameters determined from the earlier ex- periments. The density of states is calculated from the corresponding dispersion relations. 2. Dispersion Relations The wave functions for the energy levels near the bottom of the conduction band of InSb have s-like sym- metry near k = 0 and the energy surfaces are nearly spherical. The energy levels are given in Wigner-Bril- louin perturbation theory by h2/2 &” – (1) Eo (k) 2m * where mº is an energy dependent effective mass given by m . . . I |p;|* nº lºt, > E, (k) – E (0) (2) where the summation is taken over all energy levels l with energy at k = 0 of Ei(0), and where peſ” is the m component of the momentum matrix element between bands c and l at k = 0. To a good approximation we can neglect the energy dependence involving energy bands more remote than the three (six) valence bands and can write (2) in the form ºc 2 m –H 771. 2. p: e (3) 77). 177) b 171 E, (k) – E (0) valence bands This gives a cubic equation for the energy, Ec(k). The solution relevant to the conduction band can be written __Fulhº (l, I Eo (k) = 2 –– 2 (+ #) Ea h 2/2 º]" –H 2 | | + 4. 2m, TE- (4) where me is defined at k = 0 and is given by m_2 2 Eat #A Inc. m. |(s|p=|z)| En (Ey + A) (5) where A is the spin-orbit splitting of the valence band. The function f(Ec) is equal to unity at k = 0 and is weakly energy dependent /, ºut A Bºº)+º, + A 6 ' E,-HåA E,(A) +E,--A (6) The density of states is obtained from (4) in the usual way for spherical energy surfaces p(E)=4 ſ /*A (E, k) dk (7) 7T ~ J0 where A(E,h) gives the distribution in energy of the level of wave vector k. If the energy levels have negligi- ble broadening, E(k) is single valued; A(E,k)= 6(E-E(k)) and (7) becomes (lE ) (8) | 1.2 (dE p(E)=#| | (# To obtain the energy levels in a magnetic field we have to solve the effective mass equation º +A) 2m” c 1 I+ 39°S h–E.1 (ſ )=0 (9) cj where A = (1/2)H X r and 8 is the Bohr magneton. The effective g-factor, g”, is energy dependent and given by J" l! — , , ! J’ pºiſ)} - pºiſ)}. E.; – E (0) (10) 2 . Sk — ‘ s - & (Ec, j) = 2+ gro- g X. 771 l valence bands where the contribution due to interactions with remote bands is concentrated in the constant term gro and the last term involves interactions with the three valence bands. The solution relevant to the conduction band is given by Eq E, - 2 En }| m +: {1+} (2n+1) ;Bh | hºkº 1/2 =}ºpuſ, ºil (11) where |-11 !--- (12) mo m m rb me ' — o ". A ******-*. Fix (13) Eg-HåA ſº E.I.E.T.A. (14) Equation (11) provides a dispersion relation from which one can calculate the density of states with ap- plied magnetic field. The singularities which occur in the density of states at k = 0 provides for well defined optical absorption lines. Using such experiments one can determine the parameters precisely. 424 3. Determination of Parameters In figure 1 we show experimental traces of sample transmission versus magnetic field at certain fixed photon energies below and above the reststrahl energy. These samples are ~20p in thickness and reveal only the strong absorption near hoc. Near the region ha) To sº hu < holo the sample is opaque due to high reststrahl reflection. We see that the absorption in general in- volves three peaks whose relative strengths depend upon temperature. Simple arguments [1] show that the high energy peak is associated with impurity transi. tions, which we shall not discuss further here. The two lower energy peaks are associated with fundamental cyclotron resonance which is spin split due to non- parabolic band effects. For hy - hoto large broadening occurs and the three peaks are not resolved. The ob- served cyclotron resonance energies are plotted versus magnetic field in figure 2. The solid lines are our best fit of the nonparabolic band theory to the experimental points. We found that we could obtain a good fit to the lower energy part of the data if we ignored the points above hu = 18 meV. A further indication of anomalous behavior in this region is shown by the discontinuity in the data on either side of the reststrahl region. This anomalous behavior is due to polaron self energy ef- fects and have been discussed in detail previously [6]. The behavior has been shown in more detail in experi- ments involving interband transitions [8] and combina- tion resonance [9]. In attempting a best fit to the data we found that the splitting between the cyclotron resonance lines is in- sensitive to the precise choice of the magnitude of g’ between values of 40-70. This uncertainty in the value of g’ results in only a slight uncertainty in the value of m. obtained from the fit. We obtain a value m = 0.0139m = .0001 m for the band edge effective mass in InSb. 2 hy = 5.34 meV C - -5 an E T `s 2 £ hu = 24.4 meV C - 2 hy = 15.25 meV O - (ſ) O - > (ſ) 2 – <[ Crº H –– | | | | | | | O 8 16 24 32 4O MAGNETIC FIELD (kG) FIGURE 1. Cyclotron resonance spectra observed in thin samples (t = 20p) at temperatures near liquid helium such that T2 - T. 28 26H 24 |- ...” 22 H | 4 O H | | | | | | | | | O 4. 8 {2 + 6 20 24 28 32 36 4O MAGNETIC F | ELD (k(S) FIGURE 2, Variation of cyclotron resonance absorption peaks with magnetic field. The solid lines give the theoretical predictions using m =0,139 and g’=-51.3. Spin resonance experiments of Isaacson [7] for a sample with n = 3.6 × 1018 cm-3 and H - 0.5 kG yield an experimental g-factor of HE 51.3 + 0.1. Such a sample would have a Fermi level ( ~ 0.3 meV above the band edge and should yield a value of g-factor very close to the band edge value. To check this value of band edge g-factor and to check the variation of spin energy with magnetic field, we have observed and analyzed com- bination resonance transitions which involve the transi- tion E01 to E1 (i.e., a cyclotron resonance transition plus a spin flip). The shift of the resonance peaks with magnetic field is shown in figure 3 and compared to cal- culations of the energy E – Eo according to eq (11). Excellent agreement between experiment and theory is obtained for magnetic fields less than ~27 kG. We conclude that the value of –51.3 for the band edge fac- tor is a good one and that eq (11) faithfully predicts the spin energy for fields up to 27 kG. The discrepancy above 27 kG is easily attributed to polaron self-energy effects. 4. Further Tests Having determined the parameters m, and g' we are in a position to calculate the density of states. However, 425 35 O 3OH 25 H. g 20 H © 5 2 15– AOH- 5 º- | | | | | | —l- O {O 2O 3O 40 H ( k G) FIGURE 3. Variation of combination resonance peaks with magnetic field. The solid line gives the theoretical prediction. 7O 6O H. (3,0) (4,0) 5O H. (1,1) (2,1) p 3. (2,0) 4O H. S QD E * O O > -C O 3OH [] (1,0) 2O H. 1O H. | | | | O |O 2O 3O 4O H (kG) FIGURE 4. Shift with magnetic field of the absorption peaks in- volving cyclotron resonance harmonics and phonon assisted FIGURE 5. transitions. The designations describe the excited states (Landau quantum number, number of photons). The solid lines give the theoretical predictions. we first apply some further checks on the values ob- tained. Experimental results using a thick sample t = 2mm and n = 1.4 × 10° crimº” have also been obtained. This sample thickness permits the observation of weak ab- sorption on the high energy side of the fundamental cyclotron resonance. For h u = 18.75 meV the observed absorption peaks are spaced approximately equal in 1|H. In figure 4 we see that the peaks observed at 18.75 meW converge to hu = 0 at H = 0. These peaks involved transitions from the same ground state to higher and higher Landau levels (i.e., cyclotron resonance har- monics). Additional peaks are seen which converge at H = 0 to an energy about 24 meV, which is close to the known value of the LO phonon energy. These transi- tions apparently involve the emission of an optical phonon. The transition energies for several of the cyclotron resonance harmonics have been calculated using eq (11) and the band parameters determined in section 3. These results are shown as the solid lines in figure 4. Also shown are the calculated cyclotron resonance harmonics shifted by 24.4 meV. A good agreement between experiment and theory is obtained. The identification of the transitions involved is con- firmed and the fit to the data gives a value of 24.4+ 0.3 meV for the longitudinal optical phonon energy. The agreement the experimental cyclotron resonance harmonic transition energies and the calcu- lated values supports the use of eq (11), and the parameters previously obtained to calculate the mag- netic energy levels to energies ~80 meV above the band edge. between n (cm−3) | 4.5 x 16° 2x40° | | § 4.2 H. 4OH |→ | | | | | | | O 2 4 6 8 |O H2 14 46 18 2O Ú (meV) Comparison of experiment and theory for variation of spin resonance with Fermi level for InSb. The experimental points are the data of Isaacson [7] for H -- 125 and ~ 500 (;. The solid line is a composite for the theoretical predictions at 0.125, 0.500, 3 and 20 k(;. 426 According to eq (11) the spin energy depends upon Landau quantum number. In a spin resonance experi- ment one can select the Landau levels whose splitting is observed by doping the sample and moving the Fermi level through the Landau levels. Such a spin resonance experiment has been performed by Isaacson [7]. His data are reproduced in figure 6 where the Fermi energy has been calculated from the carrier concentration with the help of eq (11). The spin energies are expressed in terms of gerp which is defined at resonance as gexpº hushigh where husk is the photon energy used in the experi- ment. The solid line in figure 5 is our prediction of the variation of gerp with Fermi level. Excellent agreement is obtained for £ S 8.5 meV, which confirms the ability of (11) to predicts the spin energy. A discrepancy of ~ 5% occurs at C = 14 meV, but this may be an impur- ity effect. 5. Density of States The dispersion relation for the conduction band deduced from these experiments is shown in figure 6. The dashed curve is the result for a parabolic band with K (cm−") { 4.5 2 2.5 3 3.5 x 40° | | T/ | | 18O H. / - 46O H. / - 440 H. / - 12 O H / - 4 OO H. / -: i / 80 H. / - sol / - aol / - l - | | | | 8 AO 42 x 40% 2 ) | 6 k? (cm FIGURE 6. The conduction band dispersion relation deduced from these experiments. The dashed line gives the parabolic band result using band edge value of effective mass. an effective mass equal to the InSb band edge mass. The corresponding density of states curve is given in figure 7, where again the dashed curve is the result for a parabolic band. The InSb density of states increases more rapidly than for a parabolic band and shows a marked departure for energies greater than ~3 meV above the band edge. In figure 8 we show the density of states for InSb in a magnetic field of 20 kG. The dashed curve gives the results for a simple parabolic band. At higher energies the peaks in the density of states crowd together and increase in magnitude due to the increase in effective mass and decrease in effective g-factor. For a parabolic band the cyclotron resonance joint density of states has a singularity at the cyclotron ener- gy hoc, and is zero elsewhere. For a nonparabolic band this behavior is modified as can be demonstrated qualitatively using eq (11). For these purposes we sim- plify (11) by setting f = f; = 1 and neglect spin by setting g’=0. The cyclotron resonance energy (hu) is then given by 2hly 4 T m hºk: T ) tº # ={i+}|3; shº] - f / Eq En L. m. 2m. 4 T m h°k: 1/2 — 1 + = | – 3H + | 15 | Eq in B 2m. (15) E ( meV) 1 4 |O 20 4O 7O 4 OO A 50 200 I- | | | |- | | | 48 x 40°- - 46 H - 4 4 H. - : 12H - E RS 'E 4 O H. * Sº * 8 – - Q- 6 H - _-T 4 H. -- T - ...~" _ T ...” __ T 2 H. 2 ” - 2- | | | | | | | O 2 4. 6 8 AO 42 {4 + v'E (mev°) FIGURE 7. Density of states versus energy. The dashed line gives the parabolic band result. 427 6 x 1019– H = 20 kG InSb H = O 5 H- 2' | | 4 H | | cº-se N N || || | U \ | |\ | 3 H |\ –tº ſº PARABOLIC | | _+\ TN | \, \! BAND | \ | \, \! H = 20 kG 2 H | | y^2+ | 4 H | \s]] | | | | | | | | | | | O AO 2O 3O 4O 5O 6O 7O 8O 90 HOO E (meV) FIGURE 8. Density of states in a magnetic field of 20 kG for InSb. The density of states is also shown for a parabolic band and for H=0. From (15) one obtains the cyclotron resonance joint density of states as given by No (dhv \' p(hp)=# (#) where No is the Landau level degeneracy, which is in- dependent of energy. If ******", an 2m. 4 in B •) eq (15) can be approximated by (16) 2 hºk: En 2m. hu (kz) = hly (0)|l – 12 m 4 m {1+} p \{1+}sh) (17) The derivative is given by dhv (kz) dk2 | | h2 1 / 2 - |vo - = 12 m 4 m. 1/4 |{1+}sh)|-}n} [hv(0) – hy (k2)] /* (18) The variation in the joint density of states with energy is given principally by the factor [hv(0)=hw(k)]!”. The derivative goes to zero at hu(kz) = hly(0) giving a singu- larity in the joint density of states. Therefore, the peak observed in cyclotron resonance absorption cor- responds to the Landau level separation at k2 = 0. In ad- dition, the joint density of states is nonzero for energies less than hu(0), in contrast to the case with a parabolic band. In figure 9 we show a computer calculation of the joint density of states for InSb, at H – 20 kG. This cal- culation is based upon eq (11) without simplification using the experimentally determined parameters. The calculation displays the expected qualitative behavior. A singularity occurs at hu = hly(0) and the density at states is nonzero for lower energies. A second singulari- ty occurs corresponding to k2 = 0 for the higher energy spin state as seen in the experiments. Indeed, addi- tional singularities corresponding to higher Landau quantum numbers for the ground state occur at lower energies than these shown. The infrared absorption is proportional to the density of states multiplied by a factor involving the population of the energy levels as given in the lower part of figure 9. In the computer calculation, the Fermi level is ad- justed to give the correct carrier concentration for the given magnetic field and temperature. 428 > @ E ES – 4 '5 Ei-Ü , , Ef-t – 2 x 10" A (E-E) X | (H+)-(+} | | | |→ –2.5 –2.O — 4.5 - A.O -O, 5 [hv-ha.] (meV) FIGURE 9. InSb cyclotron resonance joint density at states com- puted using eq (11) and the experimental parameters. It is to be noted in figure 1 that the experimental ab- sorption at 20 kG is asymmetric being sharper on the low magnetic field side which is consistent with the nonparabolic density of states. From figure 9 we ob- tain a half width for the absorption at 25 K of about 0.1 meV. This corresponds to an experimental half width of ~0.3 meV deduced from figures 1 and 2. Therefore, it is apparent that additional broadening mechanisms are present. The techniques for taking the broadening mechanisms into account is discussed in the accom- panying paper by B. Sacks and B. Lax. The broadening mechanisms remove the singularity in the joint density of states, but a peak still should occur at k2 = 0, as in- dicated by the behavior of the single particle density of states calculated by Kubo [13]. 6. Discussion The density of states near the conduction band edge of InSb, both with and without a magnetic field, have been obtained from dispersion relations. These disper- sion relations are based on a nonparabolic band theory and the values of band parameters m, and g’ determined by several intraband experiments. These dispersion relations are found to hold to within 5% for conduction band energies to ~80 meV above the band edge. Presumably, the corresponding density of states are also good to this accuracy. It is interesting to inquire whether the remote band terms 1/mr, and grº in eqs (12) and (13) are necessary to explain the experimental data. An examination of the equations shows that the m. and g’ are not independent parameters in the absence of the remote band terms. We have, however, experimentally determined mº, and g. essentially independent of each other. We find that equations (12) and (13) can be reconciled with the ex- perimental results only if we assume grº = 0.lg' and (m,n)- = 0 or (mrū) = −.1(m.) and gro = 0, or some combination of nonzero values of these. Therefore, the remote band interactions are significant. We have that the intraband experiments are consistent with one another. The transition energies calculated from a single dispersion relation agree well with experiments. This consistency does not appear to apply to intraband versus interband transitions. Pidgeon and Brown [10] type of analysis of interband absorption [11] gives a value of m, that is 7% higher than the intraband value. The source of this discrepancy is currently under investigation. A simple theory of the intraband absorption in a mag- netic field would predict a single absorption line at the cyclotron resonance frequency. Most of the additional transitions we observe can be explained in terms of various band interactions. However, there is currently no explanation which is consistent with experiment for the cyclotron resonance harmonics. In calculating the density of states from the disper- sion relation we neglect energy level broadening. For H = 0 the effect of energy level broadening on the density of states would be most serious at the band edge where a tail on the density of states can be expected that ex- tends into the energy gap. The nature of this tail has been the subject of much theoretical work, but defini- tive experiments are very difficult [12]. shown various 6 x 10°H H. 5 H- !--- e-º. 4 H. | -- Lil Q- | 3 H - 2 -- |− |-- 4 --- 33 34 35 36 E ( meV tºlo — ( meV) O AO 2O 3O 4 O E (meV) FIGURE 10. Magnetic density of states for ha) = halo. The dashed curve gives the unperturbed result. The solid curve shows the modification in the density of states by electron-phonon interaction predicted by Nakayama [5]. 429 In the presence of a magnetic field, collision broadening can be expected to smooth out the sharp structure in the magnetic density of states shown in figure 10. The problem of energy level broadening of Landau levels has been treated theoretically by Kubo et al. [13] who find significant departures from theory based upon a simple Lorentzian broadening of each level associated with a value of ke. Collision broadening of Landau levels is also discussed in the ac- companying paper by Sacks and Lax. The experiments indicate the presence of an anoma- ly in the density of states when hoc = holo. In figure 10 we show how the density of states obtained from the dispersion relation is expected to be modified by the electron-phonon interaction. The dashed curve is the result of the same calculation used in figure 8 for lower magnetic field. The solid curve gives the modification of the density of states predicted by the theory of Nakayama [5]. The n = 1 Landau level is split and a zero occurs in the density of states at an energy cor- responding to holo above the lowest Landau level. The splitting in the density of states has been seen in inter- band absorption [8] and combination resonance [9] and is consistent with the fundamental cyclotron resonance results. However, it remains a challenge to experimentally observe the detailed structure in the density of states shown in figure 10. 7. Acknowledgments The authors would like to express their appreciation for discussions concerning the cyclotron resonance joint density of states with B. Lax. 8. References [1] Johnson, E. J., and Dickey, D. H., Phys. Rev. (to be published). [2] Kane, E. O., J. Phys. Chem. Solids 1, 249 (1957); also Semicon- ductors and Semimetals, R. K. Willardson and A. C. Beer, eds., Vol. 1. (Academic Press, New York, 1966) p. 75. [3] Lax, B., Mavroides, J. G., Zeiger, H. J., and Keyes, R. J., Phys. Rev. 122., 31 (1961). [4] Bowers, R., and Yafet, Y., Phys. Rev. 115, 1165 (1959). [5] Nakayama, M., J. Phys. Soc. Japan 27,636 (1969). [6] Dickey, D. H., Johnson, E. J., and Larsen, D. M., Phys. Rev. Letters 18, 599 (1967). [7] Isaacson, R. A., Phys. Rev. 169, 312 (1968). [8] Johnson, E. J., and Larsen, D. M., Phys. Rev. Letters 16, 655 (1966). [9] Dickey, D. H., and Larsen, D. M., Phys. Rev. Letters 20, 65 (1968). [10] Pidgeon, C. R., and Brown, R. N., Phys. Rev. 146, 575 (1966). [11] Pidgeon, C. R., and Groves, S. H., Phys. Rev. (to be published). [12] Johnson, E. J., Semiconductors and Semimetals, R. K. Willard. son and A. C. Beer, eds., Vol. 3, (Academic Press, New York, 1967) p. 249. [13] Kubo, R., Miyake, S.J., and Hashitsume, N., Solid State Phys. 17, 269 (1965). 430 Cyclotron Resonances of Holes in Ge at Noncentral Magnetic Critical Points J. C. Hensel and K. Suzuki Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey O7974 The anomalous, “quantum” cyclotron resonance spectrum of holes in the degenerate valence bands of Ge is analyzed utilizing the concept of a critical point in the magnetic joint density of states. Contrary to previous work done for kH = 0 (where kh is the hole wave vector along the magnetic field) our results indicate that cyclotron resonance lines originating at critical points away from kh = 0 are respon- sible for most of the prominent features of the observed quantum spectrum in Ge. Key words: Cyclotron resonances; germanium; joint density of states; Landau levels; magnetic; magnetic critical points. 1. Introduction At low temperatures and high magnetic fields (i.e., fia)/kT ≤ 1 where a) = eHolmc) cyclotron resonances of holes in the degenerate valence bands of Ge exhibit a complex “quantum” spectrum, observed first by Fletcher [1]. According to Luttinger and Kohn [2,3] this anomalous behavior results from the deviations in spacing between the valence band Landau levels bear- ing low quantum numbers from the uniform intervals characteristic of the higher “classical” levels. The ex- ceptional complexity of the resultant spectra observed presents a formidable challenge in their interpretation. Earlier attempts [4,5] to analyze the hole quantum spectra in Ge have achieved only limited success. In the present work we have made a detailed analysis of the quantum spectra in Ge utilizing the concept of a critical point in the magnetic joint density of states. This may be understood as follows. A cyclotron resonance absorption line due to transi- tions between two Landau states n and n' can be in- terpreted in terms of a critical point in the one-dimen- sional joint density of states (6enniſokn)- where enn is the energy between Landau states and kh is the wave vector along the direction of the magnetic field. This quantity plays, by analogy, the role of the three-dimen- sional joint density of states of the interband optical transitions but with two (transverse) degrees of freedom quantized by the magnetic field. This analogy for mag- netic transitions has no particular utility for simple bands; however, in more complex situations such as the degenerate valence bands where the Landau states de- pend upon kh in a complicated way, the concept of a critical point becomes a highly useful one. Here the ex- istence of a critical point and, hence, a peak in the spectrum at points away from kh = 0 becomes possible. An investigation of the behavior of enn, as a function of km locates the critical points as well as predicts the cyclotron resonance line shape from the nature of the critical point. Contrary to previous analyses [4] done for kh = 0, our results show that cyclotron resonance lines originating away from kH = 0 are responsible for most of the prominent features of the observed quan- tum spectra in Ge. We illustrate these concepts by way of an example using the quantum spectrum in Ge for Ho! [11]]. 2. Energy Levels The Landau states for a hole in the degenerate Tst valence band of Ge are given by the Luttinger effective mass Hamiltonian [3], 2 2 l 2,--" |y * -y, | (/;- P) g-en| I?) –2)3 ({Jr./ y} {k,k,l} + cp) ++ kJ Ho-Hº- whº top)| (1) fic hc where k=v- “A. (2) l, C 431 Here, Jr., Jy, and J. are the components of the angular momentum operator J(J = 3/2) referred to the cubic axes; “cp” denotes cyclic permutation; the quantities {JºJº), etc. represent the symmetrized products, i.e., {JoJº) = 1/2(JºJº + JºJº); and A is the vector potential of the external magnetic field Ho. For simplicity we represent the hole Landau levels by a conventional energy level diagram at kii = 0; this is shown in figure 1 for the low-lying Landau states ob- tained by diagonalizing eq (1) for Ho! [111]. (It should be emphasized that in the analysis of the experimental spectra the assumption k h = 0 is not made.) The values of the parameters used are: y = 13.38, yº = 4.24, y2 = 5.69, k = 3.41, and q = 0.06. The in- verse mass parameters yi, y2, and ya have been ob. tained from precision cyclotron resonance experiments [6] in uniaxially stressed Ge; and the parameters k and q, which are related to the hole g-factor, are derived from combined resonance experiments [7]. Through- out the present work these values will be taken as “fixed”; therefore, the theoretical analysis in section 4 will involve no adjustable parameters. In figure 1 we classify the Landau levels (at kH = 0) by the scheme [8] (Nn, K"). Two quantum numbers are eſ 5O H. + 2,+ 4OH 9|o ;|É 4'- Li- - O Oſ) t 2: 3OH + -) 3,4- 2 + - - 2 - * ſt É 10,-- || -- os-- ; 9.4: -º- 2O H. 8 97 2+ 2- 87 – 8s— 1+ i- * O+ 6s- 64- - + - 10H 2, -é– 5,3- 53-3- gº |+ 4.-- + 42– - O - Ooº- 3, 9– 14 O+ lo – 32– 20+ Hollſ III] Valence band Landau levels in Ge at kh = 0 for Holſ111] calculated from the Hamiltonian in Eq. (1). As customary, the sign of the energy is inverted so that the levels are plotted in an “electron” sense. The energy levels are labeled by the scheme (Nn, K") discussed in the text. FIGURE 1. necessary to uniquely identify each eigenstate: N, the Landau quantum number for the envelope function, and its subscript n, which is required to differentiate among the four states of the same N. The quantum number K (K = 0, . . . v — 1) embodies the v-fold rota- tional symmetry of the crystal about the direction of Ho (v=3 for Holl||111]). T is the parity of the envelope function. Using the quantum numbers (Nn, K") it is possible to express the cyclotron resonance selection rules [9] for e L Ho, where e is the microwave electric field. If we symbolically categorize the successive symmetry breaking interactions Žo + 3% + 3%, + 3%, where 3%0 is the axial Hamiltonian: Ž, represents the kh terms; 3% contains “warping” terms proportional to yº — ya (or q); and Ž3 combines warping and kii interaction — we get a hierarchy of selection rules for a linear polarization of e, as follows: Žío AN = + 1, ATT = yes, AK = + 1 2%, AN = + 1 AK = + 1 Ž’s AT = yes, AK = + 1 2^3 AK = + 1 In general the selection rule AK--El [understood to mean AK = + 1 (mod v)] is the only strictly rigorous selection rule; the extent of the violation of the other selection rules depends upon the magnitudes of the symmetry breaking terms in the Hamiltonian, i.e., the magnitudes of yº — ya and kH. 3. Experimental Details and Results The cyclotron resonance experiments were done under “quantum” conditions at 54,000 Mc/s and at both 4.2 and 1.2 K on ultrapure samples of Ge. The above temperatures were strictly maintained by immersing the samples directly in the liquid He bath. The cavity spectrometer (TE101 rectangular mode) which employs a superheterodyne detection scheme was operated at very low power levels, 10-8 to 10-9 watts, to avoid cyclotron resonance line saturation. Also, for this same reason the samples were mounted in the cavity in a re- gion of low microwave electric field. Carriers, both holes and electrons, were generated in the Ge sample by illumination with white light. To minimize line broadening from high carrier densities the light intensi- ty was deliberately reduced by neutral density filters. The magnetic field was measured using field markers from proton NMR signals. 432 The complexity of the hole quantum spectrum is evident from the top two traces (a) of figure 2. These cyclotron resonance spectra were taken for Holſ111] with linear microwave electric field polarization e1 L Ho. The positions of the hole lines are indicated by their effective mass values. (In taking the spectra in figure 2(a) an integration time constant of 0.1 sec was employed; at higher gain and with longer time constants many of the weaker lines can be more clearly discerned.) At 1.2 K only a vestige of the classical hole lines at m”/m = 0.0421 and 0.374 remains; instead the cyclotron resonance intensity is redistributed among the twenty or more quantum transitions shown. As ex- pected, the hole spectrum displays a marked depen- dence on temperature. The increase in intensity of some lines upon going to 1.2 K identifies them as originating from the lowest-lying Landau states. In contrast to the electron lines, which have a simple Lorentzian shape characteristic of broadening by a relaxation time process, the hole lines are frequently asymmetric and of varied widths as expected for kh- broadening. In a few cases the shapes of the hole lines, in particular the line at m”/m = 0.250, display strain broadening and therefore, depend to some degree on the choice of sample. For certain samples a line at m*/m = 0.264 appears as a partially resolved shoulder. 4. Comparison of Experimental Results and Theory The analysis of the experimental spectra was accom- plished by computer diagonalization of the effective mass Hamiltonian in eq (1) using the values of the parameters given in section 2. The infinite secular determinants generated by the Luttinger Hamiltonian with “warping” (terms proportional to yº — yº) and kii terms simultaneously present were truncated at a total dimension of 69 × 69 (i.e., 3 dets. of 23 × 23), the smal- lest size which gave satisfactory convergence of the eigenvalues for all relevant eigenstates. The intensities of all possible transitions were calculated by computer and integrated over kh to yield synthesized quantum spectra. The kir-broadened resonances were assumed to be composed of Lorentzian component lines having a width defined by the scattering time T = 2.7 × 10−10 sec [10]. The results calculated for T-4.0 K and Hol [111] are shown in figure 2(b). The salient features of these spectra can be in- terpreted by using the aforementioned principle that maximum stationary contributions to the integration occur at critical points in the magnetic joint density of states. This may be seen by referring to figure 2(c) 417–156 O - 71 – 29 | which shows m”/m for each transition plotted versus {= kn/VeHoſhc, a dimensionless form for kh. (The transitions in figure 2(c) are labelled with reference to the energy levels at kh = 0 given in fig. 1.) The “mag- netic” critical points for the transition curves in figure 2(c) are the points of zero slope ô (m”/m) ðū We can categorize critical points as being either “strong” or “weak” depending upon the curvature of m*(Q)/m at the critical point. Nearly singular critical points of sharp curvature have a low density of states and contribute little to the integrated intensity. Conver- sely, more or less “flat” critical points, which have a high density of states, give strong resonances (all other factors being equal). In passing we note that for an iso- lated resonance the sign of the curvature dictates on which side of the asymmetric kil-line the tail will fall. Critical points must necessarily occur by symmetry at C = 0 for every allowed transition; but many such transitions in Ge exhibit “cusp-like” behavior at ( = 0 and do not give lines in the integrated spectra even though their “differential” intensity at C = 0 may be ap- preciable. We note in figure 2(c) several clear-cut exam- ples of isolated transitions having cusp-like & = 0 criti- cal points—-lo -> 21, 32 → 21, and 0) → 21 — none of which show up as resonances in figure 2(b) [nor in the experimental spectra in figure 2(a)]. Transitions 20 -> 31, 31 – 42, 43 -> 54, 54 -> 65, and 00-> 43 are of a similar kind and, likewise, give insignificant contributions to the integrated spectra. It would seem then that a sub- stantial fraction of the lowest “quantum” transitions anticipated at Ç-0 do not materialize in the spectra. On the other hand, numerous critical points off (- 0, the noncentral critical points, give strong peaks in the line shape. In particular we note the secondary critical point for 32 – 43, as well as critical points for many transitions forbidden by parity at (= 0, 20 – 00, 21 -> 65, 54 – 64, 21 → 43, 42 → 54, and 00 – 42, all of which con- tribute strongly to the spectrum in the heavy mass re- gion (m’ſ m > 0.2). In the intermediate mass region (0.1 < m”|m - 0.2) in figure 2(b) nearly all features result from noncentral critical points. We should point out that the line at m”|m = 0.125, the most prominent ex- perimental feature in this region at 1.2 K, comes from the noncentral transition lo -> 54 (forbidden at C = 0). Numerous other critical points for & # 0 contribute many weak lines in this region. [The “background” ab- sorption beginning at m”/m > 0.15 stems from contribu- tions from the two series of transitions labelled 0-> 1, 0' – l', etc. throughout the range [=2 to 5 (the upper = (). (3) 433 §§ º 2 ºf z m*/rn-> - {\} tº ºf if) tºº tº tſ) [O rº C N- G) 00 N- ºr tº) 1 ap- pear in the spectrum up to Q = 6 which corresponds to a value of kh - 1.5 percent of the distance of the [111] zone boundary. From the foregoing we conclude that the overall character of the spectrum is to a great ex- tent dominated by the kh-lines. This point is further emphasized if we look at the “differential” spectrum calculated for Q = 0 which is shown in figure 3. Aside from the unrealistic Lorentzian line shapes, this spectrum (with the possible exception of the “light” holes) does not bear even qualitative resemblance to the experimental spectrum in figure 2(a). The observed structure of the absorption in the in- termediate mass region is almost totally missing in figure 3; on the other hand the two strong peaks in the differential spectrum at m”|m = 0.130 and 0.140 are either absent or extremely weak in both the integrated and observed spectra. In the same way, the differential lines at m”/m = 0.33,0.35 and 0.57 are “washed” out by integration. Their near coincidence with experimental lines is quite fortuitous as the noncentral critical points actually turn out to be responsible for the observed resonances. Finally, in the differential spectrum there is no hint of the prominent experimental lines at m”/m = 0.307, 0.408, and 0.445 which are clearly portrayed in the integrated spectrum. Some of the line assignments proposed here may be convincingly verified by experiments [6] utilizing small uniaxial stresses (along [111]) which gently decouple the valence band and shift the quantum lines in a meas- ured fashion. We briefly mention two results. First, the resonance at mº/m = 0.250 (whose sensitivity to strain broadening has been previously noted) is split into a resolved triplet of lines by uniaxial stress in ac- cordance with the structure suggested in figure 2(b). Second, by observing the behavior of the lines m”/m = 0.125 and 0.133 in figure 2(a) under uniaxial stress we confirm their assignments lo -> 54 and 10 – 21, respec- tively. Careful comparison of the experimental and cal- culated spectra, furthermore, strongly suggests that the third member 10 -> 53 of the “fundamental” triplet is the weak line observed at m”|m = 0.117. In conclusion, we add that analyses similar to that above have been completed for Ho along the [001] and [110] crystal axes [11]. In both of these cases, just as we have seen for Ho! [111], the synthesized spectra faithfully reproduce nearly all significant features of the experimental spectra in regard to position, inten- sity, and line shape and further corroborate that the dominant contributions to the cyclotron resonance quantum structure stem from the noncentral critical points. 5. References [1] Fletcher, R. C., Yager, W. A., and Merritt, F. R., Phys. Rev. 100, 747 (1955). [2] Luttinger, J. M., and Kohn, W., Phys. Rev. 97,869 (1955). [3] Luttinger, J. M., Phys. Rev. 102, 1030 (1956). [4] Goodman, R. R., Ph. D. dissertation, University of Michigan, Ann Arbor, Michigan, 1958 (unpublished) and Phys. Rev. 122, 397 (1961). Hensel, J. C., Proceedings of the Interna- tional Conference on the Physics of Semiconductors, Exeter, 1962 (The Institute of Physics and the Physical Society, Lon- don, 1962), p. 281; Stickler, J. J., Zeiger, H. J., and Heller, G. S., Phys. Rev. 127, 1077 (1962); Mercouroff, W., Physica Status Solidi 2, 282 (1962). [5] The possibility of kh-lines was first pointed out by R. F. Wallis and H. J. Bowlden, Phys. Rev. 118, 456 (1960) who con- sidered only the spherical case (yº = ya). Subsequent calcula- tions which included kh-effects were made by V. Evtuhov, Phys. Rev. 125, 1869 (1962), and Okazaki, M., J. Phys. Soc. of Japan 17, 1865 (1962). [6] Hensel, J. C., and Suzuki, K. (to be published). [7] Hensel, J. C., Phys. Rev. Lett. 21,983 (1968), and Hensel, J. C., and Suzuki, K., Phys. Rev. Lett. 22,838 (1969). [8] We depart from the conventional Luttinger scheme for Holl [ll]] (see ref. 3) so frequently used, because it does not admit to generalization either for other directions of Ho or for the case of kh #0 which is germane to the present work. [9] Suzuki, K., and Hensel, J. C. (to be published). [10] This scattering time is taken from an earlier experiment (ref. 6). [11] Hensel, J. C., and Suzuki, K. (to be published). 435 Landqu Level Broadening in the Magneto-Optical Density of States” B. H. Scicks” Department of Electrical Engineering and Computer Sciences and the Electronics Research Laboratory University of California, Berkeley, California 94720 B. Lax Francis Bitter National Magnet Laboratory and the Department of Physics, M.I.T. Cambridge, Massachusetts O2139 In this paper we derive a convolution integral expression of the optical density of states for k-con- serving transitions between broadened Landau levels in crystalline solids. The convolution is between single Landau level densities of states expressed in a form derived by Kubo. The expression includes as parameters the reduced mass, magnetic field strength, and the strength of the broadening mechanism. We present the results of numerical evaluation of this expression for various values of the parameters. Our consideration is directed to dipolar transitions between the n = 0 Landau levels of the valence and conduction bands, and we assume that the dominant broadening effects are intraband scattering processes. This assumption is reasonable in view of the comparative intraband and interband lifetimes. The resulting optical density of states function has an appearance identical to that of the single band function, but with the frequency measured from the gap energy replacing the energy measured from the band edge, the reduced mass replacing the single band mass, and a reduced broadening parameter replacing the single band parameter. Our expression for this last parameter is shown to be a consequence of a mutual consistency requirement between the single band and optical densities of states, and this same requirement also leads to the reduced lifetime expression encountered in the more conventional Lorentzian formulation. Key words: Delta-function formulation of density of states; Green effect; k-conserving transitions; Landau levels; laser semiconductor; magneto-optical density of states; optical densi- ty of states; semiconductor laser. 1. Introduction In this paper we derive a convolution integral expres- sion of the optical density of states for k-conserving transitions between broadened Landau levels in crystalline solids. The expression includes as parame- ters the reduced mass, magnetic field strength, and the strength of the broadening mechanism. We present the results of numerical evaluation of this expression for various values of the parameters. Our consideration is directed to dipolar transitions between the n = 0 Landau levels of the valence and Rººned by the Air Force officeoscientific Research otherancis Blue National Magnet Laboratory, M.I.T. and the Joint Services Electronics Program, grant AFOSR-68-1488 to the University of California, Berkeley 94720. conduction bands, and we assume that the dominant broadening effects are intraband scattering processes. This assumption is reasonable in view of the relative in- traband and interband lifetimes. 2. Single Landau Level Density of States 2.1. Density of States in Delta-Function Formulation We begin by observing that in general terms a densi- ty of energy states for a quantum-mechanical system can be expressed by the operator relation p(E) = trø(3%–E) (1) **Formally with the Francis Bitter National Magnet Laboratory, M.I.T., Cambridge, Mas- sachusetts. where 3% is the appropriate Hamiltonian. This notation, of course, implies a complete set of states between 437 which the operator on the right side is sandwiched, and the trace “sweeps” over this set. Equation (1) represents a “counting” process. The simplest demonstration of this formulation can be carried out when the set of states used is the collec- tion of eigenstates of the Hamiltonian in the operator. In the case of simple parabolic energy bands, the elec- tronic eigenstates in the presence of a magnetic field are the familiar Landau functions. || – K'ºe'ſ!!e"2* (b., (x-xO(k))) u0(r) (2) where K* is a normalizing constant whose magnitude depends upon the magnetic field strength, q), is the nth harmonic oscillator wave function, and u0(r) is the cell- periodic part of the Bloch wave function associated with the band edge. (This form is not unique, but rather is the result of the choice of the unsymmetrical Landau gauge A = (O.Hx,O) to express the vector potential of the magnetic field. We will use this gauge and its result- ing wave functions throughout this work. All physical results, of course, turn out to be gauge independent.) The energies associated with these Landau levels are given by h2|-} Ecn = (' 2 (n + 1/2) ha), + 2mo (3.a) h?/ ? E,--E})– (n + 1/2)ho, -º-; (n=0, 1, 2, . . .) * 2m (3.b) –l | F p(b)=#tr Im Ta-F, -ī, The trace sums over the quantum numbers k, and k2 of the Landau states. Thus the term “Lorentzian broaden- ing” indicates that the density of states is the sum of contributions at a given energy from the tails of Lorentzians whose centers are distributed over the whole energy range. To give this result a physical interpretation, we can consider that the perfect periodicity of the lattice is slightly perturbed by the broadening mechanism, and hence kº is no longer a good quantum number for states at a single energy. In other words, eqs (3a,b) no longer hold, and states with any given value of k, are spread over the entire energy range in a Lorentzian distribu- tion centered at the original value. Equivalently, we can say that an approximation to the wave function at any given energy is a linear combination of states of all values of kg, with coefficients deduced from the Re 47°/2 VE – (1/2) hoc — ifiſt for levels originating in the conduction and valence bands, respectively, with the zero of energy taken at the zero-field conduction band edge. For the n =0 Landau level of the conduction band, the eigenvalues are expressed in terms of kº, and the traces are readily converted to integrals over k, and kº, leading to the well-known result p(E) = trø(7% – E) dk, dk, 6 hºll, E (2T/Ly) (2T/L.) (; 5 nor ) | 2 TE y ––. (*!)" ºr | 472/2 \ h 2 E – 1/2ha), where lº = (eH/hc)-1 is the square of the cyclotron orbit radius. It should be remarked that if the delta functions in energy are replaced by Lorentzian distributions of con- stant width h|T - l h/T 8(? E)–: (Ž – E)?-- hºſt? (5) which, we note, can also be written as l h/T = Im | (6) T (? – E)2 + h^/T2 iſ (Ž–E) – ifiſt then the density of states at any given energy E is given by the expression - (*; 1/2 | *#) (7) Lorentzian formula. For the density of states at the energy of interest, we then count all the states that enter the linear combination, weighted by their Lorentzian coefficients. Summations of a related sort will appear in subsequent sections of this paper. In ap- pendix A the optical density of states is derived for transitions between such Lorentzian-broadened sets of initial and final states. 2.2. Expressions in Terms of the Resolvent Operator In order to treat the effects of scattering processes more rigorously, we must include in the Hamiltonian the scattering potential, written in general as V. How- ever, since the trace is invariant under the similarity transformation that takes the eigenstates of the unper- turbed Hamiltonian into those of the complete Hamil- 438 tonian, we will continue to use the complete set of un- perturbed Landau state functions in the calculations that follow. With the notation just introduced, the density of states is written We now wish to take advantage of the well-known identity P . . m H.-: Hiſto(s) (9) as a means of expressing the 6-function in eq (8). Hence - & l/T p(E)=tr Imlim a TſºrT. (10) This in turn bears a close resemblance to an operator called the Resolvent, used by Hugenholtz [1] and by van Hove [2] in connection with nuclear many-body theory, and into semiconductor gal- vanomagnetic theory by Kubo [3]. The resolvent of a complex number, s, is defined by | 2% — S introduced R (S) = (11) so that our density of states function can be written | * p(E)=tr Im lim # R(E+ io) (12) e—-() The properties of the resolvent have been studied in detail and are discussed in several places in the litera- ture [4,5], so we will merely present the results rele- |(n, ky. ke|V|n', ki, k! )|2 vant to our needs and omit their rather complicated derivation. Because we calculate the trace, we are con- cerned only with the diagonal elements of the resolvent. Specifically, the diagonal part of the operator R(S) is designated by D(S) and is given by I P(a)=z. Taº- (13) in the representation of which Žo is diagonal. The ef- fect of the perturbation is contained in the diagonal operator G(s), given by a series expansion in powers of the perturbation: (, (s) = {V}d – {VD(s) V}a + higher order terms (14) where {}d indicates the diagonal part of the operator in brackets in the same representation in which Žo and D(s) are diagonal. Following Kubo, we make the ap- proximation of discarding the higher order terms, and we then relocate the zero of energy so that {V}a = 0. Then eq (13) is inserted into the right hand side of eq (14) to yield an implicit equation for G in terms only of the unperturbed Hamiltonian and the perturbation V: l |r * + G – (E+ ie) cº-le)--" (15) The solutions of this equation, i.e., the diagonal matrix elements of G in terms of E and the quantum numbers of the states involved, can be inserted back into eq (13) to yield the matrix elements of D(E+ ie), also in terms of E and the quantum numbers of the same set of states. The fact that D(s) is diagonal removes one of the two summations over intermediate states involved in taking matrix elements of eq (15), and we are left with (n, hy. ke|G(E+ ie)|n, ky, ke) — X. E () f f • ? }! , Tó, Fº A simple solution of eq (16) for G(E+ ie) can be ob- tained in cases where the perturbations V are of the fol- lowing three types: the first and simplest one is the delta function potential, which corresponds physically to a highly localized or short range force acting to scatter the carriers. We will assume their distribution to be random throughout the sample. The second per- turbation is the deformation potential as related in par- ticular to the acoustic phonon density—it is an impor- tant source of scattering at moderate and higher tem- peratures. The final type of perturbation to be treated is an example of a long range force, namely the attrac- (n', k, k.) + (n', k, A. G. (E+ ie)|n'k. (16) y K!) – (E+ ie) yº tive screened Coulomb potential. This is an approxi- mate representation of the effect of a charged or polarized impurity site on an electron in the conduction band or hole in the valence band. We limit ourselves at this point to describing the method of solution and proceeding with the results – the actual calculations for the three cases are carried out in appendix B. The squared matrix element for each of the three potentials mentioned above is shown in the appendix to be independent of k, and ka' and to depend upon ky and ky' only in the combination (k,' — km). The energy of the unperturbed Landau levels, Eo, depends only on n' and 439 ka', but not on ky'; and because of the (k,' – km) depen- dence in the numerator, eq (16) has a solution for G (n which is independent of ky. Therefore, eq (16) can be rewritten as 1. | o(E+ ion)=# ſ .dk. E. where the ky' integral contains the specific information about the potential responsible for the scattering. From the results given in the appendix, we see that the k, integral can be expressed as | dºſ(4,-4)=s+ W. 27/2 (18) where W is a scattering constant with dimensions of (energy)* X (volume) and l is the cyclotron orbit radius. Using eq (3a) or eq (3b) to express E0(k, n), the k." integral is elementary and yields the result (#) 1/2 W h2 *) iſ XIVII) ioTow).I.T. }) ' (19) A final simplification, based on the assumption that the dominant scattering processes occur between states within a single Landau level, allows us to sup- press the summation over n'. We focus our attention on 20 (k, n') + Glº, ºg (E+ ie) – (E+ ie) * , , , , , , | dk, f(k,-k, ) (17) the n = 0 level; and to complete the solution for G and for the density of states, we adopt the following nota- tion: e = | – 1/2 ha), conduction band (20.a) s — (–E)- 1/2 hor) valence band (20.b) _ l (2m,\!!” B-#( #) p = C, v (21) g = 3W In each case the quantity e denotes the energy of in- terest as measured from the appropriate Landau level edge. With this notation, the equation for G, (19), reduces to G=– ([G – el-1/2 (22) from which follows that G3 — e(X2 – 3 = 0. (23) Solution of this cubic equation is straightforward, and G has the following three possible values: …o-º-º: F. Wºº-yº 6 (e. g)=3 | Vitº, + (V3++ i + Wi-Fi – «V; +. (24) , , , e_1 | "e". ©1, leº, "let, g_, leº i V3 || 3/ e3 £2 e” (* *|e” (” e° (* —H - - - - - - - - * | *-* -}= - - ºf H -- - - - - 2 Nº. 57 tº \; t , ~ \; tı (25) In order to choose the correct solution, we impose the following two conditions: G must approach zero as Ç approaches zero, since the density of states must ap- proach the unperturbed form under that condition; G. must be a solution of (22), the equation from which the cubic equation was obtained. The first condition eliminates G1(e,g). The second condition, which can be p(E) = Im tr in . RE tie) re-expressed as the requirement of a positive density of states, eliminates G3, the complex solution with nega- tive imaginary part. Finally, in order to obtain the expression for the den- sity of states function, we return to our original form, eqs (12) and (13): (12) l/T =Im lim e — () X. ſºo-H G (E+ ie) – (E+ ie) 440 As long as the summations can be carried out under the condition that G(s) does not depend upon either k, h? e—” () . I l (2,\!/? p (e) = Im in Hºſ º 47/2 On the other hand, the equation for G itself, eq (22), is l (, (e)=–li W —— & Y & (e)=-lim Bl, vi- (22) Hence (e)=–1 + 1 (, (27. £) iſ im , (e) ) The interesting form of the result is a direct con- sequence of the approximations made in neglecting the higher order terms in the original equation for G and of keeping only the scattering terms within a single Lan- dau level. The function p(e) is graphed in figure 1. Also, we call attention to the summation process indicated in eq (17); in form it is very much like the one in eq (7), ex- cept that instead of summing over tails of Lorentzian functions of energy, this summation takes place over the tails of more complicated functions of energy deter- mined by the imaginary part of G(E). These functions, shown in figure 2, indicate the manner in which states of a given k2 value are distributed in energy by the scat- tering process. This is an important consideration when k, conservation is required in the transitions, and it is discussed in the next section, where our method of cal- culating the optical density of states is presented. We would like to restate the results of this section in terms of two functions whose definitions will be useful in the subsequent section. The first one, a density of states in e and k2, is given by I | p (e., kz) = 872/2 Im º + g- (28) so that Joſe, kg) dh, H p(e). (29) And the second function, a density of states in e, kz, and ky, is given by – 1 | pe, k, ºv)=1, E.J.) To L. (30) or k, the result is directly obtained from two elementa- ry integrals as | | | — == Im lim + 8 –– v=H =lm in 6 vº. (26) so that p(e, kz, ky) dy= p(e. kg). (31) \ (£ p(#) \ FIGURE 1. Density of States functions. The dotted line indicates the unbroadened case: the thin line indicates the Lorentzian broadened case, and the thick line indicates the case considered in this paper. The energy scale is in units of Ǻ". p(3,k.) † : Energy-Dependent Distributions of k, states. The numbers atop the peaks stand for normalized values of k”, The arrows represent photon energies. FIGURE 2, Before going on to develop the optical density of states integral, we would like to examine briefly some of the properties of this single Landau level density of StateS. 2.3. The Density of States Function: Characteristics and Approximation To begin with, we write out the function explicitly, using the result given in eq (25), -º-º-o-º: p(e)=# iſ Im lim 26)–: ; (; ) (.. Ç e” (” 3// e” (” e” º ++)- 37 tº (#) Ç 57" 4 (32) 441 It is shown as a function of the normalized variable, e|{*, in figure 1. The dependence of the value of this function on magnetic field, contained in the parameter C, is quite complicated; and unlike the Lorentzian- broadened case, it varies over the range of e. We will show the explicit dependence on H in three regions shortly — here we remark only that the dependence is weaker on the low energy side and grows stronger with increasing energy. Unlike the Lorentzian-broadened density of states, whose peak is shifted, this function peaks at exactly the value of the energy variable e at which the unbroadened functions become singular, i.e., at e = 0. This can be readily established by differentiation. The value of the function on resonance can immediately be calculated from the above equation, by letting e = 0, and the result is (33) We call attention to the dependence of the peak height on magnetic field. In this region of e, the height goes as H*, since 3 is linearly proportional to H. We next examine the high energy side of the func- tion, in the region where e > *. It is convenient to rewrite eq (32) as follows, in terms of e and the small dimensionless quantity (ſe”: -Hºlwº 1 / ( \* 3|, | | | | \* (V27 p(e)=. F-5 14; (#) + e3!? l-Hi (#) — a 1-5 (#) - e3/2 (34) If only the terms under the radical sign up to first equation is simplified to order in this dimensionless quantity are retained, the ( ) *~~ l l V3 e 3 | *) ºr IF. T., 3 (35) and the cube roots can be expanded to yield the final expression _ ] I V3 e (V27 & V27) || 1 || V3 e [2G V27 | 1 , | (e) − i iſ tº ; (+ #) ( #)|=}} 2 #| 3e3!? |- 6:# (36) where we have used the relation ( = 8W to obtain the final line. This result is exactly the same as eq (4) for the unbroadened density of states. Thus, in the limits of very high energy or of vanishingly small broadening parameter, the broadening density of states equation we have presented in this chapter approaches the un- broadened result, as does the Lorentzian-broadened case we have also considered. At these values of e, the function is linearly proportional to H. However, it is on the low energy side of the peak that the distinguishing features of this form appear. First, as shown in the figure, the function vanishes for values of energy less than or equal to e= (–3|V4 (*). To show this from the equation, we note first that the terms of G2(e) (eq (25)) not multiplied by i must be real for all positive and negative real values of e. This is true since whenever the square root under the cube root sign in- troduces an imaginary part, the complex terms are the sum of a number and its complex conjugate. Then, in the terms multiplied by (i\/3/2), as soon as e becomes equal to or less than (–3|V4:28), these terms become the difference between a number and its complex conju- gate, hence imaginary. And, multiplied by (iv.32), they also give a real result so Im (, (e) = 0 for € s eO = (37). Let us now examine the function in the vicinity of e = eo. It turns out to be convenient, again for purposes of expansion, to rewrite the density of states equation in somewhat different form. By recognizing that the ex- pressions under the cube root signs are perfect 442 squares, we can rewrite p(e) as po-ºº! () ( . When e = eo, the quantity in the square root is small compared to unity and the above equation can be ap- proximated by V3 l l (*) [4 || 4e" po- (*#): Nº. 1 39 This equation is to be expanded around e = eo. Defining g|–27(? Ae = e – 4. V3 l (2/8) 4 |4/3 | 221; p(e) = ( – H. . ] . Aſ ºf Ae = + ºf 27 W 223) 3 W (28 T \/3 Thus the density of states at the bottom of the Landau level has exactly the same energy dependence as an ordinary parabolic band in the absence of a magnetic field, but with a coefficient that is many times larger and depends not only on the effective mass but also on the magnetic field and the broadening parameter. In this region the magnetic field depend- ence is weaker than in either the peak region or the higher energy region, being proportional here to H''”. However, a numerical computation shows that with typical values of W and £3, this parabolic form is a very good approximation to the complete expression all the way from the bottom of the Landau level at e = eo to the peak at e = 0. This interesting result enables one to use many of the available computa- tional techniques for dealing with carriers in parabolic bands. wº-ſeº.º.º. Kyl) X Ö (En - El - fio)6 (kyu - kyl)ö (kau - k21) X dendedkyūdkhidkaudkai where the subscripts u and l stand for “upper” and “lower,” respectively." The delta functions assure ener- gy conservation and momentum conservation; the | The subscripts u and l can be considered interchangeable with c and v, for “conduction” and “valence,” respectively. 4e.” 2} .3 4e.” 2/3 — — | | + , || – e #11) -(+N++)" (38) and inserting the resulting expression for e into the previous equation, we obtain the result 3 l (28 Wº Ae-H – 27.21% | ###): #|A. 4. |+| (41) V3 p (e) -( 2 and if we cube the quantity in square brackets and keep only the lowest order term in the small quantity Ae, the final result for the approximate density of states in the vicinity of eo is (42) 3. The Optical Density of States 3.1. Formulation of the Convolution Integral An optical (or “joint”) density of states can be defined as the number of pairs of quantum states separated by a given energy difference ha) and joined by a given selection rule. A count of these pairs is obtained as the product of the density of initial states times the density of final states to which transitions are allowed by the selection rules. For the Landau states expressed by eq (2), the appropriate quantum numbers are ky, kz, and e. It is important to reiterate here that, because of the broadening, there is no longer a one-to-one correspon- dence between k2 values and energy values. In counting pairs of states we must therefore sum over energy and momentum variables. Nonetheless, we can consider in- terband transitions that conserve momentum and ener- gy simultaneously, and the optical density of states for these transitions is expressed in the following manner: (43) latter arise from the dipole matrix element between the Landau wave functions. We can begin to simplify this expression by perform- ing the integrations over kyu and kyl, which removes one delta function and then yields a coefficient proportional to magnetic field in front of the remaining integrals. 443 Further simplification is then affected by integrations over kei and el, to remove the remaining two delta func- tions. The result of these operations is the expression l p(ho)= Lºſ || pm (E. k-)p (EM - ha), kº) dB/dkº (44) The form of eq (44) can be recognized as that of a convolution. And indeed an optical density of states sig- I (en) nifies precisely that, namely, the convolution of a final density of states with an initial density of states, for a given value of separation ha) between the two energy ar- guments. We are now ready to insert the actual expressions derived in section 2 for the initial and final density func- tions into eq (44) and carry out the remaining integra- tions. Expressing the complex function G(e) as A(e) + iT(e), and explicitly taking the imaginary part indicated in eq (28), we write T(e) | p(ho) = Ti, || dkden V.I. 472/ ( k + A (en) 2mm where el and en must be measured from the respective unbroadened band edges in order to correctly evaluate the factors in the integral. Before going further in the mathematical develop- ment, we would like to consider the physical meaning of this integration. This is best done by reference to figure 2. There it can be seen how the k, states of both the conduction and valence bands are spread into dis- tributions over energy. Unlike the Lorentzian case, the distribution function itself varies with energy, and two important features are clearly evident. First, since the numerator of the function, Im G(e) = T(e), goes to zero at e = eo, all the distributions vanish at eo; this affects low energy distributions quite strongly. The second fea- ture is that for high energies, where G(e) is rather slowly varying, the distributions again appear Lorentzi- an, but the peak heights are proportional to [T(e)] ', so they gradually increase with increasing e, at a rate of (e)=1/2. We have drawn several arrows of equal length to in- dicate direct transitions absorbing or emitting photons of equal energy, but having different initial and final - •) + Tº (e) (; (45) + A (e) - •) –H Tº (e) 2m) energies from one another. The double integral represents the summation for a given “arrow length” of all values of k, at a particular upper energy and then the sum over all allowed upper energies. The range of al- lowed upper energies is determined at the low end of the scale by the cutoff energy eon, and at the high end by the cutoff occurring in the other band. In other words, for a given value of ha) imagine the arrow to slide on the energy diagram from where one arrowhead touches one cutoff energy to where the other head touches the other cutoff energy. In this manner, the upper and lower limits on the Eu integral are deter- mined. The limits on the k-integral are taken as – Co to —Hoo. 3.2. Evaluation of the Integral and Relation to Single- Band Density of States For generality and convenience of application, we have chosen to evaluate the integral in terms of the dimensionless quantity obtained by normalizing all energies to the upper cutoff energy eon. Thus p(ha)) is rewritten l 2mm l – 2 AP – 1 || T(En) T(E)) - Aſ "Il p * * En * * •) e •) * p(ha)) H h2 * Ryº RE * OTWEji), IFVE) ( , , A(E) *) T2(E) OW O. OW (46) where: RM = Jhu l RE B 60m 60l 6tt En E #3 _ ha) – Eg-en O = RM & RE (47) 444 and where the gap energy, Eg, has the magnetically ex- panded value Eg") + hoc-H ha). The integration over k can be carried out analytically and is most easily affected by contour integration. The pole pattern in the complex plane exhibits quadrantial symmetry, with the poles occurring at k = + VA(E) = E, TTCE) k = + VA(E) =FE, ET(E.) A3 = + WAE) + Ey + iT(E) OK! (48) k4 = + WAE) + E – iT(E)) OW Our computer program began by reading in values of Ry, Re, and ha) and evaluating the residues at the four poles in the upper half plane. It then integrated the residues over Eu between the appropriate limits, as discussed above. In terms of our normalized energies, the limits on Eu run from –3|V4 up to 3/V4. Re--- (ha) — Eg)/{2/8. The result, as a function of ho), is in each case a curve having the form shown in figure 1. The general characteristics are the following: (1) The function vanishes at a value of ha) given by | ha) – Eg = - epi, (i+}) (49) (2) The function reaches a peak at ha) — Eg = 0. Thus, unlike the Lorentzian case, this peak occurs at exactly the value of energy where the singularity occurs for the unbroadened optical density of states. The height at the peak is discussed below; (3) On the high energy side of the peak, the curve diminishes and approaches the un- broadened form proportional to (ha) — Eg)-". Since the appearance of the optical density of states function is identical to that of the single band function, we are led to ask if we can define an appropriate reduced broadening parameter analogous to the reduced lifetime for the Lorentzian case discussed in appendix A. This is indeed possible, and in fact turns out to be a direct consequence of the mutual consisten- cy among the single band densities of states and the op- tical density of states. Essentially the condition for con- sistency can be stated as the requirement that the zero of the optical density of states, which defines the ab- sorption or emission edge, occur at an energy equal to the separation of the individual band edges, and similarly, that the peak of the optical density of states occur at an energy equal to the separation of the in- dividual peaks. We will show that a certain relationship must exist among the density of states parameters, the B’s and W’s, in order to fulfill these requirements. The consistency condition is expressed as follows: (50) eor = €0c -i- €or where the subscripts r, c, and v refer respectively to reduced, conduction, and valence. Equation (37) allows us to write this directly as (8, W.)” = (8.7%)2/3 + (8,70)*. (51) Also, we recall from eq (33) that at the peak the den- sity of states is given by V3 B} pu (peak) – 2T W/3 (p. = C. v. Or r) (52) On the other hand, in terms of the familiar phenomenological parameter T, the peak height of a Lorentzian-broadened density of states is given by the expression 3, IT, /3\** pu(Lor)(peak) - º † (#) Experimental observation shows that the magnitude of T is somewhat magnetic field dependent, decreasing with increasing field. If we now equate the two expres- sions for the peak height, we obtain the equation V3 4* h 2 3 T), (53) B}/87/3 = (54) and squaring it yields the very interesting result that 2 ſh 3/3]/3/3 — — a 55 B#8W: V3 T. (55) In other words, eon of h/Ta, and substituting this result into eq (51), we obtain the condition T}. To Tr This result, which holds at any value of magnetic field, restates our condition for mutual consistency among the set of broadened densities as the usual ex- pression for the interband (reduced) phenomenological collision time in terms of the single band times. In addi- tion, eq (55) above would predict, at least qualitatively, the kind of variation in the value of T, with variation in magnetic field that has been observed experimentally. A reduced broadening parameter, Wr, can now be defined directly from the consistency condition, as fol- lows: Inserting l (2m; \!/? £3, = 4T/2 ( h2 ) (57) 445 into eq (51) and doing some algebraic manipulation, we find 1/3 W#3 = (1 + Ry) 18W:/8 + ( + #) IV:13 (58) (peak) = V3 8;" V3 º (# £) \ } 27 Wº 27 | 47/2 \ hº Note that the factor in square brackets is the same as that in eq (46). To confirm the hypothesis that the opti- cal density of states is formally identical to the single band function, we should be able to equate eq (59) above to eq (46) with the integral evaluated for ho-Eg, for any chosen pair of values of RM and RE. This has been verified by our numerical integration, using values of RM and RE ranging from 1 to 20. Therefore, the pro- perties of the function, as developed in section 2, apply equally well to the optical function considered in this section. 4. Conclusions From the single Landau level densities of states in the presence of energy dependent broadening, we have developed a convolution integral to express the optical density of states for transitions between Landau levels across the energy gap. The form is identical to that of the single band function; but with the frequency mea- sured from the gap energy replacing the energy mea- sured from the band edge, the reduced mass replacing the single band mass, and a reduced broadening parameter replacing the single band parameter. This last parameter was shown to be a consequence of a mu- tual consistency requirement, and this same require- ment also leads to the reduced lifetime expression en- countered in the more conventional Lorentzian formu- lation. Hence W, depends only upon the individual W’s and the mass ratio, and is independent of magnetic field. And when Wo and We are replaced by their expres- sions in terms of 8c, 3, eoc and eon, and the entire right hand side inserted into eq (52) for the peak value of p(ha)), we obtain )"(...)". Tº The most striking contrast of these results with the Lorentzian expressions lies in the vanishing of the func- tions near the band edges. In this region, a parabolic approximation turns out to be valid, and this behavior enables us to explain the functioning of the semicon- ductor laser in the presence of a magnetic field. Further applications of this formulation to other electron transi- tion processes are currently being studied, notably the Gunn effect. R!!” 1/2(1 + Rº) /* | €0.u (59) 5. Acknowledgments The authors wish to express their appreciation to Mr. James Raphel and Mr. James Spoerl for programming assistance, and their gratitude for the services of the University of California Computation Center. Special thanks go to Mrs. Teddi Herron for preparation of the manuscript. Appendix A In this appendix we outline the calculation of the op- tical density of states for transitions between Lorentzi- an broadened sets of initial and final states. Inserting Lorentzian functions centered around the energies given by eqs (3a,b) into the integral expression (44), we can write Tº T — + 2 1.2 \ 2 2k}\? p (ha)) || || || (*) ri (E. E. lº) + r. 2mu 2mi × 6 (En – E – ho)6(kyu – kyi)6(kau - kai) × dkyūdkyidkaudkaidEudEl (A.1) where Tu and T are constants. As in section 3, this expression is greatly simplified by carrying out the integrals over El, kyi, kal, and kyu in order to remove the three delta functions and to yield a coefficient proportional to magnetic field. Thus W OC CC 2 1, 2 pho)-gºnſ a ſide. (E.-: T 2 / 2 (E. - fia) + En –H ‘...) 2mi 2 X + Tº (A.2) This double integral is most easily accomplished by contour integration. Defining new variables — W." h?/ ? y = T, it 2mm (A.3) e = ** – hit Eg (A.4) 2m, where m, is the reduced mass, we find poles in the upper half plane at y = – e -- il". (A.5) y = iſ n and so the remaining integral becomes K ſ” (Tu + [i] . p (ha)) - 472/2 |. dk e” (k) –H (T, –H T,)? (I'm + TV) – ha) + Es) –H (Tº –H TI) 2 K ſº il. -i iſ a (# 2m, V 2m, e | * 472/2 h? Vha) – Eg – iT, It is interesting to note that the sum over Lorentzians in eq (A.6) is formally identical to the sum encountered in the single Landau level density of states, eq (7). Also, the resulting optical density of states is formally identi- cal to the single level density, but with the energy mea- sured from the band edge replaced by the photon ener- gy measured from the band gap, the single band mass replaced by the reduced mass, and the single band broadening parameter T replaced by the joint broaden- ing parameter T. -- T. This last replacement is equivalent to the replacement of the lifetime T by a reduced lifetime Tr given by | l=l11. T}. Tuſ T (A.6) (A.7) Appendix B In this appendix we will evaluate the matrix element in the numerator of the right hand side of eq (16) for each of the three scattering potential functions men- tioned. B. l. Scattering by Low Energy Acoustic Phonons In examining the effect of lattice vibrations as the source of perturbation, we take advantage of the mu- find 1/2 (v)=( 2p/cv ) qEaë (k - ky– qº) ô (k - k2 - q2).Jnn' (ky, q.r, kiſ) c + q tual independence of the various normal modes of the lattice. We assume that the electrons couple to the vari- ous modes separately and do not serve as a source of mode mixing. Hence, we will treat the interaction between the electrons and a particular phonon mode of wave vector q and then sum over all the q’s to obtain the final result. Following Ziman [6] we indicate the matrix element of the perturbation by (|v|) = (; , VHop|n'a,\!') (B. 1) where &ep, the electron-phonon interaction operator, is given by h 2p/cv. 1/2 ) Va(n!"age-in r-H (n + 1)*aie-iq r.) (B.2) & op= ( where aqt and an are creation and annihilation opera- tors, respectively, of phonons with wave vector q, and Va is the deformation potential operator on the elec- tronic wave functions, no is the phonon density, va their frequency and p is the mass density of the crystal lat- tice. It is clear that na' must be equal to (na-H 1) or (no – 1) for this operator to have nonvanishing matrix elements; to be specific, let us carry out the calculation for na' = no – 1, and then add the obvious extension to the other case. Letting the phonon operators act on the phonon density part of the state, we obtain 1/2 - (h)-(+)" | "wººd, Bº and then invoking the slow variation of the envelope functions F and F' and the factor e”''' compared to that of the cell-periodic part of the Bloch function, we effect the usual separation of the integral into a product of two integrals. One is treated by deformation potential theory [6] and leads to q times a constant, usually called the deformation potential parameter and denoted by Ed. The other integral is separable into product of factors along the three cartestian coor- dinates, with the two in the y and z directions resulting in 6-functions that insure momentum conservation. Hence (B.4) 447 where we have used the notation Jun' (ky,q,r,ky') in- troduced by Argyres [7] and now used by other authors E Jº, (hy. (1.1. h) The matrix element corresponding to an initial state no" = (no + 1) would then have (nº. + 1) in the coefficient where the one just calculated has nq. We will make the further assumption of restricting our attention to the case of low energy acoustical phonon scattering. (This is of particular interest, and is certainly justified, for the moderate temperatures and narrow range of kinetic energies of the quasi-equilibri- um carriers involved in the lasing process.) Because these are low energy phonons, they are nu- ſ dxilſ, (x - [8.9] to indicate the integral along the x-coordinate: /*k,)e"r"), (x-1}). (B.5) merous enough so that na = no + 1, which is consistent with rewriting nº, a Bose-Einstein factor, as follows: hum hu, hugq eRT – 1 nq= (B.6) where v, is the sound velocity in the semiconductor. Thus the form of the matrix element for a single phonon mode to be used in eq (14) is |TE} \!/2 ..., , f f f (*)-(#) ô(k - ks-q2)6(k - ke- q.).Jnn (ky, Qa: , k/) (B.7) and the equation itself is written * - kTE% 6(k-k, - qI)^(k4 = k– – qx)|Junº (ku, ar. kº)| * - X. Xº E},..., + 6–s (B.8) where the coefficient has been multiplied by 2 to ac- count for both the processes of absorption and emission of phonons. The summation over q brings in the effects of all the phonon modes. As discussed by Callaway [10], no difficulty is introduced by squaring a matrix element containing momentum-conserving delta func- tions. We now proceed with the summation in order to ob- tain the expression for G: first the two delta functions are eliminated by integration over q, and ky', leaving ATE, |Jim'(ky. Q r. ky+ qu)|* G= X. X. Y , phºw: E%', kº-F Gn', kº 1. dy, k n", k? q r, q Since Jun'ſ" depends upon k, only in the combination x – 30(ky), it is possible to have a solution for G which is independent of ky; hence G in the denominator of the >k 27 ſ ſ dqrday/nn'Jim' (ky, Q r. km + qv) =7; ônn'6mm, allows the result to be written immediately as TE2 l l k #) (B.11) G = X. (# 271° E), -- C(s) – s f * n;kg }l ', R 2 and a simple linear transformation of variable will now lead directly to the form of the ky integral mentioned in (B.9) – S 2 right hand side is independent of ky' = k + qi, and the summation over q, need only involve the numerator. A theorem due to Argyres [7], (B.10) section 2. The factor in parentheses is a specific ex- pression for the factor we called W/2Tl2. B.2. A Random Distribution of Scattering Potentials To insert a random distribution of scatterers into the formalism we have been using, it proves most con- venient to deal with the Fourier transform of the poten- 448 tial in computing its matrix element. Thus, at any point r, the potential V(r) due to the N, scatterers distributed throughout the crystal at locations R; is V(r) = S v(r–R) (B.12) j= 1 and its Fourier transform is given by W(q) -*. ſe" ºr (B. 13) ſ (V(r)) =X WCQ) Before putting the specific expression for W(q) and proceeding to evaluate the right hand side of eq (16), let (KPG))lº), -N (P(q)7. - f ** , , ;: X 6 k . k, toºkg, R,+ag” Junº (ky. Qar. ki,).J., It is evident that only the product of W(q)'s has explicit dependence upon the distribution of scattering centers; hence that is the only part included in the brackets (), * f7;k / / N, * (V(q) V*(q')),= Vá "..., | (q)|* ſº x e^*-****),(x-k/2)×enrºl,(x-kº)dºr CI W(q)ökſ, Rytºly ökg, k 2 + q2 Junº (ky. Q r. ki, ) from which an elementary transformation of variable leads to \' | ` W(q)=jº X e-" " (q) (B.14) X. 1 j where Vc is the crystal volume and v(q) is the Fourier transform of the single scatterer's potential. The matrix element which appears in the numerator of the right hand side of eq (14) has the following form: (B.15) us square eq (16) and average over the spatial distribu- tion of scatterers *(q')), * *, *, *g, *, +, (ky, q., k,) (B.16) that indicate spatial average. Using eq (B. 14) and taking account of the random distribution of the Rj's, it im- mediately follows that (B.17) and the average of the squared matrix element simplifies to (KPO))) → S ; Hºlº / \ | 2 7 s=X. W2 |v (q)| ***, +0.8%. , 10./Jun (ky, ar, kſ) (B. 18) (I C which we proceed to insert into eq (14): Y - N, |v (q)|*|Jin' (ky, Q.r. k)|* G(s)=– ź. X. V2 E}|, |g-H G(s)—s örſ, k ſta,”g. F 2 + q2 (B.19) n', kº , k? CI C , 3 fu 2 The two delta functions are then eliminated by summa- tions over q, and ky', where the latter is carried out as in the two previous cases, under the condition that G is N, |v (q)|*|Junº (ky. Qa: , ky+ qu)|* independent of ky. The result is the following expres- sion: G(s)=– X. X. Vº n", k, qr, qu ° Before continuing, we would like to remark that up to this point no properties of the particular scattering (B. 20) E} |. H- G(s) – s 417–156 O - 71 – 30 centers have been invoked. The result has only required that the distribution be random to allow the 449 averaging to yield the Öq, q', in eq (B.17). If the scattering centers were delta-function potentials of amplitude hºo”|Tm, the Fourier transform V (q) would be a con- stant, and eq (B.20) above would be identical to eq (B.9), but with W/2Tl” given by (nº/477/2 hºo/m2). The delta function can be considered a good approximation wher- ever the force range of the potential is considerably smaller than the cyclotron orbit radius of the electrons being scattered. On the other hand, the scattering source to be con- sidered is that of the attractive screened Coulomb potential; we write: Ae-qs” !" (B.21) v(r) where A is the amplitude and (q)-1 the screening length. The screening length determines the force |2 e-j (4; + dā) range of the potential, and unlike that of the delta-func- tion potential just treated, the validity of our approxi- mation in this case depends upon this force range being large compared to the cyclotron orbit radius. The Fouri- er transform of this potential is given by A ū (QI) = −. (B.22) q* + qi where, because of the 6 function involving q2. q = (q.r. gy, kº-'k-): (B.23) and since we are restricting our attention to elastic col- lisions with attractive potentials only, kz' +k. Furthermore, we retain the assumption that only scat- tering within the n = 0 Landau level is important so we can write Jool” explicitly in simple form. Thus •) _. -- A*n, ., € (, (s) A*n, - -** > ſ durday ( k. A change of variable, t = q.” + q2/24,” enables the above equation to be rewritten as q; + qi + q})*(E)-- (, (s) (B.24) | ſ •) − dq* → —s) /2 > Bio- o “” dº I dº Performing the sum over k,' and also recognizing the exponential integral leads to the following expression for G: y A*n, l ºf ſ e-Buſ. § G(s) = /2 > E|). 4- G –s 62 . dt. (B.25) A f /*q: Y *n, /2m *\!/? e 2 G(s) = #( * ) + In dº ſº — dº ſº. . . . |-tiâ (; ) viº= (x+h ºf 4. ) 1 /2m *\!/? A*n, ln q:/* - – — e S B.26 472/2 ( h2 ) V1/2ha), H G(s) – s ( ) where y, the Euler-Mascheroni constant, is approxi- mately equal to 1/2, and our assumption that q}l” < 1 allows us to replace the exponential by unity and to ignore y and the higher powers of q,”!”. Again we arrive at an equation for G whose form is identical to eq (19) and (B. 11), and therefore the same procedure would be used in finding the density of states, with A*nslnq21% as the factor we labelled W. We point out in passing that this form of W, unlike the previous two forms, has an explicit dependence on magnetic field. 6. References [10] [1] Hugenholtz, N. M., Physics 23,481 (1957). [2] [3] Van Hove, L., in Quantum Theory of Many-Particle Systems (Benjamin Press, New York, 1961), p. 1. Kubo, R., Miyake, S., and Hashitsume, N., in Solid State Physics, Seitz and Turnbull, Editors, 17, 269 (1965). Kumar, K., Perturbation Theory and the Nuclear Many Body Problem (North Holland Publishing Co., Amsterdam, 1962). Roman, P., Advanced Quantum Theory (Addison-Wesley Publishing Co., Reading, Mass., 1965). Ziman, J. M., Electrons and Phonons (Oxford University Press, London, 1960). Argyres, P., Phy. Rev. 109, 1115 (1958). Korovin, L., and Kharitonov, E., Sov. Phys. Sol. St. 7, No. 7, p. 1740 (January 1966). Roth, L., and Argyres, P., Magnetic Quantum Effects, in Physics of III-V Compounds, Willardson and Beer, Editors (Academic Press, New York, 1966). Callaway, J., Energy Band Theory, Chapter 4 (Academic Press, New York, 1964). [4] [5] [6] [7] [8] [9] 450 DISORDERED SYSTEMS I CHAIRMEN. L. F. Mattheiss A. Kahn The Electronic Structure of Disordered Alloys" J. L. Beeby Theoretical Physics Division, Atomic Energy Research Establishment, Harwell, Didcot, Berks., England The problem of calculating the electronic density of states in an alloy is considered from first prin- ciples. Choosing a suitably simplified model potential a diagrammatic expansion is discussed within which the various existing theories can be compared. Some comments are made on the comparison with experiment. Key words: Density of states; disordered alloys; one-electron propagator; perturbation expansion: sum rule. 1. Introduction It has long been appreciated that the information ob- tainable from experiments on alloys provides a useful supplement to one’s knowledge of the pure materials. Much of this information, such as the Hume-Rothery rules, was obtained using binary alloys with similar non-transition-metal constituents. In such an alloy the electron mean free path is long and the alloy can often be regarded as homogeneous. The problem which will be discussed in this paper concerns how to calculate the properties of an alloy in which the constituents are very different. For such an alloy there arises, besides the routine difficulty of choosing an appropriate poten- tial, a major problem involving the order. Perfectly or- dered alloys can be handled just as for pure materials, but disordered alloys with strong scattering potentials have needed the development of new theoretical techniques. These methods will be discussed below via a diagrammatic expansion which appears to give im- proved insight into the whole problem of disorder. No attempt will be made to formally review the literature and no claim is made to completeness. Rather, it is hoped, readers will be better able to judge for them- selves the contents of papers in the field. The paper will begin with a derivation of the alloy potential. The intention here is to clarify, for those not already familiar with the field, what disorder is and how the theorist can describe it by an averaging process. *An invited paper presented at the 3d Materials Research Symposium, Electronic Density of States, November 3–6, 1969, Gaithersburg, Md. While concepts of this type are already common, as in statistical mechanics for example, it is clear from the current literature that many authors still misun- derstand them in the alloy context. The derivation of the potential is then completed by briefly listing the as- sumptions required with a few comments on their validity. The potential used is of muffin-tin type with in- variant potentials for each constituent. The brevity of this section should not be regarded as indicative of the trivial nature of obtaining the potential, which in itself forms a most interesting and difficult task. The heart of the paper is contained in sections 4-6 where the density of states is obtained by considering the imaginary part of the T-matrix for the alloy. The T-matrix can be ex- panded in a series involving the individual scattering centers (the muffin tins) in the usual way. The procedure adopted in section 5 concerns the way in which the series is to be averaged term by term. Each term requires the knowledge of a probability function and it is these probability functions which must be ap- proximated if the series is to be resumed. A diagram- matic expansion is given which, it is argued, formally converges like Zº' where Z is the number of nearest neighbors. In practice, however, it is the degree of fluc- tuation which determines the convergence which is anyhow at best asymptotic. Section 6 illustrates these remarks by looking at numerical solutions for different approximations. In the final sections a few examples will be cited of the interrelation between this work and experiment. These are mainly concerned with the transition metals to which the formalism is most ap- propriate. 453 2. The Medning of Disorder When an alloy sample is made there is much infor- mation which is available in principle yet is not availa- ble in practice. It is normally possible to determine the content of each constituent, the structure and certain ordering parameters. But it is plainly absurd to expect to know the actual positions of all the atoms in a disor- dered sample. Thus while an experiment is performed on a particular sample whose atomic positions may be regarded as fixed during the experiment, a theoretical calculation for this sample must proceed in ignorance of the actual positions. The formal device used to offset this ignorance consists of an averaging procedure about which one might make the following comments. (i) It is expected that all macroscopically identically produced samples will have (within experimental error) identical properties. Systems with large fluctuations in their proper- ties due to unavoidable variations in production need a different approach. Such fluctuations are usually due to variations in some macroscopic parameter not yet controlled in the production process. (ii) The detailed microscopic order is therefore only important to the extent that certain macro- scopic properties (e.g., order parameters) are satisfied. (iii) Naturally the theoretician is, in these circum- stances, at liberty to choose any one microscopic distribution which satisfies the macroscopic restraints. However, since all such distributions are equivalent, it is easier to average over them with a probability function specifying the chance that they occur. (iv) Such an approach is well known in Statistical Mechanics and works for the same reason: the number of particles involved in the average is very large. It is most important that any such averaging is made only over observable properties of the sample. As an ex- ample consider the density of states which for a given sample might be written n(E, q1, q2, ... qn) depending on certain parameters of the sample. If these parameters have a probability of occurence P(q. . . . qm) within the constraints of the sample production then the average density of states is ſ n(E, q1... qm) P(q. ... qn) dai ... dqn. If the job has been done properly this average value should differ from a typical single alloy value only to the order (1/N) where N is the number of atoms. Averaging in this way makes things a little simpler al- gebraically because, like the experimenter, the theorist can ignore all except the macroscopic features of the sample. There are some conceptual difficulties, how- ever, which are worth briefly illustrating. Take first a perfect lattice of one type of atom. Figure la shows for a finite number of atoms the energy levels of the system and the singularities which will occur in the complex E- plane for a single particle Green function. A pole will correspond to each energy level. When N becomes in- finite the energy levels pack together and can be described by a density of states n(E) as in figure 1b. At ^ E. E - PLANE a) N FINITE AN E E - PLANE —- n (E) b) N INFINITE FIGURE 1. The energy levels and complex plane singularities for (a) a finite, (b) an infinite ordered system. the same time the poles merge together to form a branch cut along a portion of the real axis. For an im- perfect arrangement the finite system again has poles on the real axis which will turn into branch cuts upon averaging. When approximations are made in the averaging these branch cuts may have been replaced by poles off the real axis; the importance of remember- ing that this is a consequence of the approximations has been particularly stressed by S. F. Edwards. A related point concerns the limit N-> Co. In the perfect lattice case periodic boundary conditions can be used and no difficulty arises. In the disordered case the limit can only be taken after the averaging procedure. To see this consider a group of N atoms which is infinitely ex- tended by repetition. The resultant crystal then has N atoms/unit cell and this is well known to yield a band structure with N-1 band gaps in general. As N is in- creased the band gaps become more numerous and nar- rower, showing that this is not a sensible treatment. 454 3. The Alloy Potential A correct procedure for obtaining an alloy potential would be as follows. (i) The positions of the nuclei and core electrons are considered known. (But will in practice be supposed to lie on a perfect lattice). (ii) The potential throughout the entire lattice is guessed. (iii) The density of electrons in the system is then obtained. (iv) A new estimate of the potential is made. Such an “in principle” self-consistent calculation must be made for a single unaveraged alloy and is plainly an impossible task. Some part of this self-consistency can be achieved in special cases, e.g., where the Friedel sum rule can be used in dilute alloys. Step (iii) to (iv) is, of course, similar to the same step in the perfect lattice except for the lack of order, but even in the perfect lat- tice case is far from easy to carry out properly. The novelty of alloy theory lies in steps (ii) to (iii), describing the electron density once the potential is known. Since this step is the primary interest of this paper the discus- sion can be greatly clarified by choosing a suitably sim- ple form for the potential. Only the one-electron ap- proximation will be considered. For a dilute alloy the best potential to use is the per- fect lattice (host) potential with an additional potential at each impurity site representing the difference between the impurity and host potentials. This poten- tial can then be used in perturbation theory or even in more sophisticated schemes. A modified version of this approach will also work quite well for concentrated nearly-free-electron alloys. For the case where at least one constituent of the alloy has a strong scattering potential with just-bound or nearly-bound states the muffin-tin approximation is best. The muffin-tin assumption is worse in an alloy than in a pure material because the local interstitial energy may vary from place to place through the alloy. It is hard to see how one can readily estimate the ef- fects of such variation. It is customary to make the ad- ditional assumption that the muffin-tin potentials for each constituent are independent of the environment. This is not generally correct; there is a spill over of the impurity potential onto neighboring sites. Unfortunate- ly neglecting this change is inevitable at this stage in the development of the theory and, it will be noticed, resembles neglect of positional relaxation about the im- purity site in defect calculations. It should be borne in mind, however, that the level of the theory at which such effects would enter is well beyond anything that will be discussed in this paper. Since it is clear that par- ticular features of the band structure may depend criti- cally upon such local effects, the most obvious example being bound states localized near an impurity atom, the defect type theories plainly have important applications here. The virtue of the nonoverlapping muffin-tin form of the potential is that in the potential free region between the spheres electrons move as in free space. The mo- tion of an electron can thus be seen as a series of scat- terings from individual sites with free electron propaga- tion in between. It is this separation of the scattering events which is so important to what follows. If the alloy constituents are at sites Ri, i = 1, . . . N then the potential just discussed may be written for a particular alloy V(r) = X. va,(r- Ry) where pºi is the type of atom at site Ri and v, (r) is the potential of the pºth atom type. The nonoverlapping restriction is that vº (r) = 0 r P ro where ſo is half the near neighbor distance. There is ac- tually no restriction that v(r) be spherically symmetric though this is often quoted as necessary. With this greatly simplified form for the potential it is possible to proceed with the formal theory of disordered alloys. 4. The T-Matrix In the discussion of the density of states in an alloy it is necessary to appeal to the concept of a T-matrix and it is therefore worthwhile to demonstrate that it is actually a simple concept. Indeed, while much of the al- gebra is rather involved, the use of the T-matrix allows almost classical mental pictures to be used and the al- gebra very largely suppressed. Consider then a single scattering center of potential V(r) with particles incident upon it. It is well known that this problem can be solved in integral equation form as eiki r-r'ſ –4 | * V(r)/(r)ar il (r) = db (r) 4T |r—r | where b(r) is the incident beam and the second term gives the wave scattered from the potential. Note that the complete wave function lºr') appears in this latter term; if bºr) were inserted here the scattering would be given in Born approximation. The T-matrix is formally defined by V(r') /(r') =ſ T(r', r")d (r")dr" 455 but is best understood in terms of the physical descrip- tion. The point is that it is very convenient to describe a potential by the scattering it induces among a set of states defined outside the potential and this is what the T-matrix describes. In the crystal these states will be spherical harmonics centered on the atomic position in question. The T-matrix for the potential describes the scattering from one (ingoing) spherical state to another (outgoing) such state. The infinite crystal potential leads to a T-matrix of different interest. The feature dominant here is that the scattering cross section is infinite for electrons incident on a potential well at an energy at which that well has a bound state. Using this in reverse one may look for the energy levels of the crystal by seeking the poles (or branch cuts) of the T-matrix. Actually the density of states is directly proportional to the imaginary part of the T-matrix and this is the link which will be adopted below. One can now utilize the fact that the total alloy potential is made up of individual scattering sites with their own T-matrices ti. All scattering processes from the alloy can be described in terms of the sequence of scatterings from these sites. Thus T=X tri-X tout; + . . . (1) i izºj where the nth term describes those scatterings from the alloy in which n individual site scatterings occur. Gij describes formally the way in which the electron moves between the scatterings at i and j and it is to be noted that two consecutive scatterings cannot be from the same site. 5. The Alloy Formalism The aim of this section is to introduce a diagram- matic technique which carries considerable insight into the nature of the disorder problem. The algebra required to set up the diagrams is given mainly because of its intrinsic interest. This algebra may otherwise largely be skipped since the meaning of the diagrams is fairly readily understood. For a single alloy the series (1) can be cast in an en- tirely real form and the imaginary part corresponding to the density of states (n(E)-X.6(E-E) for this alloy) will only emerge after the infinite series has been summed. In other words the bound states occur at the points where the series diverges. Since the averaging plainly cannot be carried out on the summed T-matrix it is necessary to resort to a term by term averaging fol- lowed by the infinite summation. To be specific con- sider a binary alloy specified by writing at the site i tycº where to, tº are the t-matrices for potential wells of type A and B respectively. Also the cº, are defined by cº-l if the atom at site i is of type A = 0 otherwise c?=1 – c = 1 if the atom at site i is of type B = 0 otherwise In the average system these cº are given by certain dis- tribution functions appropriate to the alloy composition in question. It is convenient to introduce dummy varia- bles cº, c' = 0, 1 so that c =X, cº., a. Then the T. matrix becomes cº T= X. tyc'ò cº, c; + X. tac"Gijt, c'ò.6 c”, c* + & e (2) i, cºv izºj J c”, cu l’, Al and the averaging which must be performed has been entirely concentrated into the Kronecker-delta func- tions. Consider these averages. What one requires to know are probability distributions such as c', cº, cº, C4 P (. j k l )– ( öc, c. öcz, c; , öcs, c. * öc., c) ) (3) For example, take a completely disordered binary system in which the concentration of the constituent A is c. Then the probability distribution of the cº, 's is II {cö... + (1-c)ö... } (4) all i ; i Sites Since the probability of finding atom A is indepen- dently c on every site. When this probability function is used to evaluate the average on the right-hand side of (3) the factor in braces in (4) only occurs once for each independent site in (3). This means there is not neces- sarily one such factor for each Kronecker delta in (3) but rather only one for each independent site. Thus one obtains "(ºf)- X. II'{cö... +(1-c)6... } i j k l a - ; 1 * ~ * #1. * , cº, c; etc. be, ºr . . . 6..., (5) where the prime on the product denotes the restriction that only independent sites among the i, j, k, l are included. It is this restriction which is at the heart of the disordered alloy problem. It states that when the same atom appears more than once in a term of the se- ries (2) the average cannot be taken as though each ap- pearance is independent of the others. It is to be 456 emphasized that this restriction is geometrical; it is a sort of counting problem. Obviously the magnitude of any counting error will depend on the relative size of the various terms and the precise determination of the physical parameters determining these sizes is the es- sential problem to be faced. The partial progress in this direction will later be illustrated by example but much remains to be done. A common first step in dealing with a restriction is to first ignore it and then make successively more accu- rate corrections. This procedure is readily adopted here. Consider the product over independent sites of some factor fi: TI f = II f II [f +1 — fil indep. indel). Other SiteS Sites Sites = II f {1+ X (liſi). X. all Other f tWO Other SiteS Sites Sites (1–fi) (1–f;) Ji) (liſ) . . . . ) (6) f f where the last line is obtained by expanding the second product. Clearly the correct result will only be regained after all the terms in the sum have been included. This formula allows a simple diagrammatic representation in which all the sites appearing in a term are given as dots on a line as in figure 2a. The dots themselves represent the t-matrices and the lines between the dots represent the propagators Gij. In this series of scat- terings the repetition of a given scattering center can be treated by the device of eq (6). The first term in the sum in (6) corresponds to ignoring the restriction in which case the probability distribution will be given by 1 22 2.3 1 2 3 P(; ; ; )-P(...) P() (). . . (...; k, * * * Pſ, P j P k sº where P(;)-> 0.4 (co-º-o-ooº…) This gives at once X tºrſ.")=c,+ (I–)-i cº, #4. (7) The independent averaging of a point can be denoted by a cross as in figure 2b. The next term in the sum (6) connects each repeat site with the previous occasion on which the site appeared and can be represented by a dotted line as in figure 2c. The third term contains - * - - - \ \ * = = <= assº * * FIGURE 2: The diagrammatic expansion for the T-matrix showing (a) an unaveraged line, (b) the lowest order approximation, (c) a typical lowest order correction term, (d) a second order correction term, and (e) a forbidden graph. two repeats i.e. two dashed lines and so on. Indeed, the entire series (2) may be given by the diagrammatic ex- pansion satisfying four simple rules. For each value of n, n = 1, 2, 3, . . . draw all possible diagrams having the following properties (i) There are n points on a straight line. (ii) Pairs of these points (but not a consecutive pair) are joined by dashed lines. (iii) No two dashed lines leave a point in the same direction. (iv) All points not touched by a dashed line have crosses on them. The meaning of these diagrams can be readily put into words. For example figure 2c corresponds to a term in which the electron first scatters from 6 different sites then scatters again at the fourth one and finally scatters at two more different sites. Figure 2d represents the electron returning to the first and second sites for its third and fifth scatterings. All such scattering topologies must be drawn and included in the series though it is important to note that diagrams of the type of figure 2e are forbidden by rule (iii). Finally, the dashed line does not represent the actual value of the process it 457 describes but the correction to the related process in which the restriction is ignored and the two ends of the dashed line replaced by crosses. This is illustrated in the example below. The evaluation of the contribution of a diagram is complicated by the two separate features involved. First there is the t-average which is given by the procedure outlined between eqs (3) and (6). Secondly the propagator sum must be evaluated for each dia- gram; it is this which is the limiting difficulty as the dia- gram becomes more and more complicated. Consider any diagram of the type 2C i.e. with only one dashed line. The average corresponding to the dashed line may be written X. tutº 6¢, c. 62, c} {cöer, or + (l – c) 6., c) cu, c' × [65– {cöcſ, c; + (1–c)6. cºl =ct:-- (1–c)ti-iº (8) ôij is written here as a reminder that j is exactly the same as i and was only introduced as a dummy varia- ble. This is plainly of the form of the exact term less the zeroth approximation to it. Each cross on the line is replaced by t as in eq (7). Propagator lines which do not appear inside the dashed curve occur in the combina- tion X. Giji=G. (In practice the sum is k-dependent; j(# i) (, (k) = S (, (R-R)e^***). j(#i) Between the ends of the dashed line the propagator must trace a scattering path beginning and ending at the same point so that the sum involved is S = º Gmněni. j(#i), k(#j) etc. Gijójk . . There is an additional restriction that no intermediate site can be that of the end point, otherwise rule (iii) will be violated and the correction terms overcounted. This sum is readily completed by Fourier transformation giv- Ing S= I/(1 + ti) where 1=7|ako-'kyiſh-Ticºol- (9) Here the appropriate number of t factors has been in- troduced and the integral in (9) is over the Brillouin zone of volume T. The formal derivation above yields a diagrammatic expansion of a very familiar sort on which the usual tricks can be pulled. These will be discussed in the next section but it is appropriate that some feeling for the meaning of the diagrams be obtained. It will readily be seen that the expansion is not necessarily in powers of any small parameter of a physical nature (it might be e.g. if t were very small) but is more precisely viewed as an expansion in the disorder of the system. One way of looking at this is to consider the sequence of neighboring sites, i,j,k. Of the possible values of k, one will be i and therefore (1/number of values of k) of the terms will require corrections as discussed above. If Gij is short ranged, as is the tight-binding case, this means that the proportion of correction terms is 1/Z (where Z is the number of near-neighbors) for each dashed line. An alternative is to regard the series as an expansion in the fluctuations of the system. While 1/Z appears to be small the number of terms with n dashed lines in- ceases something like n! as n becomes large so that the series is at least asymptotically convergent. This is well known to occur in certain statistical mechanical expansions to which the above procedure is naturally related. 6. The Simplest Approximations It is now possible to classify most of the theories of disordered alloys according to which diagrams they retain. This is useful in two ways. It gives information about the nature of the various approximations and it helps one to understand some of the physical parame- ters determining the convergence of the resummations. Begin with the simplest possible terms. 6.1. The average t-matrix approximation 11] Ignoring all the diagrams with dashed curves gives at once the result Nî T(k)) = —— (T(k))=[-º, ––– f — (, (k) in which the alloy is represented by a perfect lattice with identical scatterers having the average scattering of the alloy constituents. The density of states given by this approximation for a binary tight-binding alloy is shown in figure 3. The tightly bound energy levels are E4 and Eb and the various cases are found by compar- ing |EA-EB with the bandwidth. While the gap between 458 / —= n (E) b) FIGURE 3. The density of states in the average t-matrix approxima- tion. (a) when |EA-Ep || > → bandwidth, (b) when |EA-En | < * bandwidth. the band halves is reasonable when the bandwidth is much less than the energy splitting (fig. 3a), this ap- proximation incorrectly predicts the gap even when the bandwidth is very large (fig. 3b). At the same time the rest of the band is moderately well described showing that the accuracy of the approximation for a given ener- gy depends on the position of that energy within the band. 6.2. First order approximation Under the assumption that the 1/Z series converges one might next evaluate the contribution of the terms of the type shown in figure 2c. This contribution must include all possible numbers of intermediate scat- terings both inside and outside the dashed loop. But now consider the diagrams in figure 4a. These each have an intermediate denominator which we expect to be nearly singular and all this sequence must be in- cluded too. This is simply the usual manipulation to give a self-energy rather than a t-matrix which is ab- solutely necessary in this context. Now - N (t + X) - 1 – G T where X is given by the diagram of figure 4b, with all possible numbers of intermediate states and can be evaluated from eqs (8) and (9). The inclusion of this term is an improvement over the average t-matrix ap- proximation but does not resolve all the difficulties. / \ / \ / \ / \ + 1. \ 22* SS 2---SS e \ / \ / \ \ / \ / \ + A \ I \ , - SS ,---> 2^ N / \ Z \ Z \ f \ f \ / \ f \ f \ f \ + → + O © (; d) ... " --~s 2 * `N z' / N / \ | \ b) > = f \ FIGURE 4. (a) the diagrams summed to give the self energy form, (b) the definition of 2. 459 2^ N / \ F: 1– _--- ~~s 2^ S Z * = ºne N. / 2- SN N / / \ \ / \ + / f \ \ 2 ” _- - - - - - --~ >s 2^ N / N / 2 - ~ 2 - SS \ / Z \ / \ \ / / \ / \ \ —H / ſ l I \ \ _---------- *s 2^ asse sº * `ss /* sº ~ J / 2^ N. N / Z 2 * ~ \ N / / / N \ \ / / / \ \ \ —H· | I f \ A \ +- © & © O FIGURE 5. The diagrams summed to give the self-consistent propa- gator inside the dashed line. /N E —= n (E) FIGURE 6. Some density of states curves showing approximations (iii) dashed lines and (iv) full lines for c=0.5 and |EA-Ehl= A/3.9. 6.3. Self-consistency It has long been known that using the unperturbed propagator in the expression for the self-energy is not the best that can be done. The same is true here, where the motion of the electron between successive scat- terings can readily be included as if it were being scat- tered by (t+X) scatterers. This corresponds to includ- ing the infinite series of diagrams seen in figure 5. This step is necessary because one really wants to allow scattering of the electron into the true, not the unper- turbed, density of states. Once this has been done the self-consistent equations obtained can be solved when the unperturbed density of states is given by the ex- pression [2] no (E) 4. N E – T, \? . 1 #A N' ( #A ) if |E-T, 3 } = 0 otherwise. This self-consistency calculation is an immediate im- provement in the sense that the band gap now closes as the splitting |EA-Eh becomes smaller than the band- width. Figures 6 and 7 show the alloy density of states in ^ E H. O. 8 > n (E) FIGURE 7. The density of states curves for c=0.5 and E4-E, HA/0.7. The scale here is such that E4 = + 1, ER = − 1.0. 460 2% NN / \\ // \\ Z/ \\ 2- - - -s ,” SS Z \ / \ \ = 1 | ...~ * ~ .” - * / SS 2^ SS / \ / N / \/ \ + 1- \ º 2--~s 2--~ 22-sº / \, / \ / / \/ \, / \ + | W V \ + © Q O FIGURE 8. The self-energy graphs required to renormalize the scatter. ing energy levels. the approximation as a dashed line. One major error of such a density of states is that it does not satisfy the sum rule on the density of states, a difficulty that can be overcome by a further resummation. 6.4. The renormalized energy levels The work of Hubbard [2] and later workers [3,4] overcomes the sum rule problem by extending the scat- tering from any single site to include all possible processes involving only that site. These are the dia- grams of figure 8. This has the effect of renormalizing the energy levels and gives the full line density of states in figures 6 and 7. It will also be noticed that the bump in n(E) near EA and EB has flattened out. Thus overall this more sophisticated summation gives a smoother density of states. It is interesting to observe that neither 4-EP A at which the gap closes is affected by this last improve- ment. Within the numerical error approximations (6.3) and (6.4) are identical in this respect. This last approximation has allowed several alloy density of states calculations to be carried through [3,4] and is probably the best that can be done at the moment. It is rather difficult to see what to do next. There is no general rule for selecting sets of diagrams obeying the density of states sum rule and one has no insight into which of the higher order terms are the most important. Indeed different sets will probably be important in different regions of the energy spectrum. the value of the gap width nor the critical ratio 7. Dimensions and Fluctuations This section is intended to draw attention to a few of the difficulties associated with the alloy theories just outlined. Rather than dealing directly with the alloy case it is preferable to simplify to the vacancy problem. Here the second alloy species is replaced by vacancies so that only a proportion c of the perfect lattice sites are occupied by type A atoms. In the formalism of the preceding sections it is only necessary to puttg = 0. The illustrative value of the number of dimensions in this argument arises because the formed procedure just outlined is independent of the dimensionality for fixed unperturbed n(E). The physical nature of the problem, however, depends extremely strongly on the dimension principally through the importance of fluctuations. Ob- viously the fluctuations in one dimension are very large compared to those in three dimensions since the number of neighbors is so much less. This is reflected in the fact that in one dimension the density of states for the alloy problem has a strongly peaked structure, the peaks being identifiable as local groupings of a few atoms as was first remarked upon by Borland [5]. In particular, for the vacancy tight binding cases, the probability of finding a line of n occupied sites with a vacancy on either end is cº(1-c)” and the density of Pl states for such a group is X. 6(E-E). The total density i = 1 of states is then effectively a weighted sum over 6– functions and is not continuous. Obviously in two and three dimensions the same is true for sufficiently small concentrations but not in general. Plainly when fluctuations are this important there is no chance that the theory described previously can hold. Another point which can be readily demonstrated is that in such a vacancy case there exist states outside the bands calculated by any of the models just discussed. Consider a simple square lattice in two dimensions. This has coordination 4. A square of side n fully occupied by atoms will have a mean coordina- tion number 4n” — 4 Pl, *— I l 4n.” n since atoms at the sides only have coordination 3. In a tight binding model the mean coordination number is a least estimate for the bandwidth of a group of atoms compared to that for the full lattice. This can be readily seen by a variational calculation with trial function V n2 =X Viln. Now the theories described above give a i- 1 bandwidth or (approx. (6.1) or Vc (approx. (6.4)) so for 461 the latter case there is a probability cº = c(i-Vº)-2 of groups of atoms with energy levels outside the calcu- lated band. Such states of course tail off rapidly but nevertheless exist with positive density of states right out to the full perfect lattice bandwidth. The same ap- plies in three dimensions where the tail is smaller but still there. Figure 9 illustrates this point. Obviously such states are very difficult to spot in the type of theory discussed earlier. They certainly exist at the out- side edges of the alloy bands but whether or not they exist inside the band gap is a more difficult question which will undoubtedly be the object of more study. 9. The Experimental Comparison It is natural enough that when developing a theory of alloys one begins with the simpler, one-body quantities such as the density of states. Other properties, such as conductivity, Hall effect, etc. really require more so- phisticated theoretical treatments. The relevant experi- ments are well dealt with elsewhere in the symposium and so will not be discussed in detail here, but it would be wrong to completely omit commentary on them. Calculations of the density of states yield a function of energy and a proper experimental comparison would require that function to be observed. Only in the optical type measurements, particularly photoemission and soft x-ray work, is this possible even in principle. In practice there is a good deal of ambiguity in separating out the n(E) curves from other energy dependent varia- tions typically in the matrix elements and many-elec- tron effects. It is clear that for a general comparison with experiment the theoretical work must be pushed to the point of predicting directly the observed data. This is a step which at the moment looks to be just about possible though complicated by the need to take into account local effects through the matrix elements. Those experiments which measure densities of states at the Fermi surface are complicated by many-body fea- tures which, for example, can cause the density of states to be enhanced by quite large factors ( ~ 1.5 or more). There is the additional difficulty that the pure transition metal band structures lead to density of states curves with a good deal of structure on them so that detailed comparison with experiment requires precise knowledge of potentials and the Fermi energy. Beck and coworkers [6] have overcome some of these objections by using transition metal alloys which seem to conform very closely to the rigid band model and in which the parameters vary only with electron concen- tration. It is still questionable whether theory can yet / / I | | I | EA-A/2 -> n (E) FIGURE 9. Density of states in the vacancy case showing the limits of the perfect band and the tails extending to those limits. predict those cases where Beck’s attack will work and in particular where and why it breaks down in the transition metal alloys. The need for an indirect step in all these experimen- tal analyses has hitherto prevented a satisfactory com- parison between the calculated and the measured den- sity of states. It now seems possible that since so- phisticated yet fast band structure calculations have been developed and better theoretical understanding of the alloy problem is being rapidly gained the time should soon come when the comparison will become direct. Finally one might expect some rewards by looking for the gap predicted by the theories when the energy levels are well split as in figure 3a. However, this par- ticular gap is strongly reminiscent of those predicted in the more general disordered system theories and still argued about at length. It is very difficult to observe the difference between a gap and a very low flat minimum even though the distinction is theoretically very impor- tant. Perhaps the tunnelling experiments can help here. 10. Conclusions The theory of disordered alloys is at last coming to the point where it can truly be regarded as a theory rather than being a collection of ad hoc methods. The 462 problem of a disordered alloy can thus be treated in a parallel fashion to many-body theories and in con- sequence considerable experience taken over from the many-body theorists. At the same time there remains a real difficulty in producing a satisfactory dialogue with the relevant experiment work. Overall, however, the picture is one holding promise of good develop- ments in the near future which may have a catalyzing effect on the theoretical understanding of disordered systems. ll. References [1] Beeby, J. L., Phys. Rev. 135, A130 (1964). [2] Hubbard, J., Proc. Roy. Soc. A281,401 (1964). [3] Soven, P., Phys. Rev. 156,809 (1967); Onodera, Y. and Toyozawa, Y., J. Phys. Soc. Japan 24, 341 (1968). [4] Velicky, B., Kirkpatrick, S., and Ehrenreich, H., Phys. Rev. 175, 747 (1968). [5] Borland, R. E. and Agacy, R. L., Proc. Phys. Soc. 84, 1017 (1964). [6] See for example, Cheng, C. H., Gupta, K. P., van Reuth, E. C., and Beck, P. A., Phys. Rev. 126, 2030 (1962). 463 Local Theory of Disordered Systems” W. H. Butler” cind W. Kohn Department of Physics, University of California, San Diego, La Jolla, California The most striking characteristic of crystalline solids is their periodicity. As a result of this feature, theoretical descriptions of physical phenomena in such systems are usually given in wave number or momentum space. The reciprocal lattice of a crystal and the Fermi surface of a metal are examples. In a disordered system, on the other hand, there is no such periodicity and momentum space descriptions are much less natural. However, in such systems, physical conditions near a point r, in coordinate space, become independent of the conditions at a distant point r", provided that |r'-r is large compared to either a characteristic mean free path or some other appropriate length. This suggests that one can analyze a macroscopic disordered system by averaging over the properties of microscopic neighbor- hoods. In the present paper we report some details of such a program. Although the point of view is of quite general applicability we have, for the sake of definiteness, studied so far only one type of system: Noninteracting electrons moving in the field of interacting, disordered scattering centers. We have focused especially on the electronic density of states. The macroscopic system is represented by an average over small neighborhoods. If one did not take special precautions, one would encounter one class of errors of the order of d! L where L is a characteristic dimension of the neighborhood, and d is a characteristic atomic dimension; and another class of errors of the order of 1/N where N is the number of ions. Both are too large to be tolerable for practical purposes. However, by an appropriate treatment of the statistical mechanics of the scatterers and by periodic repetition of the small neighborhoods, these errors can be avoided. The remaining errors are exponentially small in the ratio y(L/R) where y is of order unity and R is the smaller of the electronic mean free path or the deBroglie wavelength of the electrons. This exponential convergence of the small neighborhood theory promises to make it a useful practical method for the study of disordered systems, especially very highly disordered ones. Numerical examples are presented and discussed. Key words: Binary alloys; density of states; disordered systems; periodically continued neighbor. hood. 1. Introduction The physical properties of strongly interacting disor- dered systems are in general difficult to calculate since simplifying symmetries, present in crystalline materi- als, are not present. In this paper we present and develop to some small extent a viewpoint which ap- pears to provide a useful line of attack on the theory of disordered systems. We shall demonstrate and utilize the rather plausible fact that in a disordered system the physical charac- * An invited paper presented at the 3d Materials Research Symposium, Electronic Density of States, November 3-6, 1969, Gaithersburg, Md. teristics at a point r depend significantly only on condi- tions inside a rather small neighborhood |r'-r| lºſſ (r) lyn (r') e E – En (2.5) 11. Of special interest is the contracted function G(r, r, E) which determines the density of states by means of the following relation, n(E)=– lmſ drcr. r; E + iO). (2.6) 3 Among fairly recent papers we mention the following: H. Schmidt, Phys. Rev. 105,425 (1957): S. F. Edwards, Phil. Mag. 3, 1020 (1958): ibid. 6,617 (1961): Proc. Roy. Soc. A267, 518 (1962); J. L. Beeby, Proc. Roy. Soc., A279, 82 (1964); Phys. Rev. 135A, 130 (1966): P. Soven, Phys. Rev. 151,539 (1966): ibid., 156,809 (1967). We see that we may regard the unintegrated quantity l * n (r, E) === Im G(r, r, E + iO) (2.7) as the local density of states density. We would like to estimate the effect of introducing the weak additional potential v, at the point r", on n(r, E) at the point r. The quantity of physical interest is then (#5 or re-o) = (G(r, r"; E-H iO) G(r', r, E + iO)), 0 (2.8) where the brackets () denote configuration average. The equation (2.9) follows directly from the equation of motion of G. Now it is well known that, for weak random poten- tials Va, l f e i VE|r-r' ſe – r-r' ||21(E), T|r—r" (G(r, r"; E + iO)) ~ 1 (2.9) where lſb) is the mean free path of an electron of ener- gy E. Similarly, one can show from (2.8) that for large r-r' (#700, re-o) Thus we see that when |r'-r => l'E), a change of poten- tial at r" has a negligible effect on the quantity n(r, E). Next we show a similar locality effect, this time not due to a finite mean free path but due to elevated tem- perature. We consider a system of independent elec- trons and use Boltzmann statistics for simplicity. We ~ e —|r—r"||l (E) (2.10) take as Hamiltonian H = Ho-H v. (2.11) where, in comparison with (2.1), we have eliminated the disordered scattering potentials. We write the partition function as Z = ſ dr Z(r), (2.12) where Z(r) = (re-Billr). (2.13) Our interest now is in the influence of a small perturba- tion v at r" on Z(r). By standard perturbation theory one can show that, for large r" — r|, TT5/2 84 (r). Tº 1633/2|r–r" € öv (r') – |r – r" | */A; 9 (2.14) 466 where A, is the thermal wavelength A = 3112 (= h/(2m kT) 112). (2.15) Of course at high temperature and with disorder, the characteristic “range of influence” R will be either a representative mean free path l, or A, whichever is the smaller. 3. The Periodically Continued Neighborhood We shall now develop a concrete method of calcula- tion, based on the locality principle of the previous sec- tion. We shall concentrate on the density of states n(E). The most straightforward way would be as follows. We imagine the large disordered system as given. We choose at random a large number of points R1 and sur- round them by spheres of radius p considerably larger than the “range of influence” R, but not too large. Each sphere is surrounded by an infinite wall (see fig. 1). The electronic Hamiltonian for each sphere is given by H = Ho-H V (3.1) where V= | X. V., (r—ro) | r – R sp (3.2) —H co r > p We then calculate for each sphere the density of states density at its center, n(R1,R). Because of the rela- tively small size of the neighborhood this is a much more manageable problem than n(E) for the macro- scopic disordered system. In view of the locality princi- ple, n(RI,E) is only insignificantly affected by the presence of the infinite wall. Hence, the density of FIGURE 1. Spherical neighborhood centered at R. states for the macroscopic system is given by n(E) = Qn(R1,R), (3.3) where the bar denotes an average over many Riº The practical drawbacks, for finite p/Ro, are two: (1) The replacement, in the outside region, of the actual potential, X. Va., by an infinite repulsive barrier is a quite drastic change and unless plk is very large, will cause sizeable errors. (2) This method does of course not lead to the exact results in the special case of a vanishing or periodic potential. These drawbacks can be largely overcome by using, instead of a finite and bounded neighborhood, as in figure 1, a periodically continued finite neighborhood as indicated in figure 2. We choose a fundamental cell, say the cube, of volume QL = Lº, ; |x|, |y|, |z| < L => 2R (3.4) and construct the space-lattice generated by it. Let us call T") the lattice translation vectors. Then we place any number of scatterers in some definite configuration c in the fundamental cell and populate the other cells in the identical way (see fig. 2). The electrons now move in a periodic cubic lattice. Their energy eigenvalues *— L —- O O O (T) oo O O O O O O O O O O FIGURE 2, Periodically continued neighborhood. * If we would use the total densities of states of the spherical neighborhoods, rather than the quantities n(RI,E) computed at their centers, we would incur large errors behaving as LT', due to the presence of the infinite wall boundaries. 467 are, because of the relatively small size of the unit cell, much more amenable to calculation than the eigenvalues of the macroscopic disordered system. Let us call the density of states, corresponding to the con- figuration c and total volume (), no(E). Then our approx- imation for the density of states of the actual macro- scopic system will be n(E) = X wene (E). (3.5) where we are weights, which we shall presently discuss, and the approximate equality, ae, will signify accuracy to within terms exponentially small in L/2R.” We shall now describe a suitable choice of we and later demonstrate that it leads to the claimed accuracy. Let us suppose that in the actual, macroscopic system under consideration there are given two-body forces between the scatterers, ‘pag = pag (ra-ra), (3.6) which vanish beyond a range a which is much smaller than L. Let us suppose further than the scatterers obey classical statistics. Thus, in the macroscopic system the probability of a given configuration c = (r1, r2, . . .) is given by the grand canonical weight function, X pag- > p;N; o #8 j where A*) is the normalization constant, puj is the chemical potential of scatterer of type j, and Nj is the total number of scatterers of this type." A suitable choice of wo, for the periodically continued neighborhood, is obtained as follows. We take the unit cell and associate with each original position vector r, the infinite set of vectors r(t) = r + T (*) (3.8) This corresponds to the topology of a three dimensional torus and is illustrated in figure 3 in one dimension. The weight we is then determined by the equation we=A exp |-6 |: > eas (Fog) -> ww.) (3.9) 5 Provided the forces between the scatterers are sufficiently short range. * For the macroscopic system, one could of course equally well use a canonical distribu- tion. However, in our local neighborhood theory, this would lead to unacceptable errors of order 1/N, where N is a mean number of scatterers in Qi. ſiz ſ4 FIGURE 3. Schematic representation of the toroidal topology. The circumference is L. where rag is the shortest vector contained in the set ra")- rg"). For example, r12 is shown schematically in figure 3. The normalization constant A is chosen so that (3.10) X w.e-1, C where the sum goes over all configurations, including all possible numbers of the scatterers. It can be shown that, with the choice (3.10), the cor- relation functions, n,(r. . . . . r. L), in the finite toroidal system, differ negligibly from those of the infinite system, n,(r1, . . . . rs|oo), for values of re-rels L/2 and s up to ss L/a. The error of a given correlation func- tion behaves as exp [-oslla], where os is of order unity. This fact assures that, if L> a, the statistical distribu- tions of the ions in any neighborhood of size - L|2 are practically identical in the ensemble of periodically continued neighborhoods and in the macroscopic system. Hence, in view of the locality principle demon- strated in section 2, the density of states n(E) of the in- finite system may be determined, via eq (3.5), from the density of states in the periodically continued neighbor- hoods. The error will be exponentially small in the quantity L/R or L/a, whichever is the smaller. Typically both a and R are of the order of 1 A, so that one must work with neighborhoods of dimensions of several in order to obtain quantitatively useful results. It is evident that this method will give exact results for perfectly periodic systems. Since it is also very ac- curate for highly disordered systems (small R), it should give good answers for most intermediate situations. The ensemble of systems defined by eq (3.9) is a grand canonical ensemble in which the volume QL(=L*) is fixed while the numbers of particles Nj assumes all possible values. For a single species of atoms it has been found practically preferable to work with a periodically continued isothermal-isobaric ensemble, in which N is fixed but the volume QL is variable. Here 468 a unit cell of volume QL, with the V atoms in a configu- ration c must be given the weight wº-á, exp |-6 |} X. eas (rag) –Roºm) O. , 8 where P is the pressure. The density of states of the ac- tual macroscopic system is approximated by n (E) –ſ dO. S. wo, on (E: Qt, c) (3.12) where n(E; QL; c) is the density of states, per unit cell Qi, in the periodically continued neighborhoods. The normalization A of wol, c | do, X wo, c= Not/N (3.13) C where Nio is the total number of scatterers in the macroscopic system. Again the convergence of (3.13) to the exact result is exponential, as for the case of the grand canonical ensemble. 4. Numerical Illustration To illustrate the theory of the previous sections we have numerically studied the following model of a one- dimensional alloy: d? H--jars No. 6(3 – Od) (4.1) where the distance between potentials, d, was taken as 1, and Na was taken, with equal probability, as -2 and –4. (This corresponds to a high temperature limit for the two different kinds of scatterers, an idealized model for crystalline Cu Au well above the ordering temperature.) Here the grand canonical method was appropriate. Two calculations were performed, with L = 6 and L = 10. Let us take the case L = 6. A typical, periodically continued neighborhood is shown schematically in figure 4. Each configuration of potentials in the fundamental interval 0 < x s 6 has the same weight, 27°. In this simple example all correlation functions are evidently exact, as long as the crystal sites considered are all contained in a single unit cell. <- d > * L T L —z- I FIGURE 4. A typical periodically continued neighborhood of the illustrative model. X = -2 \ = — 4 The density of states curves was calculated for each of the 2° configurations and then averaged. For L = 10 a sampling procedure was used. The results are shown in figure 5, together with the exact result obtained for the infinite system using the Schmidt method.” Our ap- proximation reproduces quite accurately all the details of the density of states structure, even for L = 6. To ob- tain similar accuracy from a single randomly populated chain would require a length of the order of L = 10°. 0.70 I | I | -T- | I | I T- I- 0.60 H – 0.50 H. A * = º: 0.40 H / * £ | a” * § 0.30 – ſ ==mſ C/D 0.20 H. 4– [] N = |0 - ge A N = 6 -- 0.|0 H / -4 O ~. ––––––––. – 6.0 – 5.0 — 4.0 – 3.0 – 2.0 – || 0 O ENERGY FIGURE 5. Integrated density of states for binary alloy. The solid line is the exact result. 5. Concluding Remarks It is commonplace to emphasize the theoretical dif- ficulties caused by disorder. In the present paper we draw attention to a favorable feature: For a highly disor- dered system the physical properties near a given point depend only on circumstances in a small neighborhood, whose dimension is of the order a mean free path. Con- sequently, a macro-system may be treated, so to speak, neighborhood by neighborhood and the macro-problem can be reduced to an ensemble of micro-problems. We have shown that the errors of such a procedure can be made to vanish exponentially with the size of the neighborhood. A numerical model calculation bears out these considerations. A more complete account of this program will be published elsewhere. 469 Discussion on “Local Theory of Disordered Systems” by W. H. Butler and W. Kohn (University of California) D. Redfield (RCA Labs.): In a system with local varia- tions in densities of states, it would seem there should be local variations in the mean free path. Does this treatment imply that the assumed smallness of this path makes these variations unimportant? W. Kohn (Univ. of California): Qualitatively speaking yes. Having established this general locality principle in which a representative mean free path appears just guides one in one’s further thinking. One does not ac- tually introduce a mean free path into the calculation. I am not sure we are in communication. J. Tauc (Bell Telephone Labs.): It seems to me that this locality principle, if it is applied in the sense that you first mentioned, is different from what you were speaking about later. If you say that it is only the en- vironment of an atom which is important, then, of course, considering only a few atoms, you get a discrete energy spectrum; the energy spectrum for the whole body would be continuous by averaging. But an elec- tron at a certain place would have a discrete spectrum, and this would have very important consequences for the conductivity and optical properties. Do you think that an electron at a certain place sees a discrete spec- trum, or rather a continuous density of states? W. Kohn (Univ. of Calif.): The locality principle holds only for the statistical average. Your question is an in- teresting one because it brings out this point. Let me answer this in two ways. First of all, in terms of the first rudimentary model, which actually we are not follow- ing, but I think it is perhaps best for answering your question. That is the model where we simply surround the four or five scattering centers by an infinite wall. And now we put the electrons in there. Obviously the energy levels are discrete. Now imagine that this sphere is rather large so that the spacing between the discrete states becomes small. Now average over all positions of the scatterers. Then for each configuration you get a discrete spectrum. When you average over all you get a very large number of discrete spectra which in fact results in a continuous spectrum. And it is that average continuous spectrum whose density relates to the density of the infinite system. I think I did forget a point in discussing my viewgraph. You notice that there are these brackets here; I apologize, I did not speak about them – these are configuration averages. And the locality principle holds for the average quantities. Now what you are interested in finally for the alloy is the density of states of an infinite system that is equivalent to averaging within the individual small cells over all possible configurations. In this way you do in fact end up with a continuous spectrum. In the second method that we use, and which is much better from a quantita- tive standpoint, namely extending the individual cell periodically, then obviously we get a continuous spec- trum even for an individual configuration. A. Williams (IBM): It seems to me, if I understand cor- rectly what is being suggested, that the proposed method holds a great deal in the way of computational promise. It seems to me that you are suggesting that we can do several band calculations for a lattice with a ba- sis. The question then becomes how many members shall we have in the basis and over what statistical en- semble of bases must we average, but nontheless the fundamental calculation is something we have learned to do quite well for a very realistic system. W. Kohn (Univ. of Calif.): Yes, I agree fully with that, and just to go out on a limb I would say there will be quite a few systems where you could expect an accura- cy of 2% by working with a basis in three dimensions with as few as 2, 3, or 4 scatterers. 470 DISORDERED SYSTEMS II CHAIRMEN. L. M. Roth H. P. R. Frederikse RAPPORTEUR. M. H. Cohen Density of Electron Levels for Small Particles” L. N. Cooper and S. Huº Physics Department, Brown University, Providence, Rhode Island 02912 The density of electronic levels for small particles is calculated. This differs from the usual expres- sion which is valid as the volume of the sample becomes very large. The leading term of the correction is proportional to the surface/volume of the sample. Depending on the environment the density of elec- tronic levels may be increased or decreased. Key words: Diffusion equation; electronic density of states: “muffin-tin” potential; small particles; two dimensional classical membrane. 1. Introduction The commonly employed formula for the number of levels of a quantum system in the momentum range dºp (/N = () dºp hº comes from an expression derived by H. Weyl which is valid asymptotically (as the volume of the sample becomes very large). For small samples this is modified. The leading additional term can be expressed as a surface/volume correction. Recently Kac [1] has written down the first few terms for the density of levels for a two dimensional classical membrane, asking “Can One Hear the Shape of a Drum?” We have generalized Kac's result for the leading correction term to three dimensional specimens for two boundary conditions: for the first the wavefunc- tion is zero at the boundary while for the second the derivative of the wavefunction is zero at the boundary. In the latter case the density of electron levels is in- creased. 2. Development In this section we outline, very briefly, a method for obtaining the density of states developed by Kac. The expression S. e-A"|l,(r)| (l) })} = 1 *Supported in part by ARPA and the National Science Foundation. **Present address: Northeastern University, Boston, Massachusetts. gives the number of eigenfunctions in some range of A using |d, S. e-A"|l,(r)| }}) = 1 * }}ll — ºc – A 2." A hit | N(X)e-AdX (2) where N(\) is number of levels per unit interval of A. This is related to the usual expression by d'A V(E)=# N(A) (3) where A = mEſh”. Kac observes that Tº. e-Amt ń. /* in ( / 2. lſ. (Fo) lyn (r) (4) is a solution of the classical diffusion equation *(r.t) l ºr, a –––. V*U(r, t) = 0 (5) with the initial condition U(r, t = 0) = 6 (r-ro) (6) if ! -º 2 Vºlſ, - — Amºl/m. (7) But the solution for the classical diffusion equation far from any boundary should be the same as the solution 473 in an infinite volume; this can be obtained by assuming periodic boundary conditions and in two dimensions has the following well known form: } |r º | * - —— am sºm-ºsmº-wº- Uo (r. t.) 5. exp | 2f (8) Evaluated at r=r, this gives X. cº",(r)==== (9) }}} = 1 27t which when integrated over the area yields TAC area. X. e-Anit - —— 27t (10) })} = 1 This argument is obviously equally valid for any number of dimensions. In particular for a three dimen- sional sample one obtains: S. cº-- *- (11) }}] = 1 (27t)3/2 where () is the volume of the sample. This using (2) yields r () /)\\ 1/2 V(x)=; (..) OT V(E) =#mºmeº (12) which is the usual expression. To take account of a boundary Kac rederives the solution of the classical diffusion equation, approxi- mately, in the following way. If one is close enough to the boundary, to the first approximation the boundary will appear as a line. As he says “for small t, the parti- cle has not had time to feel the curvature of the bounda- ry.” He then writes down what is the solution of the classical diffusion equation with the initial condition (6) and with the boundary condition that the solution go to zero at the boundary and obtains for r = ro | — e–25°ſt U1 (r- ro, t) = (13) 27t where 6 is the distance of the point r from the boundary. This result can easily be generalized to any number of dimensions and for the opposed boundary condition, that in which the derivative of the solution is zero at the boundary. One then obtains for a three dimensional specimen | + e-26°ſt l, (r= ro, t) = (27t) 32 (14) where the E signs correspond to the wavefunction or its derivative going to zero at the boundary. When (14) is integrated over the volume of the sam- ple, one obtains for a three dimensional particle JC – A hit — () - X. 27t e - Tº –– # X. (27t).3/2 (27t) 32 4 }) ] = 1 (15) where Q and X are respectively the volume and surface of sample. 3. Conclusions Using (2) the above expression yields the following density of electronic levels for a three dimensional sam- ple: Th | 4 V2m E where Noſ E) is the usual density of levels as given by the Weyl asymptotic formula and X/Q is the ratio of the surface area to the volume of the sample. The two signs correspond to the boundary conditions: N(E) = No (E) (1 T +. . .) (16) à lſ – 0 at boundary e-> minus il' = 0 at boundary – plus [The next term in the expansion which is related to the curvature of the surface bounding the sample has been evaluated by Kac for various special two dimen- sional closed curves but has not been generalized to higher dimensions. For a two dimensional circular sam- ple of radius R evaluated near the Fermi level, one ob- tains for l =0 at the boundary: N(Ep.) = No (Ep.) (l - (kºk)-1--É (kp R) -* + e = ± ). Thus it is reasonable to expect that for most samples this next term would be much smaller than the second.] Near the modified Fermi surface (16) becomes X. T N(E)=\,(E,00+; ºt. † ..) (17) Referring the ratio of surface to volume to that for a sphere we have (X/Q) Nº-Nºrod-snººk)- sº Sphere + . . . ) (18) For an irregular metallic sample we might guess that X/Q could be an order of magnitude larger than that for a sphere so that one could expect the second term to be 474 as large as 10% of the first for particles of the order of 10−6 cm in radius. For a smaller sample or for a material like a semiconductor with smaller kº the effect would of course be larger. Such an alteration in the density of electronic levels would be expected to produce dramatic effects in the various density dependent physical quantities. Unfortu- nately under ordinary conditions it will very likely be obscured because the physical boundary condition will fall between the two extremes above (l=0 and ly' = 0 at the boundary) diminishing the effect of the second term. In a properly arranged environment, how- ever—for example, one might consider a metal-vacuum boundary to simulate ili = 0 at boundary, or a metal- other material boundary to simulate lº'-0 at the bound- ary — the effect might become visible and possibly important. 4. References [1] Kac, M., American Math. Monthly 73, No. 1, Part II, p. 1 (1966). 475 Discussion on “Density of Electron Levels for Small Particles” by L. N. Cooper and S. Hu (Brown University) A. Yelon (Yale Univ.); For a linear combination boun- dary condition, does the effect tend to disappear? L. N. Cooper (Brown Univ.); Yes, it may well be that one does not see the effect normally. It may add to con- fusion in work with small particles if one is comparing results in one environment with results in another en- vironment and having changes in the density of states due to that. H. H. Soonpaa (Univ. of North Dakota): The effect described in this paper has been observed. We have measured an increase (by 2.3 eV) in the contact poten- tial of 40 A thick semimetal film as compared to the bulk material. A similar effect was observed by Batt and Mee [1]. They observed a decrease in the work function by about 0.2 eV when an Al film used in their photoemission work was made thinner than about 50 Å. These changes are related to the Fermi energy, which increases with decreasing thickness, as predicted by the equations derived for size effect quantization. [1] Batt and Mee, Journal of Vacuum Science and Technology 6, 737 (1969). 476 One-Dimensional Relativistic Theory of Impurity States” M. Steslicka,” S. G. Davison, and A. G. Brown” Quantum Theory Group, Departments of Applied Mathematics and Physics, University of Waterloo, Ontario, Canada The one-dimensional Dirac equation is solved for the Kronig-Penney model containing a 6-potential impurity. Depending on certain existence conditions, the impurity states can be classified into two types called relativistic and Dirac impurity states. In the nonrelativistic limit, the Dirac states disappear and the relativistic ones become the ordinary impurity states. A detailed discussion is given of the complete energy spectrum. Key words: Impurity states; Kronig-Penney model; one-dimensional Dirac equation. 1. Introduction An important problem in the electronic theory of semiconductors is that of impurity doping. Up to now, only nonrelativistic (NR) investigations [1-4] have been carried out on the effect of substitutional impurities on the electronic properties of crystals. During the last few years, however, a relativistic theory of heavy atomic solids has been developed. Most of the calculations have involved the solution of the Dirac equation for the “muffin-tin” potential [5]. In order to study localized states, within the framework of the relativistic theory, the simpler Kronig-Penney (KP) potential [6] has been employed [7–10]. In this paper, the behavior of relativistic electrons in the region of an impurity is ex- amined, by means of the Seitz model [11]. Com- parisons are made with the NR results in the final sec- tlOn. 2. Plane Wave Solution of the Dirac Equation For the k-region of constant potential Vſ., the 2-com- ponent form of the Dirac equation is [12] ific or, Vrds = – (e– W.) (b. 1, e = E – mocº, (1) ihc or, V., (b) = -(e- V. --2mocº) (b2, (2) where 0-(?) ,-(*). Decoupling these equations yields o'-(' !) (3) V; (bj=-pî q j: j = 1 or 2, (4) in which p} = (e– V.) (e–V, +2mocº)ſhºcº. (5) The general solution of (4) is (bj — Ajeo ſº + Bye-ſpºr, (6) Aſ and B; being 2 × 1 matrices. From (1) and (6), it fol- lows that A = yor, A2. B =– yor, B2, (7) with y = (e – Vſ.)/hcpſ. (8) Thus, the 4-component plane wave solution to the Dirac equation can be written as d (x) =( %. ) = ( ſ".) Age'Piº + (;". ) B3e-iplº, (9) *QTG article: S-156 **Permanent address: Department of Experimental Physics, University of Wroclaw, Po. land ***Work supported by the National Research Council of Canada and the University of Waterloo Research Committee where I is the 2×2 unit matrix. 477 3. Relativistic Crystals States The simplest relativistic theory of solids is based on the KP model in a 6-function limit. Initially, the volume [7] and surface [8] states of such heavy atomic systems were analyzed by adopting a continuity condi- tion in which only the 2-component spinors were matched across potential discontinuities. However, the choice of this continuity condition means that there is a nonmatching of the component slopes. Furthermore, the resulting mathematics are somewhat complicated. Since electron velocities in solids are not extremely high, the contribution of the small component d is not as significant as that of the large component (b. Thus, another set of suitable continuity conditions is the equating of the large component and its derivative across the potential discontinuities. This, of course, im- plies a nonmatching of the small component and its derivative. The problem of boundary conditions has been discussed at length in a recent paper [9]. The latter type of continuity condition is adopted here, because it not only leads to a simplification of the Subsequent mathematical analysis but also is a very good approximation, as was shown in [9]. In this ap- proach, the relativistic Kronig-Penney (RKP) relation in 6-function limit takes the form [9] COS pla = cos pg a -i- p is * (10) where p3 = e(e–H 2mocº)/hºcº (11) and – pſ, = lim # pi ab, (12) () — () p3 -> x. (b is the width of the potential wall), while the bulk large component wave function in the unit cell aſ 2 s x s 3a/2 is given by [9] (b2 = O(2 [eip2(r-a/2) + \ne-ip2(*-alº) | • (13) where An = (1 – e-i(-0.2)")/(e-i(-02)” – 1). (14) The subscripts 2 and 3 refer to the two regions II and III of different constant potential in the KP model. 4. Relativistic Seitz Model 4.1. Impurity Energy Expression The potential field in the vicinity of an impurity atom in a linear crystal is represented schematically in figure M x) | | | II I II | | | | - - | - X - O H O | —W | FIGURE 1. 6— Barrier representation of the potential field in a linear crystal (II) containing an impurity atom (I). The difference between the crystal and impurity potential strengths is – W. 1 [2,11]. The procedure for analyzing the problem of impurity states is to match the wave function in the un- perturbed periodic part of the crystal to that in the per- urbed region near the impurity atom. With the system being symmetrical about x = 0, it is sufficient to per- form the matching process at the potential discontinui- ty at x = a12 only. In region I, the solution to the Dirac equation is even or odd, i.e., df = ori (eit *-Ee-'01"), (15) p1 being defined in (5), where V1 is the strength of the impurity potential. Matching the wave functions (13) and (15), and their derivatives, at x = a12, leads to a determinantal equa- tion in o.º. and O2. Setting the determinant equal to zero gives (1 + \º)/(1 – An) = - p 17*/ips, (16) where Tº = -tan ; pia, (17) TT = cot # pia. (18) After some simple rearranging, (14) and (16) give e'" – cos pea + (p 17-ſp2) sin p2d. (19) For localized states to occur, p. has to be complex [1,11], that is, of the form p = n Tſa + ič, & real - 0. (20) With the aid of (20), combining (10) and (19) yields cot p2a = 02a || + (tºpia - 2p (Tp 1)/p; a 1/2pm, (21) 478 where, for convenience the superscript + has been omitted from T. Equation (21) gives the relativistic im- purity state energy (via p2) in terms of the impurity strength (pi) and the crystal potential (pi). It should be noted that for T" or TT eq (21) has solutions in alternant bands only, thus, to obtain the complete energy spec- trum it is necessary to include both the even and odd solutions. 4.2. Existence Condition Subtracting (19) from (10) using (20) gives ( – 1)" sinh (a = | (pl. - Tp la) sin p2a ||pga. (22) For the (n + 1)th forbidden energy gap (FEG), which lies in the range nT ~ (n + 1)T, the RKP relation (10) shows that sign (**)-(–)- £)2(t (23) Thus, since ( - 0, it follows from (22) that DR × Tp 1 a, (24) which is a necessary condition for the impurity states to exist (i.e., existence condition). This inequality imposes a restriction on the values of the parameters for which the solutions of (21) correspond to the impurity state energies. 5. Analysis of Relativistic Effects 5.1. Energy Correction The NR analog of (21) is obtained by taking c → do to give cot p!a = [p]a +T"p! (T"p"a - 2p")/p!]/2p", (25) where the zero superscript denotes the NR limiting value of the corresponding parameters. Solving (21) and (25) numerically, enables the impurity state energies p2 and p20 to be determined, respectively. The effect of the relativistic corrections on the energy is shown graphi- cally in figure 2, where Ap (= p,"-p2) is plotted against n, the band number. At low energies, the relativistic shift in the impurity level is sufficiently small to be ignored. However, for high energies, the shift becomes of the order of the FEG and, therefore, cannot be neglected. 3 - FIGURE 2, Variation of relativistic energy correction (Ap) with band number (n) for pn= p = 0.25, a = 6 (= 3A) and V = + p^2. 5.2. Classification of Impurity States The NR counterpart of (24) is p" > T"pºa. (26) It is convenient to introduce the relations pſ, F mp", T = vT", p1 = kp', (27) where m, v and k are positive and equal to unity in the NR limit. Inserting (27) in (24) leads to p"+ R - Tºp! a (28) where the relativistic correction term R = (1 – vk/m)T"p"a (29) can be positive or negative, depending on the particular band being considered, and the values of the parame- ters v, k, m. If (28) is always valid, then p" > T"p"a (30) depending on the sign and magnitude of R. Comparing (30) with (26), shows that the upper (lower) inequality is identical (opposite) to the NR situa- tion. Thus, the impurity states satisfying the upper in- equality are called relativistic impurity states (RIS), since they become the usual impurity states in the NR limit. However, the states for which the lower inequali- ty holds violate the NR existence condition (i.e., disap- pear altogether in the NR limit). Hence, these states 479 arise solely because of the Dirac formulation and, for this reason, are known as Dirac impurity states (DIS). It is worth noting that, near a band edge, if R → 0 (R < 0) in the existence condition (28), then the presence of this relativistic correction term enhances (hinders) the possibility of an impurity state occurring compared with the NR case. 6. Conclusions A relativistic investigation of the impurity states of heavy atomic crystals has been made, using a simple model, which was proposed initially by Seitz [11]. In deriving the expression relating the impurity energy and strength, only the so-called large component was taken to represent the relativistic wave function in the crystal lattice. The form of the energy relation is identi- cal to that of the Schroedinger approach. The sub- sequent analysis showed that both the existence and lo- cation of the impurity states in the energy spectrum are sensitive to the relativistic correction terms, especially at high energies. The impurity states appear in two categories, namely, relativistic and Dirac states. In the NR limit, the former become the ordinary impurity states, while the latter have no analog. 7. References [1] Saxon, D. S., and Hutner, R. A., Philips Res. Rep. 4, 81 (1949). [2] Friedel, J., Advs. Phys. 3,445 (1954). [3] Koster, G. F., and Slater, J. C., Phys. Rev. 95, 1167 (1954). [4] Breitenecker, M., Sex!, R., and Thirring, W., Z. Phys. 182, 123 (1964). [5] Loucks, T. L., Augmented Plane Wave Method (W. A. Benjamin, New York and Amsterdam, 1967), for literature sur- vey and collection of relevant reprints. [6] Kronig, R. de L., and Penney, W.G., Proc. Roy. Soc. A130,499 (1931). [7] Glasser, M. L., and Davison, S. G., Intern. J. Quantum Chem., in press (1969). [8] Davison, S. G., and Steslicka, M., J. Phys. C. (Proc. Phys. Soc.) 2, 1802 (1969). [9] Steslicka, M., and Davison, S. G. (to be published). [10] Davison, S. G., and Levine, J. D., in Solid State Physics, F. Seitz, D. Turnbull and H. Ehrenreich, Editors (Academic Press, New York, in press). [11] Seitz, F., The Modern Theory of Solids (McGraw-Hill, New York and London, 1940), p. 325. [12] Davydov, A. S., Quantum Mechanics (Pergamon Press, Oxford, 1965), p. 222. 480 Discussion on “One-Dimensional Relativistic Theory of Impurity States” by M. Steslicka, & S. G. Davison, and A. G. Brown (University of Waterloo) L. Roth (General Electric): (1) How does the M. Steslicka (Univ. of Waterloo): (1) We have cal- relativistic impurity energy shift compare with the shift culated only the difference between the relativistic as of the edge of the forbidden energy gap? (2) How many compared with the non-relativistic effect. We did not bound states are there in the forbidden gap? calculate how it is related to band edges. (2) In one gap at most one level (zero or one). 417–156 O - 71 - 32 481 The Influence of Generalized Order-Disorder on the Electron States in Five Classes of Compound-Forming Bindry Alloy Systems E. W. Collings, J. E. Enderby,” and J. C. Ho Metal Science Group, Physics Department, Battelle Memorial Institute, Columbus Laboratories Columbus, Ohio 4320] The influence of generalized order-disorder (including solid-state order-disorder, and melting) on electronic structures will be discussed for various types of binary intermetallic compounds which, for the purposes of discussion, will be arbitrarily subdivided into five classes. Classes A and B exhibit solid- state order-disorder (O-D) reactions. In the first of these the atomic potentials are sufficiently similar that the use of low-order perturbation theory at all concentrations is valid. The effect of O-D on such al- loys will be described in relationship to the density-of-states, as measured by low-temperature specific heat and magnetic susceptibility, and to the electrical transport properties, particularly the conductivity and Hall coefficient. We then consider from the same standpoint a second type of system in which the potentials are sufficiently different as to produce bound states which appear in the ordered form. The disappearance of these bound states when the alloy disorders gives rise to characteristic electronic behavior. The effect of O-D on the experimental parameters referred to above will be compared for the two types of systems. In particular, published data for Cu-Au, as a representative example of the first type of system, will be contrasted with new electronic property data for Ti-Al, and compared with recent ex- perimental results for Pt-Cu, which occupies an intermediate position. In class C, which is metallic in the solid, and in classes D and E which are semiconducting in the solid, structural order persists up to the melting point. For D the liquid is metallic, and data are presented for BigTes, a typical system of this class. The conditions for the existence of intermetallics of class E are extreme, and give rise to non- metallic behavior in the liquid. These various systems will be discussed in terms of differences in atomic potentials. The major problems involved in giving precise estimates of the required differences will be outlined and a critical account of the use of concepts like the electronegativity parameter will be presented. Key words: Binary alloys: BiºTes: CaSb: copper-gold (Cu-Au); Cu3Au; CuPt: Cu3Pt; ductility: elec- trical resistivity: Hall effect: magnesium bismide (Mg3 Big); mechanical behavior; melting: model potential of Heine and Abarenkov; nickel aluminide; order-disorder; Peierls barriers: Pt-Cu: silver, tellurium (Agº"Te): Ti-Al: TiCo.; Tife: TiNi: Tl2Te. 1. Introduction It is reasonably well established that when one ele- ment is dissolved in another of sufficiently similar atomic potential, relatively minor perturbation of the electronic states occurs, even at high concentrations [1,2]. In terms of a screening model, the ions are screened independently or linearly. Provided that the atomic potential differences and the atomic size dif- ferences are favorable, pairs of elements will exhibit *On leave from the University of Sheffield. Present address: Physics Department, The University, Leicester, England. unlimited mutual disordered solid solubility, and the electronic structures of the alloys will tend to follow the rigid-band prescription. Examples are consecutive pairs of transition elements near the middle of the 4d and 5d series, respectively. Within the low-perturbation region, increasing devia- tions from ideality will manifest themselves near the stoichiometric compositions, by the occurrence first of all of short-range order, and eventually of long-range order. However, in the linear screening model, changes of structure brought about by order-disorder (O-D) or even melting will produce only small electronic effects. 483 TABLE 1. Five distinct classes of intermetallic compounds Major Type of electrical conductivity Major Represent- | Solid state changes in changes in Class ative order- electronic electronic compound disorder properties Solid Liquid properties with order- with melting disorder A Cu3 Au Yes No B Tiº Al Yes Yes (. NiAl No Metallic Metallic Yes” Ti4Sn D BigTes No Semiconducting | Metallic Yes E Mg3 Big No Semiconducting | Semiconducting No T],To Agº"Tc * Measurements through the melting point have not been made on these systems, but it is expected that these compounds will be more nearly free-electron-like in the liquid than in the solid. On the other hand, as the atomic potentials of the two species become increasingly different, the electronic properties become increasingly structure sensitive as the linear screening approximation breaks down. This effect is discussed with respect to long-range or- dering systems, commencing with Cu5Au as the representative member of our first class of compound (class A, table 1). In Cu5Au the atomic potentials of Cu and Au are sufficiently similar that Cu, Au undergoes O-D with practically no change of density-of-states. In contrast to this are compounds of elements of suf- ficiently different atomic potentials that significant changes of electronic structure take place during solid state O-D. For example in Ti-Al, near the composition Tig Al, experiments in which the degrees of order were controlled by suitable heat treatments, demonstrated that for TigA1, n(EP)aisord./n(EP)ard. - 2. Frequently, however, it is observed that intermetallic compounds, which may be either metallic or semiconducting in the solid, are structurally stable up to the melting point and therefore do not undergo O-D prior to melting. To discuss the effect of structure on electronic properties of this type of material we must therefore consider the solid-liquid transition. Metallic intermetallic com- pounds invariably behave when molten like conven- tional liquid metals. In semiconducting intermetallics, on the other hand, two types of melting behavior can be distinguished. In most cases a metallic description of the electron states in the liquid is appropriate; but a few intermetallic semiconductors are characterized by nonmetallic behavior when in the liquid phase. We will set out to show that the wide variety of behaviors exhibited by intermetallic compounds can be correlated through an appropriate electronic screening model. 2. Experimental Results Described below are a representative selection of ex- perimental results based on measurements which we and other authors have carried out on the five classes of intermetallic compounds, the properties of which are summarized in table 1. 2.1. Class A Intermetallic Compounds The low temperature specific heat of Cu3 Au, the representative class A compound, has been measured for both the ordered and disordered states by Rayne [3] and Martin [4] who agree that the change in elec- tronic specific heat coefficient (y) accompanying O-D is extremely small. According to Martin Yaisord./yord.F 1.04.1 The results of measurements of total magnetic susceptibility [5], taken in conjunction with the density-of-states data outlined above, indicate a small change of electronic diamagnetism with O-D. Recogniz- ing that the interband diamagnetism [6,7] usually makes a significant contribution to the susceptibility of a nontransitional metal or alloy, the magnetic results suggest again that relatively small changes in the elec- | y is proportional to the density-of-states at the Fermi level, n(EP), according to the rela- tion n (EF) or y(1+Vn (EF)]-'; where V n(EF), a correction term for electron-phonon effec- tive mass enhancement, is small for a nonsuperconductor. This results in an approximately second-order correction to the density-of-states ratio. 484 tronic states accompany O-D in Cu3 Au. The Hall coeffi- cient changes from RH(disord.) = – 0.64 × 10−12 Q- cm/oe to RH(ord.)=+ 0.17 × 10−12 Q-cm/oe: both values being well within the range normally associated with good metals.” (This change in Hall coefficient is small compared to that encountered in class B compounds (fig. 5)). In contrast to the behavior of y and RH, the residual resistivity of Cu5Au is extremely sensitive to O-D, responding to the change in structure-factor which controls the scattering [9]. Cu-Pt may be regarded as transitional between class A systems typified by Cu3 Au above and the subsequent classification. The low temperature specific heats of or- dered and disordered CuPt have been measured by Roessler and Rayne [10]. At the time, the possibility of a magnetic transition seemed to obscure the interpreta- tion. However, recent magnetic measurements in this laboratory have confirmed the absence of a transition to a cooperative magnetic state on ordering, permitting the results of the above authors, viz Yaisord./yord. = 1.56, to be interpreted as indicating simply a density-of- states change of that ratio. We have also measured the low temperature specific heat of the O-D compound Cu3Pt and studied the electrical resistivities and Hall coefficients of a series of Cu-Pt alloys. The calorimetric results are summarized here for the first time. For Cu3Pt, Yaisord./yora. = 1.15. The results of measurements of the room temperature resistivities of disordered? Cu- Pt alloys as a function of composition were in satisfac- tory agreement with those of Schneider and Esch [11] after extrapolation from the high-temperature disor- dered region. As expected the resistivities of the or- dered compounds Cu3Pt and CuPt were found to have dropped to relatively low values. The results of the elec- trical resistivity and Rh measurements are summarized in figure 1. The measured Hall coefficient corresponds to an effective change in carrier concentration of 0.5 — 1.0 × 10% electron/cc; although it is expected that at least a two-band model would be required to describe this system. The ordering reactions of both Cuºpt and CuPt are accompanied by small positive-going incre- ments in RH. It is clear that this system exhibits no wide departures from good metallic behavior. These results will be discussed later in the context of class A and class B intermetallic compounds. 2.2. Class B Intermetallic Compounds Systems of this type, in which major changes in elec- tronic properties accompany solid-state O-D, although * For example, RH (Cu, experimental) = – 0.55 × 10-1” Q-cm/oe; and Rn (Cu, free electron model)= — 0.74 × 10-12 Q-cm/oe. * Rolled strips from arc-melted buttons were measured in the unannealed condition to avoid the possibility of short-range ordering, as discussed by Kim and Flanagan [12]. |OO | | 8O E P E 3 6O S. Q- P # 40 Uſ) wº Q) ſº 2O º * | I # g St. Sº Q) # 5 0 t | § E O -C O O – | - N £ o - 2 | | | | | O |O 2O 3O 4O 5O 6O Atomic percent Pf FIGURE 1. Resistivity and Hall coefficient for Cu-Pt alloys. Filled points – room temperature; open points–4.2 K. OO– “disordered” (cold-rolled from arc-melted buttons). AA – after a long ordering heat treatment (Cua Pt-step-cooled from 500 °C over a period of 2 weeks; CuPt – step-cooled from 850 °C during 18 days). Analyzed compositions of the compounds were respectively, 25.7 at.% and 50.5 at.% Pt. Broken line refers to the room-temperature “ordered” alloy resistivity data of reference [11]. Arrows indicate the response to ordering heat treatment which, as the resistivity data shows, is still not complete. not common, are of considerable importance both prac- tically and from the standpoint of this discussion. TigAl is taken as an example, and some new experimental results are presented below. Figure 2 shows the results of room-temperature magnetic susceptibility measure- ments. Using Blackburn's [13] equilibrium phase dia. gram as a guide (see inset to fig. 2) the degrees of long- range order present in the small (~ 150 mg) suscepti- bility specimens were controlled over wide ranges" by suitable heat treatments followed by rapid quenching into iced brine. The chief components of susceptibility in a transition-metal alloy are Xspin, the spin paramag- netism which is approximately proportional to n(EF), and Xorn, the orbital paramagnetism [14]. Clearly figure 2 demonstrates that major changes in electronic struc- ture accompany O-D in the Ti-Al system and particu- larly in Tig Al itself. The electronic specific heat results are shown in figure 3. Because of their bulk (30-40 g) the specific heat specimens were not able to be quenched as rapidly as were those used for magnetic susceptibili- ty measurements. For example, the Ti-Al (28 at. 9%) specific heat specimen could not be retained in the dis- ordered form. However, the results of figure 2 help to validate the extrapolation procedures used in figure 3 for estimating yaisord. For TigAl it follows that n(EP)aisord/n(EP)ard. = 2.1. Some of the results of elec- “An exception is Tia Al itself, in which the ordering reaction is so rapid that it proceeds al. most to saturation, with respect to density-of-states properties, during ice-brine quenching. 485 | | | | | | | | T S. Ti – A | 3 3.2 H. *- E Qt) S. \tº O— > 3. I ºm [] * *E [] E y º 3.O H. disordered a - O wº =} (f) .9 ) 'a, 2.9 H. - Q- U, o E 3OO l § 2.8 H a +/3 Y S g look-bcc(3) cy Sº 2.7 H. O (2) -, S. 900 t; § s É hcp (q) +G2 RS: G2 (3) # 2.6 H. # whi (l) * §: (3) 8: 999 sto : 3 2.5 H *- 300 - 5 IO is 20 25 3O # at. 9% Aſ H 2.4 H. | | | | | | | | | O 5 |O |5 2O 25 3O 35 4O 45 Atomic percent aluminum FIGURE 2, Average room-temperature magnetic susceptibility (Xav.) of Ti-Al alloys. Xav = 1/3 (Xr + X) + Xs). Inset is the equilibrium phase diagram (0-25 at.% Ah due to Blackburn [13]. The arrows indicate the effect of improving the degree of long-range order. D – annealed at 50 °C below of (a + 8) and quenched, except for Ti-Al (30) which was heat treated as for Ti-Al (20). O – quenched into iced brine from bec field. () to G) — quenched from 1260, 1,000, 800, and 700 °C, respectively. RCF – quenched from 1100 °C. A – as cast. © -long step-cooling anneal to promote maximal long-range ordering. trical resistivity measurements on Ti-Al are shown in figure 4. This type of measurement, and particularly residual resistance ratio, is well known” to be much more responsive to the degree of long-range order than are n(Ep)-sensitive measurements such as specific heat and magnetic susceptibility. Although Tiº Al in the as- cast condition may from the point of view of n(EP), and for most practical purposes, be regarded as fully or- dered, the antiphase domain size is still sufficiently small [13] that grain boundaries make a significant contribution to the electron scattering. As a con- sequence, the resistivity-concentration curve for as- cast Ti-Al shows only a dip at 25 at.% Al characteristic of partial ordering (fig. 4 cf., also [15]) instead of the otherwise expected sharp drop to the baseline. The results of the present Hall coefficient study are sum- marized in figure 5 (cf., [16]). Considerable scatter of the data was encountered for alloys of compositions close to 25 at. 9% Al. This was attributable to microcracking, which always occurred even when spe- cial preparation procedures were followed." Of crucial importance for an interpretation of the electronic property data presented above are the * For example, relatively large changes of residual resistance ratio brought about by cold work or very dilute alloying are frequently accompanied by negligible changes in y or x. "The end of a suspended “finger-ingot” of Tiaal was remelted by r.ſ. induction heating in a gettered argon atmosphere, and cooled to room temperature in about 8 hours. OH- --~" - .5H / Disordered hcp , C. 4 2. 5 2.3 OO Ordered, dz — O 5 |O 15 2O Atomic Percent Aluminum FIGURE 3. Low-temperature specific heat of disordered and ordered Ti-Al Filled symbols represent single-phase (cy or oº) material, while open symbols represent alloys which are known or suspected to be two-phase (or + Cº.) (see inset to fig. 2). Circles: as cast: Triangles: quenched from 50 °C below cºſ (or + 3) transus; Squares: prolonged low-temperature anneal; Diamonds: quenched from 8 field. results of x-ray structural measurements by Gehlen [17] of a maximally-ordered single crystal of Ti-Al (26.7 at. 9%). The results of that work, whose physical sig- nificance has been the subject of a preliminary note [18] are best described with reference to figure 6. Gehlen's work has demonstrated that Tig Al possesses a hexagonal DO19 structure, but with the Ti atoms slightly displaced “inwardly,” within the basal planes, toward a hexad axis passing through the Al atoms. This results in the formation of the chains of - Al- Tiº – tetrahedra delineated in figure 6. The spatial arrangement of ordered Tiº Al, in con- junction with the electronic property evidence as a function of O-D, suggests (a) that part of the bonding in ordered Tig Al is covalent in type with some fraction of the total number of otherwise-available conduction electrons removed from conducting states; and (b) that such an electronic arrangement, favored by the struc- tural long-range order, becomes smeared out in the dis- ordered lattice. In terms of our model we would say that the ordered state is characterized by nonlinear screen- ing, which requires a self-consistent readjustment in response to a change of structure. 2.3. Class C Intermetallic Compounds This is a large class of compounds. Members of this category (table 1) may exist within a finite composi- tional range about stoichiometry and are generally suf- 486 Ti - Al , 4.2 °K © As-cost O Annedled | | |O 2O 3O 40 Atomic percent Al FIGURE 4. Electrical resistivity of Ti-Al at 4.2 K: © — as cast; O— annealed for 14 days at 900 °C and furnace-cooled. A comparison of the magnetic susceptibility and specific heat data for Tiº Al with the resistivity data presented here shows that, while the density-of-states is practically un- affected by annealing (figs, 2 and 3), the improvement in the degree of long-range-order has a profound effect on the electron scattering. 5O | | 7.O *- 6,O mº Q) 5.O º * S Q) () º | g 4.O O N --- E & O Q 3.O - i IC 2.O *m: Cº. # 'o |.O - tº- * g O —O— Room temp. O - — —(D- — 78°K E { I isolated points — refer q) I.O to caption - .P. t; § 2 2.O --- 3.O | | | O |O 2O 3O 4O Atomic percent Al FIGURE 5. Hall coefficient of Ti-Al at room temperature (-O-) and 78 K (-(D-). The isolated points refer to specimens which were more or less imperfect through qracking: O-Ti-Al (22.3 at 9%) (RT); O. O. Q-Tiº Al specimens 1, 2, and 3 resp. (RT); ) – Tia Al specimen 3 (78 K). Microcracking in Tiº Al was unavoidable. The data for Ti-Al (22.3 at.%) indicates that the presence of cracks drastically lowers R iſ . This fact, together with the behavior of the surrounding data, suggests that R n (Ti, Al) should, in a perfect specimen. be much higher than the measured values. The data compares favorably with that of reference |l 6|. ficiently stable to resist disorder prior to melting. In re- gard to both electronic and mechanical properties they are metallic solids; and of course are metallic when | / \ | | | | ſ == sº L^ | | l | | / * * * FIGURE 6. Regular hexagonal DO19 structure. In Tia Al the Ti atoms (O) are displaced, in the directions indicated by the arrows, towards an axis passing through the Al (O) atoms. The resulting chains of tetrahedra are indicated by the heavy lines. liquid (disordered). The ductility of class C intermetal- lic compounds is poor at temperatures low compared to the Debye temperature, but improves at higher tem- peratures. An example is 8-NiAl for which a considera- ble amount of published data are available.7 Figures 7 and 8 summarize the results of recent measurements by Yamaguchi et al. [19,20] and Jacobi et al. [21] of the electrical resistivities and Hall coefficients of Ni-Al and related systems. The resistivity curves exhibit sharp minima at the stoichiometric composition, characteristic of metallic * Paraphrasing [21], [3-NiAl has a homogeneity range of approximately 40-55 at 9%. Al at room temperature, and on the basis of x-ray measurements it probably remains ordered up to the melting point. The aluminum-rich compound is a (Ni) vacancy structure. 487 I | | | | | | 40 4OO _e_Ni-Al Refſ2] E o Co-Al Ref [19] E C) º £ 30- —H3OO E 5 YS / :5 S SS / S 3. : l - ‘5 º Q- Q- > * 'S s gº mºre (ſ) # & \ i * / OLl | i | | | | | 4O 42 44 46 48 50 52 54 Atomic percent Al FIGURE 7. Electrical resistivity of Ni-Al[2]|and Co-Al[19]. (OO– room temperature: ---77 K). conduction in a long-range ordered structure. The depth of a residual resistivity minimum is a measure of the degree of ordering, which is controlled mainly by the precision with which exact stoichiometry can be achieved. The Hall coefficient rises sharply to a posi- tive value at the composition NiAl, the effective charge- carrier concentration still appearing to remain well within the metallic regime. Recent measurements in this laboratory have shown the Rii versus composition curve for Ti-Sn to be comparable to that of Ti-Al (fig. 5) but of the opposite sign in the vicinity of the first inter- metallic compound. The electrical and magnetic properties of Tife, TiCo, and TiNi have been discussed by Allgaier [22] and Bu- tler et al. [23], but because of their complicated and uncertain metallurgical properties, these compounds cannot be unequivocally placed in the simplified clas- sification scheme described here. 2.4. Class D Intermetallic Compounds This is a relatively populous class. Here we refer to intermetallic compounds, semiconducting in the crystalline state, which revert to metallic behavior when molten. Some of the many semiconducting com- pounds which exhibit this behavior (e.g., CaSb, ZnSb, BigTes, and Sb2Tea) have been discussed elsewhere [24,25]. We offer here, as an example of class D, the com- pound BigTea and the results of some new measure- ments of RH through the melting point (fig. 9), which demonstrate the transition from the semiconducting to the metallic state on melting. O.6 H Ref. [20] R.T. -O- T. 4.2 °K -O- O.5.H. Ref. [2] R.T. -º- O. 4. i O 3 o:–§| ; Atomic percent Ni FIGURE 8. Hall coefficients of Ni-Al. Data points are from references [20] and [2]] (excepting that data from [21] has been shifted slightly in composition, bringing the peak to 50 at 9% Ni). 24O 22O H. * 2OO H. Biz Tes - | 80 H. mº |6O - * m.p. |40 H |2O H. * # |OO }* - 8O }* •mº 6O H. smº 40 H. ſ "El | | | | | | 42O 46O 500 54O 580 62O 66O 700 Temperature,”C | — FIGURE 9. Hall coefficient of BiºTeº showing the transition to the metallic state on melting. 488 28 H | Taſe - 24 H - g is 20 H - º É o |6|H - QN ! C ... 12H - Dr. 8 H - 4 - “.. ! BizTes "........ Bi º- -º- I {3– | H]−1 +++ -ė. Te |O 20 30 40 50 60 7O 80 90 100 Atomic Percent T1 or Bi FIGURE 10. Hall coefficient as a function of composition for liquid Te-Tl showing a singularity at the composition of the liquid semi- conductor Tl2Te. The dotted portion of the curve for Te-Tl refers to the two-phase region. The Hall co- efficient of liquid Te-Bi, which is metallic at all compositions, is shown for comparison (after reference [25]). 25OO 2OOO T E O | É S. |500 b P |S 3 IOOO 5 5 O Reference [27 C) O Present dotd 5OOH- - O | | | 54 56 58 6O 62 Atomic percent Mg |8 | T |6 H. FIGURE 12. Electrical conductivity of liquid Bi-Mg near the com- |4 H — 84O position of the liquid semiconductor Mg3 Big. g Aq Te O Open circles are data from reference [27]. Filled circle represents the minimal conduc- - 12!— g • —|820 tivity obtained in the present experiment [26] after enriching the alloy Bi-Mg (63 at.%) with 5 O Bi (through evaporation of Mg, and by direct addition of Bi). # IoH —º- —|800 5 -º- I sº s!— --> —780 5 T J. O - - e º - - * e 6H O º * Mg3Big currently under investigation in this laboratory se º - Or 4 H. sº Ti” [26]. Figure 12, most of the data for which are due to O e e 2H -1749 Ilschner and Wagner [27], shows the conductivity of | | | | | | | . . e — 1 gº-Hº-Hº-Hº-Hºº-º-º-º, the liquid alloy dropping to less than 60 (0-cm)−1 near Temperature, *C what must be assumed to be the stoichiometric com- - - * º osition. FIGURE 11. Transport properties, RH and or, of the liquid semicon- D ductor Ag'ſe as functions of temperature (after reference [25]). 2.5. Class E Intermetallic Compounds Compounds in this class are not common. They pos- sess an extreme type of behavior in which their non- metallic character in the crystalline state persists into the liquid. We cite as examples Tl2Te and Agº Te which have been discussed in the recent paper of Enderby and Simmons [25]. Again the Hall coefficient is quoted as a useful measure of the degree to which a given material can be characterized as nonmetallic. Figure 10 shows RH for liquid Tl-Te rising to a singularity at the composition Tl2Te. Figure 11 reproduces the results of transport property measurements on AgTe. Both RH and O are out of the range of values usually associated with metallic conduction. As a final example we quote 3. Discussion In the data presented above an overall pattern can be discerned in the properties of intermetallic compounds. The chief features of this pattern have been sum- marized in table 1. In proceeding further it is of con- siderable heuristic value to focus attention on the cohesive energy and note how this quantity varies as we proceed from A to E. We introduce the partial cohesive energies [28] such that &con = 3. ài. For weak pseu- dopotential metals only 31 (the structure-independent term) and & 2 (the term depending on pair potentials) are important. Under this condition rearrangement of the atoms at constant volume will not significantly change ãcoh. If the pseudopotential is weak the ions may be screened separately. Movement of such ions, with self- consistent adjustment of the screening results effec- 489 tively in a screening charge that accompanies the ions (or neutral pseudo-atoms as they may then be called [29]). That is, in the linear screening regime of weak pseudopotentials, the electron states are approximately independent of structural order. This situation obtains for (a) pure metals with weak local pseudopotentials, and (b) alloys of metals whose differences in atomic potentials (to be defined) are small (cf., Stern [1,2]). As the differences in pseudopotentials become larger, a new aspect enters the problem. There are many equivalent mathematical descriptions of this. For exam- ple, the concept of nonlinear screening on one hand, as discussed by Phillips [30], or the breakdown of pertur- bation theory on the other (cf., Stern [1], Beeby [31]). In terms of our basic approach it means that à (i > 3) become of increasing importance, i.e., the ionic screen- ing becomes increasingly nonlinear.” Although no sin- gle piece of evidence is conclusive, the trends in the data discussed above are unmistakable and are out- lined as follows: 3.1. Mechanical Behavior A loss of ductility occurs as we proceed from class A to class E intermetallic compounds. This is associated with a high resistance to shear (i.e., high Peierls bar- riers) brought about by noncentral forces. For example, Cu, Au is ductile but, as expected, is easily work. hardenable [4]; whereas the class C compound 3-NiAl is already extremely brittle [21]. 3.2. Hall Effect It is not possible to calculate from first principles the Hall coefficient of solid-solution alloys. Conversely it is usually difficult to make a satisfactory detailed physical interpretation of an experimentally-measured value of RH. If, however, during alloying charge carriers became immobilized, as bound states begin to form, a signifi- cant increase in Rul may occur. We see this just beginning to take effect in class B compounds (fig. 5): whereas class E compounds are characterized by singu- larities in Ru at stoichiometry indicative of a non- metallic state (fig. 10). However, in multiband conduction, the possibility of which must be considered in any intermetallic com- pound, positive and negative components of Rh may partially cancel. Poor metallic characteristics in a solid intermetallic compound are therefore necessary, but not sufficient, conditions for appearance of a large “In an alloy it is useful to regard this as a description of incipient bound state formation. |6O |4O |2O E C) | E -- O S IOO : s º $ 80 Or. 6O 4 T | | | | | | | | | O 2 4 6 8 |O Solute Concentration, af. 9% FIGURE 13. Influence of small additions of (a) Al. (b) Nb, and Zr, on the resistivity of Ti- after Ames and McQuillan, Acta Met. 4, 61 () (1956). The scattering Gross-section of an “adjacent” transition element is seen to be relatively small. value of Ruſ. Such measurements in the solid must therefore be reinforced by other electronic property measurements when studying the bonding behavior of alloys. With liquids (in which only negative Hall coeffi- cients have been observed experimentally) the in- terpretation seems more straightforward. 3.3. Electrical Resistivity for Low Solute Concentrations A marked increase is observed in the resistivities of dilute alloys (per atomic percent solute concentration) as the atomic potential difference between solute and solvent increases. This is illustrated in figure 13 in which the resistivities of dilute Ti-Zr and Ti-Al alloys are compared. Ti-Zr may be regarded as an “ideal” solid solution alloy while Ti-Al leads eventually to a class B intermetallic compound. 3.4. Electrical Resistivity near Stoichiometry In classes A and B compounds, disordering can be achieved either by heat treatment or by varying the stoichiometry; while in classes C through E com- pounds, only the latter technique is available in the solid state. Bearing this in mind we make the general observation that in classes A, B, and C the resistivity decreases on ordering through structural considera- 490 tions in spite of a lowering electron density (brought about by the bound state formation which may produce an increase in Rul). On the other hand in classes D and E the bound state effect dominates, and both the re- sistivities and Hall coefficients have maximal values at the stoichiometric compositions. 3.5. X-Ray Studies Direct crystallographic evidence also reinforces the above picture. As has been pointed out in detail in sec- tion 2.2 the results of x-ray structural studies on TigAl attest to the existence of noncentral bonding forces. Similarly electronic density contours, such as those derived from CoAl and NiAl by Cooper [32] demon- strate graphically the existence of directional charge distributions in class C intermetallic compounds. 4. Conclusions Given a binary alloy how can we determine ab initio which of these categories represents an appropriate description? Clearly the gross features can be deter- mined simply by looking at the periodic table. The further apart the elements are, the more the alloys or compounds tend to E-type behavior. The next degree of sophistication is to use the concept of electronegativity in either its traditional [33] or its modern [34] form. The estimates that these give of the differences in potentials sometimes fail in detail; for example, they would predict that the class D compound AugTe falls in class E. The reason for this is that such methods do not take into account the self-consistency of the conduction However, the tronegativity approach to bonding [33] and the theories of structural stability propounded by Brewer and his collaborators [35] are linked by considerations relating to the gaseous atomic energy levels. However, these levels form the basic input data for calculations of pseu- dopotentials of the type described by Heine and co-wor- kers. Such potentials are known to be useful for a variety of applications. In our view the successes of both the electronegativity concept and the approaches used by Brewer, which at first sight seem highly empiri- cal, can be fully understood from this point of view. The pseudopotential in its k-space form is not well suited to this present discussion since it relates to the electron screening. Pauling elec- first few lattice vectors in reciprocal space, whereas to discuss bound state formation and the localization of electrons we need to consider the first few atomic spacings in real space. The pseudopotential in k-space deals with the long range part of this potential (i.e., screening) in a satisfactory way, but the influence of the core is distributed through k-space and is not accessi- ble. The Engel-Brewer theory emphasizes the im- portance of the core but does not take sufficient ac- count of screening and the requirement of self-con- sistency that the screening imposes on the problem. What we seek is a synthesis of these two points of view, and our starting point is the model potential of Heine and Abarenkov [36]. This potential in real space has the correct asymptotic behavior, and though ill-defined near RM, the core radius, is fixed inside the core by terms from the spectroscopic data. Table 2 lists the energy parameters of the Heine- Abarenkov model potential. It is apparent that large dif- TABLE 2. Selected parameters for the screened model potential of Heine and Abarenkov [36]* Aft A A 2 Li 0.336 0.504 0.455 Na .305 .339 .402 K- .240 .256 .368 Rb+ .224 .226 .384 Cs .205 .207 .366 Be2+ 1.01 } .22 1.48 Mg2+ 0.78 0.88 0.99 Ca” . .54 .50 1.49 Ba2+ .45 .34 1.07 Zn” .99 1.14 0.98 (.d3 .88 ().98 1.78 Hg?" .97 l. 1 | 0.85 Al3 1.38 1.64 1.92 (Saº 1.44 1.58 1.4] In 3 1.32 1.46 1.1() T]3. 1.44 1.51 0.98 Si3+ 2.08 2.39 2.44 (;e"+ 2.10 2.34 2.09 Sn++ 1.84 2.04 1.62 Pb2+ 1.92 ***(2.00) 0.90 As?" 2.7] (3.08) (2.0) Sbºx: 2.42 2.66 (1.8) Bià 2.38 2.58 0.25 Sebt 3.42 (3.77) (3.0) Teti 3.04. 3.32 (2.80) *After A.O.E. Animalu and V. Heine, Phil. Mag. 12, 1249 (1965). **The Ai (which are defined in the above- mentioned references) are in atomic-energy units; 1 atomic unit=2 Ry. ***The numbers in parentheses are ob- tained by “extrapolation” from one point. 491 ferences can exist between the Ao energy parameters for pairs of metals. For example Ao(Sb)-A0(Mg) is fairly extreme on the scale of atomic potential differences. Accordingly Mg3Sb2 is a class E compound. On the other hand ColSb is in class D. This example demon- strates that, contrary to some previous suggestions, valence difference alone is insufficient to describe the effect but that the type of bonding depends in detail on the core energy levels. When comparisons between the data of table 2 and experiment are possible the correct trends are observed. One outstanding problem is the proper way of describing the core states in transition metals. The evidence that we have from this work (see also [28]) is that they can be treated on roughly the same footing as normal metals. But we know that a simple model like that of Heine and Abarenkov would be completely inap- propriate. For transition metals even the published electronegativity values are of little help since they do not show sufficient variation. The recent theoretical work by Harrison [37] might form the basis of an at- tempt to resolve this difficulty. On the experimental side, in view of the results already obtained for liquids of class E, it is clear that useful information on the rela- tive potentials of metals can be obtained from suitably designed experiments on molten alloys and that this type of work should be extended to include transition metals. 5. Acknowedgments We wish to acknowledge Messrs. R. D. Smith and G. W. Waters for technical assistance, and the following two agencies for financial assistance: The Air Force Materials Laboratory, Wright-Patterson Air Force Base, Ohio; and the Division of Research (Metallurgy and Materials) U.S. Atomic Energy Commission. J. E. Enderby wishes to acknowledge Battelle Memorial In- stitute for the award of a Battelle Institute Fellowship, during the tenure of which some of the ideas submitted here were developed. 6. References [1] Stern, E. A., Physics 1,255 (1965). [2] Stern, E. A., Phys. Rev. 144,545 (1966). Rayne, J. A., Phys. Rev. 108,649 (1957). Martin, D. L., Can. J. Phys. 46,923 (1963). Airoldi, G., and Drosi, M., Phil. Mag. 19, 349 (1969). Verkin, B. I., Svechkarev, I. V., and Kuźmicheva, L. B., Soviet Physics, JETP 23,944 (1966). Misra, P. K., and Roth, L. M., Phys. Rev. 177, 1089 (1969). Komar, A., and Sidorov, S., J. Tech. Phys. (USSR) 11, 711 (1941): J. Phys. (USSR) 4, 552 (1941). Ziman, J. M., Electrons and Phonons, (Oxford 1960). Roessler, B., and Rayne, J. A., Phys. Rev. 136, A1380 (1964). Schneider, A., and Esch, U., Zeits. Elektrochem. 50, 290 (1944). Kim, M. J., and Flanagan, W. F., Acta Met. 15, 735 (1964). Blackburn, M. J., Trans. AIME 239, 1200 (1967). Kubo, R., and Obata, Y., J. Phys. Soc. Japan ll, 547 (1956). Kornilov, I. I., Pylaeva, E. N., and Volkova, M. A., in Titanium and its Alloys, Publication No. 10, Israel Program for Scien- tific Translations, Jerusalem (1966) p. 76. Grum-Grzhimailo, N. V., Kornilov, I. I., Pylaeva, E. N., and Vol- kova, M. A., Dokl. Akad. Nauk, SSR 137, 599 (1961). Gehlen, P. C., Proc. Intern. Conf. on Titanium (London, 1968) to be published by Pergamon Press (1970). Ho, J. C., Gehlen, P. C., and Collings, E. W., Solid State Com- munications 7,5ll (1969). Yamaguchi, Y., Kiewit, D. A., Aoki, T., and Brittain, J. O., J. Appl. Phys. 39,231 (1968). Yamaguchi, Y., and Brittain, J. O., Phys. Rev. Letters 21, 1447 (1968). - Jacobi, H., Vassos, B., and Engell, H. J., J. Phys. Chem. Solids 30, 1261 (1969). Allgaier, R. S., J. Phys. Chem. Solids 28, 1293 (1967). Butler, S. R., Hanlon, J. E., and Wasilewski, R. J., J. Phys. Chem. Solids 30, 281 (1969). Enderby, J. E., and Walsh, L., Phil. Mag. 14,991 (1966). Enderby, J. E., and Simmons, C. J., Phil. Mag. 20, 125 (1969). Enderby, J. E., and Collings, E. W., Proc. Intern. Conf. on Amorphous and Liquid Semiconductors, to be published in J. Non-Crystalline Solids. Ilschner, B. R., and Wagner, C., Acta Met. 6, 712 (1958). Collings, E. W., and Enderby, J. E., in preparation. Ziman, J. M., Advances in Physics 13, 89 (1964). Phillips, J. C., Phys. Rev. 166, 832 (1968); 168, 905 (1968); 168,912 (1968). Beeby, J. L., Phys. Rev. 135, Al30 (1964). Cooper, M. J., Phil. Mag. 8,811 (1963). Pauling, L., The Nature of the Chemical Bond (3rd edition), (Cornell University Press, Ithaca, N.Y., 1960). Phillips, J. C., Phys. Rev. Letters 22,645 (1969). Brewer, L., in Phase Stability in Metals and Alloys, P. S. Rud- man, J. Stringer, and R. I. Jaffee, Editors, (McGraw-Hill 1967), p. 39. Heine, V. and Abarenkov, I., Phil. Mag. 9,451 (1964). Harrison, W. A., Phys. Rev. 181, 1036 (1969). 492 Localized States in Narrow Band and Amorphous Semiconductors D. Adler” and J. Feinleib% Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Massachusetts 02:139 The electronic density-of-states is discussed in situations where some of the states near the Fermi energy are localized, due to either intraionic Coulomb repulsion or disorder. When localized states are present, the Franck-Condon principle necessitates separate electrical and optical densities-of-states. In the case of ionic Mott insulators, it is shown that doping or nonstoichiometry drastically affects the ener- gy-band structure. For the particular example of NiO, introduction of Lit impurities or excess oxygen leads to a large upward displacement of the 2p band associated with the oxygen ions, moving it suffi- ciently near the Fermi level that hole conduction in the 2p band predominates above 200 K, in agree- ment with the available experimental data. In the case of amorphous semiconductors, it is shown that introduction of electron-electron and electron-phonon interactions results in a shift of the relative posi- tion of the localized parts of the valence and conduction bands, as well as shifts of the localized states relative to the itinerant states. However, the qualitative features of the model of Cohen et al. are preserved. Key words: Amorphous semiconductors; Anderson transition; augmented plane wave method (APW); chalcogenide glasses; electronic density of states; Franck-Condon prin- ciple; localized states; Mott insulator; NiO; optical density-of-states; photocon- ductivity; photoemission; polaron. 1. Introduction The existence of localized electronic states in a crystalline solid has been recognized for many years, and methods have been developed for representing these states on an effective one-electron density-of- StateS diagram. In particular, donor, acceptor, and ex- citon states in ordinary semiconductors are familiar. These states are represented in such a way that the energy to excite an electron, a hole, or an electron-hole pair, either optically or thermally, is the correct value. Since the states are localized, certain additional rules must be borne in mind, in order to use the resulting density-of-states diagrams. In particular, these states generally do not contribute to electrical conductivity, and the simultaneous presence of both a spin-up and a spin-down electron in such a state results in a large Coulomb repulsion, and this possibility must be ex- cluded from consideration. These rules significantly af. *Dept. of Electrical Engineering and Center for Materials Science and Engineering, M.I.T., Cambridge, Mass. 01239. Research sponsored by the Advanced Research Projects Agency. **Research sponsored by the Department of the Air Force. fect the statistics, and consequently modify the tem- perature dependence of the conductivity. It is also clear that many other localized electronic states exist in crystalline materials. The core electrons are much more reasonably treated in a Heitler-London approximation, as always moving with their cor- responding nucleus, than as itinerant Bloch electrons moving in the periodic potential of all the nuclei. How- ever, it can be shown, for closed-shell configurations, the two opposing treatments give identical results [1]. The fact that this is not the case for partially-filled shells was first emphasized by Mott [2,3], who showed that electronic interactions in narrow energy bands could lead to the localization of outer electrons. It is now recognized that many transition-metal and rare- earth compounds are Mott insulators, nonconducting because electronic correlations localize their outer electrons. A quantitative version of Mott's model has been presented by Hubbard [5-7]. Hubbard introduced the effects of correlations into ordinary band theory by adding a term to the Bloch Hamiltonian which in- creased the energy of the system by a constant, U, whenever two electrons were simultaneously present 493 on a particular ion core. U represents the intraionic Coulomb repulsion between two electrons, and can be estimated in the atomic limit, or limit of infinite lattice parameter, as the difference between the ionization potential and the electron affinity of the atom. In a real solid, the value of U is much smaller than in the atomic limit, due to the effects of screening. However, in many crystals, such as NiO, this screening does not appear to be very large [8]. The result found by Hubbard [7], for the case of a half filled s band, is that the solid will be a Mott insulator provided U × 0.87 A. (1) where A is the electronic bandwidth. If the condition (1) is not fulfilled, the material is metallic. For degenerate bands, A must be replaced in (1) by 12J, where J is the relevant overlap integral. Mott [9,10] has recently called attention to the lo- calizing effects of lattice disorder. A quantitative treat- ment of an analogous problem was performed by An- derson [11], who considered the diffusion of a single electron out of a state in one of a series of equally- spaced potential wells of random strengths distributed over an energy range V. Anderson found that if V > 5 A (2) the electron would not diffuse away. This result in- dicates the occurrence of an “Anderson transition” in which localization of an electron can be brought about by sufficient disorder. Mott [10] has suggested that a disordered lattice has energy bands whose states are itinerant if the density-of-states, g(E), is greater than a critical value, but are localized if g(E) is smaller than this value. This proposal has obtained some quantita- tive confirmation from the calculations of Ziman [12], Edwards [13], and Neustadter and Coopersmith [14]. The sharp edge between itinerant and localized states has been termed a “mobility edge” by Cohen et al. [15]. It seems clear that in situations in which both the energy spread due to disorder, V, and the intraionic Coulomb repulsion, U, are important, a condition analogous to (1) and (2), such as [U2+0.03 V*] /2 - 0.87A (3) will result in localization. This combination of correla- tion energy and disorder are most likely responsible for the suppression of metallic conductivity in the high temperature phase of Cr-doped V2O3 found by MacMil- lan [16] and by McWhan et al. [17]. These effects are complicated in ionic materials by the strong coupling between the electrons and the lon- gitudinal optical phonons, which leads to polaron for- mation. It is a reasonable assumption that in the itinerant states the overlap integral is sufficiently large that large polarons will form. However, in the localized states the overlap is sharply reduced by the localiza- tion, leading to the likelihood of small-polaron forma- tion. In nonpolar substances, the possibility of a Jahn- Teller stabilization of degenerate localized states must not be overlooked. Finally, the Franck-Condon principle suggests that ionic rearrangements do not occur at optical frequen- cies. This does not affect the effective density-of-states to be used in analyzing the results of transport experi- ments, but has important consequences as far as the optical density-of-states is concerned. In this paper, we shall be concerned with localized states in crystalline Mott insulators, such as NiO, and in amorphous semiconductors, such as chalcogenide glasses. In section 2, we discuss the optical density-of- states of pure and doped Mott insulators. In section 3, we analyze the electrical density-of-states of the same materials, pointing out the drastic effects that doping or nonstoichiometry can have on the mechanisms for electrical conductivity in transition-metal compounds. In section 4, we discuss the electrical and optical densi- ty-of-states in amorphous semiconductors in terms of the theory developed in the previous two sections. The conclusions are summarized in section 5. 2. Optical Density-of-States of Crystalline Mott Insulators For simplicity, we begin with a discussion of the opti- cal properties of pure, stoichiometric single crystals. We assume that the Franck-Condon principle applies, and that no ionic motion accompanies an optical transi- tion. We restrict ourselves to photon energies of 20 eV or less. It is convenient to illustrate our major points by a concrete discussion of a single material. We shall use NiO as the prototype Mott insulator, primarily because more experimental data exist for NiO than for any other transition-metal or rare-earth compounds. However, all our conclusions and techniques can be applied equally well to any other Mott insulator. Restricting ourselves to energies within 20 eV of the Fermi energy, we need consider only the 2p band associated with the oxygen ions, and the 3d and 4s bands associated with the nickel ions. Since the covalency parameters of NiO have been shown to be less than 4% [18], a good starting point for determination of the band structure is the energy levels of the fully-ionized atoms, Ni2+ and O2-, in the limit of infinite separation. We then must take into account the 494 stabilizations and destabilizations due to the Madelung potential, the crystalline field potential, large-polaron formation, and various many-body effects such as elec- tronic screening of the electrostatic Coulomb interac- tion and induced ionic polarization. The evidence is now quite convincing [4,19) that the 2p and 4s bands in NiO are made up of itinerant states, and thus A/U => 1. On the other hand there is much evidence that the 3d band is very near the atomic limit, in which A/U → 1 [19]. Thus the 3d states are very lo- ca,zed, and NiO, with 83d electrons per Ni2+ ion, is a Mott insulator. The possibi, transitions which we must treat in detail a T62 . 308 — 30.8% (4) 3d 8 + 3d 8 – 307 –– 3d 9 (5a) 3d 8 + 3d 8 – 3d 7 – 30.9% (5b) 3d 8 + 3d 8 – 307* + 309 (5c) 3d 8 + 3d 8 – 307* + 3d 9° (56) 3d" -> 3d" + (electron in 4s band) (6a) 3d” – 3d” + (electron in 4s band) (6b) 3d” + (electron in 2p band) → 3d” (7a) 3d" + (electron in 2p band) → 3d” (7b) (electron in 2p band) → (electron in 4s band) (8) Reaction (4) is just a localized excitation on a particular Ni2+ ion, the notation 3dº indicating an excited crystal- line-field or multiplet split 3d” configuration. Reactions (5a-d) represent transitions between localized states, with two Ni2+ ions being excited into a Nit — Niºt pair in their ground or excited states. Reactions (6a, b) represent the excitation of a localized electron on a Niºt ion into the 4s band, leaving behind a ground-state or excited Niºt ion. Reactions (7a, b) indicate the possibili- ty of exciting an itinerant electron in the 2p band into a localized state on a nickel ion, leading to the formation of a Nit ion in its ground state or an excited state, and leaving a hole in the 2p band. Finally, reaction (8) represents the usual interband excitations between itinerant one-electron states. The crystalline-field and multiplet splittings of Niºt states in NiO can be determined either experimentally [20] or theoretically [21], and are in good agreement. They lead to a series of optical absorption peaks in the 1-4 eV range. The energy range of reaction (5a) can be estimated from the difference between the ionization potential and the electron affinity of Ni2+ as requiring 18.6 eV [22]. This figure, however, takes no account of any screening of the intraionic Coulomb repulsion. This screening could result from either polarization of the surrounding O2- ions and covalency effects, or it could be caused by the presence of other 3d and core elec- trons on the Ni2+ ions themselves. The latter effect is one which must occur in free Ni2+ ions, and can be esti- mated from the fact that the experimental Slater-Con- don parameters are about 15% smaller than those cal- culated from Hartree-Fock wave functions [23]. The effects of polarization and covalency can be estimated from the reduction in the multiplet splittings of Niºt in MgO compared to those in free Niºt ions, an effect which amounts to approximately 20% [24]. Finally, we should also expect some screening from overlap between 3d electrons on nearest-neighbor Niºt ions. This can be estimated from the experiments of Reinen [20], who found that the Racah parameter, B, which is a measure of intraionic electronic repulsion, decreases 7% from dilute solutions of Ni2+ in MgO to pure NiO. Taking into account all of these screening effects, we find that the intraionic Coulomb repulsion for the 3d electrons in NiO is reduced from 18.6 eV to approxi- mately 12 eV. We still must take into account the differences in crystalline-field stabilizations of 2 Niºt ions as com- pared to one Nit and one Niºt ion. It is likely that Niºt is in a high-spin state in NiO [19]. Assuming this to be the case, and using the estimated values [25] of the crystalline-field parameter, D4, we can conclude that reaction (5a) requires approximately 13 eV. This is the effective value of U which should be used in condition (1) to determine whether or not the 3d electrons in NiO are localized. Since the 3d bandwidth in NiO can be estimated as being of the order of 0.3 eV [26] or con- siderably less [19], there is little question that condi- tion (1) is fulfilled, and the 3d electrons are extremely localized. Assuming the same values of D4 for Niºt and Nit as used in estimating the crystalline-field stabilizations, we find that the d7 excited states extend over a 5 eV range, while the dº configuration is 1 eV wide. Thus, reactions (5a-d) should contribute to optical absorption in the 13-20 eV range. In order to estimate the energies of reactions (6-8), we must take into account the finite bandwidths of the 2p and 4s bands. We expect both bands to be in the Bloch limit, with effective values of U small compared. to the bandwidths. It is a reasonable approximation that 495 the APW calculations should be quite good in deter- mining the bandwidths and relative separations of these bands. Despite the differences in assumed poten- tial, both the APW results of Switendick [27] and Wil- son [28] are in agreement that the 2p band of NiO is 4 eV wide, while the 4s band is 6 eV wide. Furthermore the bottom of the 4s band is about 5.5 eV above the top of the 2p band. There is a small band measuring due to large-polaron formation [19]. Assuming that these values are good approximations, the 2p-4s transitions, reaction (8), should contribute interband optical absorp- tion between 5.5 and 16 eV. We can estimate the energy range for reaction (6) by noting that the free-ion process, 3dº → 3d, 4s, takes 7 eV in Ni2+ [29]. Assuming that the effective value of U is negligible in the 4s band, we can conclude that in the NiO crystal, the 6 eV wide band spreads symmetrically around the free-ion 4s level. This reaction (6a) should contribute to the optical absorption in the 4-10 eV range. Since the 3d" configuration is 6 eV wide, reac- tion (6b) extends the energy range of this d -> s absorption up to 16 eV. Finally, the energy range of reaction (7) involves elec- tron transfer from the oxygen to the nickel ions. The free-ion process, Ni2+ + O.”--> Nit + O-, is ex- tremely exothermic, since Niºt has an electron affini- ty of 17.5 eV and O- has a negative electron affinity of 9 eV [22]. Thus, as free ions, the process would release 26.5 eV of energy. However, in an NiO crystal, the Madelung potential, which is 24.0 eV [22], stabilizes both the Ni2+ and the O2- ions relative to Nit and O-. Thus the net stabilization of the free Ni2+ and O2- ions in NiO is 22 eV. Assuming the same screening and crystalline-field stabilizations as used previously, we can estimate the average energy of the process which creates a Nit – OT pair as 16 eV. Since the 2p or- bitals can be assumed to spread into a band 4 eV wide, with an effective value of U small compared to the bandwidth, reaction (7a) should contribute to opti- cal absorption in the 14-18 eV range. Reaction (7b) should give a further contribution between 15 and 19 eV. - We have not yet considered the excitonic contribu- tions to the optical absorption. The most significant of these should be the Mott-type excitons representing Nit – Niºt bounded pairs, or the excitons in which a hole in the 2p band is bound to a Nit ion. Estimates of the excitonic binding energies give maximum values of 0.8 eV in the former case and 1.1 eV in the latter case [19]. Thus, we might expect one exciton peak in the vicinity of 12 eV and another near 13 eV. Reactions (4) and (5) are d-d transitions which should be somewhat suppressed by the parity selection rules. However, the crystalline-field splittings, (4), represent the only contributions to absorption on the 1-4 eV range, and thus should be quite observable. Reaction (6) requires a change of 2 in the orbital angular momen- tum quantum number, and also would be strongly sup- pressed, were it not for hybridization between 2p and 3d orbitals, which is very significant in NiO [27]. Reac- tion (7) represents an itinerant-localized allowed transi- tion, and should give an important contribution to ab- sorption. Finally, reaction (8) is an allowed interband transition and should produce the most effective photon absorption of all. A sketch of the expected opti- cal absorption as a function of photon energy is shown in figure 1. The antiferromagnetic peak at 0.24 eV which appears below the Neel temperature [30] is in- dicated in the sketch. The experimental absorption below 4 eV [30] is shown in figure 2, and the absorption above 4 eV, as determined from reflectivity measure- ments [25], is shown in figure 3. Our interpretation is that the edge at 4 eV is the onset of reaction (6a), and the strong absorption above 12 eV represents the com- bined effects of (5), (7), and (8). The large peak at 17.6 eV thus represents the maximum contribution of the al- lowed absorptions, reaction (7), estimated as in the 14- 19 eV range. The peak at 13.8 eV is most easily in- terpreted as due to Nit – (2p hole) bound excitons. The shoulder at 13.0 eV can be associated with the Nit- Niët bound exciton. Our interpretation of the optical results is also consistent with the observed photocon- ductivity [31], since the lowest energy excitation to a conducting state is the 4 eV edge of reaction (6a). Figure 1 represents the possible optical transitions in pure stoichiometric NiO. However, it is not an effective single particle density-of-states diagram in the usual manner. We have previously suggested a method for such a representation when localized states are present [19,32]. The basic idea is to employ a split density-of- states plot, with itinerant states drawn to the left and lo- calized states drawn to the right. The left-hand side can thus be treated as ordinary one-electron band states, and free carriers on the left contribute to conductivity in the normal manner. However, states on the right- hand side are not one-electron states and can be treated as such only if certain rules are borne in mind. The positions and number of states on the right must vary with the occupation numbers, and partially-filled bands contribute to conduction only by means of thermally-ac- tivated hopping. Furthermore, it is necessary to set the energy of one of the states on the right relative to that of one of the ones on the left. Exactly how this is done should depend on the experiment being interpreted. 496 p?--dº -- pºt d? p –- s *. d’Hô’ i º M | | | | | | | | | O 2 4 6 8 |O | 2 |4 |6 |8 20 ENERGY (eV) FIGURE 1. Sketch of the predicted optical absorption of pure, stoi- chiometric NiO as a function of photon energy. 6 10 I I | | | | | I | | | I | I | i | I | I | I | | | | | alo' 5x10°E- ,--~~~~~~ –5x10% º - f --- – o 300°K HALIDE DECOMPOSITION | - • 77°K HALIDE DECOMPOSITION | s] a 300'K OXIDIZED NICKEL ! 6 E | E 5x10" ! —5 x 10° ! - I - - ! - 10°– —10° lil - - -> !--- - Cº. 3|T - 3 5 X | *H –5x10" . -- I s - - # 10°E- —10° ; 2.O 300°K CURVE ſº = # NiO ABSORPTION º (SEE SCALE AT LEFT): –5x10° § i. 8 – COEFFICIENT - & 2 H-4 --- 35 --> - F. is 1.6 {3.8 }- - & cº à 9 1.4| 10°E. - 103 * H. É | I ă 1.2|- 50 H, — 5x10% É T].3 | - # I.OH —Q a C \ o -- O |\ | 77 TK CURWE : os º (SEE SCALE AT RIGHT ) -- z v. o F o f C | 2 # 0.6 |O —10 g “T I § - O4 – 5 H. —|50 - O.2}- g - O l i | | | | | | i ſ | l " _ _ –––––––––––––––– 2 4 6 8 |O || 2 || 4 ||6 || 8 20 22 24 26 | | | | | | | | | | | |O 15–Ho-Hs—go-zis—so--sº-Ho-Hº-Ho-Hº-Hºo--Es—t PHOTON ENERGY (eV) PHOTON ENERGY (ELECTRON WOLTS - e * & g ( ) FIGURE 3. Experimental optical absorption of an NiO crystals FIGURE 2, Experimental optical absorption of NiO crystals below above 4 eV, as determined from reflectivity measurements (data of 6 eV (data of Newman and Chrenko [30]). Powell [25]). 417–156 O - 71 - 33 497 ENERGY (eV) | AE----------- EEE)* +3d” 1 –––– = =========== !, — 3d’ + 3d” | : 3d 7 + 3d 9 4 s” ; | 3. O iſ’ | ſ: | O. O l 4 s 4.O 3.6 L T T.T.T.T.T. Tri = ---> — — — — — — — — — — — — - - --> º, amº - “º " 3d 83- i. I E=EEE========== O.O 3d6 - 1.5 2p - 5.5 FIGURE 4. Optical density-of-states of pure, stoichiometric NiO. Itinerant states are drawn to the left, localized states to the right. States which are filled at T= () are shaded. For photoemission experiments, the work function of the material is a convenient reference point for all states. Electrical conductivity is most easily analyzed if the Fermi energy is fixed on both sides. In this sec- tion we are concerned with optical absorption measure- ments. The most convenient choice would appear to be one which sets correctly the energy of the lowest-ener- gy optical transition between a localized and an itinerant state. This is just the technique used in han- dling donor, acceptor, and exciton states in a conven- tional semiconductor. Such an effective density-of-states diagram is given for pure, stoichiometric NiO in figure 4. The zero of energy has been taken to be the energy for the upper- most state which is occupied at T=0, a localized 3dº state of the Ni2+ ion. States on the right and left are connected by setting the minimum energy of reaction (6a), 3d" -> 3d" + (electron in the 4s band), cor- rectly. All other transitions across the center line are then approximations. However, in the case of NiO, the only other low-energy set of transitions between local- ized and itinerant states, reaction (7), is correct to within 0.5 eV. As figure 4 is presented, only one special rule need be introduced. We must require that states drawn with a dotted line be available for transitions only from filled states on the right. Thus the crystalline-field split states, 3dº”, can only be excited from the 3dº ground states. The excited transitions given by reaction (6b) can then be easily handled by introducing a multiplicity of dotted 4s bands, labeled 4s” in the diagram. This is to be interpreted as the excitations of electrons from the 3dº ground state to the 4s band which leave excited 3d” states of the nickel ion. With this rule in mind, figure 4 can be used to obtain an excellent approxima- tion of the optical absorption spectrum of pure, stoichiometric NiO, figure 1. Extension of this discussion of the optical properties to impure or nonstoichiometric NiO is straightforward. In fact, the optical absorption spectrum above 1 eV need not be modified at all, since the strong absorption continuously present in the 1-20 eV range will mask any effects of doping and vacancy concentrates up to a few percent. The most common dopants in NiO are monovalent ions, such as Lit. Lit enters the NiO lattice substitu- tionally for Ni2+, and in order to preserve charge neutrality in the crystal, a Niºt ion is formed for each Lit ion in the material. The lowest-energy state of the doped material clearly is that in which all Lit ions have a nearest-neighbor Niºt ion, forming an effective “elec- tron-hole” bound state, similar to an exciton. The max- imum binding energy of this pair can be estimated as 0.4 eV [19]. A series of optical transitions are possible in the doped material which were not present in pure crystals. These represent excitation of an electron from any of the Ni2+ ions to the bound Niºt ion. The domi- nant optical transitions of this type are from Niºt ions far removed from the Lit center, and can be looked as a photon-assisted freeing of the hole (Niºt ion) which was bound to the Lit center. This transition should thus require about 0.4 eV. The other transitions, to more weakly-bound Lit - Niºt states, where the Niºt is located farther from the Lit center than the nearest- neighbor Ni2+ sites should give weak absorption in the 0.2-0.4 eV range. Clear evidence for such Lit-induced optical absorption has been found with a peak at 0.43 eV [33]. Another possible absorption in Lit-doped, but not in pure, NiO can be represented by the reactions 3d" + (electron in 2p band) → 3dº (9a) 3d" + (electron in 2p band) → 3d” (9b) This additional absorption can occur because of the presence of bound Niêt (3d") ions in the doped material. Assuming that the vast majority of Niºt ions are bound to Lit centers, reaction (9) can be shown to lead to ab- sorption between 0.7 and 4.7 eV [19]. There is evidence for a Lit-induced background absorption which increases from 0.2 eV through at least 2 eV, with 498 a peak at about 1.0 eV [34,35], which may well be a combination of excitations from both bound and free Niºt ions of type (9). The peak at 1.0 eV, if real, alterna- tively could represent the lowest crystalline-field peak of the Ni3+ ion. 3. Electrical Density-of-States of Crystalline Mott Insulators As pointed out in the Introduction, when localized states are present, we cannot use the same density-of- states diagram for analyzing the electrical properties as we did for the optical absorption. This is because trans- port of electrons through the lattice occurs at times suf- ficiently long for the ions to relax around the new charge distribution, and thus ionic motion can no longer be neglected. We must thus take into account small- polaron formation and perhaps Jahn-Teller distortions in the localized states. We can, however, use the optical density-of-states as a starting point for the analysis of electrical transport. As in section 2, we shall confine our remarks to NiO, whose optical density-of-states is shown in figure 4. The lowest energy excitations are the 3d”- 3d” tran- sitions, which are completely localized and do not contribute to conduction. The lowest energy excitations which do produce intrinsic conductivity are those given by reaction (6), which creates an itinerant electron in the 4s band and leaves a localized hole in the 3dº band. Optically, this transition requires about 4 eV. The cor- responding electronic transition will require somewhat smaller energy, since a lattice distortion will be induced around the 3d" (Niët) ion, lowering the energy of the final state. As discussed in section 2, the 3d electrons in NiO are very near the atomic limit, and the corresponding elec- tronic bandwidth, A, is expected to be negligibly small [19]. It is thus clear that the effects of the electron-photon interaction on the 3d electrons must be analyzed by nonadiabatic small-polaron theory [36]. Using the value for the strength of the elec- tron-phonon interaction which gives a formal equiv- alence with large-polaron theory [37], and the parameters appropriate to NiO, we can show that the small-polaron binding energy is then of the order of 0.01 eV [19]. In this case, the transition tem- perature above which conduction by thermally ac- tivated hopping of small polarons dominates polaronic band conductions is small compared to the Debye tem- perature [19]. It is clear that the band like conductivity of the electron excited into the itinerant 4s band dominates the phonon-assisted hopping of the 3d hole at all temperatures. Using large-polaron theory [38] for the 4s band, we can estimate the polaron binding ener- gy as 0.2 eV. Thus the intrinsic thermal energy gap should be about 0.2 eV lower than the intrinsic optical gap. The intrinsic conductivity must be n-type, and requires an activation energy near 2 eV. Thus intrinsic conduction should be observable only at extremely high temperatures and only in either extremely pure or highly compensated samples. There is some experi- mental evidence for this intrinsic process. In one rela- tively pure crystal of NiO, an activation energy of 1.9 eV was found between 700 and 1200 K at atmospheric pressure [31]. In a number of highly compensated polycrystalline samples, activation energies equal to approximately 1.8 eV have been measured [39]. In section 2, we showed that neither doping nor non- stoichiometry results in a major modification of the op- tical absorption spectrum of NiO. On the other hand, neither has a profound influence on the electrical pro- perties. Once again, let us first consider the effects of doping with Lit impurities. As discussed in section 2, this will lead to the formation of Niët ions. A Ni3+ site is much like a hole on a Ni2+ ion, and this hole can move through the lattice by hopping to adjacent Ni2+ sites. The nar- row bandwidth of these hole states allows for a local- ized deformation of the lattice around the hole, and thus the formation of a small polaron. The polaron ini- tially will be electrostatically bound to the Lit centers. Theoretically and experimentally, this binding energy can be estimated as 0.4 eV. Once thermally freed from the Lit center, the small polaron will conduct by means of thermally activated hopping, but because of the small value of the polaron binding energy, the mobility will be only weakly dependent on temperature. Thus this process should contribute a term to the conductivi- ty equal to OT = NA €AL06 —(0.2e I’)/kT (10) where NA is the density of Lit centers and po is the mo- bility of the hopping process. In the view of normal band theory, the holes just described would be simply empty states in the 3dB band of figure 4. But in the present context of localized states, we must take into account the fact that the separation between the 2p band and the 3dB states is determinded by the energy to add an electron to the 3ds states. This results in it being much easier to excite a 2p hole in the Lit-doped material than in the pure material. The reason for this is that doping with Lit automatically results in the presence of Niº ions. Since the electron affinity of Niêt is 18.6 eV greater than that 499 ENERGY (eV) I == Vo i. 3.4 : : Lit ACCEPTORS K-ºs- F Z Z ZººZººZººZººZº. 3d : # V Ni 4 EF 2p –4.5 FIGURE 5. Electrical density-of-states of Li doped NiO. Itinerant states are drawn to the left, localized states to the right. States which are filled at T= () are shaded. V., and Vºl refer to singly and doubly ionized X-vacancy levels. respectively. of Niºt, reaction (9a) takes much less energy than the analogous intrinsic process, reaction (7a). The sig- nificance of this point with respect to the density of states is that Lit doping raises the 2p band up to the vicinity of the Fermi energy. Because of this, conduc- tion by means of free holes in the 2p band cannot be neglected. This process, which can be represented by reaction (9), was shown to require a minimum photon energy of 0.7 eV. However, thermally, this minimum energy is significantly reduced by large polaron forma- tion about the 2p hole. This reduction in energy can be estimated as 0.25 eV [19], leaving a thermal activation energy of the order of 0.45 eV. This value depends criti- cally on differences between large numbers, such as the Madelung potential and the ionic energies, and should not be expected to be very accurate. However, the important point is that it is comparable in mag- nitude to the activation energy for conduction by hopping in the 3dº band. Since the Fermi energy is of major importance in determining transport properties, an electrical density- of-states diagram should refer all states, both localized and itinerant, to Ep. Such a diagram is constructed for primarily Lit doped NiO in figure 5. The acceptor levels corresponding to singly and doubly ionized nickel and oxygen vacancies are also shown. There is much evidence for partial self-compensation in Lit doped NiO [4,40,41], and this is indicated in the diagram by the presence of the Fermi energy in the Lit acceptor band. The compensating donors appear to be oxygen vacancies [19]. At low temperatures, when the free hole concentration is small compared to the donor con- centration Np, 2p hole transport will contribute a term to the conductivity equal to [19] W. – Wi) O = Weepwo - e-(0.3eſ/RT) W1) (11) where N = 2(m^kT/2Th”)”, po- 0.3 cmºſſ-sec, and m” is the large polaron effective mass, which can be estimated as 6 free electron masses for the 2p band of NiO. It was assumed that optical phonon scattering dominates the mobility in this temperature range. The activation energy is affected by interactions between the Li impurities. At temperatures sufficiently high that the free hole concentration is large compared to ND, the contribution of 2p hole conduction becomes O = N, ; NI : eptoe-(0.18eºlk"). (12) It can be shown that the observed conductivity of Lit doped NiO is given by eq (11) from 200 to 500 K, and by eq (12) from 500 to 1000 K in well characterized sam- ples [19]. We can conclude that conduction in Lit doped NiO is dominated by the transport of free holes in the 2p band above 200 K. It might be expected that small-polaron hopping con- duction will dominate at sufficiently low temperatures, because of the somewhat smaller energy necessary to create a free 3dº hole than a free 2p hole. However, there is much evidence for the predominance of impuri- ty conduction in the Lit acceptor band below 150 K |33.40,41]. Thus, it appears that small-polaron hopping is unobservable in NiO in do measurements. Bound small-polaron hopping has been observed in ac conduc- tivity experiments [42,43], and is in agreement with the calculation presented here [19]. 4. Density-of-States of Amorphous Semiconductors Mott [2,3] and Cohen et al. [15] have presented qualitative band models which have had great success in accounting for experimental data on covalent amorphous semiconductors, such as the chalcogenide glasses. At first, this appears to be surprising, since electron-electron interactions and electron-phonon in- teractions, which should be of the utmost importance in these systems, are completely neglected. An essen- tial feature of the model of Cohen et al. [15] is a broad tailing of localized states up from the valence band and down from the conduction band into the energy gap. In 500 CONDUCTION l BAND l | VALENCE BAND CONDUCTION BAND VALENCE BAND CONDUCTION BAND VALENCE BAND − ey - FIGURE 6. Density of states, g(E), disorder potential, V, and intra- atomic Coulomb repulsion, U , as functions of energy for a covalent amorphous semiconductor. the chalcogenide glasses, it is assumed that the com- bination of positional and compositional disorder creates such a large density of these localized states that the valence and conduction band tails cross somewhere in the middle of the original gap. Thus there is no gap in the density-of-states of these materials. However, as discussed in the Introduction, a “mobility gap.” exists between the itinerant states in the two bands. Since this model is based on a one-electron approxi- mation, electronic correlations are ignored completely. However, they can be introduced into the model in the same approximation used by Hubbard [5-7], and discussed in the Introduction. We consider only cor- relations between electrons located on the same atom, and introduce the intra-atomic Coulomb repulsion, U. Since it is known that the crystalline phases of most elemental and covalent amorphous semiconductors are wide-band materials, U is not expected to be an impor- tant quantity in the itinerant states. Let us consider the valence and conduction band densities-of-states shown in the left third of figure 6. As we move out in energy from the main part of each band into the tail, the as- sociated disorder, V, defined in the Introduction, in- creases. This is shown in the center of figure 6. When V becomes sufficiently large that the Anderson condi- tion, (2), is satisfied, all states further out in the tail become localized. At this point, the mobility edge, the effective difference in ionization potential and electron affinity U, suddenly becomes significant. Thus U. must behave qualitatively as in the right third of figure 6. It is V and not U which is responsible for the localization, but once this Anderson transition takes place, U. stabilizes this localization considerably. This is a reasonable explanation for the sharpness of the mobili- ty drop at the edge. If the valence and conduction band tails overlap, some states in the unoccupied conduction band have lower energy than states in the occupied valence band. It has consequently been suggested [15] that the material will reduce its ground state energy by undergo- ing a redistribution of electrons in these states, creating equal numbers of positively and negatively charged lo- calized states deep in the gap. However, this suggestion fails to take into account the strong dependence of lo- calized states on occupation numbers, due to the finite value of U. The redistribution of electrons can be represented by the process X + Y → X + + Y-, (13) in which an electron is transferred from a state local- ized on atom Z to one on atom Y. If there is a large dif- ference in the ionization potential of atom X and the electron affinity of atom Y, reaction (13) can be highly endothermic, even if the one-electron state correspond- ing to an electron on X is much higher than that cor- responding to a hole on Y. The effective difference between the ionization potential of X and the electron affinity of Y is just equal to U. If the difference in one- electron energies were smaller than U, the redistribu- tion would increase rather than decrease the total ener- gy of the system, and thus would not occur spontane- ously. However, we have still ignored the electron-phonon interaction. Reaction (13) can be thought of as the crea- tion of a localized electron-hole pair. The redistribution of electronic charge density which accompanies this process couples strongly to the longitudinal optical phonons, and can lead to lattice distortions in the vicini- ties of both atoms X and Y. Since the states under discussion are localized, the effective overlap integral is small, and we would expect small polarons to form. 501 If the binding energy of these small polarons is suffi- ciently large, reaction (13) could well be stabilized, despite a large U. Another possible stabilizing force could be the reduction in energy around Xt and Y- brought about by a Jahn-Teller distortion of the sur- roundings of the ions. The condition for the spontane- ous occurrence of the redistribution is thus A > U – 2Eſt - 2Esp. (14) where A is the difference in one-electron energies of X and Y-, E/T is the Jahn-Teller stabilization energy, Esp is the small polaron binding energy, and U is the effec- tive value of the difference between ionization potential and electron affinity, in which the effects of electronic screening are taken into account. In elemental and simple compound amorphous semiconductors, such as Ge. Si, and As2Sea, there does not appear to be qualitative differences between the band structures of the amorphous materials and the corresponding crystalline materials, except for a smear- ing out of the bands brought about by the positional dis- order [44]. For these materials, the right-hand side of (14) can be approximated from the experimentally known differences in the first and second ionization energies of multivalent donors in the corresponding crystal. This is a reasonable estimate because the same effects enter when a donor is doubly ionized as when a redistribution such as (13) takes place. The difference between ionization potential and electron affinity of the singly-ionized donor atom is analogous to U, and small- polaron formation (important only if the crystal is par- tially ionic) and Jahn-Teller distortions can stabilize the doubly-ionized donor in the same way that it can stabil- ize the redistribution, (13). For example, the difference in the first and second donor ionization energies of Au in Ge is 0.2 eV [45]. Using this as an estimate for U- Eyr-Esp., we can conclude from (14) that a redis- tribution such as Ge (1) + (;e (2) → (;e (1) + Ge (2) (15) will take place spontaneously at T=0 only if the one-electron energy of the filled state localized on Ge(1) exceeds the one-electron energy of the empty state lo- calized on Ge(2) by at least 0.2 eV. For a material such as amorphous Ge, in which only positional disorder is present, it is extremely doubtful that the valence and conduction band tails overlap by 0.2 eV. Thus, we should not expect any charged traps to be present at T=0 in an equilibrium. On the other hand, the posi- tional and compositional disorder present in the chal- cogenide glasses makes an overlap of this magnitude quite possible. Thus, the major features of the model l | CONDUCTION BAND EV - (Esp- ELP) VALENCE BAND FIGURE 7. Electrical density-of-states for a covalent amorphous semiconductor. Itinerant states are drawn to the left, localized states to the right. Es, is the sum of the small polaron binding energy and the Jahn-Teller stabilization energy. of Cohen et al. [15] are preserved when electronic correlations and electron-phonon interactions included in this manner. The question remains whether or not this modified model can be represented by an effective one-electron density-of-states diagram. This can be done, in much the same way as was discussed for the case of NiO in sections 2 and 3. Once again, the Franck-Condon prin- ciple forces us to draw separate diagrams for interpret- ing the optical properties and the electrical properties. Figure 7 shows a plot of the electrical density-of-states for a glass with positional and compositional disorder. As in figure 5, itinerant states are drawn to the left and localized states to the right. The possibility of large- polaron formation in the itinerant states is taken into account by a downward shift of these states relative to the Fermi energy. The small-polaron binding energies and Jahn-Teller stabilization energies result in downward displacements of the localized states. The finite value of the correlation energy, U, in the localized states must be represented by a relative separation of the localized parts of the valence and conduction bands. are 5. Conclusions We wish to emphasize the following conclusions: (1) When localized states are present and the electron- phonon interaction is important, the Franck-Condon principle forces us to use different density-of-states dia- grams for interpreting electrical and optical properties; 502 (2) When localized states are present, the intra-atomic Coulomb repulsion, U, cannot be neglected. This makes it convenient to separate localized from itinerant states on a density-of-states plot; (3) In the case of ionic transition-metal or rare-earth compounds, doping with an ion of different valency or preparation of non- stoichiometric material leads to the formation of minority valence states of the transition metal or rare- earth ion. If these materials are Mott insulators, the vastly different ionization potentials of the majority and minority valence states can drastically change the rela- tive positions of the energy bands. In particular, in NiO, creation of a significant density of Niºt ions, either by Lit doping or introduction of excess oxygen, results in a large upward displacement of the oxygen 2p band relative to the localized levels. The 2p band moves suf- ficiently close to the Fermi energy so that 2p hole con- duction dominates the do conductivity from 200 to 1000 K; (4) The different relative positions of the bands in pure, stoichiometric crystals and in doped or non- stoichiometric material necessitates the use of at least four density-of-states diagrams, rather than the two in- dicated by the Franck-Condon principle; and (5) In amorphous semiconductors, electronic correlations and electron-phonon coupling lead to changes in the rela- tive positions of the localized and itinerant parts of the valence and conduction bands. However, provided the overlap of the band tails is sufficiently large, the essen- tial features of the model of Cohen et al. [15] for the chalcogenide glasses are preserved. 6. Acknowledgments We should like to thank N. F. Mott, M. H. Cohen, and F. S. Ham for the valuable discussions. 7. References [l [2] [3] [4] [5] [6] [7] | Wannier, G. H., Elements of Solid State Theory, (Cambridge U. Press, 1959) pp. 161-162. Mott, N. F., Proc. Phys. Soc. A62, 416 (1949). Mott, N. F., Phil. Mag. 6, 287 (1961). See, for example, D. Adler, Solid State Phys. 21, 1 (1968); F. J. Morin, Bell Syst. Tech. J. 37, 1047 (1958). Hubbard, J., Proc. Roy. Soc. A276, 238 (1963). Hubbard, J., Proc. Roy. Soc. A276, 237 (1964). Hubbard, J., Proc. Roy. Soc. A281, 401 (1964). [8] [9 [10] [ll] [12 [13] [14] | | [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [43] [44] [45] Adler, D., and Feinleib, J., J. Appl. Phys. 40, 1486 (1969). Mott, N. F., Phil. Mag. 17, 1259 (1968). Mott, N. F., Festkorperprobleme 9, 22 (1969). Anderson, P. W., Phys. Rev. 109, 1492 (1958). Ziman, J. M., J. Phys. C 2, 1230 (1969). Edwards, S. F., J. Non-Cryst. Solids, to be published. Neustadter, H. E., and Coopersmith, M. H., Phys. Rev. Letters 23, 585 (1969). Cohen, M. H., Fritzsche, H., and Ovshinsky, S. R., Phys. Rev. Letters 22. 1065 (1969). MacMillan, A. J., Tech. Rept. No. 172, Lab. Insulation Res., M.I.T. 1962 (unpublished). McWhan, D. B., Rice, T. M., and Remeika, J. P., to be published. Fender, B. E. F., Jacobson, A. J., and Wedgewood, F. A., J. Chem. Phys. 48,990 (1968). Adler, D., and Feinleib, J., to be published. Reinen, D., Ber. Bunsenges. Physik. Chem. 69, 82 (1965). Orgel, L. E., J. Chem. Phys. 23, 1819 (1955). Kroger, F. A., J. Phys. Chem. Solids 29, 1889 (1968). Watson, R. E., Phys. Rev. 118, 1036 (1960). Low, W., Phys. Rev. 109,247 (1958). Powell, R. J., Stanford Electronics Laboratory Technical Re- port No. 5220-1, 1968 (unpublished). Austin, I. G., and Mott, N. F., Adv. Phys. 18, 41 (1969). Switendick, A. C., Quart. Progr. Rept. No. 49, Solid State and Molecular Theory Group, M.I.T., 1963, p. 41 (unpublished). Wilson, T. M., Res. Note No. 1, Quant. Theoret. Res. Group, Oklahoma State U., 1969 (unpublished). Moore, C. E., Atomic Energy Levels, NBS Circular 467, Vol. II (1952). Newman, R., and Chrenko, R. M., Phys. Rev. I 14, 1507 (1958). Ksendzov, Ya, M., and Drabkin, I. A., Soviet Phys. Solid State 7, 1519 (1965). Feinleib, J., and Adler, D., Phys. Rev. Letters 21, 1010 (1968). Ksendzov, Ya. M., Avdeenko, B. K., and Makorov, V. V., Soviet Phys. Solid State 9, 828 (1967). Austin, I. G., Clay, B. D., Turner, C. E., and Springthorpe, A. J., Solid State Commun. 6, 53 (1968). Austin, I. G., Clay, B. D., and Turner, C. E., J. Phys. C 1, 1418 (1968). - Holstein, T., Ann. Phys. (NY) 8, 343 (1959). Holstein, T., Ann. Phys. (NY) 8, 325 (1959). Allcock, G. R., Adv. Phys. 5, 412 (1956). Pizzini, S., and Morlotti, R., J. Electrochem. Soc. l 14, 1179 (1967). Bosman, A. J., and Crevecoeur, C., Phys. Rev. 144, 763 (1966). Austin, I. G., Springthorpe, A. J., Smith, B. A., and Turner, C. E., Proc. Phys. Soc. (London) 90, 157 (1967). Snowden, D. P., and Saltzburg, H., Phys. Rev. Letters 14,497 (1965). Kabashima, S., Kawakubo, T., J. Phys. Soc. Japan 24, 493 (1968). Stucke, J., Festkorperprobleme 9,46 (1969). Woodbury, H. H., and Tyler, W. W., Phys. Rev. 105, 84 (1957). 503 Discussion on “Localized States in Narrow Band and Amorphous Semiconductors" by D. Adler and J. Feinleib (MIT) H. P. R. Frederikse (WBS): You left out the anti-fer- romagnetism of NiO. Is that not important? Switendick had to invoke the anti-ferromagnetism in order to ob- tain an insulator rather than a metal. D. Adler (MIT): In our model, NiO is an insulator rather than a metal because of the intraionic Coulomb repulsion, independent of the anti-ferromagnetic order- ing. Anti-ferromagnetism should modify the densities of states presented here by energies of the order of kTy, or 0.04 eV for NiO. There is an optical absorption in the i.r., at 0.24 eV. apparently due to the magnetic order- ing, which can be included on our optical density of states diagram above the 3dº ground-state band. Since it refers to a localized excitation, it does not contribute to the electrical conductivity. D. F. Barbe (Westinghouse Aerospace Div.); What would you expect to happen to the energy bands in say NiO as positional disorder is introduced? D. Adler (MIT): Positional disorder always introduces a tailing of the itinerant energy bands, and an Anderson localization of some of the state in the tails. Con- sequently, in amorphous NiO, the 2p and 4s bands would tail into the energy gap, and some of these states would be localized. On the other hand, the 3d electrons are already localized because of correlation effects, and positional disorder is largely irrelevant. The crystalline- field and multiplet 3d levels are insensitive to positional disorder, since they are near the ionic limit. A slight broadening of these narrow 3d bands should take place, but since no significant mobility changes are induced by positional disorder, this broadening could not be ex- pected to be observable experimentally. The major measurable effect should be the broadening of the opti- cal 2p → 4s absorptions. 504 A Cluster Theory of the Electronic Structure of Disordered Systems ” K. F. Freed” cind M. H. Cohen James Franck Institute, University of Chicago, Chicago, Illinois 60637 The equation of motion for the averaged Green's function in an alloy couples the latter to the Green’s function for which the average is restricted so that the composition of one atom is held fixed. The average Green’s function may be regarded as the Green’s function for a zero-atom cluster, and it is coupled to the Green’s function for a one-atom cluster. There is thus an infinite hierarchy of equations of motion in which the n-atom functions are coupled to the (n + 1)-atom functions. The coherent potential approximation of Soven corresponds to truncation in the equation of motion of the one-atom function. We have generalized the coherent potential theory to a theory the n-atom functions with truncation in the equation of motion of the (n + 1)-atom function. The formalism is developed and a few of the results obtained thus far are presented in this paper. Key words: Amorphous semiconductors: band tail of localized states; cluster theory; coherent potential approximation; disorder systems; electronic structure; Neel temperature. Apart from strongly ionic or molecular materials or from simple metals, our present knowledge of the elec- tronic structures of condensed materials is rudimentary except for crystals. There, structural periodicity leads to the remarkable simplicity of the electronic wavefunc- tions expressed in the Bloch-Floguet theorem and to the existence of energy bands. Our present concern is with the corresponding universal features of the elec- tronic structures of disordered materials. Soven and others [1] have developed the coherent potential ap- proximation (CPA) into a quantitative tool for the study of the electronic structures of simple alloys. It yields, however, only bands of extended states with sharp edges and shows signs of inaccuracy at the band edges. In amorphous semiconductors, the band edges must play a central role in determining the electronic proper- ties. Moreover, on both theoretical and empirical grounds it seems highly plausible that the electronic structure of disordered materials consists of bands of extended states with tails of localized states which may, in fact, overlap [2]. The character of the wave func- tions changes from extended to localized at an energy Ec near each band edge, where the carrier mobility drops abruptly. One of the central tasks of the electron theory of disordered materials is to substantiate or cor- *Supported by ARO(D), NASA, and ARPA. **Alfred P. Sloan Foundation Fellow. rect these models. Accordingly, we have addressed our- selves to improving the CPA to the point where it can conceivably contain such features as tails of localized states and mobility edges [3]. The localized states appear to be associated with fluctuations (in composition, local order, etc.), whereas the CPA averages over all such configurations. It is necessary to solve for the electronic structure before carrying out the averaging over the configurations im- portant to the formation of localized states. The equa- tion of motion for the one-electron Green’s function is (E-Ho-V)% (E) = 1, (1) where Ho is a simple reference Hamiltonian, free-parti- cle for a liquid or amorphous material, or crystal for an alloy, and V is a random potential. We suppose V can be decomposed into a sum of contributions from each site O., Va., in a crystalline alloy or from each atom o in a liquid or amorphous material, V=X. V. (2) The CPA concerns itself only with the average Green’s function Go (E) = (%(E)) (3) where the brackets indicate an average over the ran- dom potential V. We examine here Green’s functions in 505 which the averaging is incomplete, a cluster of n of atoms being fixed either in composition and/or in posi- tion. The potential V can be decomposed into a contribu- tion from the cluster V, and one from the remaining atoms WN_n , W = W., + VN_n. (4) Let ()" imply an average only over the set of atoms m and ( ), one over all atoms excluding the cluster I so that ()=()^' (5) The cluster Green’s function is then defined as Co- (4), - (4)". " (6) G, now appears as the lowest member of a hierarchy of cluster Green’s functions. By averaging eq (1) we ob- tain the corresponding hierarchy of equations of mo- tion. (E - Ho - V., ) G, -X. ( V.C.) = 1. &fn (7) Here no is that cluster n + 1 composed of the cluster n and the atom ov. It is convenient to rewrite (7) in terms of the proper self energy on (E), (E – Ho - Vn – or,) Gn = 1. (8) and to define the T-matrix T. for the scattering of an electron by the specified atom o in the presence of a specified cluster n by Gna- G, H G, T;G. (9) From (7), (8), and (9) it follows that o,+ X. oº), (10) cºn o'º)= (V, -i- V.G, T; ) ", (11) T;= U; [l – G.U.]-, (12) U;= Wa-H orna- orn. (13) Truncation of the hierarchy can be affected by a rela- tionship between orna and orn, leading to Tº, being a functional of or, via (12) and (13) and hence to closure via (10) and (11). Equivalently, we may observe that Gn = (Gna) CY (14) which, from (9), requires that (T: ) * = 0, all n and oftn. (15) The coherent potential approximation follows from (15) for n = 0 together with the truncation condition o'º)= orº), all 9 = 0, (16) i.e., Orn for n = 1 is related to or n for n = 0. We can im- mediately generalize the CPA so that the truncation oc- curs for an nth instead of for a zero order cluster, o'º - O'º), all Béna, IT (Y (17) and consequently U*= V.-gº). (18) Inserting (18) into (12) and (12) into (15) gives us an equation which can be solved selfconsistently together with (8) for Orn and Gr The CPn approximation has a number of important properties, among which are: (1) Consistency for clusters of lower order. (T,) = 0, 0 < m = n, all mºn. (19) (2) Locality of Orº) and T. If Wa is local in the sense that it is site diagonal, (8|Waly) = V268,6ay, (20) then so are or;” and T. (3) Asymptotic limits of G, and O'º. If we assume that all of the eigenstates of Ho + V are either extended with a short phase coherence length or localized, as in the simple models, then Gm., (m s m + 1) must have a finite range R in the sense that ( o'G,B) → 0, re-rel - R. (21) This implies that for |ry – rel) R or |rs – rel) R, where Bem. Similarly, we have (BGnaly) → ( 3|Galy ) (23) |rs, y- ral) R., 6én, land o;)— o'ſ" independent of (a) (24) Öen. |ra – ral) R, (4) Bound states occur in Gn provided on (E) is real outside a finite arc of the real E axis and the statistical fluctuations in Va are large enough. 506 (5) For sufficiently disordered systems, conver- gence in Go obtained by averaging down from successively higher Gn may occur at values of n which are tractable for numerical calcula- tions. Even for G obtained by averaging down G derived from the CPI approximation, the proper self energy is nonlocal and in that sense a significant improvement over the CPA. Our final approximation to (%) would then be obtained by averaging down a Gn from the CPn 1S G=X. P.G, (25) where the sum on n is over all possible clusters n, each having the probability of occurrence Pn. References [1] Soven, P., Phys. Rev. 156, 809 (1967): Velicky, B., Kirkpatrick, S., and Ehrenreich, H., Phys. Rev. 175, 747 (1968); Soven, P., Phys. Rev. 178, 1136 (1969); Velicky, B., Phys. Rev. (to be published). [2] Mott, N. F., Adv. Phys. 16, 49 (1967); Cohen, M. H., Fritzsche, H., and Ovshinsky, S. R., Phys. Rev. Letters 22, 1065 (1969). { ] [3] Cohen, M. H., J. Non Cryst. Solids (to appear). 507 Discussion on “A Cluster Theory of the Electronic Structure of Disordered Systems” by K. F. Freed and M. H. Cohen (University of Chicago) J. L. Beeby (Atomic Energy Res. Establishment): Can it be proved that the density of states sum rule is satisfied in your procedure? M. H. Cohen (Univ. of Chicago). It is satisfied auto- matically at each stage in the approximation. R. M. More (Univ. of Pittsburgh): If V is not site diagonal, can (TRA) = 0 be satisfied even off-shell? M. H. Cohen (Univ. of Chicago): You just have an ad- ditional series of equations to take care of the addi- tional set of variables in the proper self energy. It is possible in principle. One can say the following: if one does not have a site diagonal potential but a potential of finite range, then the T-matrix will have a range which is some compromise between the range of the potential and the range of the Green’s function itself. It is in- fluenced by both. So the T-matrix itself has finite range and all the asymptotic theorems still go through. The overall structure remains the same: the difficulty then is in satisfying the condition of the average T-matrix which must be 0. You have both the diagonal elements of that equation then the off-diagonal elements of that equation and this gives you an additional set of condi- tions which are satisfied by varying instead of just the diagonal elements of the proper self energy now vary- ing the off-diagonal elements as well. 508 T Matrix Theory of Density of States in Disordered Alloys — Application to Beta Brass M. M. Pant cind S. K. Joshi Physics Department, University of Roorkee, Roorkee, India The T matrix theory of electronic states in disordered systems, has been used to determine the spectral density for states of various symmetries, for binary alloys. Soven's averaged tº matrix procedure is improved by retaining the distinction between the t matrices of the constituents, and introducing par- tial Greenians of the Pant-Joshi theory. Information about the pair-correlation function obtained by criti- cal neutron scattering method is used to evaluate the partial Greenians, as well as the crystalline poten- tials for the constituents. In order to facilitate computation of the t matrices, these potentials have been replaced by energy-dependent model potentials. The parameters of the model potentials are determined by the requirement that they yield logarithmic derivatives (of the radial wave function at the muffin-tin spheres) identical with those generated by the real potentials. The scheme has been applied to disor. dered beta-brass. The separation in energy of the peaks of the spectral density of states at the high sym- metry points of the Brillouin zone, are compared with experimental results, and with the results ob- tained by the virtual crystal approximation. Key words: Alloys; brass; delta-function potential; disordered systems; Green's functions; Korrin- ga-Kohn Rostoker (KKR): “muffin-tin” potential: short-range-order parameters: spectral density of states: t-matrices. 1. Introduction The object of this paper is to outline a t matrix ap- proach to determine the spectral density for states p(E.k) in disordered alloys, and apply it to the disor- dered substitutional binary alloy 3-brass. For the case of a perfect lattice, p(E,R) plotted against E, has 6-func- tion peaks at the band energies. The presence of dis- order results in the broadening of these peaks and their widths give an indication of the departure of the alloy wave function from Bloch-like character. An investiga- tion which bears on this aspect is that of Soven [1], who determined the spectral density for some states in O-brass, by employing the averaged tº matrix approxima- tion. This approximation consists in placing at every site of the alloy lattice, a scatterer whose t matrix is the average of the t matrices of the constituents. The averaged t matrix method yields spurious band gaps in the case of one-dimensional alloys [2]. The approach presented in this paper retains the distinction between the constituents of the alloy. The method relies heavily on the work of Beeby [3], and the calculation utilizes the model 6-function potentials of Soven [1] and the in- complete Green’s functions of Pant and Joshi [4]. Any short-range order present in the alloy is accounted for, by the occurrence of the short-range-order parameters in the expressions for the potentials as well as the in- complete Green’s functions. There are two basic ap- proximations in the theory. The first is the approxima- tion of the actual potential around a lattice site, by a potential of the muffin-tin form. The other is the geometric approximation introduced by Beeby [3] to enable a summation of the infinite series expressing the T matrix of the system in terms of the t matrices for the individual scatterers. For an ordered alloy, this approxi- mation gives the exact result. The method has been applied to the disordered 8- brass to determine the spectral density for states of various symmetries. The results of the experimental measurements of the short-range-order diffuse scatter- ing are available for this alloy [5]. The band structures of the constituents are well understood and an attempt at studying the electronic structure of 8-brass has been made within the framework of the virtual crystal ap- proximation [6]. These considerations prompted us to study the spectral density of states for disordered 8- brass at a few symmetry points. 509 2. Formalism The spectral density of states for noninteracting elec- trons in the presence of a system of potentials Ua is given by p(E, k) = - , Im (JTG, x) | (E – k+)-7( exp [-ik (x-x') |dzdx'), which we write as | p(t, k)=-VF-ºn Im (T(k)). (l) In these expressions, () is the volume of the assembly, the angular brackets denote an average over the disor- dered system of potentials and Im indicates the imagi- nary part of the expression that follows it. The T function for the assembly is given by the series T=X to + X to gots-H > to 40te 40ty-H . . . (2) ( \| (x + £3 (t + £3 (3 + Y where to is the t function corresponding to the potential U, at the oth site and is defined by to (x, y) — Ua (x) 6(x-y ) + | UA (x)? 0(x - z) to (z, y) dz. %0 (x - z) is the free particle propagator. In order to obtain a matrix representation, we make angular mo- mentum expansions of the t functions. It is convenient to use the real spherical harmonics Yi (x) of the angles of x, where L is a compound subscript denoting both l and m, so that t(x, y)= S tºx, y)), (x)Yi (y). (3) In the case of a disordered binary alloy, any site may be occupied by either of the two types of atoms. We use the superscripts 1 and 2 to denote the two types of atoms. The T matrix series may then be split into four parts, such that T= X. Tss'. (4) S = 1, 2 s' = 1 .. 2 We have Tº =X. tº + X. tº 40th-F X. tºol B %0t!...+ nº ( \| (t + 3 ck + (8 B = Y T12 = X tº ot; + X tº got 3% ot}+ . cy H. B (* = B B = Y T21 = X. tº @ot}+ X. tº 9%0ts got' + º cy + [3 (x + 3 8-| Y Tº =X. t; + X. tº gotá-F X. tº 40ts 30t; + . {{ (t + £3 (t + £3 * B = Y (i)) Here toº is the t function corresponding to the potential U* at O. Tº corresponds to that part of the total T function in which the electron scatters firstly off an atom of the sth type and lastly off one of the sth type. The intermediate scatterers may be of either type and are represented by tºs without any superscript in the above series. The Fourier transformation T(k) implied in eq (1) may now be carried out separately for each term of the series (5). The first term of T11 or Tº gives ſ exp [-ik- (x -y)] X tº (x, y)dxdy= (47)*N, X. ſ j(kx).j(ky)t;(x, y)xile yºdy Y,(k)Y (k), (6) { { 1. where N, is the number of potentials of the type s. The calculation of a general term involves angular integra- tions of the type (sº), -i- ||Y(x),..6-2+R.-R) exp [-ik (R-R))) (zuolo. (7) besides the radial integrals involved in ti and a summa- tion over L. Ra, Rg, etc. denote the positions of the o, B, and other sites. A typical term in series of eq (5) there- fore contains products of the form & l, • . . " `i o f o f f , f f : , , , f f f |X sº X sº X sº". (3 + cy Y= B a) + \lſ where the superscripts on S take the values 1 and 2 de- pending on the type of atoms at locations specified by the subscripts. The problem is to sum an infinite series with terms of this nature, and then to average such sums over all configurations. The simplest way to manipulate this is to replace S㺠= X S㺠by some S* w ==tly 510 which does not depend on li. This is the “geometric approximation” and may be seen to be exactly true for a perfectly ordered alloy. We then have Y , , , . " | . . . ' | s # " , a - * Z. A [sº]º-yº X. [Sºlº-Rº X. ſt -') (y), (y-z+R,-Rº) exp [-ik (R.-Rº)]), (z)do, do. * 3 + a * {3 + c, f (8) We can now identify X. %0 (y – z + Ra – Rg) This may therefore be expanded in terms of the spheri- (3+ cy cal Bessel functions j as done in eq (16) of reference 4. exp [-ik (Ra - Re)] as the incomplete Green's function of reference 4, with the O. = 8 term omitted. X 4 00 = z+R,-Rº) exp [-ik (R.-Rº)]=X iſ - Gº.j(ky).j (kz)Y, (y)Y, (z). [3+ cy LI. ' Therefore, G*'. The radial y and z integrals now involve only Bessel functions and the radial tº functions. Their |S; ]. - Giż, j(ky) jº (kz), (9) most general form is Y - • fº ºf t * , , , , / t; (p, q) = | ji (px) tº (x, y).j x*dx y” dy, (10) where k = VE if E > 0 and i V-E if E -< 0, and Cº. f(p, q) = ſ.j. (px)tā (x, y) ji(qy) y- ay are related to the Bjj, of reference 4. The Gº, are in- with p and q taking values of k or k. We use tº to denote dependent of y and z and will be collectively denoted by tſ (k. k.). In this notation, we have for the series of eq (5) Tº = (47)*N. S. Y (k), (k) 00, A.) 61.1, + |*(i. room LL' + X. (, 1878(, si -- X. G | STSG'ss'Ts' (, 8' -- . . . }*(e. o, } S = 1, 2 S = 1, 2 | , 2 F- Tº = (47)-N. S. Y(k), (k)|tº o'cº- S. Cºrcº LI. ' S == 1, 2 –H X. (, 1878Css'Ts'G's"? -- . . tº o, (11) 2 with similar expressions for Tº" and T**. On performing the summations, we arrive at Tº = (47)*N. S. Yi (k)Y, (k) {t} (k, k)öll- |*. k)M LI. ' × {C} + (1 — Gºt?) (G1272) - G 11) tº (k, blº }, tº-(+)-N. S. Yºdoy (loºk, ow, ſcº LL' + (1 — Gºt?) (G1272) - C12} tº (k, or. (12) T22 and T* are obtained by interchanging the super- The above set of eqs (10), (12), and (13) enable us to scripts 1 and 2 in the above expressions. Mi and M2 are determine the spectral density of states. The only defined by the following expressions problems we face at this stage are the calculations of M = (1 – G2272) (G1272) – 1 (1–G | Tl) — Gºt", the matrix elements of G*' and of the evaluation of the Y & 2 * h a t matrices. It is clear that the calculations of G* M., E (I – G 11+1) (G21+1)-1 (1 – Gºt?) – C1273. (13) require a detailed knowledge regarding the relative 511 positions of the atoms. In the case of a disordered alloy, the short-range-order parameters may be used to esti- mate an average distribution pattern of the con- stituents, thus enabling us to calculate the Gº'. A complete discussion of the use of the short-range-order parameters to determine the matrix elements of G* has been given in reference 4. The approximation in- troduced in order to calculate these, is that the short- range-order extends only up to a certain neighborhood, Dº'-m, D,---- m, 6/4 ki– S exp (ik Ry) × [n (KRy) -ij(KRy)]Y (Ry)[P*(Ry) – mºl. VAT Y ~ Or In this expression, my is the atomic concentration of atoms of sth type and the DL without superscripts are the familiar structure constants of the ordered crystal which occur in the Kohn-Rostoker method. n is the spherical Neumann function. Pº'(Ry) denotes the probability of finding an atom of the s'th type at a posi- tion Ry with respect to an atom of the Sth type. This probability can be expressed in terms of the short- range-order parameters as discussed in reference 4. The summation in eq (15) runs through a neighborhood in the direct space, and the prime on the summation in- dicates that the term with y = 0 is to be omitted. The ôLL, term is introduced to compensate for the fact that the calculation of DL for the perfect lattice does not exclude this term. The matrix elements of G* are then directly obtained from eqs (14) and (15). We discuss the calculation of the t matrix in the following section. V*(x, x') = X. YL(x) L where rm is the radius of the muffin-tin-sphere and vrºſ B) are energy-dependent potential amplitudes. The vº(E) are chosen to yield the same logarithmic deriva- tive of the radial wave function as generated by the ac- tual potential. We then have vj(E) = r. [y; (E) ºmº Kj}(krm)|j (Krmſ)] 2 (17) where j} is the derivative of Bessel function and y; (E) is the logarithmic derivative of the radial wave function (for angular momentum l and energy E) at the sphere radius rint. With this form of the potential, the angular momentum components of the t matrix can then be written as 6(x - rmt) 6(x' * rmſ) 2 2 rYnt r }}\t tl (x, x') = ti 2 (18) so that t = viſl — vigil T', (19) º vj (E) 6(x -rm) YL(x'), 7nt beyond which the occupation probabilities are those of a randomly occupied lattice. If O is the order of signifi- cant neighborhood we have for the matrix elements of Gss' Gº-4T X Diº Cºlº. (14) L!' Here CLL,L, are related to the Clebsch–Gordon coeffi- cients and (15) 3. Potentials and Evaluation of the t Matrix The type of crystal potential which has been found to be most successful in describing the band structures of noble and transition metals is obtained from a super- position of atomic potentials on neighboring sites [7]. This is the Mattheiss prescription [8]. It seems reasonable therefore to construct the muffin-tin poten- tials for the constituents of the alloy, along the lines of the Mattheiss prescription. The difference from the procedure for pure metals is that the overlap contribu- tion from the yth neighboring Cu (or Zn) atom has to be multiplied by the probability of its occurrence. This probability may be determined from a knowledge of the short-range-order parameters [9]. In order to facilitate the calculation of the t matrix, Soven [1] suggested the replacement of the muffin-tin potentials by 6-function potentials of the form (16) råt where gi = Gi(rmſ, rm) is theilth component|in the 8 Il- gular momentum representation of G(x–x). We then arrive at the following simple expression for the matrix elements in eq (10) of t tl (p. q) = tiji (print).j (qrmt) g (20) The eqs (17-20) completely define Tº'(k) in terms of Gºs' and the logarithmic derivatives of the radial functions at the muffin-tin radius. The spectral density of states is then obtained from | p(*, *) -- (E-ſººn Im S. Tº (k). S = 1, 2 s' = 1, 2 (21) Soven [1] has shown that the use of energy-dependent model potentials necessitates the use of the correction dviſ E) dE factor || – X. Ins with eq (12). Our calcula- S= 1 3 *- 512 tions showed no significant difference in the peak posi- tions due to this term. However, all the results pre- sented or discussed in the following are for p(E, k) cal- culated with the correction term included in eq (12). 4. Application to £3-Brass Walker and Keating [5] found that it was not possi- ble to assign unique values to the short-range-order parameters in 8-brass, because of the long-range nature of the short-range order. But the short-range-order parameters could be fitted to a Zernike type expression |o (r)|=0.540 exp (-0.400r')/r', where r =2nla and a is the lattice constant. We have used this expres- sion to calculate the short-range-order parameters employed in the calculation of the muffin-tin potentials and Gº'. Although the disordered 3-phase is found for a range of zinc concentrations in the vicinity of 50-50 stoichiometry, we have chosen the concentrations of the Cu and Zn atoms to be equal. The relevant parame- ters regarding the calculation are shown in table 1. TABLE 1. Parameters for 8-brass calculation Lattice parameter, a = 5.6521 a.u. Radius of inscribed sphere = 2.4472 a.u. Radius of muffin-tin sphere for both copper and zinc, rm = 2.4148 a.u. Constant part of muffin-tin poten- tial Uc = –0.9152 Ry, Order of significant neighborhood, or = 10 We have carried out numerical calculations for the points T, H., P and N to determine what Soven calls the reduced spectral density p(E, k) = X. p(E, k + K). K Where K is a reciprocal-lattice vector and k is confined to the first Brillouin zone. The curves for the reduced spectral density p(E, k) plotted against E for some of the states are displayed in figures 1 to 3 and the peak positions in p(E , k) for all states at the symmetry points T. H., P and N are tabulated in table 2. 5. Discussion In this section we compare our results with experi- mental data for 8-brass. In the case of disordered al- loys, electronic states are probed rather indirectly, through analysis of the optical measurements, positron 10 |- w 8F ſaE ſ2 6H 4 – E $2 < 2 - > OC ^ Uſ) 92 g 1 0 16H § 8F (\] c; 6 H > -> *(E)=THWH) E— —l One final comment may be interesting. One often likes to contemplate the possibility that, while V is so large that the Born approximation is unjustified, T may be still quite small. This is a rather generalized expres- sion of the “pseudopotential” idea. Even if this were true, we would expect such fortuitous cancellation of large terms to occur only for a limited range of values of V. Therefore, reasoning from formula (7), inclusion of “excluded” terms appears to be especially dan- gerous in such a situation. Appendix Edwards and Beeby [2] evaluate 6p1) by first com- puting the quantity ôp (k; E) = – Im Gä(k; E)Tº (E) They urge that, except on a set of measure zero, one has ôp (k; E) =–4 G}(k; E) Im Tº (E) T However, the contribution from the set of measure zero is proportional to a delta function, and their procedure leads to an erroneous result. From formula (12) we see l våp (E) T Im Tºk (E) = 1 + trºv; – 27twoE which is independent of k. Interpreting the square of Go as real, we have dºk G} k; E dº k = | Hº-- | #( ) D (E – ek)* T and therefore Tvåp(E) 1 + Tºvá–2TwoF ôpbeeby (E) = This disagrees with the result (13) above. In particular, it has the wrong analytic behavior; it misses the 517 threshold singularity. It is unhappy to have to complain that a nondivergent answer is wrong, however! We think that this error is actually related to a famous error in the application of the multiple scattering series to nuclear many-body theory [5,7]. The result we have labelled Öppeal, is not compatible with the Friedel sum rule, whereas (13) above is. References [1] Anderson and MacMillan, in Theory of Magnetism in Transition Metals, Proceedings of International School of Physics “Enrico Fermi,” Course XXXVII. [2] Beeby and Edwards, Proc. Roy. Soc. (London) A247, 395 (1962). Beeby, in Lectures on the Many-Body Problem, Vol. 2, Edited by Cainiello (Academic Press, 1964). [3] Soven, P., Phys. Rev. 156,809 (1967). [4] Velick W. Kirkpatrick and Ehrenreich, Phys. Rev. 175, 747 (1968). [5] (, oldberger and Watson, Collision Theory, (Wiley, 1964). [6] Slater and Koster, Phys. Rev. 96, 1208 (1954). [7] Reisenfeld and Watson, Phys. Rev. 104,492 (1956; DeWitt, B., Phys. Rev. 103, 1565 (1956): Fukuda and Newton, Phys. Rev. 103, 1558 (1956). 518 Discussion on “On the Terms Excluded in the Multiple-Scattering Series” by R. M. More (University of Pittsburgh) J. L. Beeby (Atomic Energy Res. Establishment): The shifted from free electron form. The remarks of More expression for the density of states derived by Beeby in his appendix concern a single scatterer problem and and Edwards is specifically for a dense array of scat- hence the criticism of the Beeby and Edwards’ paper is terers in which the band structure is not markedly incorrect. 519 On Non-Localization qt the Centre of a Disordered Bound Band F. Brouers” H. H. Wills Physics Laboratory, Royal Fort, Bristol It is shown for a three dimensional model of tightly bound electrons with cellular disorder that the electronic states at the middle of a continuous band cannot be strictly localized. This conclusion is just the opposite of what has often been suggested. Nevertheless, an approximate calculation of the electron localization “life-time” suggests that with increasing disorder the localized character of the electron states become more and more pronounced. It is argued that in such a system there is no sharp transition between localized and non-localized regions of the energy spectrum. Key words: Cellular disorder; disordered systems; electronic density of states; localization life- time; quasi-localized state; tight-binding. 1. Introduction In a number of papers (for references see the review papers of Mott [1] 1967 and Halperin [2] 1967), it has been suggested that under certain circumstances the solutions of the Schröedinger equation are localized for an electron moving in a non-periodic field. If the Fermi level lies in a region of the density of states where the states are localized, the do conductivity is supposed to vanish at T=0. This result has application in the theory of liquids, amorphous semiconductors, impurity bands, alloys, etc. This assumption is generally discussed in terms of the wavefunction of the “disordered electron,” a con- cept which is questionable when one is concerned with macroscopic quantities such as the density of states, the electrical conductivity or the optical properties which have to be averaged over all possible values of a random field. If in doped and compensated semiconductors, one can admit intuitively that the electrons can be localized at low impurity concentrations, in the case of liquid metals, amorphous and liquid semiconductors and dis- ordered alloys, the statement that some regions of the continuous density of states correspond to localized states is much less obvious. Several authors (Taylor [3] 1965, Bonch-Bruevitch [4] 1968) have questioned the validity of the various ar- *Royal Society E. P. Research Fellow 1968-1969. Permanent address: The lnstitute of Physics of the University of Liège (Belgium). guments and concepts used in the theory of electronic states in disordered systems. This paper is a contribu- tion to this discussion. Ten years ago, Anderson [5] (1958) applied to the discussion of impurity conduction a formalism set up to investigate the absence of diffusion when the atomic potentials vary randomly. So far this model is the only one which can be treated qualitatively in three dimen- sions. Anderson's theorem is that at sufficiently low densities transport does not take place and that the exact wavefunctions are localized in a small region of Space. These conclusions have been extended to the case where there is a random potential on each site of a regu- lar lattice (cellular disorder). This model is supposed to yield some insight into the behaviour of electrons in a tightly bound band of an amorphous system or a liquid. In that case the generalization of Anderson’s theorem leads to the following statement: If we consider a uniform distribution with width W, the whole spectrum becomes localized when the energy of the individual atomic states varies randomly over a range which is somewhat greater than the width of the band that would be produced by their overlap. Ziman [6] (1969) has written a clear and simplified version of Anderson’s arguments and formalism ap- plied to this problem in a critical review of various ap- proaches used in this field. We refer to this paper and to the references quoted in it for a discussion of the physical interest and limitation of this model. 521 On the other hand, very recently Lloyd [7] (1969) has shown that if we choose for the random electrostatic potential a Lorentzian distribution, this model can be solved exactly and a theorem is derived proving that lo- calization is never possible. Lloyd was chiefly concerned with the discussion of localization in the tail of the density of states. In this paper, attention is concentrated upon the situation con- sidered in the paper of Ziman, states at the centre of the band and a bounded flat random distribution. Our conclusion is that the Anderson theorem which must be valid at sufficiently low concentration of impu- rities cannot be generalized to the continuous disor- dered tightly bound band of an amorphous or liquid semiconductor. It is shown, however, that when the width of the distribution increases, the electron gets more and more localized but no sharp transition between localized and non-localized states appears in the theory. 2. The Model We use the model of cellular disorder in a tightly bound band. It is assumed that in a three dimensional regular lattice, each site & is occupied by an atom with a single tightly bound state |^). The energy eigenvalue e is supposed to be a stochastic variable with a proba- bility distribution P(e) characterized by a width W. The atomic states interact through a potential W22. This in- teraction may itself be a statistical variable but here is supposed constant and equal to V when Č and Č' are nearest neighbors and zero otherwise. Coulomb and statistical correlations, though probably important in disordered systems, are ignored. The Hamiltonian can be written in the base of the set of localized atomic orbitals |^) (single site representa- tion) (1) 4 & / 22, “A “2. H=H.-H...->e a'a +X V, a'a AA' where a, and aſ are respectively the annihilation and creation operator for an electron localized at site 6 and obey the usual anticommutation rules. In the ordered system, all energies e can be put equal to zero and the first term drops out, the Hamil- tonian Ho is then a quadratic form and can be diagonal- ized in the reciprocal space: Ho-X. V.A.A. (2) k with A = 2. exp (ik?)a). (3) 2, * Note that aſt) is not the probability amplitude for the physical diffusion process; for this quantity one must take account of the Pauli exclusion principle. The eigenvalues of Ho give rise to a band Vl, = ZW's (k) (4) where Z is the number of next nearest neighbors. The function s(k) is defined in the first Brillouin zone. It de- pends on the symmetry of the crystal and for cubic lat- tices is such that max.s(k) = 1 and min.s(k)= –1 (Mott and Jones [8] 1936). The width of the ordered band is thus B = 2/V. In the disordered system described by H, because of the lack of translation invariance, there is no dispersion relation similar to (4). If we are interested in the energy spectrum and more precisely in the density of states, the averages over Green functions are made (Matsub- ara and Kaneyoshi [9] 1966, and Ziman 1969). 3. Conditions for Localization Now following Anderson, we ask how fast, if at all, does an electron put onto a site & diffuse away from this site? The probability amplitude 24, (t) is simply defined by & (t) = iſola,(t)a(0)|0) =iG. (t) for t > 0 (5) The state |0) is the true vacuum and we are concerned with non-interacting electrons." The time dependent Green function Ge (t) is the Fourier transform of: G. (t) = ſc. (E)?-iedE (6) where G2(t) is the diagonal part of the Green function in the single site representation. It satisfies the equation ôu, | * –H V. tf (, y 2;g 7 E – el E – el 2. A/ A'z ( ) GA, (E) = The retarded form G+(E) means that we have to add to E a small imaginary part ið in order to fulfill causality requirement. Equation (7) can be solved by successive iterations for the diagonal part G2(E) and one obtains: | 6. (P)-ET, AIE) (8) where E = E + iB2 is a complex variable and the self- energy is given by a Brillouin-Wigner expansion: _ ºn Wee Wee Wºe Wee Were **)->}=#"> Fº ==l (9) The terms in the series represent all possible succes- sive virtual transitions which start out from the site & 522 and propagate throughout the disordered system until the electron returns to the initial state. The condition for the amplitude to be finite when t → co (see condition for non-transport in Van Hove [10] 1957) is that G(E) should have a pole on the real axis. Thus the conditions for having a localized state with energy E are simply: Im A (E1) = 0 (10) and – el-H E – Re A (E1) = 0 (11) If Im A (E1) is different from zero, its value gives the decay rate of the state and we can define a lifetime for the electron state. In the case of an ordered lattice: VII, VI, f :/ 1 / 1 ff w iff A}(E)=X. ll 4-H S ºr , * - - (12) l' + | ſ', ſ” - ==/ is equal to A}(E) = E – (G)(E))- (13) where G/(E) is the Green function of the ordered band. This function can be expanded along the same lines as A"(E). In the k representation it can be written in the form: | l 0ſ F \ = — NY — o'E)-xx. Fºr (14) For a tightly bound band, this function is well described by a model function (Hubbard [11] 1964): G}(E)=2|[E – VE2 – 1] if B/2 is the energy unit. In the disordered case, the quantity A(E) has to be averaged over all the random configurations of the dis- ordered system. The difficulty arises from the stochastic nature of the function A(E). In eq (9) we have to average products of site locators (15) (S.S.S… . . . ) with | The averaged Green function (16) must be considered as a conditional averaged since the energy of state |I) is fixed and equal to ea. The function G(E) is different from the ensemble averaged Green function: y | (º)-(z=zºº) (17) which must be used to calculate the density of states, for instance. It has been shown by Lloyd (1969), that the condi- tional average Green function C/E) can be calculated directly without involving any series expansion if the distribution is a Lorentzian one: | T P(e)===F, (18) This is due to the fact that for such a distribution, the averaged product (S.S., . . . S, ) is exactly equal to (SAE)-(ºil) One can see that all the cumulants (Kubo [12] 1962) ex- cept the first are zero and the average can be per- formed for each S2 independently. Physically this means that there is no correlation between different site locators. For the particular distribution (18), the self-energy is given by: (19) A: (E) Vitº Vul Vu, Viru Vun - F #+ S +++++ . . . = A9 (E+iT) X ###" > *ś A ==l (20) the density of states p’ (E) ==}lm (,00E-Hiſ') ==#F#Im METTFI (21) T T and a- | Głę)=F (22) — e.g-A%(E+ iT) which must be used to calculate (6). This expression is equivalent to eq (39) of Lloyd’s paper, if use is made of relation (13). We can see immediately from (13) and (15) that Im A"(E+iſ)#0. This gives the theorem proved in a more complicated way by Lloyd (1969). For any value of E and T no localized state can occur. 523 In particular, if we are interested in states at the mid- dle of the band, one has Im A" (iT) = A" (iT) = 0 (23) 4. Non-Localization at the Centre of the Band We turn now to the case considered by Anderson (1958): a rectangular distribution with width W. Ziman (1969) reproducing Anderson’s arguments and conclu- sions has assumed in the discussion of the convergence of (9) that: (S,S, . = exp [log (S/. . . S.) . . . S.)] = exp [n(log S,)] This approximation means that all the cumulants ex- cept the first are zero or that the site locators are uncor- related. It is equivalent to assume that: (S, , . . . S, ) = (S, )." (24) (25) The full argument of Anderson’s paper is more com- plicated. Correlations are included in the theory by replacing el in the locators by a renormalized energy but qualitatively Anderson’s conclusion on localization at the centre of the band is not very dependent on this refinement. For a uniform distribution of width W, one has E– W/2 iT ... (W_ ####| # * (* E) (26) (sº))=# log where m(x) is the usual step function. Though in this approximation, the ensemble averaged (G" (E)) is simply G"(1/(S, )), it does not seem possible to find a simple expression for the con- ditionally averaged G" (E) in terms of (S2) because of the restricted summation in (9). But for E=0, i.e., in the middle of the band, the result is simply: (27) and one necessary condition for localization is that A"(iW/T) should be equal to zero. It is the same condition as for a Lorentzian distribu- tion (eq 22) except that we replace T by W/TT. This result is physically meaningful. If we neglect correla- tions, the electronic states at the center of the band are not very dependent on the form of the random distribu- tion of individual site potentials. We conclude that for states at the middle of the band and in the case of statistically independent site loca- tors, a situation described in Ziman’s paper (1969), no state can be strictly localized in Anderson’s sense. 5. Localization Life-Time We can now, using (13) and (15) evaluate the value of the imaginary part of the self-energy at the centre of the band: | Im (A (0)) = A" (iT)= T- -------- 28 (iſ) giviºr, º and the localization “life-time”: T(T) =– h|y (29) with - Ao(iT) "IEEFT) | (30) - ð E E = () and |-(***) |=} | 6 E E = () 2 Vl +Tºſv/l +T2 – T. (31) In the limit T = 0, i.e., in the case of the ordered band, the life-time is two times the inverse of the or- dered band width. If T or (W/T) is increased, the func- tion (28) decreases and tends to zero as T – Co. But for any finite T, no state can be strictly localized. This result is in agreement with Lloyd’s theorem. In figure 1, we have plotted the localization life-time defined by (29) with respect to (or W/TT). For every value, we can give a measure of the electron localiza- tion at the middle of the band. For instance for T = 5 (in B/2 units), the localization corresponds to that of an electron in an ordered band of width B/20. It is also possible to obtain directly a closed formula for the self-energy (9) by using the Matsubara and Kaneyoshi (1966) technique, if we neglect the restric- tion (l', l'', . . . A l) in the summation. In this case, one has: l (Avº)-xx Al. (E) (32) with Al (E) = V}(G.I. (E)) (33) where V, and Gºº are the Fourier transform of the over- lap integral V2, , and the disordered Green function. 524 If we discard correlations between site locators, it can be shown easily that | | * | TVS, (E)) + (S, (e))* Go ( (SAE) ) The expressions corresponding to (28) and (31) are then written as (AM (E)) = (34) Im (Ay(0))=–T(1+2ſ”) +2ſ” VT*+ 1 (35) ). |-(**) Ó E | + T2 — T –2+2"|*H* | *rtr–Mºri V1 + T2 (36) It is interesting to note that this expression almost coincides with (28) for T ~ 0.5. But for T ~ 0.5, the lifetime increases and is infinite for T = 0. In the vicini- ty of T = 0, the present discussion is not quite valid because the decay in the ordered band is an algebraic decay due to a cut in the ordered Green function. 6. Discussion There is a discrepancy between these conclusions and the statement on localization in disordered tightly t bound band reproduced in review papers by Mott (1967) and Ziman (1969). In the discussion of the self-energy expansion con- vergence, Anderson eliminated singular factors in averaging the terms of the series for (A(E)). In this case, to have a localized state Re(AI(E)) must converge and this convergence is a criterion for localization. The same procedure would give a non-continuous disordered density of states and can be justified when the concentration of impurity is so low that there is no banding effect (Miller and Abrahams 1960 [13]). But in the model of a disordered tightly bound band when the distribution of random potentials is continuous, the consideration of the singularities in the expansion of (A(E)) when the average is performed is essential to rediscover the exact result of Lloyd in the case of a Lorentzian distribution and to find a continuous density of states (eq 21). It turns out that when the terms of the series are summed, the small imaginary part of the locators gives rise to a finite imaginary contribution for (A). We have shown that the condition Im (A) (0)) = 0 is never fulfilled in the model considered in this paper. Moreover it is possible to define a closed form for Re (A(E)) at the centre of the band. The conclusion of this paper is that it is impossible with conditions (10) and (11) to define regions in a continuous disordered tightly bound band where elec- tronic states are strictly localized. This conclusion is thus in agreement with a general theorem of Bonch- 12 16 20 T ( } ) 70 : 50 T 30 f 4. 8 FIGURE 1. Localization “life-time” (eq 29) as a function of the width of the random distribution of random atomic potentials. The units are B/2 for l’ and (2/B) h for T. 525 Bruevitch (1968) on conductivity stating that the electri- cal conductivity of the macroscopically homogeneous system at zero temperature is non-zero if and only if the Fermi level lies in the region where the density of states is non-zero and continuous. This conclusion is also consistent with the results of Nakai and Flawiter (1968) [14]. These authors show using a quite different approach in a model where the randomness is included in the hopping distances that localization for an impurity band appears when the spectrum becomes discrete and dispersed in a wide region. Nevertheless, as was suggested by a rough calcula- tion of the localized electron lifetime in section 5, when the width W of the random distribution increases, the localized character of the electron state becomes more and more pronounced. This discussion suggests, however, that if in a con- tinuous band some states are more localized than others, it is hard to imagine a sharp transition between them. A relevant theoretical problem would be to try to define this concept of “quasi-localized state” on a more rigorous basis. The lifetime of this quasi-particle should be calculated for all the energy spectrum and compared with other characteristic lifetimes of the system. Without this knowledge, it seems difficult to reach definite conclusions concerning the behavior of the electrical conductivity when the Fermi level varies across the density of states distribution. 7. Acknowledgments I am grateful to Professor Ziman for suggesting this problem. He is acknowledged as well as Drs. B. Berger- sen, D. Herbert, P. Lloyd and K. Thornber for informa- tive and stimulating discussions. 8. References [1] Mott, N. F., Adv. Phys. 16, 49 (1967). [2] Halperin, B. I., Advances in Chemical Physics 13, 123 (1967). [3] Taylor, P. L., Phys. Lett. 18, 13 (1965). [4] Bonch-Bruevitch, W. L., The theory of condensed matter, Inter Atomic Energy, p. 989 (Vienna, 1968). [5] Anderson, P. W., Phys. Rev. 109, 1492 (1958). [6] Ziman, J. M., J. Phys. Chem. 2, 1230 (1969). [7] Lloyd, P., J. Phys. Chem. 2, 1717 (1969). [8] Mott, N. F., and Jones, H., The theory of the properties of metals and alloys (1936). [9] Matsubara, T., and Kaneyoshi, T., Progr. Theor. Phys. 36,695 (1966). Van Hove, L., Physica 23, 441 (1957). Hubbard, J., Proc. Roy. Soc. A281. 401 (1964). Kubo, R., J. Phys. Soc. Japan 17, 1100 (1962). Miller, A., and Abrahams, E., Phys. Rev. 120, 745 (1960). Nakai, S., and Flawiter, W. F., Phys. Lett. 27A, 393 (1968). [10] [11] [12] [13] [14] 526 The Half-Filled Narrow Energy Band * L. G. Caron” and G. Kemeny” Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02:139 The antiferromagnetic and paramagnetic states of a half-filled narrow energy band are investigated using a t-matrix approach. This method is justified despite the high particle density in the system. A phase diagram including the Mott state is given. Employing the gap of the antiferromagnetic insulator as a variational parameter, it is shown that the increase of the band width potential energy ratio leads to a first order phase transition into the paramagnetic metal state, nearly where Mott has estimated it to OCCUIT. Key words: Antiferromagnetic insulator; first order phase transition; half-filled narrow energy band; Mott state; paramagnetic metal; t-matrix. 1. Introduction The Mott insulator [1] on the Hubbard model [2] has been the subject of investigation in two papers by the authors [3,4]. In the paper presented at the first In- ternational Conference on the Metal-Nonmetal Transi- tion [4] some of this research was reviewed. Parts of this paper and the discussion that followed were devoted to the paramagnetic and antiferromagnetic ranges of the half-filled narrow band, and to the com- petition of the long range antiferromagnetic order with the short range Mott order. We have outlined the physi- cal motivation and mathematical methods we felt were necessary to treat these problems. In this paper we present the solutions of these problems. The results in- dicate that the problem is not as complex as we have anticipated and the self-consistent pair correlation method is unnecessary except in the Mott insulator range. Other than that our physical expectations seem to be justified by the results. Apart from the Mott insulator we consider two possi- ble physical states of the half-filled narrow energy band, with the Hubbard Hamiltonian [3] H — X. Tijci,c], –– J X. Ili + Ili – . ijor i (1.1) These are the paramagnetic and the antiferromagnetic states. In the paramagnetic state the electrons scatter *Supported in part by the Office of Naval Research, Contract No. 78721 and by the Canadian National Research Council. **Permanent address; Department of Physics, University of Sherbrooke, Quebec. ***Permanent address; Department of Metallurgy, Mechanics and Materials Science, Michigan State University, East Lansing, Michigan 48823, on each other. This will be handled by a t-matrix ap- proximation. As we have argued in [4] an expansion in powers of the t-matrix could handle the scattering problems. Our results indicate that already the first term should be a good approximation. If the interaction is strong enough the antiferromagnetic state becomes stable. Part of the effect of the interaction is the establishment of the long range order. The rest of the effect is taken care of by the same t-matrix calculation but now the zero order wave functions are antifer- romagnetic rather than paramagnetic. The stability of the paramagnetic versus antifer- romagnetic states is decided by the minimization of the total energy in the t-matrix approximation with the an- tiferromagnetic gap as a variational parameter. When the gap so determined turns out to be zero the paramag- netic state is stable. A semiquantitative argument is used for the competition of the antiferromagnetic and Mott insulator states near the atomic limit. We shall use Martin-Schwinger [5] Green’s functions in a manner similar to Reynolds and Puff [6]. The general formulas are evaluated for a simple cubic lattice in the tight-binding approximation. 2. Paramagnetic State: Hartree Approximation In the limit of small intra-atomic interaction, J -> 0, one would expect the correlation effects to be small. One might even be tempted to neglect them complete- ly. The energy would then be approximated by first- order stationary perturbation theory on the noninteract- 527 ing electron system. This is the Hartree approximation. It is best worked out by first Fourier transforming the ch, - N-1/2 Seikº. °io. (2.1) k one finds H =X, X e(k)n, + JN- X. k or kikgkºk. where e(k) = S Teikº. (2.3) I The unperturbed ground state of the half-filled band is obtained by filling all k states for which e(k) < 0. This makes up the Fermi sea whose surface is defined by e(k)=0. The Hartree energy is then simply X. h., "... kikº “kr (H) = 2 S e(k) + JN- kº kr. = 2 X. e (k) + NJ/4. (2.4) k's ki. 3. Paramagnetic State: Redction Matrix Approximation We wish to study the behavior of the Hubbard model as a function of the strength of the interaction. It is ob- vious that correlation effects will be of importance for larger values of J. One might try adding a few more terms in the perturbation series expansion of the ener- gy. The danger is that this series diverges over a paramagnetic ground state as J becomes large. So, a priori it seems to be necessary to get rid of this pending divergence by summing up judiciously chosen terms to all orders. It is known from the Breuckner-Goldstone work [7] that those terms which involve only two elec- trons — the reaction matrix expansion — are easily summed. This means the binary collision problem is solved exactly and this in turn leads to a well behaved treatment of the short-range intra-atomic interaction. ô(k –H kg - k3 *s ki) c Hubbard Hamiltonian in the reciprocal space of the crystal structure. This diagonalizes the motional part of the Hamiltonian eq (1.1). With + ſ^+ kā- C. °k. & *k, (2.2) 8,O----O 8, *(); 30. + k, FIGURE 1. Reaction matrix correction to the noninteracting energy in the paramagnetic system. --- K. Kº *(): 30, + k, tº gº k, “. tº gº º ºs º- K. K. FIGURE 2, Reaction matrix correction to the Hartree energy in the paramagnetic system and to the Matsubara energy in the anti- ferromagnetic system. Using time dependent perturbation theory and the adiabatic approximation it is possible to deduce a dia- gram representation equipotent to the perturbation se- ries expansion [7]. In this diagram representation, the reaction matrix correction to the noninteracting elec- tron energy is given in figure 1. In these processes, any two opposite-spin electrons of the Fermi sea ki,k2 undergo all possible scatterings on one another. The ex- clusion principle is partly taken into account by restricting the scattered electrons above the Fermi sea. If we wish to compare the final energy to the Hartree energy we are left with figure 2 where the first term is missing because it is already included in the Hartree energy. Following Day [.7], we write down the wavevector for two such electrons in momentum space as: . |V (kikº) > = |d (k) d (kg) > - Q/ev|V (kikº) > = |d (k) d (kg) k{k, P ke |d (k) d(k) = – «d (k)(b (kg)|vld (k)(b(k2)2]. (3.2) Translating this into configuration space (b. (R) (b. (R2) "...(tº)=a º (º), X. Tin-ºn-Tºm × S dº (R') dº (R)p(R,R)\,... (R,R) (3.3) R{R. and AE = S. [Nº. (R,R)p(R,R)d, (R)0. (R.) – d. (R)d. (R.) × v(R,R)d, (R)d. (R.)]. (3,4) kık, sº k, R R2 Since e(k)= — e(TT-E k), we can transform the summa- v (R,R) = Jöss. (3.5) tion above the Fermi sea to one below, by changing the and sign of the corresponding energies. Substituting dº (R) = N-11-eikº, (3.6) into eq (3.3) we find \ſſ R. R., ) = N-lei (k, R + kg R2) X W–1e iſk; R, + k, R2) ..º-Yº" " " ...?' - Hº-Hº-Hºm × N-1 X. e-i(k+k) Rºx Vºl. (R,R) (3.7) R; from which \lſ R R ei(k, + kg) R. ºr (R,R) 1–JN- X. 6(ki + k2 - k - k.) (3.8) º, ſe(k) +e (kg) + e(k) + e(kz)] is obtained. The reaction matrix energy correction is then J J AE = f f - - .*. |-IN- X. 6(ki + k2 - k – k.) | 4. (3.9) kºk, ‘kº [e (kſ) + e (kg) + e(k) + e (kg)] the Brillouin zone. In this limit the paramagnetic state is a bad trial state on which to build the perturbation ex- pansion since it confines the electrons below the Fermi level. A ferromagnetic or antiferromagnetic trial state would be better suited to a perturbation treatment. It is known the half-filled band in the Hubbard model can- not be ferromagnetic [8,9]. We then look at the antifer- romagnetic state in the large J limit. A band theory of antiferromagnetism was discussed by Matsubara and 417–156 O - 71 – 35 529 Yokota [10]. They reduced the inter-electron interac- tion to a single particle spin symmetry breaking poten- tial X. X. Agni, i or in which Ajo is self-consistently defined as the Hartree potential seen by an electron Aid - Jni–0. (4 .l.) This potential has one value on one sublattice and another one on the other. The Hubbard Hamiltonian in the Matsubara approximation is then HM – X. X. Tuc,c), + X. X. Aignia & (4.2) ij or i or Transforming into momentum coordinates HM -- Ao. + > X. (eſk) –H %) Who. –H > X. 5 “...ºra, + ‘….'...) (4.3) where J_ e Aoi. T5- + A for up spins Aq = J (4.4) A0–- 5T — A for down spins each k state is seen to be coupled to a k-H TT state. The eigenstates of this Hamiltonian are going to be linear superpositions of states below and above the Fermi level of the noninteracting system. The electrons will then be spread out throughout the Brillouin zone as wished. The Matsubara Hamiltonian is diagonalized by the following canonical transformation: el, Albiº, = Bibolº, ekiro FFB, bika + Akbok, (4.5) the upper sign being for up spins and the lower for down spins, and where *-NC-º). *-Nſº Since there are two solutions for each spin or, we labeled them with the same value of k chosen in the noninteracting Fermi sea, differentiating them with a second quantum number taking on two possible values (4.6) 1 and 2 for the lower and higher energy solutions, respectively. The eigenfunctions and eigenvalues are: duº, (R)=N-1/*e” (A. H. B.e"") (4.7) E(k)=-VA, FIEF- (4.8) dº (R)=N-ºe"(B.E.A.e") (4.9) E2(k) = VA, FIRE tº (4.10) In the half filled band case, the lower energy band is filled and the higher energy one empty. The self-consistency condition is then J +A+;=J X |dº (0)| k < kº (4.11) OT | JA 2. VA* + e(k)” There are two possible solutions to this self-consistency equation. The first one is trivial: A = 0. A = (4.12) (4.13) This is the paramagnetic Hartree solution already stu- died. The second one is the Matsubara condition for an antiferromagnet: J S = Hº-1 k < kp VA” + e(k)? (4.14) The energy of the Matsubara state is (H) = X. XX. Tºdi, (R)-bill, (R) k < kº, O iſ +J X. X | bit (R)|*x|diº (R)|*. k < kº i (4.15) In the tight binding approximation this becomes (H)=2 > (4:-Boeſk)+/ > (1–44;b;). k < kº k < kº (4.16) 5. Antiferromagnetic State: Redction Matrix Approximation The Matsubara approximation is surely very decent in the J-P Co limit. For not so large J's one could again 530 attempt a perturbation series expansion, this time using the Matsubara states as single electron eigenstates. This series would not diverge since opposite spin elec- trons already undergo minimal interaction in the Mat- subara approximation. But in order to treat the antifer- romagnetic solution on equal footing with the paramag- netic one, we will again perform a reaction matrix anal- ysis. We add an extra degree of freedom to the approxi- mation by leaving the degree of polarization of the wavefunctions as a parameter with which to minimize the total energy. The logic behind such an expansion on long-range or- dered wavefunctions can be understood using ther- modynamic Green’s function theory. This requires the addition of a chemical potential term J T5 X. X. Plior (#-h.) G!" (lo, 1.) = 6(1-1') ôt (5.4) where hio Gº" (lo, 1.) = (A.-:) Gº" (lo, 1.) + X. Tj6" (1 + j : 1.). j Following Martin and Schwinger [5] we define . () ( #-h.)=c(...)" (5.6) z to the Hamiltonian in order to be consistent with the grand canonical ensemble involved in Green’s func- tions. Let us also add and subtract a Matsubara poten- tial. We can then subdivide the Hamiltonian into a one- electron part and an interacting part: / J H -º-;xxº-h-h (5.1) H1 = X. Tijc. Co-F X. X. (A. -:) /lior (5.2) ij i Or Hi-JX. mi., ni--XX. Along. i i Or (5.3) The one-electron propagator’s time evolution is then controlled by Hi in which the Matsubara part of the in- teraction is already included. Its equation of motion is: (5.5) The equation of motion for the two particle Green’s function is then G"(11)-G"(2 )-'{G. (1.2-, 1/2") – Gº (11: 1.) G|"(2, 2’)}-Gº (11: 1.) [A, G. (2 : 2") + i.J.G. (2 2.; 2/2+)] - Gº" (2–; 2') × [A.G. (11; 1) + i./G2(1,1-, 1/1+)|}=A. A. G. (11.2 : 1/2") + i \,, JG (112–21; 12'21) + iº JG (l, 1-2-, 1/1+2)-J*G, (1, 1–2–21; 1,112'21) + iſ G. (11.2 :12')6(1–2). (5.7) If a Hartree approximation is made i.JG 2 (2–21; 2'2+) = −A2. G (2–; 2ſ.) (5.8) i.JG 2 (11 1 ; 1, 1 t) = - A11C (11; 1.) (5.9) i.JG3 (112–21; 112'21) = - Ag G2 (11.2 : 1.2 ) (5.10) i.JG3 (11 1–2–; 111t 2') = - All G2 (11.2 : 1' 2".) (5.11) Jº G. (11 1–2–21; 1,112'21)=J*Gs (11.2 : 1/2") × G. (1–21; 1121) (5.12) == G2 (112–; 1' 2".) A 1A2– (5.13) we get G"(11) - "G"(2–)-'IG2 (112–; 1.2%) — Gº" (11; 11)G"(2–; 2')] = i.J.G. (11.2-, 1/2.)6(1–2). (5.14) 531 k K. K. k "\ }k & Jº * >2 FIGURE 3. Power series expansion in J. The first hole-hole scattering diagram is of third order in J. This is similar to the Q approximation on a long-range ordered state. Air now serves as a variational parameter with which to minimize the energy. The reaction matrix is that part of the Q matrix which includes only electron-electron interaction. Since it involves lower order terms to the energy correction than any hole-hole scattering—as shown in figure 3– this equation also serves as justification for the reaction matrix approxi- mation. It is interesting to note that neglecting the right hand side of this equation one gets back the Matsubara result since it is for Air =Jni—g that the energy is minimized. In the diagram representation, the reaction matrix correction to the Matsubara energy would be as given in figure 2. The first diagram of figure 1 is here again missing since it is already included in the Matsubara energy. This is in analogy with the paramagnetic case. The difference is in the electron propagators which can here be adjusted variationally to yield the minimum energy. As in the paramagnetic case, we write down a wavefunction for any two opposite-spin electrons: ba, (R ) bel. (R2) (R2)-J k{k, sº ke V. (R. R2) = (bu. (R) dº [E2(k’) + Ey (k.) — E(k) – E (kg)] XX biº (R) biº (R)\!'... (RR). (5.15) R The energy correction to the Matsubara energy as given in eq (4.16) with Air as a variational parameter is: AE=J X. X. [Vºl. (RR)(b), (R) dº (R)-(bº, (R) dº (R) × b ...(R)(b), (R)]. k i kº < Rp. R (5.16) Since the wavefunction amplitude must be antiferromagnetically modulated, we can write Vºl. (R. R2) = \'nº' (R. –R2) –H eit RV.1.1, (R — R2) (5.17) *} Substituting into eq (5.15) the expressions for the wavefunctions in eqs (4.7) and (4.9) it can be deduced that [A.A. – Bl, Bill \pſ , , , (0) = K T IC 2 K ITT K2 (5.18) ikºº ) [l T. P. lot Q.....T. Q.-, [A.B. - B.A..] \ſſ, , (0) = | IV 2 I tº 2 5.19 2k.tº' ) |lt Q, lot Pºrt Pºk.-- | ( ) with •) , y - ' — / - – A A w 2 Pºll = N-'J X. * * * k k!) [B], Big ki 1. (5.20) - " " - k, k, ‘ ke |Eº (k) + E2(k.) – E1 (k1) – Ei) k2)] •) , , – ! – k.) [A, B, -B, A, lº Q. =N-J X. **** k k!) [A, B, - B,4,] (5.21) | IV. 2 tº : k . k. ‘ k, [E2(k’) + Ey (k.) – E (k) – E (k2)] The energy correction will then become: AE=J X. kikº - ke {\'nº (0) [A.A. - B.B.) –H V, i (0) [A, B, B.A.) - [A.A. - B, B, lº - [A, B, - B.A..]*} (5.22) 532 6. Density of States The electron transport properties are governed by the density of states at the Fermi level. It would surely p (E) = T-' lim Im [G : (11: 1 - |v)]+, , = E e — () + where the zero of energy is at the chemical potential. Because of the energy gap occurring in the antifer- romagnetic state, its density of states at the chemical potential will surely be zero. We then need only concen- (, ) (11) - (, ; (11: 1) = 6(1-1') — i.JG 2 (1, 1 ; l', li) – J/20 (11; l') from which we deduce G, (11; l') = G"(11; 1.) – iſ ſqi (,"(11; II)[G, (1,1-, 1.1L) – G. (T1: 1.) G, (T_; T.I.)]. The knowledge of the two-particle Green’s function is required. This we get from the Q approximation. Since G2 (1.2-, 1.21) = G, (11; 1) G. (2–; 2) + iſ ſail G}(11; 11)G"(2 : T ) Go (IIT : 1,2'). We define the Q matrix by the following equation: JGs (l, 1 ; 1/2") = ſal() (1:1)G, (11; 11)G, (T_; 2') which with eq (6.4) leads to Q(l; l')=Jö(1–1') + i.J.ſ diCº (11; T.) Gº (1–5 T-)0(1; l') be of interest to calculate this quantity for the Hubbard model within our reaction matrix approximation. The electron density of states can be related to the spectral representation of the single-electron Green’s function (6.1) – i.e. trate on the paramagnetic case. The equation of motion for the one-particle Green’s function in the paramag- netic state using the Hamiltonian in eqs (5.1), (5.2), and (5.3) with A11= A2 =J/2 is: (6.2) (6.3) the density of states requires a better approximation than the energy we use the more accurate form [5]: (6.4) (6.5) (6.6) If we now Fourier transform the space and time variables into the momentum and frequency domain, with Odd (, ; (11: 1.) = 1/NT X. X. eik, (R-R)e-i(7”/ſ)(ſ-t')C (kv) (6.7) k v and G}(kv) =–– (6.8) 1 Tv/T – e (k) º and also () (1: 1) = 1/NT X. X. eik (R-R) e-i(tw/1)(t-t') () (kv). (6.9) P k we get the following equation for the Q matrix º Odd 6(k - k1 – k2 Q(kv)=J+i(J/NT) X. X. ( 1 ) Q (kv). (6.10) v 1 kikº [Twiſt – e (k1)][77 (v — v1)/T – e (kg)] 533 But then, from the Poisson sum rule, the summation over v1 yields 6(k - k – kg) | } | () (kv) = J– JN- — — 6.11 (kv) * |Tv/T-e (k) – e (kg)] l l -- e Belk) 1 + e- Beſk.) (6.11) X () (kv) At zero temperature, 3–2 Co, and the term in brackets matrix approximation where only electrons interact, is either +1 or – I depending on whether k and k2 both this implies we must restrict the scattered electrons are below or above the Fermi surface, respectively. above the Fermi sea. This brings us to a reaction or t- Since we are restricting ourselves to the reaction matrix whose equation is 6(k-ki - kg) T(kv) = J--JN- .* ... [Tut-e (k)-e (k2)] T(kv) (6.12) that is - J T(kv) = | l–J/N X. ô (k - k1 – kg) } (6.13) ... [Tut-e(k)-e(k)] We can now substitute this value for the t-matrix into eq (6.3) for G, and get: Odd G1(kv)= C(kv) – iſNT X. X. eit"/"I Gº (kv)G, (kv) ly k' × T(k + k', v-H v') G1 (k'u') — (J/2) Gi (kv) Gº (ku). (6.14) In order to perform the summation over v', we must transform the t-matrix into its spectral representation: S1 (ka)) T(k)=S(k)+| do (Tv/T – oy (6.15) where So(k)=J (6.16) S1 (kv) = 1/T lim Im T(ku)] mºr-o-is (6.17) J2N-1 X. 6(k-ki - k3)6(a) – e (k1) – e (k2)) kıkg - ke - —1 ô(k - k3 – ka) 2 2 — I - ºg sy - - - | JNº. 2, ***Eºn) + T |N .x, aſk-k-koo-sº eſko) || (6.18) 534 It is seen from eq (6.18) the t-matrix has a branch cut in the positive real axis of the complex frequency plane. If we now perform the summation over v' using the spectral representation and approximating Gr(k", v') by G10(k', p'): Odd I/NT X. X. T(k+ k", v-- v') Gº (k'v')e(7")" k u" – Odd ºf: S1 (k+ k", a)) ei(Tv'ſſ)0+ | l/NT 2. > | || da) [T(v-H v')/T-6)] [Tv'ſ T-e(k’)] (6.19) — ... I } { * 2C S1 (k+ k", a ) = i J/2 + iſ W 2. da) |Tv/T – a + e (k’)] (6.20) =-i/2+ iſN X. [T(k+ k", v")],…, - Tv/T-Fe(k’) (6.21) we then get k' < kº JG}(kv) G1(kv) w – ('0 G1(kv) = Gº (kv) "..?. 1 — J 6(k+ k" – k1 – kg) ...” kE [Tv/T + e (k’) – e (k1) – e (k2)] –JG 1 (kv) Gº (ku) (6.22) Before pursuing the calculation further, one notices the energy spectrum for G, is not symmetric with respect to electrons and holes, i.e. G1(k,w) # – GI(Tſ — k, - v). This was to be expected since only electron- electron pair scattering has been considered. One would have to include hole-hole scattering to restore electron-hole symmetry. Since the hole-hole contribu- tion to the energy occurs only in higher order terms as Q(kv)=J–JN- S. kikº sº ke shown in figure 3, and since convergence of the per- turbation expansion is rather rapid in the paramagnetic domain, we can safely restore electron-hole sym- metry without upsetting the reaction-matrix approxi- mation. We then define a hole reaction matrix which is the result of all possible sequential hole-hole scattering within the Fermi sea. From eq (6.11) its Fourier transform is: ô (k - k – kg) [Tv/T-e (k) – e (kg)] (6.23) Going through a similar analysis as with the electron reaction matrix we would arrive at the final expression for G1: J G1(kv) = Tv/T – e (k) .2. | 1–JN- S. kikº P ke –– k P kr |-JNº. 6(k+ k" – k1 – kg) (Tv/T + e (k’) – e (k1) – e (k2)) J — 1 ô(k+ k" – k – kg) | | (6.24) *** *** (Tv/T + e (k’) – e (kr) – e (kg) 535 From eqs (6.1) and (6.24) the density of states at the Fermi level can be shown to be: ô(ki + k2 - k3 – k+) ºf . (e.(kg) – e (kg) – e (k))* p (EP) - X. X. 6(E) {1 + Jº ki i kººkº | - J ô(ki –H k2 - k; - kg) | käki-kº (e (kg) – e (k;) - e (kg)) (6(k1) + k2 - k3 – k+) *... (e (kg) – e (kg) – e (k))* + Jº lº, (e(k2) 3) – e (k)) 5) – 1 (6.25) lººkſ: | + J ô (k1 + k2 - k3 – kg) | t kškºkº (e(kg) - e (kā) --- e (kg)) where E; - e (k1) — J Q € ( K1 2. | --- ſ ô(k –H k2 - k3 - ki) | * käki-kº (E; –H e (kg) - e (kº ) - e(k)) J - (, –H kg - ke | –H J ô (k –H k2 - k3 - ka) | - () (6.26) * , º, (E = e(kz) = e(k) = e(k)) 7. Results authors [10-12] generate only the two latter curves In this section we present the results of the above quantitative analysis together with some semiquantita- tive and qualitative arguments to round out our con- siderations. Figure 4 shows the energy of the four states in question as a function of c which is defined by c = 4T/J (7.1) In these units the width of the tight binding band for a simple cubic lattice is 12T. We see that the t-matrix energy is always lower than the corresponding Hartree and Matsubara energies. The calculations of previous ;I-I-I- O H-T-T— -1. H- #: se | . . -. i". ! : | . 1O H . º ! . \ . ! : E || : *=== \ NT |\ . \ 5 H- \ - \ . -1 \ . \ ". \, ', \ ". \, ... | l 1. | I l | - 1.5 | | l O 5 .5 1 1. 1.5 which never intersect. The antiferromagnetic state al- ways has lower energy than the paramagnetic one in this approximation. The t-matrix approximation gives a different result. The corresponding energy curves in- tersect at about c= 1.4. For lesser c, i.e. nearer the atomic limit, the antiferromagnetic state has lower energy. For larger c, i.e. nearer the band limit, the paramagnetic state has lower energy. We see then that the short range correlations introduced by the t-matrix not only lower the energies of the respective zero order states but also change the predictions for the ground state of the system. Figure 5 exhibits the energy gap in the Matsubara and the variational FIGURE 4. Energy per electron in units of T in the . . . para- magnetic Hartree paramagnetic t-matrix anti- ferromagnetic Matsubara and antiferromagnetic variation t- matrix states. 1 ‘. “Sº . ~. ~ . | | s. *s '. S. ". S. ". ^. 2A '. S. --- '. S. º ~ J '. S. '. S. e ^ 5 H- & S. •= S. >. O | | O .5 1 1.5 FIGURE 5. Energy gap 2A in units of J in the – – antiferro- magnetic Matsubara antiferromagnetic variational t- matrix and . . . . Mott insulator states. 536 t-matrix approximations for the antiferromagnetic states. The variational t-matrix gap is smaller than the Matsubara gap. Apparently it is energetically worthwhile to exchange some long range correlations for short range correlations. We see that at c = 1.4 the antiferromagnetic t-matrix gap and the sublattice mag- netization, which are related by Aſ J = n – 1/2 (7.2) and where A is half the gap energy, collapse into the paramagnetic t-matrix state. Since the order changes discontinuously, this is a first order phase transition. Thus the density of states in the gap is suddenly raised to a finite value, which is shown in figure 6. This con- clusion agrees with that of Edwards [11], who found that the density of states across an energy gap lifts at once if the long range order fails. The phase diagram is shown in figure 7. We have now two solutions in the range c < 0.42. They are the Mott insulator and the long range antiferromagnetic solutions. The question is which one is more stable or does one possibly have to consider some combination of these two states. For the latter case the system would possess both short and long range order. Our considerations are only qual- itative because we do not produce the combined state, neither do we know how to calculate the Mott in- sulator at elevated temperatures. It is known that the long range antiferromagnetic order and the energy gap decrease with increasing temperature and they both disappear at a second order phase transition into the paramagnetic state [10]. But this can happen only if the paramagnetic state is stable. For c < 0.42 probably the Mott insulator is stable as we shall see below. Thus in this range one expects the gap to decrease from its value at zero temperature to the Mott value at the Néel temperature, while the long range order diminishes and the short range order increases. At the Néel tempera- ture the long range order completely disappears. The system is an insulator both below and above the Néel temperature and the phase transition manifests itself only in the disappearance of long range order. Thus one expects a second order phase transition from the mixed insulator state to the Mott insulator state. If the tem- perature decreases to zero in the mixed insulating state the long range order will not become perfect except at the atomic limit c=0. Thus the short range order shall have room to operate and lower the energy. Therefore we expect the phase transition line between the mixed and antiferromagnetic insulators not to cut the c axis somewhere at c < 0.42. It is possible that the axis should be cut at 0.42 < c < 1.4 because we cannot ex- clude the possibility that the antiferromagnetic insula- tor retains some short range order for a range of C values at which the bound states of the Mott insulator do not exist. We have to realize that the short range order in the mixed insulator does not necessarily exist in the form of bound states. Above the Néel tempera- ture the paramagnetic state provides the correct nonin- teracting one particle Green’s functions. If c < 0.42 and the temperature is zero the Mott insulator is stable with respect to the paramagnetic metal. The question is, will the elevated temperature disrupt the Mott state and cause transition to a metal? We do not believe this will happen near the Néel temperature. Since the Mott state requires only short range order the increasing temperature will not affect it as drastically as it affects the antiferromagnetic state. If a given atom is prin- cipally occupied by an up spin electron and down spin hole, the nearest neighbor atoms are most likely occu- pied by pairs of opposite spin. Thus some short range antiferromagnetic order prevails. Increasing tempera- ture can take its toll on this short range order. Note, however, that this is not just a local effect, since each pair is spread over the entire crystal. Thus probably the average binding energy per pair would decrease with == - -, * * * - * * eas - • * - * * ~ * * FIGURE 6. Electronic density of states at the Fermi level for the . paramagnetic Hartree - - antiferromagnetic variational t-matrix states. - - paramagnetic t-matrix and NMOtt SS insulator ^. ^. (l) `s - Q- paramagnetic –5 Ps N metal +-> ^ | ^. Q | Ss (1) mixed > < ^. Cl *~ insulator | Y S S 8– | > < 9 antiferromagnetic | insulator | 1 1– O .5 1. 1.5 FIGURE 7. Phase diagram. Solid lines: first order phase transitions. Broken lines: second order phase transitions. ^ N. 537 increasing temperature. In addition to this, free pairs will also be created at elevated temperatures. Ex- perience with the density of states at the antiferromag- net-paramagnet transition indicates that the Mott insu- lator paramagnetic-paramagnet transition will also be first order as a function of c. Mott's original discussions of this subject matter did include both the bound-pair insulator free-pair and antiferromagnetic insulator- paramagnetic metal transitions, although it was not made clear that these are two different transitions. Mott estimated [1] that in a hydrogen lattice the transi- tion from antiferromagnet to paramagnet occurs at an interatomic distance of 4.5 Bohr radii. The other transi- tion was not calculated. On the basis of Slater's [12] solution of the hydrogen ls integrals, we can evaluate the intra-atomic interaction strength as J = 1.25 Rydbergs and the hopping integral, as a function of the in- teratomic distance R, as T | a (r) (–V}–2|ria – 2|rib)b (r) dr = — e-R (3 + 3R + R*/3) Rydbergs (7.3) where aſ r), b(r) are ls hydrogen orbitals centered on nearest neighbor atoms, and where electron-electron interaction two center integrals in the Hartree-Fock ap- proximation have been neglected as by Mott. Our esti- mate of the interaction is then R = 3.8 Bohr radii which is close to the estimate of Mott. 8. Discussion One may object to using a t-matrix approach in a high density system without going to higher orders in the ex- pansion. The results indicate that the higher order terms are not necessary. Starting with the paramag. netic state as the ground state near J = 0, a phase transition into the antiferromagnetic state occurs be- fore the interaction could have drastic effects within the paramagnetic state. When the interaction could have drastic effects in the antiferromagnetic state the system is so close to the atomic limit that additional correlations have no room left to operate in. Thus it seems likely that once the correct zero order paramag- netic or antiferromagnetic wavefunctions have been chosen, the additional correlation effects could be han- dled by a power series expansion in J, i.e., by ordinary perturbation theory. Since we have not anticipated such simple results we did not use this approach. There is one more merit to the t-matrix method. It is necessa- ry in the low density system where even with strong in- teractions phase transtitions are hard to achieve. This is due to the fact that in a low density system particles can keep out of each other's way most of the time and a collective state does not easily form. Thus the zero order wavefunctions are not so sensitive so the strength of the interaction has to be handled with the t- matrix. Thus our present results will join better with the low density calculations we plan to perform if the same method is used. 9. Acknowledgments The authors are indebted to Professor G. W. Pratt, Jr., for the hospitality extended to them at the Mas- sachusetts Institute of Technology. The initial stages of this work were performed at Ledgemont Laboratory, Kennecott Copper Corporation, Lexington, Mas- sachusetts. The numerical calculations were performed at the University of Sherbrooke Computation Center. 10. References [1] Mott, N. F., Proc. Phys. Soc. (London) 6.2, 416 (1949); Canadi- an J. Phys. 34, 1356 (1956); Nuovo Cimento Supp. 7, 312 (1958); Rev. Mod. Phys. 40, 677 (1968). [2] Hubbard, J., Proc. Roy. Soc. (London) A276, 238 (1961); A281, 401 (1964). [3] Kemeny, G., and Caron, L. G., Phys. Rev. 159,687 (1967). [4] Kemeny, G., and Caron, L. G., Rev. Mod. Phys. 40, 790 (1968). [5] Martin, P. C., and Schwinger, J., Phys. Rev. 115, 1342 (1959). [6] Reynolds, J. C., and Puff, R. D., Phys. Rev. 130, 1877 (1963). [7] Day, B. D., Rev. Mod. Phys. 39, 719 (1967). [8] Slater, J. C., Phys. Rev. 52, 198 (1937). [9] Kemeny, G., Phys. Letters 25a, 307 (1967). [10] Matsubara, T., and Yokota, T., Proc. Int. Conf. Theor. Phys. Kyoto and Tokyo, p. 693 (1953). Edwards, S. F., Phil. Mag. 6,617 (1961). Slater, J. C., Quantum Theory of Molecules and Solids, Vol. I (McGraw Hill, New York, 1963). des Cloizeaux, J., J. Phys. Rad. 20, 606, 751 (1959). Penn, D. R., Phys. Rev. 142, 350 (1966). [11] [12] [13] [14] ll. Appendix A: Numerical Technique The various expressions for the energy we have derived for the Hubbard model all have in common the following typical summation in momentum space: X = N-1 6(k - k - k2)f(e(ki); e (k2)) kikº-kp (A.1) 538 To evaluate such a summation we went over to the con- tinuum representation and calculated a density of states p,(EE2) such that: N- S ki kºskſ. () 6(k - k – kg) eſ dEidExpk (EE2) (A.2) Thus () A = | dE1dBºp! (EE2) f(E1; E2) (A.3) The two energy variables in this density of states were subdivided into 50 intervals each. There were 64000 values chosen for the relative momentum in the Bril- louin zone and 75 values of the total momentum k chosen in the S.C. symmetry element. In order to further smooth out fluctuations in the density of states the spread of the energies around each value of k was estimated using the first derivative of these energies. The paramagnetic Hartree energy then becomes: () (H) – owſ dEid Expo (EE2) El -- NJ/4 (A.4) while the t-matrix correction to this is: AE = () dE1dExpº (E1 E2) - J > ſ. | _j [" dºidºspºſº) | NJ/4 _. (El + Ey + Ey + E.) (A.5) On the other hand, the Matsubara energy is: () () (H) = 2N | dE/dE.E. [A (E1)* – B (E1)*] +JN | dE/dExpo (El E2) × [l – 4A (E1)*B (E1)*] (A.6) where I – E| VA2 LE2 A (E) 2 = º + E (A.7) – E/ VA2 - EP B(E) = } E/ º + E (A.8) The correction to this energy from the t-matrix is: AE = J > || dE dExpº (El Es){\Vm. (El Es) [A (E1)A (Es) – B(E1)B (E2)] + Vol. (E1 E2) [A (E1) B (E2) – B (E1)A (E2)]– [A (E1)A (E2) — B(E1) B (E2)]? – [A (E1) B (E2) — B(EI)A (E2)]?} (A.9) with — A (E1)A (E2) – B (E1) B (E2) "...(º)-TTFºEE, (A.10) - A (E1) B (E2) – B (E1)A (E.) ****)-Haſiºn (A.11) and p, (pp.) , ſº *p, *)[4(P)+(P.) P(E)^{P3)]; A. 12 “"“” "J-2 (VATE-VAZTE; VATE IVATE) (A.12) " dE/dE. p. (E. E.) [A (E.) B (E.) — B (E.)A (E.) |2 Qi. (E1 Eg) =J ( dE.p. (E. E.) [A (EA) B (E.) B (Eſ)A (E.)] (A.13) 539 The overall accuracy of this scheme was estimated at three significant digits for the Hartree and Matsubara energies and two significant digits for their t-matrix cor- rections. The resulting accuracy on the position of the energy crossover is somewhat poor because of the small energy differences involved. The critical value of c could possible by off by as much as 20%. But knowledge of the exact value of c at the crossover from the antiferromagnetic state to the paramagnetic state is not too critical, its existence being of primary im- portance. - () •A)-> || deaf p. (EE)|-2Eile k — ºc 12. Appendix B: c → do Limit We will show that in the limit of very large c’s, i.e., J → 0, the paramagnetic state has lower energy than the antiferromagnetic state. In this limit, the self-consistent value of the gap should be very small. As a matter of fact, in the Matsub- ara State lim A = 0 and lim A/J = 0 (B. 1) (– ºc ( → ~c This parameter A could then serve as an expansion parameter for the energy. The expression for the ener- gy of the antiferromagnetic t-matrix approximation as a function of A is: +/ (1 - Aºſe leg-H E Eg/e leg) + (l − A*/e leg – E Eg/e leg) | (B.2) 2(1 + P. (El Es) + (), (El Es) + (), , (E1 Eg)) 2(1 + (), (El Es) + Pl, , (El Es) + Pi—, (E1 Eg)) * with * ---> 0 dB/dE.p. (E/E.) A” . E.E. Pl, (EE2) = J/2 ſ". (e) + e2 + e ſ -- e.) (l ge," efe: ) (B.3) EE) -12ſ + (I––%) B.4 Qi. (EE2) = –2 (eſ -Fe2+ eſ + eſ; ) eſe eje; (B.4) and e = V/A2 + E2 As J – 0 we can approximate | IIE,(EE).To...ºf) IO, IEE) - I - P. (Bºb) - 0- (Pº) - 0- (Pº) (b.5) Taking the derivative of the energy with respect to A we get to first order in A' with (Eſe = 1): Öe (A) () 2J 2J lim 6A :S→ A > ſ". dE/dExpº (EE2) (2. - €162 +ix [P. (EE) + Pº , (EE2) + Pº (EES)] +J (1/e} + 1/e}) [P}(EEA) – P., (EE) – Pº, (EE)] –Jr. (EE) (B.6) where " dE/dE pº (EE.) PQ ..) | F l 2P k 1 *-* 2 ! (EE2) Jſ. (e) + eg -- eſ + e.) (B.7) 0 dB/dE pº (E/E.) P. (EE)=–1 | H++++ [1/eſ -- 1/e. -- 1 lſeo | (EE2) J — , (e) + e2+ eſ + e.)” [1/e} + 1/e} + 1/e -- 1/es) (B.8) 540 Since it is those integrals containing the factor 1/e," or 1/e} which will dominate as A → 0, we conclude: dE/dEx ; 6. * () lim º = 2.J.A X. | p. (EEA) [P (EE2) - Pºla (EEA) — Pº a (EE)] (B.9) C— CC k — OC But since = Tö(E1) (B.10) li li A In — = 11m — •) •) •) A.O. ei Aº Aº + E: we finally get: Öe (A () lim º = TJ > ſ". dEp,(0, E2) × [P(0, E2) –P., (0, E2) – P! (0, E2)] (B.11) This integral was evaluated numerically and it is found The energy then increases for A > 0 which implies any that: antiferromagnetic state has larger energy than the lim **) 2-S-. º > 0. (B.12) paramagnetic one A = 0. 541 ALLOYS; ELECTRONIC SPECIFIC HEAT I CHAIRMEN: J. R. Anderson J. H. Schooley Electronic Structure of Gold and Its Changes On Alloying E. Erlbach” City College of CUNY, New York, New York 10032 D. Beaglehole University of Maryland, College Park, Maryland 20740 We have measured the changes in reflectivity upon alloying small amounts of Ag, Cu and Fe into Au. By means of a Kramers-Kronig analysis, we have deduced the changes in e2 produced by this alloy- ing. We relate these changes to shifts in the position and character of the electron energy bands in gold. Key words: Electronic constant; electronic density of states; gold (Au); gold-copper alloys (Au-Cu); gold-iron alloys (Au-Fe); gold-silver alloys (Au-Ag); optical constants; reflectivity. The optical constants of gold have been measured by several methods [1,2] and feature an edge at 2.5 eV fol- lowed by a second peak which starts at 3.5 eV (fig. 1). The first edge has been identified as being due to transitions from the top of the d-bands to the Fermi sur- face. The second edge has recently [3,4] been as- sociated with transitions from the Fermi surface to the upper conduction band (L2' → Lj). This identification is supported by its similarity to the second edge in copper, where this identification has been verified by piezoreflectance measurements [5]. Additional struc- ture is seen at 4.5 eV which has been assigned [2] to transitions from the d-band to the Fermi surface at X(X; → X4') and another peak in e2 is seen around 7.5 eV. Upon alloying Ag, Cu and Fe into Au, we see struc- ture at all the above energies, and also at some addi- tional energies (figs. 2-4). The experimental method has 7.O 6.O , ) 4 5.O O a : + § 7 C. : 3.2 °/o # $40 H - O 4 ſº. ENERGY (EV) |- uſ & 2 § ° O } # t { Å i t T-_, 3,O H. 2 W 4. 5 6 7 8 9 jū ; : {N} g \} – O4 - C = O.4% 2.O H. – O 8 !, O F – 12 –. 6 {}{} i t #. i § ...? § R l |O 2 O 3.0 4 O 5.0 6.C. Y.O 3.O 9 C |O | Et, ERGY (EV) – 2 O g * * ... .24 FIGURE 1. The real part of the dielectric constant, e.g. for pure gold. *=-m- FIGURE 2: The normalized changes in e2 for two Au-Ag alloys. *Work performed while on a Sabbatical at the University of Maryland. The concentration of Ag are as indicated. 417–156 O - 71 - 36 ENERGY (eV) | ? # 4 5 6 7 } f -T- O *—r & º, & & Aé2 & - & . & • * & & § & & 4 * 4 & * & C "... : . . . . . . " ' " rº . . . . ci - O dº • *a 5 ° o o Au * Fe *** * ‘. tº to o tº a tº C - •. * a : ex o O.45 of 9, .. . . . . " " .." £º ſh O.9 s." * e a tº e ‘’ - O. A 2. te iº © o gº • 4.3 *" o * ... • - O.3 *...* * e o ' FIGURE 3. The normalized changes in e2 for four Au-Fe alloys. This data is taken from Phys. Rev. Letters 22, 133 (1969). The concentrations of Fe are as indicated. A C = 2.6% FIGURE 4. The normalized changes in e2 for four Au-Cu alloys. The concentrations of Cu are as indicated. been partly published already [3,4], and a more detailed account will be published in the future. Briefly, we evaporate, simultaneously two films, one of pure gold and one containing the desired impurity, and then we measure the difference in reflectivity of the two films. A Kramers-Kronig analysis is used to convert these reflectivity differences into changes in e2 which are then normalized to 1 at. 9% concentration by divid- ing by the impurity concentration and plotted as a func- tion of incident photon energy. Structure referred to in the following refers to these Aeg|c curves. If alloying causes a shift of a band edge to higher energy, then the curves of Aegſc will be proportional to —deº/dE of the pure material, and a peak will be found at the point of maximum slope of e2. If alloying causes a broadening of a peak, then Aeoſc will exhibit a disper- sive-resonance-like shape, going through zero at the peak of deº/dE. - Referring to our experimental curves in figures 2, 3 and 4, we find peaks in Aeg|c at 2.5 eV, which we have associated with a shift of the band edge upon alloying. The edge moves to higher energies for Ag [4,6] or Fe [3] alloys and to lower energies for the Cu [4] alloys. In the Ag and Cu alloys, the shifts have been explained as due to smooth changes in the relative energies of the d-bands and Fermi surface as one moves from Au to the other noble metals. These shifts are determined by two effects, perturbations in the average potential caused by the impurity atoms, and by changes in lattice con- stants [4]. - We have found an additional sharp structure in Aeg for Au-Cu alloys at an energy of 2.9 eV (fig. 4); the peak at about 2.9 eV is evident in the curves for all four con- centrations of copper. When one subtracts the background, one finds that the amplitude of the peak of Aeoſc, when plotted vs. the concentration, c, decreases roughly linearly with c, being approximately given by Aes/c = 0.039 – 0.009c (c in atomic percent) (1) A larger concentration is relatively less effective in in- creasing e2 in this region than a smaller concentration. A possible explanation for this will be given later. If the curves for Aeº/c for Au-Ag and Au-Fe alloys (figs. 2 and 3) are examined closely near 2.9 eV, one notices a change in slope occurring at this energy. Furthermore, this change of slope can be seen in the curve of the derivative of e2 with energy. It is likely that both this change in slope and the peak in the Au-Cu curves have the same origin. Thus we believe the evidence indicates that the peak in gold at 2.5 eV ac- tually consists of two peaks, which are not resolved in the e9 curve for pure gold nor in the Aeg curves for Au- Fe and Au-Ag. The fact that the edge in gold is not as sharp as the equivalent edge in copper would tend to corroborate this interpretation. The addition of copper to gold enhances the peak near 2.9 eV more than the peak at 2.5 eV, thus allowing it to be resolved from the main peak. The reason for the particular effectiveness of copper in enhancing this peak may be the following. One can 546 E E. E n \ \_7 E- 9| \ as \ A #2 /. 5 EF 5 EF t i º *% -2.7 Cu Ag A u FIGURE 5. Schematic diagrams of the band structure of gold, silver and copper. plot the bands of the noble metals schematically as in figure 5. The Fermi levels have been placed at their free electron values, and the top of the d-bands at an energy E1 below, where E1 is the energy of the first ab- sorption edge for the pure metals. The widths of the d- bands have been taken as 2.8 eV for copper (from piezoreflectance measurements [5]), as 4 eV for gold (from our results) and 3.7 eV for silver (estimated by Lewis and Lee [7]). If one alloys copper into gold, one expects a mixing of the d-levels since they overlap, and the greatest perturbation is expected initially in the re- gion of overlap. Since the bottom of the d-band of copper fails at an energy of 0.4 eV below the top of the gold d-band, one would initially expect the greatest per- turbation near this energy. This is just the energy needed to affect the level responsible for e2 at 2.9 eV in gold since this is 0.4 eV below the absorption edge of 2.5 eV in gold. Thus, the addition of copper would be expected to greatly affect the 2.9 eV level and we see that this is indeed the case. As one increases the con- centration of copper, one expects the perturbation to spread more evenly through the d-bands of gold and the enhancement per unit concentration of the 2.9 eV level should decrease with c. This may be the reason for the linear decrease in the peak mentioned previously. The addition of silver to gold would not be expected to sin- gle out the level at 2.9 eV more than the 2.5 eV level since the overlap is mainly with the lower part of the gold d-band. This would explain the absence of a large effect in Au-Ag at 2.9 eV. While the existence of the double edge seems to be established, the nature of the splitting is not deter- mined. Since gold is expected to have a spin-orbit splitting at the edge of the zone, and the order of mag- nitude of this splitting would not be inconsistent with a value of 0.4 eV the doubling of the edge may well be due to this splitting. Recently, Jacobs [8] has calcu- lated the band structure of gold and suggests that the first edge of gold is shifted by 0.5 eV if spin-orbit coupling is included. This is close to our measured value. Other experiments and calculations are needed to confirm this identification. We now turn our attention to the structure at 3.5 eV. We see that for the Au-Cu alloys, for the two lowest concentrations there exists a peak in Aeg at 3.5 eV. Since des/dE has a peak at this same energy, we as- sociate the structure in the alloy data with a shift of the edge to lower energies. This shift seems to be linear at these low concentrations, but changes drastically when c is greater than 1.0 at. 9%. Above this concentration, there is still a small peak which has moved to a slightly lower energy but since the peak in deg/dE occurs on the descending edge of the structure, it is probably now better to consider it as a dispersive resonance. This suggests that at higher concentrations the edge is broadened by alloying rather than shifted. The rate of shift at low concentrations is about — 0.03 eV/at. 9%. If one now examines the data for Au-Ag alloys one notices a pronounced peak at 3.5 eV and a dip at 3.65 eV in the low concentration sample, both becoming smaller for the higher concentration sample. One notes that the structure in deo/dE peaks at 3.5 eV thus in- dicating that the addition of Ag to Au causes a shift of the edge to lower energies. The rate of shift is approxi- mately — .023 eV/at. 9% for the lower concentration sample and —.007 eV/at. 9% for the higher concentra- tion samples. The presence of the dip at 3.65 eV, how- ever, makes it difficult to be certain that this interpreta- tion is correct, and further study is required. As outlined above, we also found nonlinearity of the 3.5 eV structure for the Au-Cu system, and the previ- ously reported data [3] for Au-Fe is also nonlinear in this region (fig. 3). In the Au-Fe alloy system a sharp peak is clearly seen at 3.5 eV for the lowest concentra- tions, and this peak decreases in size for higher concen- trations. Thus, all three alloy systems exhibit this characteristic behavior, i.e., a reduction or even disap- pearance in the shift of the edge per unit concentration as the concentration increases. The interpretation of this variation is not clear. The 3.5 eV edge has been attributed to transitions at L from the region of the Fermi surface to the higher lying L. state. The equivalent edge occurs at 4.3 eV in Cu and at 4.1 eV in Ag. Thus, the simplest guess one would make for the motion of this edge is that it moves to higher energy for both Au-Ag and Au-Cu alloys. The average shift would be expected to be .006 eV/at. 9% for Ag and .008 eV/at. 9% for Cu. In both silver and copper, 547 however, the experimental shift is in the opposite direction. Furthermore, effect of a lattice constant change which explained the anomalous variation of the 2.5 eV edge in Au-Cu is also in the wrong direction. This can be estimated as follows. Since piezoreflect- ance measurements have not been made on single crystals of gold, we must use the piezoreflectance measurements on unoriented films [9]. The results for gold and copper are very similar, except that the magnitude of Aeg is about a factor of two lower in Cu compared with Au. We thus assume that we can use the single crystal results for copper, with a reduction by a factor of two for gold. This predicts a shift of +.01 eV/at. 9% of Cu for the 3.5 eV edge of Au using the published lattice constant parameters [10]. The experimental shift to lower energies is thus incon- sistent both with the stress produced shifts and with the expected smooth shift of the same edge from its position in gold to its position in copper. When one notes that (1) all the alloys exhibit the same type of nonlinearity for low concentrations, and that (2) the edge at 3.5 eV seems to be most affected, one concludes that the energy levels associated with this transition are particularly sensitive to small amounts of impurities. Since these effects do not occur for the level at 2.5 eV, which also involves the Fermi surface, it is likely that the Lſ" conduction band level is the one which is being perturbed nonlinearly. This level is the one which is split from the L2' level primarily by s—d interaction with the L' level at the bottom of the d- band (see fig. 2 of ref. 13, p. 662) [5,11-13]. One might expect that this splitting, which can be affected by changes in any of the other levels via changes in matrix elements or by changes in hybridization through orthogonalization, could be nonlinear with impurity concentration. One would then also expect that an edge caused by transitions from the lower L' level to the Fermi surface would also be nonlinear with concentra- tion. This edge should occur around 6.5 eV if the d- band is 4 eV wide. Examining the data near this energy, one sees in the Au-Cu alloys that one does indeed have nonlinear structure near 6.1 eV. This structure is strong for the low concentrations and much weaker for the higher concentration. It thus appears that the Li' - F. S. edge occurs at 6.1 eV. The L1' – L3" separation – the width of the d-band at L– is thus 3.6 eV. Our conclusion is thus that the nonlinearities in the edges at 3.5 and 6.1 eV are due to changes in the splitting of the two Li levels which are very sensitive to changes in potentials. A detailed calculation of the ef. fect of impurities on the LI” – L' splitting would be most interesting. The structure around 4.5 eV is more difficult to analyze. The data for eº for pure gold shows only a slight shoulder at this energy; des/dE showing an asym- metric dip. The added absorption in this region has been attributed to X, − X,' transitions [2] but the peak expected from this is obviously somewhat obscured by the background absorption from other points in the zone. The most prominent feature of the curves for Aeg|c for all gold alloys is a sharp rise beginning at 4.5 eV. This rise is also seen in des/dE and indicates that Aeº caused by alloying is roughly proportional to deo/dE in this region. The data thus indicate a shift to lower energies for the states responsible for this transition. Since it is difficult to separate out the part of des/dE contributed by this edge alone, we have not estimated the rate of shift of this edge. In pure copper the X5 — X4' edge is at 3.9 eV [5], while in pure silver, the edge is at 5.4 eV [7]. Thus adding copper to gold shifts the X5 – X4' edge toward the copper value, while adding silver to gold shifts it away from the silver value. Once again, hybridization effects must subtly determine the way these energy levels vary. Additional structure is seen in the form of a large dip in Aeg for the Au-Cu alloys at 8 eV. The curve of e2 for pure gold also has structure near this energy, resulting in a small negative dip in deo/dE around this energy. The alloys data is not as reliable in this region as it is below 6 eV, because of experimental difficulties in the vacuum ultraviolet and because of uncertainties in the Kramers-Kronig analysis near the limit of the available data. But, within experimental error, we do not find any nonlinearities in the dip in this region. Further, the dip is a relatively large one, and it is not found in the Au-Ag alloys. The structure is thus particularly strongly per- turbed by copper impurities, which move the edge to lower energies at a fairly high rate. If one examines the band structure calculations for copper and gold, the only critical points which seem TABLE 1. Levels Separation in eV FS – L. 3.5 eV L|- F.S. 6.1 eV L– Lºſ 9.6 eV L! – F.S. 2.5 eV L– L. 3.6 eV (d-band width at L) X5 – X. 4.5 eV XI – X. 8 eV XI – X5 3.5 eV (d-band width at X) 548 capable of producing an edge near 8 eV are X → X'. [2] (We are grateful to J. C. Phillips for this suggestion.) If [3] this identification is correct then we can calculate the width of the d-band at X, i.e., XI – X5, using the previ- [4] ously established value of 4.5 eV for X5 -> Xi'. We thus get XI – X5 as 3.5 eV, nearly the same width as we [5] found at L. These energy levels are summarized in table 1. [6] We gratefully acknowledge conversations with N. F. Berk, J. C. Phillips and T. J. Hendrickson: This work was supported by ARPA and the NSF. References [9] [1] Cooper, B. R. Ehrenreich, H., and Philipp, H. R., Phys. Rev. 138, A494 (1965); Hodgson, J. Phys. Chem. Solids 29, 2175 [10] (1968): Theye, M. L., Proc. Colloquium on Thin Films, Bu- dapest, 1965, p. 251 (1966): Canfield, L. R., Hass, A., and [ll] Hunter, W. R., J. Phys. (Paris) 25, 124, (1964); Beaglehole, D., [12] Proc. Phys. Soc. (London) 85, 1007 (1965). [13] Pells, G. P., and Shiga, M., to be published. Beaglehole, D., and Hendrickson, T. J., Phys. Rev. Letters 22, 133 (1969). Erlbach, E., and Beaglehole, D., Bull. Am. Phys. Soc. II, 14, 322 (1969): Beaglehole, D., and Erlbach, E., U. of Mol, techni- cal report No. 1000, to be published. Gerhardt, U., Beaglehole, D., and Sandrock, R., Phys. Rev. Let- ters 19, 309 (1967): Gerhardt, U., Phys. Rev. 172,651 (1968). Wessel, P. R., Phys. Rev. 132, 2062 (1963): Fukutani, H., and Sueka. O., in Optical Properties and Electronic Structure of Metals and Alloys, F. Abeles, Editor (North Holland Publish- ing Company, Amsterdam, 1966). Lewis, P. E., and Lee, P. M., Phys. Rev. 175, 795 (1968). Jacobs, R. L., J. Phys. C 1, 1296 (1968). Garfinkel, M., Tiemann, J. J., and Engeler, W. E., Phys. Rev. 148, 695 (1966). Pearson, W. B., Handbook of Lattice Spacings and Structures of Metals, (Pergamon Press, New York, 1958). Mueller, F. M., and Phillips, J. C., Phys. Rev. 157, 600 (1967). Dresselhaus, G. D., Solid State Comm. 7, 419 (1969). Mueller, F. M., Phys. Rev. 153,662 (1967). 549 Discussion on “Electronic Structure of Gold and Its Changes on Alloying" by E. Erlbach (City College of CUNY) and D. Beaglehole (University of Maryland) W. E. Spicer (Stanford Univ.); We have a measure- ment of the d-band width from photoemission data and it appears fairly definitely to be 5 to 5 1/2 volts wide, which I think is in fair agreement with the calculations. E. Erlbach (City College of CUNY): Well it is unlikely that I’d get 5 electron volts even when one understands that the energy of the maximum at L and the maximum at X don’t have to be the same. And therefore it’s en- tirely possible you would get 4.6 possibly 5. You would be unlikely to get the 5 1/2 that you apparently get from your photoemission measurements, but it’s possible also that the minimum does not occur at X or L. It could occur somewhere else. I do not know. It does seem that the widths at the X and L are smaller than the total widths of the d-bands in any case. I don’t know if our results are inconsistent with yours although they don’t seem consistent automatically. W. E. Spicer (Stanford Univ.); But the widths you give there can be so much smaller than the density of states widths. E. Erlbach (City College of CUNY): That’s also right. 550 Density of States of Ag/Au, AgPd, and Agln Alloys Studied by Medns of the Photoemission Technique P. O. Nilsson Chalmers University of Technology, Göteborg, Sweden The density of states of AgAu, AgPol, and AgIn alloys have been studied by means of the photoemission technique. General trends of the results are compared with the predictions from simple models of alloys. The rigid-band or virtual-crystal approximation cannot explain the results, while model calculations in the coherent potential approximation reproduces the observed density of states. The Friedel screening theory explains the shift of the Fermi level on alloying. Key words: Coherent potential approximation; electronic density of states; Friedel screening theory; photoemission; silver-gold alloys (Agâu); silver-indium alloys (AgIn); silver-palladium alloys (AgPoi); virtual crystal approximation. 1. Introduction One of the earliest theories of the electronic proper- ties of disordered alloys was the rigid-band approxima- tion, which was introduced by Mott and Jones [1]. This first order perturbation theory predicts that the shape of the density of state curve does not change on alloying but is only rigidly shifted. Similar results are obtained in the so called virtual-crystal approximation in which the crystal has a periodic, concentration weighted mean potential. Although these theories have been widely used to interpret experimental results they do not in general describe the properties of disordered al- loys correctly. For large band separations the theories are of course not valid at all: separated bands are formed which may be treated individually. More refined theories have to be used to describe inter- mediate cases. Korringa [2] and Beeby [3] used a mul- tiple scattering description [4] to derive a t-matrix ap- proximation. This theory was an important contribution but gave even for small band separations a spurious band gap. This is not present in the coherent potential approximation (CPA) introduced by Soven [5]. The free electron Green’s function is here modified to con- tain an energy dependent self energy, which is self-con- sistently chosen so that the perturbation potentials do not contribute further to the scattering on the average. Velicky et al. [6] showed that the CPA properly inter- polates between the extreme limits of small and large band separations. Model calculations in three dimen- sions were also performed, to which we will return below. We have studied by means of the photoemission technique three silver based alloys: AgAu, AgPol, and AgIn. These alloys have overlapping, slightly over- lapping, and separated d-bands, respectively, and con- stitute suitable testing cases for alloy theories. A more extensive report on these measurements will appear elsewhere [8,9]. 2. Results and Discussion When one substance is dissolved in another the new density of states can be said to depend on three factors, namely the atomic potentials, the atom valence, and the lattice spacing. Both Ag and Au have almost the same lattice constant, 4.07 Å and 4.08 A respectively, and the same valency, +1, so we have the largest contribution from potential effects. Figure 1 shows a summary of the photoemission results for this alloy system. Here the electron energy distribution curves are shown for the highest available photon energies, approximately 10 eV. Except for the cut-off at low electron energies, due to the so called escape function, the spectra are sup- posed to be a picture of the optical density of states below the Fermi level. We shall not discuss for the time 551 º N (E) I —I- | I —T- —T O C Au C; 5 10 x 16"| - O -C Cl QU Cl. 9 | Uſ) C S f Toº 8 H S i. -C, E 7 H -5 Z 6 - 5 H- - 4 H 3 H Ag - 2 H } H O | I | l | | – 6 - 5 - 4 – 3 –2 – O Reduced electron energy, E. g. -hv (eV) FIGURE 1. Electron distributions obtained from photoemission experiments on AgAu alloys. being the relative importance of nondirect versus direct optical transitions [10]. It suffices to say that because of the low dispersion of the d-bands this question is not of importance when studying the main behavior of the bands on alloying. We observe in figure 1 that when Au is dissolved in Ag (the Ag3Au-alloy) the Ag d-band is lar- gely unaffected while the gold electrons seem to set up a band which extends above the Ag band. This is in contradiction to the rigid-band and virtual-crystal ap- proximations. To study the predictions of the CPA we use the equations derived by Velicky et al. in their § 20 | | I- | | I g U) º 2. J, 5 ~5 15 H. × t= 10 H 5 - e” .." ‘... N -- T … '.. \ --~.........: “” 10xim: “...N O 2:... “‘’’. l –– 1 –– –8 –7 -6 –5 –4 –3 -2 Electron energy relative the Fermi Level (eV) FIGURE 2: Model calculation of the density of states in a 7.5–25 percent alloy. The equations derived in CPA by Velicky et al. [6] were used (6= 0.70), p * and p" is the component density of states and 2 the self-energy. model calculations [6] and calculate the alloy com- ponent density of states with appropriate parameters. Of course such a comparison is very crude (a single band Hamiltonian is used, in which all elements are in- dependent of alloy composition. Thus we cannot take the real band structure into account). In the com- parison we also disregard interference from conduction electrons. However, the results, presented in figure 2 tell us that in fact a Au band is expected at the top of the Ag band, where it was observed experimentally. In the decomposition of the electron densities we observe that the Au states in reality extend down to the bottom of the Ag band and vice versa. The electrons at the bot- tom of the Ag band can still be described by quasiparti- cles. At the top of the Ag band and in the impurity band the spectral density, however, will have a certain non- negligible width. The imaginary part of the self energy in figure 2 tells us that we have a damping of up to 0.6 eV halfwidth. The spectral density is found to be large over a range of k-values which means that the Au-state is “half-localized.” On increasing the Au content (in the AgAu alloy), the Au band increases further in strength and width, again in agreement with CPA. As regards the Ag Aug alloy, the pure Au-spectrum is observed at the high energy part of the d-band. This is the same behavior as for the Agº Au alloy but with the con- 552 Ag h y = 10 : 1 eV Energy FI(; U. RE 3. stituents exchanged. Influence of the Ag-atoms at the bottom of the band is not observed in our experiment because of masking by the escape function. Turning to the AgPo alloys, we expect, according to the CPA, a gap in the density of states for low Pa-con- tent because of the relative large band separation. In figure 3 is shown the photoemission spectra for pure Ag and for Ag0.85Ino.15 [11]. As seen, the qualitative predic- tion is borne out. By choosing appropriate parameters in the model calculation it is also possible even for this alloy system to obtain quantitative agreement. The decomposition of the density of states shows that even for split bands we have a contribution from one kind of atom in the band of the other kind of atom. This can be viewed in the following way. Electrons scattered from, for instance the Pd atoms, will propagate through the lattice and resonate strongly with other Pol atoms. Because of the fact that the Ag atoms have a potential not differing too much from that of the Pol atoms they ( e.V.) Photoelectron distributions from Ag and Agº, Pds. also contribute to the scattering in the Pol band but not, however, with the same strength. (A ratio 1:6 is ob- tained for p"9:p” in the Po band). We also know that the band gap introduces strong fluctuations in the real part of the self energy, Rex. (In fact, a pole is present if the band gap is large but we cannot with certainty say if this is the case here.) Because of this, Im} will have appreciable strength at Pd energies (2.5 eV at the bot- tom and 0.5 eV at the middle of the band), causing strong damping and a deviation of the spectral density from Lorentzian form. The energy versus k curve is flat and extends over all k values showing the strong lo- calization of the d electrons. For a more complete description of the PC states the interaction with the conduction electrons must be considered. The electron states are then said to be virtual bound [12]. Increasing the Pa content the band gap disappears at about 30 at. % Pd again at least qualitatively in agreement with the CPA. 553 –7 -6 - 5 - 4 –3 - 2 – 1 O Electron energy relative to the Fermi level of Ag (eV) FIGURE 4. Photoelectron distributions from Ag and Agº, In 15. Finally we consider the AgIn alloys. The d-band of In is about 10 eV below that of Ag and the interaction is expected to be small. As seen in figure 4 the main effect on alloying is a smearing out of the fine structure. We assume that the upper part of the d-band does not shift very much. This is a reasonable assumption because the effect of decreasing bandwidth and the repulsion ef- fect between the two d-bands is small and of opposite sign. We then note a shift of the Fermi level of about 0.3 eV. A pure filling of the conduction band gives a shift of the Fermi level of 1.6 eV if we assume a density of states at the Fermi level of 0.275 eV-1 atom [13]. How- ever, taking the effect of screening of the In atoms into account [14] gives a shift of 0.27 eV in good agreement with the present experimental result. For the Ag0.85Pdo.15 alloy a downward shift of the Fermi level of 0.14 eV is calculated while no change is observed in separation between the d-band and the Fermi level. This may be due to the fact that the decreased band- width and the downward repulsion effect of the Ag d- band now sum up and contribute just as much as the Fermi level is shifted. 3. Conclusions The theories for the electronic structure of disor- dered alloys give different predictions about the general shape of the alloy density of states. Among the theories the CPA seems to be the most general and ap- plicable one. It gives the right predictions about the general shape of the density of states for the three alloy systems we have studied, while other theories fail. By using model calculations in the CPA even finer details in the density of states can be studied and interpreted. We have found that for the AgAu alloys contributions from the two kinds of atoms can be followed through the alloy compositions although the density of states is strongly overlapping. For the Ago.85Pdo.15 alloy the two constituents form two split bands with a contribution one in the other of about 15%. The Ago.85Ino.15 alloy has small interaction between the d-bands and it is possible to determine the shift of the Fermi level approximately. The observed value of 0.3 eV is in agreement with the screening theory by Friedel. 4. References [1] Mott, N. F., and Jones, H., Theory of the Properties of Metals and Alloys (Dover Publication, Inc., New York, 1958). [2] Korringa, J., J. Phys. Chem. Solids 7, 252 (1958). [3] Beeby, J. L., Phys. Rev. 135, A130 (1964). [4] Lax, M., Rev. Mod. Phys. 23, 287 (1951). [5] Soven, P., Phys. Rev. 156,809 (1967). [6] Velicky, B., Kirkpatrick, S., and Ehrenreich, H., Phys. Rev. 175, 747 (1968). [7] Soven, P., Phys. Rev. 178, 1136 (1969). [8] Agau, AgIn: Nilsson, P. O., to be published. [9] AgPol: Norris, C., to be published. [10] Nilsson, P. O., Norris, C., and Wallten, L., Solid State Comm., in print. Norris, C., and Nilsson, P. O., Solid State Comm. 649 (1968). See e.g. Friedel, J., Nuovo Cim. Suppl. 7, 287 (1958) and Myers, H. P., Proc. of the Chania Conference on Magnetism, June, 1959, and references therein. Kittel, C., Introduction to Solid State Physics (John Wiley & Sons, 1956). Friedel, J., in Advances in Physics, N. F. Mott, Editor (Taylor and Francis, Ltd., London, 1954) Vol. 3, p. 461. [11] [12] [13] [14] 554 Discussion on “Density of States of AgAu, AgPd, and Agln Alloys Studied by means of the Photoemission Technique” by P. O. Nilsson (Chalmers University, Göteborg, Sweden) H. Ehrenreich (Harvard Univ.); I would like to point out that largely as result of the work by Kirkpatrick and Velicky that we’ve been able to extend the coherent potential approximation calculations to systems having degenerate bands and s-d hybridization effects such as in those described in the last two papers. We have ap- plied this only to copper-nickel so far, but it’s quite clear that the rigid band model does break down. These kinds of calculations are actually done fairly easily and I would hope that more people will undertake them. There is just one other point I wanted to make. The d- band width in the various noble metals can be quite dif- ferent. For example, the gold d-band may be as much as 2-3 volts wider than that of Cu. Accordingly, the models involving overlapping or non-overlapping d- bands discussed in the preceding paper may be somewhat over-simplified until one knows more about what the widths really are. A. Williams (IBM): Can the data be analyzed with comparable success by simply averaging the EDC's for the pure materials? P. O. Nilsson (Chalmers Univ.); The experimental results cannot be obtained by taking the concentration weighted sum of the density of states of the con- stituents. Consider, e.g., the Ag-Au alloys. In pure Ag and Au the d-bands begin at about 4 and 2 eV, respec- tively, below the Fermi level. In, e.g., the Ag3Au alloy, however, the d-band begins at 3 eV; that is, 1 eV lower than the value one would expect for an averaged densi- ty of states. H. Ehrenreich (Harvard Univ.); Our calculations also indicate the failure of simple averaging. W. E. Spicer (Stanford Univ.); I would just like to reinforce that. Seib has just completed a very detailed photoemission study where he looked in detail at this type of analysis. You just cannot add copper and nickel data of pure materials and get the photoemission results actually obtained from the alloys. It’s a very in- teresting thing because there is a paper in the literature on soft x-ray emission where the authors stated that they were able to use successfully such an analy- sis — add copper and nickel and get the soft x-ray result of the alloy. That would indicate that there is a basic difference between the sort of information you get out of the two measurements. 555 Density of States Information from Low Temperature Specific Heat Measurements” P. A. Beck and H. Claus University of Illinois, Urband The calculation of one-electron density of state values from the coefficient y of the term of the low temperature specific heat linear in temperature is complicated by many-body effects. In particular, the electron-phonon interaction may enhance the measured y as much as twofold. The enhancement factor can be evaluated in the case of superconducting metals and alloys. In the presence of magnetic mo- ments, additional complications arise. A magnetic contribution to the measured y was identified in the case of dilute alloys and also of concentrated alloys where parasitic antiferromagnetism is superim- posed on an over-all ferromagnetic order. No method has as yet been devised to evaluate this magnetic part of y. The separation of the temperature-linear term of the specific heat may itself be complicated by the appearance of a specific heat anomaly due to magnetic clusters in superparamagnetic or weakly ferromagnetic alloys. Key words: Alloys; density of states; low temperature specific heat; magnetic specific heat; many- body effects; superconductivity. 1. Introduction In the Sommerfeld-Bethe theory of metals the elec- tronic specific heat at low temperatures is linear in tem- perature in first order approximation. The lattice specific heat in the low temperature approximation is proportional to Tº so that, in the absence of other con- tributions, the total specific heat C = yT+3T3 (1) If C is known as a function of T, the two terms can be separated by making use of the linear variation of CIT with T2 and by extrapolating to T- 0. The intercept of the extrapolated line with the ordinate axis gives the temperature coefficient of the electronic specific heat y. In the simplest case, y is proportional to the elec- tronic density of states at the Fermi surface, N(EF): y = (1/3)Tºok*N(EP) (2) where k is the Boltzman constant and o. is a numerical *An invited paper presented at the 3rd Materials Research Symposium, Electronic Density of States, November 3-6, 1969, Gaithersburg, Md. factor determined by the units used for y, N(EF) and k. Unfortunately, in a very large majority of cases, the simple procedure just described cannot be used, or at least it does not give reliable results. Many-body effects and, in some alloys, magnetic effects may make the determination of N(EF) from low temperature specific heat data more complicated than implied by eqs (1) and (2), or even impossible at the present state of the art. 2. Many-body Effects In recent years it has become known that many-body effects, in particular the electron-phonon interaction, require renormalization of the effective mass of the electrons at the Fermi surface. This increases the meas- ured electronic specific heat coefficient over the one- electron “band structure” value by the enhancement factor (1 + \). For Na, Al and Pb, it was possible to determine the value of this factor [1], by comparing the “band structure electronic specific heat,” calculated from the known band structure and the topography of the Fermi surface, with the measured electronic specific heat. These values: 1.25, 1.45 and 2.00, respec- tively, were found to agree quite well with the enhance- 557 ment factors calculated from band structure, Fermi surface topography and phonon dispersion curves, on the basis of the electron-phonon interaction [1]. Unfor- tunately, for most other metals calculations of this sort cannot be made at present since at least some of the required data are not yet available. For superconduct- ing metals the electron-phonon couping constant N has been recently calculated by McMillan [2], using the following equation which he derived from the strong coupling theory: {} Te=115 exp=| 1.04(1+ \) | A — put (1+0.62X) (3) where Tc is the superconducting transition temperature and 6 is the Debye temperature. The electron-electron interaction constant put was assumed to have a value of 0.13 for all transition metals. The values of A calculated by McMillan [2] for superconducting metals are given in table I. TABLE I. The Electron-Phonon Interaction Coefficient A and “Band Structure” Density of States N(EF) for Superconducting Metals.” Element º % N evº,hº Be...................... .026 1390 .23 .032 Al...................... 1.16 428 .38 .208 Zn...................... .85 309 .38 .098 Ga..................... 1.08 325 .40 .09] Col..................... .52 209 .38 . 106 In...................... 3.40 | 12 .69 .212 Sn...................... 3.72 200 .60 .238 Hg..................... 4.16 72 1.00 .146 Tl...................... 2.38 . 79 .7] .182 Pb..................... 7.19 105 1.12 .276 Ti...................... .39 425 .38 .5l. V....................... 5.30 399 .60 1.31 Zr...................... .55 290 .41 .42 Nb..................... 9.22 277 .82 .9] Mo..................... .92 460 .4l .28 Ru..................... .49 550 .38 .46 Hf...................... .09 252 .34 .34 Ta...................... 4.48 258 .65 .77 W...................... .012 390, .28 .15 Re..................... 1.69 415 .46 .33 Os..................... .65 500 .39 .35 Ir....................... .14 420 .34 .51 It is now clear that the enhancement factor (1 + \) of the electronic specific heat due to the electron-phonon interaction can be as high as 2, or more. This interac- tion affects only the electrons whose kinetic energy is close to the Fermi energy. The density of states at lower levels, that is for most of the electrons in the metallic band, may be assumed to correspond to the one-elec- tron “band structure” situation. Hence, the lower, “band structure density of states” values must be used in determining the band width, for instance, rather than the density of states enhanced by electron-phonon in- teraction, as obtained from low temperature specific heat measurements. Using the electron-phonon coupling constant A, for instance the values given in table I, the “band structure density of states” at the Fermi level N(EP) can be calculated from the experi- mentally determined value of the low temperature specific heat coefficient y' as follows: 3 N(Br)=#End TV, y' (4) For most of the nonsuperconducting metals and alloys the value of A is at present unknown and, as a result, the “band structure density of states” cannot be calcu- lated from the low temperature specific heat. As seen in figure 1, the experimental electronic specific heat coefficient y' for the b.c.c. 3d-transition metals and their alloys as a function of electron concen- tration [3] shows prominent maxima and minima in the range of eſa from 4 to 9. Since in the region of the minima and of the second maximum the alloys are not superconducting, the “band structure density of states” cannot be calculated at present. Thus, the in- teresting question whether the prominent features of these curves are due to changes in the electron-phonon enhancement factor upon alloying, or indeed these fea- tures are characteristic of the electronic band structure of the transition metals concerned, cannot be answered 4O } }\ 3 5 \ 2-6 >O. | | . 2 O f t \ \ ^l Á, Ti V Cr (Mn) Fe Co 4. 5 6 7 8 9 FIGURE 1. Coefficient y' or y” of the low temperature specific heat term linear in temperature vs electron concentration eſa for b.c.c. alloys of 3d transition metals [3]. Points marked by filled squares represent data for close-packed structures. 558 with certainty. However, the work of McMillan [2] al- lows the conclusion that the electron-phonon coupling constant (and, thus, the enhancement factor) depends primarily on the phonon frequencies, rather than on the electronic properties. Since the elastic constants and, therefore, the phonon frequencies are not known to undergo drastic changes with the composition in such solid solution alloys composed of metals near one another in the same row of the periodic table, it may be concluded with a reasonable degree of probability that the prominent features mentioned of the y' versus ela curve of figure 1 are in fact resulting from correspond- ing variations in the “band structure density of states,” even though the relative magnitude of the various minima and maxima may be appreciably altered by the gradual changes in the coupling constant with composi- tlOn. 3. Magnetic Effects Considerable difficulties are often encountered in determining the value of y' for solid solution alloys of ferromagnetic with antiferromagnetic or nonmagnetic metals. For instance, it was found [4] for the random solid solution alloys Mn-Ni that, in addition to the elec- tronic specific heat coefficient y', the measured coeffi- cient y” of the term of the low temperature specific heat linear in temperature includes also a magnetic con- tribution yn: y"= y' –– Tym. (5) The alloy Mn Niš can be ordered by thermal treat- ment and, in the well-ordered condition, the coefficient of the linear term of the low temperature specific heat is approximately half that for the disordered alloy of the same composition. This lower value is substantially free of the magnetic contribution ym and it may be con- sidered as approximately equal to the real experimental electronic specific heat y' of the alloy. On the other hand, the larger y” value for the disordered alloy in- cludes yn. Similar magnetic contributions to y” were identified in a number of other f. c.c. solid solution alloy systems [4] and in b.c.c. Fe-Al alloys [5]. It is signifi- cant that in the same alloy systems, and at similar com- positions, magnetic measurements by Kouvel [6,7] de- tected the appearance of an asymmetrical hysteresis loop after cooling in a magnetic field through the Curie temperature (“exchange anisotropy”). In addition to this effect of field cooling on the magnetic properties, in several instances an effect of field cooling on ym was also detected [4,8], figure 2. i T* in DEG? FIGURE 2, (C-A)/T vs Tº (where C-A is low temperature specific heat less magnetic cluster contribution, see eq (5)) for alloy Nio.48 Cuo.52 cooled without a magnetic field (top graph), cooled in 14 k0e field from 300 to 4.2 K with the field turned off during the measurements (graph m) and field-cooled with the field on during measurements (graph m—m) [8]. The occurrence in the same alloys of “exchange anisotropy” and of a magnetic contribution to the low temperature specific heat term linear in temperature suggests that these two phenomena may be associated with the same structural condition. This expectation is further supported by the fact that ym is normally also af- fected by field cooling. According to Kouvel's highly successful model [7], the structural condition responsi- ble for “exchange anisotropy” is a spatially in- homogeneous magnetic state, e.g., the superposition of local “parasitic antiferromagnetism” on net overall fer- romagnetism. Overhauser [9] and Marshall [10] con- nected the magnetic contribution to the linear term of the low temperature specific heat with the location of a sufficient number of spins in a near-zero field. In Overhauser's theory this condition arises at the nodes of the static spin density waves of an antiferromagnet. Marshall pointed out that the required condition may arise in dilute spin systems, where the average distance between neighboring spins is sufficiently large, so as to make the interactions weak, as in dilute Cu-Mn alloys. The alloys considered above are neither antiferromag- netic nor dilute. However, because of the peculiar, complicated spin arrangement, resulting from the su- perposition of local parasitic antiferromagnetism on net overall ferromagnetism, it may be expected that many spins are located in small regions where ferromagnetic and antiferromagnetic exchange interactions nearly cancel each other locally, so that the average field in such regions is near zero [4]. In accordance with this “local field-cancellation” model, the effect of field cool- ing on yn may come about if the application of an exter- nal magnetic field during cooling through the Curie temperature increases or decreases the number of spins located in near-zero field. Both increase and decrease [(Mn/Nia [4], Nio,480.uo.52 [8])] were in fact observed. It is easy to visualize that the change in yn as a result of field cooling may also happen to be negligibly 559 small, even though the value of the magnetic contribu- tion yn itself is large. Thus, while the occurrence of a measurable effect of field cooling on the temperature- linear term of the low temperature specific heat may be considered as a proof for the existence of a magnetic contribution to this term, the absence of such an effect does not prove that ym is zero, or that it arises through a mechanism different from the “local field cancella- tion.” If the experimentally determined coefficient of the temperature-linear term of the low temperature specific heat includes a magnetic contribution, it is at present not possible to derive from such a y” value the “experimental electronic specific heat coefficient” y', which is free from yn. This is well illustrated by the Ni- Cu alloys, for which the coefficient of the linear term has a maximum around the composition Nio,480 u0.52 [8]. A detailed study of the properties of these f. c.c. solid solutions at compositions in the vicinity of the maximum [11] shows that the experimental values of the coefficient do in fact include a magnetic contribu- tion. It is, therefore, not possible to tell whether the maximum is entirely due to ym, or whether y' itself has a maximum, which is merely increased by the addition of yn. A maximum in y' has been expected on theoreti- cal grounds [12] because of enhancement due to the electron-paramagnon interaction [13,14]. The theory cu 24 g O Cu |- x Nix - w * (Cul-x Nix)09 Aloi 7 'a 16 |OO O E E ºr 8 5O o S |5 /\ =-||5 , 10-y: \ . #|ſo 5 Fº-s o'e- 5 2. Sº --~~~ Y- . S. / O.O5 SOCr N O Ne." O.4 O5 O.6 O7 O.8 O.9 X FIGURE 3. Coefficient y” of the temperature-linear low temperature specific heat term, Curie temperature Tc, temperature independent specific heat term A and coefficient 8 of the Tº term, obtained by least squares fitting to eq (5), for Cul-, Nir (solid lines) and for (Cul-e Nir)0.9 Alo.1 (dashed lines) alloys vs x [11,22,24]. * (VT-x Fex)o 9 Alo. o 24 Ho VI-x Fex t cy, Q) ~5 : 000 (as by interband Landau level transitions). In both types of experiments, mc * can be measured accurately. No electron-phonon enhance- ment has been observed in mo” for certain carriers in bismuth and arsenic. A. Baratoff (Brown Univ.); Well, I would expect that in bismuth, which is a semimetal, where there are com- paratively few electrons, the electron-phonon enhance- ment would be rather small and maybe you would not notice it at all. J. Schooley (NBS): The same sort of thing occurs also in aluminum. J. Callaway (Louisiana State Univ.); When you analyze the specific heat of a ferromagnetic metal, do you make allowance for the contribution of the spin waves to the specific heat? P. Beck (Univ. of Illinois): In these measurements, no such analysis has been made. For iron this was done by Rayne and Chandrasekhar [1] by using their own elastic constant measurements and our specific heat data. They found very little effect on the electronic specific heat. Most of the spin wave contribution ap- pears in the “lattice specific heat.” F. J. Blatt (Michigan State Univ.); I should like to ask if, possibly, you had extended your measurements to sufficiently low temperatures so that you could esti- mate the Einstein temperature for what obviously can- not really be a constant contribution to the specific heat? P. Beck (Univ. of Illinois): Yes, magnetic cluster con- tribution is indeed not constant. It is only nearly con- stant in the temperature range from 1.4 to 4.2 K. We did not make measurements at lower temperatures, but Scurlock [2] did and he found a decrease starting just below the range at which we are measuring. He made measurements down to about 0.3 K, and he found values quite compatible with an Einstein function. The Einstein temperature appears to be somewhere usually around 1 K. N. M. Wolcott (NBS). With regard to the comment of Prof. Blatt, we have measured the specific heat of a Cu- Ni 60-40 at. 9% alloy down to Heº temperatures and have observed the decline in the constant term in the specific heat. Fitting our results to an Einstein function gives a value of 6 - 1.5 K. [1] Rayne, J. A., and Chandrasekhar, B. S., Phys. Rev. 122, 1714 (1961). [2] Scurlock, R. G., and Wray, E. M., Phys. Letters 6, 28 (1963); and Proctor, W., and Scurlock, R. G., Proc. 11th Conf. Low Temp. Phys. (St. Andrews) 1969, p. 1320. 563 Electronic Density of States Determined by Electronic Specific Heat Measurements T. Mamiya and Y. Masuda Department of Physics, Nagoya University, Chikusa-ku, Nagoyd, Japan The superconducting transition temperatures, electronic specific heats, and Debye temperatures have been recently measured by us for 5d transition-metal alloy series Ta-Re. By making use of these data and the theoretical predictions by McMillan, we have deduced the electron-phonon coupling con- stant and bare electronic density of states. The density of states is compared with the theoretical one derived from band-structure calculations of Ta using the augmented-plane-wave (APW) method by Mattheiss. Key words: Augmented plane wave method (APW); Debye temperatures: electronic density of states; electron-phonon coupling constant: electronic specific heat: superconducting transition temperatures; tantalum (Ta); tantalum-rhenium (Ta-Re) alloys; Ta-Re alloys. 1. Introduction McMillan has recently calculated the superconduct- ing transition temperature, using the strong-coupling theory assuming the appropriate phonon spectrum of density of states, as a function of the coupling constants for the electron-phonon and Coulomb interaction [1]. Our recent experiments provide detailed information about the superconducting transition temperature Tc, the Debye temperature (), and the electronic heat capacity coefficient y for the bec 5d transition metal al- loys of Ta-Re [2]. By making use of McMillan’s theoretical predictions and our experimental data, we can find empirical values of electron-phonon coupling constant and the band-structure density of states in the 5d band. The theoretical density of states of Ta was cal- culated by Mattheiss using the augmented plane wave (APW) method [3]. Our empirical values of the band- structure density of states of Ta-Re alloys were com- pared with the theoretical ones. 2. Experimental Procedures and Results The detailed descriptions of the experimental equip- ments and procedures will appear elsewhere [2]. Our new data for the alloy series Ta-Re are reproduced in table 1 and figure 1, in which the superconducting transition temperature T., Debye temperature (), and electronic heat capacity coefficient y are listed as a function of the number of electrons per atom n. The detailed measurements on the superconducting proper- ties of Ta-Re alloys, especially the temperature depen- dences of the critical field show that they are the weak or intermediate-coupled superconductors, depending on Re concentration. In the light of McMillan’s theoretical model, we cal- culate the electron-phonon coupling constant A and the bare density of states No(0), from experimental values of T., (), and y in the alloy series Ta-Re. The method of obtaining the values of Coulomb coupling constant put will be mentioned below. TABLE 1. Superconducting transition temperature Tc, Debye temperature (3), and electronic specific heat y of bcc 5d transition metal alloys of Ta-Re series. % Second n T. (3) Sample metal ſelectrons (K) (K) (m.J/mole ( at Om ) K2) Ta 5. () 4.463 250 6.15 TaBe 2.5 5.05 3.458 261 5. 70 5.0 5.1 2.77 277 5. 12 7.5 5.15 2.08 285 5.05 10 5.2 1.49 296 4.10 15 5.3 ().75 307 3.75 20 5.4 .21 317 3.00 25 5.5 <.06 330 2.42 30 5.6 <.06 345 1.90 40 5.8 <.06 361 I , () 565 4 OO 35 O * 3OO 250 2O O FIGURE 1. The superconducting transition temperature T., the Debye temperature 0, and the electronic specific heat coefficient y, versus the number of valence electrons per atom n, for Ta-Re alloy series. According to McMillan [1], Te was given as functions of the electron-phonon coupling constant A and Cou- lomb coupling constant p": 1.04 (1 + X) | T-19- |- "TI-45 °P |TNL (ITOEN) (1) This equation holds well not only for strong-coupled su- perconductors but for the intermediate coupling ones in question. It is well known that the electronic specific heat y is enhanced by electron-phonon interactions. y is therefore written in terms of the bare density of states No(0): 27°k? y=== No (0) (1 + X). (2) If we want to find No(0), it is necessary to know the values of A and y. The empirical values of put can be found from the superconducting isotope effect. McMil- lan has determined empirically the average value of put = 0.13 for the transition metals. However, the isotope shift can be measured in only a few metals. In the case of superconductors for which the isotope effect was not determined, we can derive put from the measured electronic specific heat by making use of the expression of Morel and Anderson [4], |M. * – * TTI, in (Elſon): (3) where Eb is the electronic band width and oo is the maximum phonon frequency. Using a Fermi-Thomas model, p. was given by | A =; a” ln (1+...) ((- (1* = 3e”h°y * Arºnº p' (4) where EP is the Fermi energy and m” is the effective mass of the electron which is given by m” = (1+ \)mo. For Ta-Re alloy series, EP ranges from 6.8 to 7.4 eV, which were derived from the theoretical band structure for W obtained by Mattheiss [3]. In order to obtain put for Ta-Re alloys, we must determine put first, using A which was obtained from McMillan’s value put = 0.13 as a first approximation. Substituting this m”, the mea- sured value of y, and EP into eqs (2), (3), and (4), the new value of put for Ta was determined to be 0.111. As the Re concentration increases, the value of put decreases very slightly and becomes to 0.107 for TaoisBeo.9. Ac- cordingly, put was taken to be 0.11 through the whole Ta-Re alloy series. Substituting the value of put = 0.11, the experimental value of Tc, and the Debye tempera- ture () into eq (1), one can obtain the phonon coupling constant A. Using eq (2) together with the values y and A, the bare density of states No(0) thus can de derived. The values of A and No(0) determined are summarized in table 2 and plotted as a function of the electrons per atom ratio in figures 2 and 3, together with the other data on bec 5d transition alloys [5]. To determine the values of N for alloys which do not show any superconducting behavior in the temperature range examined, we assumed the linear relation between A and No(0) and calculated the values of A from TABLE 2. Electron phonon coupling constant A and bare density of states at the Fermi energy No(0) of bcc 5d transition metal alloys of Ta-Re series. Coulomb interaction parameter p * was assumed to be 0.11. p.” = 0.11 Sample 9% º 71. tat met SLa LeS \ No (0) (i. e †) Ta 5.0 0.62 ().80 TaBe 2.5 5.05 .56 77 5.0 5.1 .52 . 72 7.5 5.15 .48 . 72 1() 5.2 .45 .60 15 5.3 .39 .57 20 5.4 .34 .48 25 5.5 (.29) (.40) 30 5.6 (.23) (.32) 40 5.8 (. 13) (. 19) 566 |.. O | | - O TO – Re - \ • Hi-To Ta-W.W-Re -- A |nterpolated from Ta-W, W-Re - – © - — 22 O.5H - |-- – `... Aº tº ! }ºm \, f - \ { tº." O | | 4 5 6 7 H f TG n W Re FIGURE 2: The electron-phonon coupling constant A versus the . number of electrons per atom n, for a Ta-Re alloy series. Open circles shown by dotted line show the estimated values for which the interpolated value of T. was used. the measured values of y. This relation holds well in al- loys in the concentration between pure Ta and Taos Reo.2, for which we can easily measure Tc. These values of A and No(0) are shown by the dotted circles in figures 2 and 3. The corresponding values of Te which are obtained using these procedures are extremely low compared with the transition temperatures for 4d series Nb-Mo alloys [6]. The reason for the extremely low transition temperature may be that at small A values, No(0) is not always proportional to A and also Coulomb interaction parameter put can not be taken to be 0.11. 3. Discussion Mattheiss [3] has calculated the band structures of Ta and W using the augmented plane wave (APW) method. Relativistic effects have been included in Ta but not W calculation. The potentials he used were derived from superposed atomic charge densities which were determined from self-consistent calcula- tions involving a (5a)*(6s)” atomic configuration for Ta and (5a)*(6s) for W, respectively. The energy bands were determined at a total of 1024 uniformly distributed points in the bec Brillouin zones. These results were converted to the density of states by a method of weighted average of the APW eigenvalues or an inter- polation scheme. The theoretical density of states for electrons has been compared by McMillan [1] with the experimental density of states for bec 5d transition metal alloys. However, the experimental density of states he used is not complete enough but includes the |.. O | | * L O TO – Re amºr E 9 s • Hf-Td, Ta-W, > - º - § W-Re º A lnterpolated 5 - &S from -- * Ta-W, W-Re 30.5– — 2 *- § — \ \. K-ºs, w - A \ \ --- \,: - A 2. | | 4. 5 6 7. Hf TO n W Re FIGURE 3. The bare density of states No (0) versus the number of electrons per atom n, for a Ta-Re alloy series. The dotted circles have the same meaning as in figure 2. interpolated plots for several alloys of bec transition metals (Ta-W and W-Re), whose superconductivity was not detected. The theoretical densities of states for Ta and W which were calculated by Mattheiss and the experimen- tal result of Ta-Re alloys are shown in figures 4 and 5 together with other data of Hf-Ta, Ta-W, and W-Re al- loys. The agreement of the theoretical density of states for Ta metal and experimental one for Ta-Re alloys (Ta rich side) is spectacular, but those for W and W-Re alloy (W-rich side) are less exact. On the other hand, the experimental density of states for Ta-Re alloys is 10 to 20% larger than the theoretical one calculated for W, but the experimental data for W and W-Re alloys are in good agreement. It is therefore concluded that the ex- perimental data for Ta rich alloys agree well with the band calculation for Ta, and data for W rich alloys show good agreement with the theoretical prediction for W. As described by Mattheiss, the interpolation scheme yields crude density of states because the number of points in the Brillouin zone are limited and therefore some of the peaks in the density of states may be over- looked. Meanwhile, more accurate values of the Fermi energy and the density at the Fermi energy can be ob- tained in such a way that the square of the wave vector is expanded in lattice harmonics. As Mattheiss also points out, the accurate density of states for Ta, 0.65 states/ew atom does not agree well with the experimen- tal data of 0.77, but the crude estimate of about 0.80 states/ew atom agrees much better. It is interesting to note that the experimental data for not only Ta but also 567 2O O TO – Re > Mºº- • Hºf- Ta, Ta-W, S 16H- W W – Re S L (ſ) §: In 12 H CD O > K- Dr. ū § 8 F H. <ſ - H. (/) 2. 4 H. % hº O | | | _l | | -I- | | | | O5 |.. O ENERGY (RYDB ERGS) FIGURE 4. The theoretical bare density of states for W by Mattheiss and the experimental data for a Ta-Re alloy series. Ta-Re alloy series are in good agreement with the crude density of states. 4. Acknowledgment It is a pleasure to thank L. F. Mattheiss for his per- mission to use figures 4 and 5, and for communication of his results prior to publication. 5. References [1] McMillan, W. L., Phys. Rev. 167,331 (1968). [2] Mamiya, T., Nomura, K., and Masuda, Y., J. Phys. Soc. Japan (to be published). [3] Mattheiss, L. F., Phys. Rev. 139, A1893 (1965) and ibid. (to be published). ELECTRONS/ATOM 6 --- O Ta-Re - e Hf-Td, Ta-W, W-Re O,2 –––. | | | L–1 | |--|-- O 4. 8 |2 | 6 2O 24 28 STATES OF ONE SPIN/RYDBERG-ATOM FIGURE 5. The theoretical bare density of states for Ta by Mattheiss and the experimental data for a Ta-Re alloy series. [4] Morel, P., and Anderson, P. W., Phys. Rev. 125, 1263 (1962). [5] Morin, F. J., Maita, J. P., Phys. Rev. 129, 1115 (1963). [6] Weal, B. W., Hulm, J. K., Blaugher, R. D., Ann. Acad. Sci. Fen- nicae A210, 108 (1966). 568 ELECTRONIC SPECIFIC HEAT II; KNIGHT SHIFT, SUSCEPTIBILITY CHAIRMEN: A. Narath H. C. Burnett RAPPORTEURS. J. Rayne I. D. Weismcin Low-Temperature Specific Heats of Hexagonal Close- Packed Erbium-Thulium Alloys A. V. S. Satya” and C. T. Wei Michigan State University, East Lansing, Michigan 48823 The specific heats of hexgonal close-packed erbium and thulium metals, and three of their isostruc- tural alloys were measured in the liquid-helium temperature range between 1.3 and 4.2 K for examining the validity of the localized 4f-band model, on which the current theories of the rare-earth metals are based. Barring possible uncertainties in the magnetic properties of the samples and their impurity con- tents, the coefficients of the specific-heat component linear in temperature calculated from the present data range in values approximately two to twenty times the constant electronic specific-heat coefficient predicted by the above model for all the hexagonal close-packed rare-earth lanthanides. Possible ex- planations for such discrepancies are discussed. An itinerant 4f-band model based on the one-electron- band model suggested by Mott is proposed for the lanthanides as an alternative to the localized 4f-band model. Key words: Anti-ferromagnet; anti-phase domain; augmented plane wave method (APW); Curie temperature; electronic density of states; enhancement factors; erbium; erbium-thu- lium alloys; ferro-magnetic spiral structure; gadolinium; itinerant 4f-band model; lanthanides; low-temperature specific heat; rare-earth lanthanides; specific heats; spin-wave theory; thulium. 1. Introduction The rare-earth lanthanides, characterized by their in- complete 4f shells in the atomic state, have been tradi- tionally viewed as consisting of trivalent atomic cores including the partially-filled and localized 4f shells, with three 5d. - 6s2 electrons per atom forming nearly- free-electron type conduction bands [1-5]. Recent band calculations [6-10] show, however, that the Fermi surfaces of the rare earths are considerably different from those predicted by the nearly-free-electron model. Dimmock and Freeman [6] calculated the band structure of gadolinium using the nonrelativistic aug. mented-plane-wave (APW) approximation, and ob- tained a very narrow (0.05 eV wide) 4f band about 10.9 eV below the bottom of the 5a-6s conduction bands. Their results also indicate that the 5a band has a width of about 6.8 eV, and that it resembles the d band in the transition metals. Extending the APW calculations, Freeman et al. [7] computed the Fermi surface for thu- lium as being largely determined by the 5d electrons. The position of the 4f band was found to be strongly de- *Present address: IBM Components Division, East Fishkill, N.Y. pendent on the crystal potential assumed. Herring [11] has some doubts about the reliability of the values of the widths and positions of the 4f bands predicted by the one-electron calculations. He believes that there is a narrow group of 4f-like bands in the lanthanides ap- preciably hybridized with the s-p-d bands. Based on the Hall-coefficient data and a room-tem- perature specific-heat analysis, Gschneidner [12] con- cluded that the 4f electrons occupy either discrete energy levels, or very narrow one-electron bands as proposed by Mott [13]. Where the overlap between atomic orbitals is small, as is probably the case for the 4f inner-shells in the lanthanides, Mott [13] suggested that an inner band would split into sub-bands of energy levels containing only one electron per each atom. Lounasmaa [14-20) measured the low-temperature specific heats of all the lanthanides except Pm and Er. Parks [21] determined the specific heats of Dy and Er. In evaluating the various contributions to the specific heats, they both used a more or less constant electronic specific-heat coefficient based on the localized 4f-band model for all the hexagonal close-packed lanthanides. Similar measurements were reported by Dreyfus et al. [22] in a summary form for Pr, Sm, Tb, Ho, and Er, but 571 without using the above model in their analysis. While the seven localized 4f electrons per atom in Gd appear to account for the major part of the 7.5 pºp saturation magnetic moment observed in this metal, [23] the elec- tronic specific heat predicted by Dimmock et al. [6] for Gd is only 40% of the value measured by Lounasmaa [15]. This discrepancy has been attributed to an elec- tron-phonon enhancement in the metal. The concept that the 4f shells are partially filled in the lanthanides and yet the 4f electrons contribute neither to the conduction band nor to the low-tempera- ture specific heats, referred to above as the localized 4f. band model, is similar to that proposed by Mott and Stevens [24] for the transition metals in which the d electrons would be localized. Based primarily on the results of the low-temperature specific-heat work of Beck and co-workers [25], this localized d-electron model was corrected by Mott [26]. If the localized f. electron model for the rare earths is vindicated, it would be a unique case in all metals. If the 4f electrons are indeed localized, and hence do not contribute to the Fermi surface, then isostructural alloys of the Er-Tm, Tm-Yb, and Tm-Lu systems (for example) should have similar Fermi surfaces as it was assumed by Dimmock et al. [6,7] and Lounasmaa [14- 20]. All such alloys should show a constant electronic specific-heat coefficient. On the other hand, if the 4f electrons do form a band in the usual sense, and hence do contribute to the Fermi surface, then alloying thuli- um with erbium, which have complete solid-solubility in each other, should gradually increase the number of f electrons in the conduction band. The alloys should then show variations in the electronic specific-heat coefficients. The only alloy system of the lanthanides that has been investigated with the low-temperature specific-heat method is the Gd-Pr system by Dreyfus et al. [27]. They did not try to establish the localized 4f. band model, but used such a model to evaluate the hyperfine coupling constants for the alloys. The pur- pose of this work is to test the validity of the localized 4f electron model for the lanthanides by measuring the specific heats of hexagonal close-packed Er-Tm alloys at liquid helium temperatures. 2. Experimental Details Erbium and thulium metals of 99.9% purity were ob- tained from Messrs. Gallard Schlesinger Chemical Manufacturing Corporation. Pure erbium metal of ap- proximately one-fifth of a mole was arc melted under a helium atmosphere in a water-cooled copper crucible. Due to the high vapor pressure of thulium at elevated TABLE 1. Some impurity analyses of the samples in ppm by weight Element Er | Ero. 75'Tmo.25 | Ero.5Tm 0.5 | Ero.25Tmo.75 | Tm Ca.............. 18 550 93 260 170 Fe.............. 400 19 30 43 3] Na.............. 13 3800 10 29 510 Nd.............. 110 390 580 940 2000 Ni.............. 3 35 46 98 600 W............... 140 700 500 620 750 Y............... 1100 520 400 150 20 C............... 86 36 140 200 850 F................ 6300 8600 4500 3800 320 N... ... 1300 54 2] 7 1100 O... 120 470 240 490 1400 temperatures, the Er-Tm alloys and the thulium metal were induction melted in tungsten crucibles under a purified argon atmosphere. The analyses of these sam- ples are listed in table 1 as measured by Messrs. Atomergic Chemetals Inc. using the mass-spectro- graphic method. The heat capacities of the samples were measured in the liquid-helium" temperature range in an experimental set-up only slightly modified from that described by one of the authors [28]. The temperature of the specimen was monitored by means of a carbon resistor embedded in the specimen as- sembly. The carbon thermometer was calibrated against the liquid-helium vapor pressure prior to each heat-capacity measurement. The resistance R of the thermometer and the corresponding temperature of the specimen T were found to satisfy the Keesom-Pearlman relation [29] N (log R/T) 1/2 = X. Cn 1 (log R)” (1) N = () with N=1, with a scatter of less than 10 milli-degrees in all the experiments. Figures 1 through 3 show the specific-heat data plotted against temperature for the Er and Tm metals and three of their alloys with 25, 50, and 75 wt. 9% Trm. The accuracy of the present set-up was discussed by Tsang [30]; and an overall accuracy of + 2% can be expected in the specific-heat measure- ment. 3. Andlysis of Results The specific heat of the rare-earth metals and their alloys can be expressed as Cp = C, + Cae Ce -- CL-H CM + Cy-H Ca (2) 572 2OO 1–I-1–1–1–1– | |- 2 150 H - S- |- Q) Tö E w - * (ſ) |- - TS © + M- - É loo H Ero.75Tmo.25 - g K- LL] I wº- 9 |- - Hi- O ºw- Lil 0. On 50 H. - M- ERBIUM O l 1 l | l | I— O l 2 3 4 - TEMPERATURE (*K) - FIGURE 1. Specific heat versus temperature curves for Er and Ero.75Tmo.25. 2OO —I- I I I I | I I 2 150 H O ^ Q O 5 tº G © + 5 loo H & Lil sº 92 |- O Lu 0. Oſ) 50 H. O t | I | 1 | I O l 2 3 4. TEMPERATURE (or) FIGURE 2, Specific heat versus temperature curve for Ero.5 Tmo 5. where Cp and C, are the specific heats at constant pres- sure and volume, respectively, separated by the dilata- tion term Ca (negligible at low temperatures), CE = 'y'T is the electronic specific heat, CL= o(T3, the lattice specific heat, CM, the magnetic specific heat, and CN= vT *, the hyperfine contribution to the specific heat. A proper analysis of the specific-heat data for the sepa- 2OO I I n | I | I | - 2 150 H -: Q ^ J2 © E ^ gº É # Fro.25 ſmo.75 - 5 100 H. – g LL I O THUL|UM E O # - ° 50 – - O I | I | -1– | l I O | 2 3 4. TEMPERATURE (or) FIGURE 3. Specific heat versus temperature curves for Ero.25Tmo.75 ration of the various specific-heat contributions de- pends on the magnetic specific-heat contributions as dictated by the magnetic properties of the samples. Koehler and co-workers [31-33] found that erbium transforms into a ferromagnetic spiral structure below its Curie temperature of 19.6 K and that thulium adopts an anti-phase domain-type structure below 40 K. Lou- nasmaa [19,20] reported that the magnetic specific heats of thulium are 1.5 Tº and 1.98 Tº millicals/ mole/K in the 0.4 to 4.0 and 3.0 to 25.0 K tempera- ture ranges, respectively, in contrast to the Tº dependence predicted by the spin-wave theory [34,35] for the ferri- and ferro-magnetic materials, and the Tº dependence for the antiferromagnets. It may be noted that the experimentally determined magnetic specific- heat values are close to that predicted by the spin-wave theory for the antiferromagnetic materials. In the case of holmium metal, which has a ferromagnetic spiral type of structure similar to erbium below 20 K, Lou- nasmaa [14,20] obtained CM = 0.3 Tº millicals/mole/K in the 3.0 to 25.0 K range and a Tº dependence for CM in the 0.4 to 4.2 K range. Kaplan [36] suggested a linear dispersion relation between a (q) and q for small values of the wave vector q in the ferromagnetic spiral case, and treated it as being similar to that of an anti-fer- romagnet. Bozorth and Gambino [37] found that the Curie temperature decreased in the Er-Tm system up to 12% Trn, and after a discontinuous rise, the thulium- type Curie temperature followed a slowly declining 573 T I to ERBIUM m & 8 H – * # TOTAL ELEC + MAG * * 6H - o © 1. É ELEC 2 t 4 F MAG > C) NUC 2 H. * LAT O I Y T T O 4. 8 12 16 2O T2 (ok?) FIGURE 4. Cy/T versus Tº curve and analyses for Er. —I I t I 25 – Fro.75Tmo.25 — & 20H * X Q * Jº g * 15 - *: o © + 5 10 H * |- * > O 5|- sm: LAT O l —l- O 4. 8 12 16 2O T2 (ok2) FIGURE 5. Cy/T versus Tº curve and analyses for Ero.75 Trno.75. I -T I I so- Ero.5 Tmo.5 smº Ş. 4O - * © * Cº # S 30 H. - tº) Tº © 4. = TOTA 5 20– L * |- Y. O ELEC (mox) |O H. -sº Nuc ELEC (min) \ LAT O i ri =E= O 4 8 12 16 2O T2 (ok?) FIGURE 6. Cy/T versus Tº curve and analyses for Ero.5Tmo 5. —I- T I I ao. Fro.25"mo.75 gº ELEC(mox) - & 3OH O ^ Gº # TOTAL * * cº iſ 20+ - 5 ELEC(min) t cº 1OH - O | i | I O 4 8 12 16 2O T2(ok?) FIGURE 7. Cy/T versus Tº curve and analyses for Ero.25 Trno.75. | I —T -I THULIUM 25 H - &T 2OH- * 3. * ELEC. 4- MAG, # ^ TOTAL to 15H - ; ELEC É E 10- * > O MAG l * NUC LAT _amºus--" O ſº-ºm º O 4. 8 12 16 2O T2 (°K2) FIGURE 8. Cy/T versus Tº curve and analyses for Tm. trend again with increasing Trn content. One may, therefore, expect a Tº dependence of the magnetic specific heat for all the present samples. The specific-heat data obtained with Er, Tm and their three alloy samples were analyzed by a least- squares fit to CJT = y + (a + p.)Tº + vT-”, (3) where y, O., p, and v are respectively the electronic, lat- tice, magnetic and the nuclear specific-heat coeffi- cients, with no assumption made on the nature of the 4f band. The analysis was straightforward for Er, Ero.75Tmo.25, and the Tm samples as shown in figures 4, 5, and 8. The CoſT versus Tº curves for the Ero.5 Trmo.5 and Ero.25Tmo.75 samples, shown in figures 6 and 7, appear 574 TABLE 2. Specific-heat contributions of the present samples in milli-cal/mole/K Sample Electronic Lattice” Magnetic Nuclear Er 4.1T 0.063T3 1.23T3 11.73T-2 Ero. 75'Tmo.25 | 2.5T 0.0615T3 | 1.4T3 23.97-2 Ero.5 Tmo.5 8.7 -- 5.3T 0.060T3 2.4--0.87'3 9,297-2 Ero.237’m 0.75 23.3 + 7.97" 0.059T" |..................l................. Tm 13.3T 0.058T3 0.42T3 1.76T-2 *Computed from the Debye temperatures taken fi m Lounasmaa's work [14, 20). to concave downward in the middle, similar to that ob- served in Gdo.23Pro.77 by Dreyfus et al. [27]. None of the known theories seem to be able to account for such a feature. In order to obtain an upper and a lower limit of the electronic specific-heat coefficient, the Co/T data of the Ero.5Tmo.5 sample was analyzed by first fitting the values below 2.5 K to evaluate the nuclear contribution, which was then subtracted from the CoſT values for the entire temperature range. The remainder, consisting of only the linear and the Tº terms in Co., was then analyzed by treating separately the data below and above 2.8 K for obtaining ymin and ymar, respectively. The true electronic specific-heat coefficient would probably lie between these two values. For the Ero.25Imo.75 sample, ynin and Ymar were obtained by ex- trapolating the two branches of the CoſT curve to 0 K. The various specific-heat contributions thus obtained are listed in table 2. The electronic specific-heat coeffi- cients are plotted against the composition as shown in figure 9. 4. Discussion The results indicate that the y values of the samples range from 2.5 to 23 + 8 millicals/mole/K”. The constant values of 1.0 and 2.5 millicals/mole/Kº respectively pre- dicted by the APW calculations [6], and assumed by Lounasmaa [14-20] based on the localized 4f-band: model, are also shown in figure 9 for comparison. Dif- ferences between theoretically predicted and experi- mentally determined values of the electronic specific heat have long been observed in the rare-earth metals [14-20,22], but the magnitude of the discrepancies ob- served in the present alloys is astonishing. The discre- pancies have been so far attributed to the electron-elec- tron enhancement [9], the electron-phonon and elec- tron-magnon enhancements [5], and the impurity con- tents of the samples the different workers [10] used. 2 5 |- *= PRESENT WORK |l2 O5O 5 VALUE ASSUMED BY ~ LOUNASMAA_______ VALUE PREDICTED BY DIMMOCK ET AL O | L 1 O 25 5O 75 IOO % THULIUM FIGURE 9. Electronic specific-heat coefficient versus weight percent thulium. Kasuya [5] estimated that for Gd the electron-phonon enhancement may amount to 30%, and the electron- magnon enhancement, 20% of its y value. Such enhancements can account for at most a factor of two in the electronic specific heats. Lounasmaa [19] pointed out that the discrepancies in the low-temperature specific-heat data of the dif- ferent workers are not uncommon for the rare-earth metals below 4.0 K, possibly due to the differences in the impurity contents in the samples. Crane [38] re- ported an increase in the specific heats of Gd by as much as a factor of two due to the presence of 0.1% ox- ygen by weight. If similar impurity effects are present in the Er-Tm system, a factor as great as four can be in- cluded in all the corrections. This is still insufficient to bring down the y values of the Tm, ErosTmos and Ero.25Tmo.75 samples to the 1 millical/mole/K” range predicted by the localized 4f-band model. If the localized 4f-band model were valid, the 5a- band should be the major contributor to the Fermi sur- faces in the rare earths, and to their electronic specific heats. The 5d band, having a width of 6.8 eV [6], is comparable to the 3d band of a width of 5 eV, as calcu- lated by Belding [39] for bec paramagnetic Cr using the tight-binding approximation. A close resemblance can be noted between Belding's result and the experimen- tally determined 3d energy band of Beck and co-work- ers [25]. On the other hand, all the experimental evidence from the present work as well as the results of Lounasmaa [14-20] and Dreyfus et al. [22] indicate much larger y values than what might be predicted by the localized 4f-band model. A localized 4f-band model is not necessarily needed to explain the low-temperature magnetic structures in 575 N(E)? 4f Ef 5d 6s O I- \ E—- N(E), FIGURE 10. A schematic itinerant 4f-band model. the rare-earth metals any more than a localized 3d-band model is required to explain the magnetic properties of the transition metals. Instead, it is possible that a nar- row 4f band could split into an up-spin and a down-spin half-bands, which may or may not overlap. The relative positions of the two half-bands with respect to the Fermi surface may vary from one rare-earth metal to another depending upon such factors as the crystal structure, the number of 4f electrons per atom, and the exchange interaction between the electrons. One may modify Mott's one-electron band model [13] so that each half-band, not necessarily localized, is built of seven overlapping one-electron bands. The density of states of each of these half bands may contain peaks and valleys. When there is an integral number of 4f electrons as in a metal, the Fermi surface is most likely to be near a valley. Such a model would explain the nearly uniform electronic specific-heat coefficients of the pure rare-earth metals as well as the large varia- tions in the y values of the alloys observed in the present work. A schematic diagram of such an itinerant 4f-band model is shown in figure 10. Further work such as the room-temperature specific- heat measurements of the isostructural rare-earth al- loys may help to confirm the proposed model. Should this itinerant 4f-band model be established, then it would not be necessary to resort to using the electron- phonon type enhancements in explaining the discre- pancies between the theoretically predicted and the ex- perimentally obtained electronic specific heats of the rare-earth lanthanides. 5. Acknowledgment Thanks are due to the National Science Foundation for making this work possible with its grant #GK-2224. 6. References [1] Kasuya, T., Prog. Theor. Phys. (Kyoto) 16, 38, 45 (1956). [2] Kasuya, T., Prog. Theor. Phys. (Kyoto) 22, 227 (1959). [3] Yosida, K., and Watabe, A., Prog. Theor. Phys. (Kyoto) 28, 361 (1962). [4] Elliott, R. J., and Wedgewood, F. A., Proc. Phys. Soc. (London) 81,846 (1963). [5] Kasuya, T., Magnetism, IIB, 215, Edited by Rado and Suhl (Acad. Press, N.Y., 1966). [6] Dimmock, J. O., and Freeman, A. J., Phys. Rev. Letters 13, 199 (1964). [7] Freeman, A. J., Dimmock, J. O., and Watson, R. E., Phys. Rev. Letters 16, 94 (1966). [8] Watson, R. E., Freeman, A. J., and Dimmock, J. O., Phys. Rev. I67,497 (1968). [9] Kim, D.J., Phys. Rev. 167, 545 (1968). [10] Andersen, O. K., and Loucks, T. L., Phys. Rev. 167, 551 (1968). [ll] Herring, C., Magnetism, IV (Acad. Press, N.Y., 1966). [12] Gschneidner, K. A., Jr., Rare Earth Research, II, 153, Edited by Eyring (Gordon and Breach Inc., N.Y., 1965). Mott, N. F., Phil. Mag. 6, 306 (1961). Lounasmaa, O. V., (Ho:) Phys. Rev. 128, 1136 (1962). Lounasmaa, O. V., (Gd, Yb:) Phys. Rev. 129, 2460 (1963). Lounasmaa, O. V., (Pr, Nd:) Phys. Rev. 133, A211 (1964). Lounasmaa, O. V., (Lu:) Phys. Rev. 133, A219 (1964). Lounasmaa, O. V., (Ce,bu:) Phys. Rev. 133, A502 (1964). Lounasmaa, O. V., (Tm:) Phys. Rev. 134, A1620 (1964). Lounasmaa, O. V., and Sundstrom, L. J., (Gd,Tb, Dy, Ho,Tm:) Phys. Rev. 150,399 (1966). Parks, R. D., Rare Earth Research, Edited by Nachman and Lundin (Gordon and Breach, N.Y., 1962), p. 225. Dreyfus, B., Cunningham, B. B., Lacaze, A., and Trolhet, G., Compte Rendu (Acad. des Sciences, 1964), p. 1764. Nigh, H., Legvold, S., and Spedding, F. H., Phys. Rev. 132, 1092 (1963). Mott, N. F., and Stevens, K. W. H., Phil. Mag. 2, 1364 (1957). Cheng, C. H., Wei, C. T., and Beck, P. A., Phys. Rev. 120, 246 (1960). Mott, N. F., Adv. Phys. 13, 325 (1964). Dreyfus, B., Michel, J. C., and Combiende, A., Prox. IX Intl. Conf. Low Temp. Phys. (Plenum Press, N.Y., 1965), p. 1054. Wei, C. T., Ph. D. Thesis, Univ. of Illinois, 1960. Keesom, P. H., and Pearlman, N., Encyclopaedia of Phys. 14, 297, Edited by S. Fluge (1956). Tsang, P. J., Ph. D. Thesis, Michigan State University, 1968. Koehler, W. C., Child, H. R., Wollan, E. O., and Cable, J. C., J. Appl. Phys. S32, 48 (1961). Koehler, W. C., Phys. Rev. 126, 1672 (1962). Koehler, W. C., Child, H. R., Wollan, E. O., and Cable, J. C., J. Appl. Phys. S34, 1335 (1963). [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] 576 [34] Kronendonk, J. V., and van Vleck, J. H., Rev. Mod. Phys. 30, [36] Kaplan, T. A., Phys. Rev. 124, 329 (1961). 1 (1958). [37] Bozorth, R. M., and Gambino, R. J., Phys. Rev. 147,487 (1966). [35] Gopal, E. S. R., Specific Heats at Low Temperatures, (Plenum [38] Crane, L. T., J. Chem. Phys. 36, 10 (1962). Press, N.Y., 1966). [39]. Belding, E. I., Phil. Mag. 4, 1145 (1959). 417–156 O - 71 – 38 577 Low-Temperature Specific Heats of Face-Centered Cubic Ru-Rh and Rh-Pd Alloys” P. J. M. Tsang ** and C. T. Wei Department of Metallurgy, Mechanics and Materials Science, Michigan State University, East Lansing, Michigan 48823 The specific heats of Rh, Pa, and a number of face-centered cubic Ru-Rh and Rh-Pol alloys were determined between approximately 1.4 and 4.2 K. Whereas the C/T vs. Tº plots for the Ru-Rh alloys show a straight-line behavior, a low temperature anomaly is observed in similar plots for Pd and the Rh- Po alloys below 2.2 K. This low temperature anomaly appears to be most pronounced in the alloy Rho.78 Polo.22, and diminishes with increasing Rh or Pa. The electronic specific heats of these alloys are generally high with a minimum occurring at Ruo.30Rho.70. A portion of the total density-of-states curve for the outer electronics in face-centered cubic transition metals is derived numerically from the present results and those available in the literature as a first approximation. Such a curve shows qualitative agreement with the first peak below Fermi level of the theoretical totals-d energy band of Pd calculated by Janak et al. Key words: Electronic density of states; electronic specific heat; palladium (Pd); Rh; Rh-Po alloys; Ru-Rh alloys; specific heat; tight-binding approximation. 1. Introduction The electronic structure of a metal is of importance in understanding the physical properties of the metal and its alloying behavior. Significant progress has been made in recent years in both the theoretical calculation of the electron energy bands and the experimental in- vestigation of the Fermi surfaces in metals. However, an accurate experimental method for determining the detailed structure of the electron energy bands in a metal is still lacking, and low-temperature specific heat measurement remains to be a useful method by which some knowledge of the electron energy bands in metals can be obtained. The low-temperature specific heat of a metal has two main contributions: the electronic contribution which is linear in T, and the lattice contribution which is pro- portional to T3. C = yT+ 3T3 (1) The coefficient y of the electronic specific heat is, ac- cording to Sommerfeld’s theory [1], proportional to the density of states N(EP) at the Fermi surface. *This paper is from part of a thesis presented by P. J. M. Tsang to the graduate school of Michigan State University, East Lansing, Michigan, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Metallurgy. **Now at IBM, Hopewell Junction, New York 12533. 2 y== T*k?N(Er) (2) The coefficient 3 of the lattice specific heat is related to the Debye characteristic temperature (3) [2]. 3= º T4K(S)-3 (3) By measuring the specific heats of a number of iso- structural alloys of neighboring elements in the periodic table it is in principle that a plot of y vs. the number of outer electrons per atom would reveal the qualitative nature of the electron energy bands in the metals. Such a plot can be converted to an N(E) vs E curve numeri- cally by using a rigid-band model as a first approxima- tion, and compared with the results of band calcula- tions for any agreement, or the lack of it. Keesom and Kurrelmeyer [3] first made a syste- matic investigation of the specific heats of a number of face-centered cubic Fe-Ni and Ni-Cu alloys of the 3d transition series. Further work has been carried out since with foc Pa-Ag alloys by Hoare and Yates [4], and by Montgomery [5]. Their results are in agreement with one another. The specific heats of foc Rh-Pol alloys were determined from pure Pd to Rho.5Pdo.5 by Bud- worth et al. [6]. Perhaps the most extensive investiga- tion has been that of Beck and co-workers [7] carried 579 G. F. KO Ster O | | | —l. 5 – 1.0 - 0.5 0 0.5 l. 0 l. 5 Tl Belding --- Wei, Cheng, Beck 1959 | l N- 0 0. 5 l. 0 l. 5 2.0 E in eV FIGURE 1. Density of states of the 3d bands in the transition metals for the body-centered cubic structure. out with alloys of the transition elements of various combinations and crystal structures. Figure 1 shows the total density-of-states curve derived from the specific heat data of Beck et al. [7] for the 3d bands in the body-centered cubic structure as compared with Koster's calculation [8] using the tight-binding approx- imation (inset). Belding [9] modified Koster’s calcula- tion by taking into consideration the second nearest neighbors, and obtained the stepped curve. A smooth curve drawn through Belding's curve is in qualitative agreement with the experimentally derived one as it is clearly indicated. Pessall et al. [10] repeated the measurements with bec alloys of the 3d transition elements except with the addition of 10% Al. Their results confirmed those ob- tained originally by Cheng et al. [7]. The investigation has since been extended to alloys of the 4d and 5d series and reviewed by Bucher et al. [11]. The parallel- ism in the y vs. the number-of-outer-electrons curves for the three transition series suggests that the 4d and 5d bands are similar to the 3d bands in the transition metals for the bec structure, and a somewhat rigid- band behavior in these metals. For the foc structure, the results of Hoare and Yates [4] with Pd-Ag alloys and that of Budworth et al. [6] with Rh-Pol alloys show characteristic variations in the y vs electron concentration plot that a high peak occurs at Rho.05Pdo.95, and the y value decreases sharply on both sides of this peak when more Rh or Ag is added to Pa. These features have been confirmed by Mont- gomery [5], and are believed to be characteristic of the tail portion of the 4d bands in the foc structure. The results of Keesom and Kurrelmeyer [3] obtained with foc Fe-Ni alloys and those obtained by Walling and Bunn [12], and by Gupta et al. with foc Co-Ni alloys show variations parallel to each other and to the results of Budworth et al. [6] between the outer-electron-con- centration range from 9 to 10 per atom. However, between the electron-concentration range from 8 to 9 the results of Gupta et al. obtained with foc alloys of Fe- Ni, Mn-Fe, and Mn-Ni show disparities which are at- tributed by these authors to a possible ordering and complications of a magnetic nature. The purpose of this work is to extend the measure- ment of the low-temperature specific heats of foc alloys of the second long period transition elements to the ex- tent of the electron concentration range in which such alloys exist for furthering the understanding of the na- ture of the d bands in the transition metals. 2. Experimental Procedure The specimens used in this work were melted in an inert mixture of argon and nitrogen using Ru, Rh, and 580 TABLE 1. Spectroscopical analyses of the Ru, Rh, and Pd metals used for making the alloy specimens and that of a typical specimen with the nominal composition Rho.7Pdo.8 (weight percent) Element Ru Rh PG Rho.7Pdo.3 Ag.......................... 0.001 0.001 0.02 0.0005 Al.….... 0.001 .001 .003 .0005 Au...…. .00] B...…. .001 Cu........…. .005 .001 .01 .0002 Fe............….... .01 .01 .05 .02 Ir............…. .02 .005 Mg......................... .001 .0002 .00] .0005 Mn......................... .001 .00] .001 Mo......................... .001 Ni.......................... .005 Pd.......................... .005 .005 balance | 28.0 Pt.................~ .00] .02 0.03 Rh.......................... .003 balance .003 71.9 Ru.......................... balance Si.......….... 0.001 .1 0.005 W........................... .04 Pol metals of 99.8+% purity obtained from Gallard- Schlesinger Chemical Manufacturing Corporation. Each specimen was made into two halves with each half melted at least three times to insure the homogeneity of the alloy. The total weight of each specimen was approximately one fourth of a gram- mole. The compositions of the alloys were controlled by their weights before and after the melting. It was found that the maximum loss in the weight of a specimen due to evaporation and sputtering during the melting opera- tion was less than 2%, with a more or less even evapora- tion rate for all three metals. Thus the uncertainty in the composition of each alloy was less than 2%. The al- loys were sealed in vacuum in quartz tubes, and each was annealed at 1,080 °C for at least 24 hours and then water quenched. The results of spectroscopical analyses of the metals as received and a typical alloy sample are shown in table 1. The complete equilibrium diagram of neither Ru-Rh nor Rh-Pol system has been reported. Rh and Pa are both foc and completely intersoluble at high tempera- tures [13]. Below 845 °C a concentrated alloy may separate into a phase mixture of Rh and Pa rich solid solutions. This transition is very sluggish, and can be suppressed by quenching from the single-phase region [14]. Five Rh rich Rh-Pa specimens together with one for each of pure Rh and Pa were prepared. The alloys were examined metallographically and with x rays after heat treating and found to be single-phase foc alloys. Ru has a hexagonal close-packed structure. Its solubili- ty in foc Rh has not been established exactly. It was found that the alloys Ruo.40Rho.60 was a single foc phase, and the alloy Ruo.42Rho.55 was a mixture of two phases. Thus, seven Rh-rich foc Ru-Rh specimens were made with Ru less than 40 at. 9%. The specific heats of the fourteen specimens were determined at liquid helium temperatures between ap- proximately 1.4 and 4.2 K using a calorimetric method described by Corak et al. [15]. A heater-thermometer assembly, consisted of a carbon resistor embedded in a copper disk to serve as the thermometer and a heat- ing coil of approximately 300 ohms, was sandwiched between the two halves of a specimen. The carbon ther- mometer was calibrated against the vapor pressure of the liquid helium bath during the cooling cycle of each experiment. The heat input was determined by measur- ing the heating current, the resistant of the heating coil at the particular temperature, and the heating time. The corresponding temperature change was recorded in terms of the resistance change of the thermometer. The resistance R of the thermometer as a function of temperature was found to agree well with the equation Vlog R/T = A + B log R (4) suggested by Keesom and Pearlman [16]. The parame- ters A and B in the equation were determined by a least-squares fit of eq (4) to the experimental data. The overall design of the cryostat and the instrumentation were similar to those used before by one of the authors [7]. The detailed experimental procedure and a discus- sion of the experimental accuracy were described in reference 7. The probable error in the measured elec- tronic specific heat coefficient is estimated to be ap- proximately + 2%. 3. Results Figures 2 to 5 show the C/T vs. Tº plots for the four- teen specimens measured. The curves for the Ru-Rh al- loys and the pure Rh specimen show a normal straight- line behavior with well defined y and 8 values accord- ing to eq (1). The y and (3) values are listed in table 2. All the Rh-Pol alloys and the pure Pd specimen show a low- temperature anamoly below approximately 2.2 K. Several of these Rh-Pol alloys were measured more than once, and the results were found to be reproduci- ble within the experimental accuracy. It appears than an additional low-temperature contribution to the specific heat other than those described by eq (1) is present. This anamoly is most pronounced in the alloy Rho.78F’do.22 2 and diminishes toward Rh and PC]. 581 2 | O Ruo.os Rho.es 2 O H. <[ y S. | > (1) t ^. 3.0 i 2 Cl- (ſ) LL] 2. 9 * { 2.O º H (ſ) 3H Li H 2 2H I.OH- 'ſ J. F.JANAK,ET AL. (PD) \ - !------- ** - - - –Z i i ! i •s *} -3 2: - | o=Ef od l —l | -Q 8 - O.6 –O.4 •O.2 0.0=Ef O.2 O,4 O.G O.8 E (eV) FIGURE 7. Density of states of the d bands in the transition metals for the face-centered cubic structure. Tight-binding approximation (TBA) is generally used to calculate d bands. Using TBA methods, 3d bands of paramagnetic Ni was calculated by Koster [8] and 4d bands of Pa by Lenglart et al. [2]]. Recently, a hybridized s-d band was thought to be more truthful describing the energy bands of transition metals [25]. In these calculations, [20,22-24], the TBA was used as a base to treat d electrons whereas APW or OPW method was used to treat s electrons and the s-d hybridization was achieved by certain interpolation scheme. The effect of s-d hybridization on 3d bands was that the location of the lower peaks (in reference to Eſ) of the band were altered and their magnitudes reduced, as in the case of Ni calculated by Hodges et al. [20] and by Mueller [22]. On the other hand, as shown in the total s-d band of palladium calculated by Janak et al. [23], s-d hybridization strongly enhanced the middle peak of the 4d-bands. However, there is a common fea- ture of the 3d and 4d bands that was not altered too much by s-d hybridization, namely, the strong peak in the vicinity of Fermi level which spans about 1.0 to 1.5 eV and can accommodate about two electrons per atom. And, due to the solubility limit of Ru in Rh or of Ru in Pa, this is the portion of 4d bands that can be in- vestigated by electronic-specific measurements. In figure 7, the total energy band of Pol calculated by Janak et al. [23] was inserted for comparison. As can be seen, very good qualitative agreement was indeed found between present experimental N(E) vs. E curve 584 and the first peak of the theoretical 4d bands. Good agreement was also found between present experimen- tal N(E) vs. E curve and the 4d bands of Pol calculated by Mueller [24]. The energy band calculations in the present stage of sophistication are a close approxima- tion to a true description of the electron energy states in the transition metals. We observed that the low-tem- perature-specific-heat method for investigating the energy bands in the transition metals may have some validity when suitable alloys are available. 5. Conclusions (1) The specific heats of Rh, Pol and twelve face-cen- tered cubic Ru-Rh and Ro-Pol alloys were determined between 1.4 and 4.2 K. The C/T vs. Tº curves show a straight-line behavior for all Ru-Rh alloys, but show a low temperature anomaly for Pd and Pol alloys below 2.2 K. (2) Aside from some uncertainty in the evaluation of the electronic specific heat coefficients of the Ro-Pol al- loys due to this anomaly, a plot of y vs. alloy concentra- tion together with the data available in the literature can be converted numerically to a density-of-state curve as a first approximation. The resulting curve is in agreement with recent band calculations using the tight-binding approximation for treating the d electrons in the transition metals. (3) The present results and those reviewed in this paper seem to indicate that for the foc crystal structure, a strong peak in the vicinity of Fermi level exists in both 3d and 4d bands and that the tight-binding approxima- tion is essentially the correct approach in treating the d electrons in the transition metals. 6. Acknowledgments The authors wish to thank Professor F. Seitz, Pre- sident of Rockefeller University for his interests in this work and his help. They are also grateful to the Division of Engineering Research and its Director, Mr. J. W. Hoffman, for their continuous support in this work. Thanks are also due to Drs. A. R. Williams, J. F. Janak, D. E. Eastman, and F. M. Mueller for their kind permis- sion of quoting their data prior to publication. This work was supported financially in part by National Science Foundation through the grant NSG GK (475). 7. References [1] Sommerfeld, A., Ann. Physik 28, 1 (1937). [2] Debye, P., Ann. Physik 39,789 (1912). [3] Keesom, W. H., and Kurrelmeyer, B., Physica 7, 1003 (1940). [4] Hoare, F. E., and Yates, B., Proc. Roy. Soc. (London), A240, 42 (1957). [5] Montgomery, H., unpublished, see article by F. E. Hoare in Electronic Structure and Alloy Chemistry of the Transition Elements, edited by P. A. Beck (John Wiley & Sons, New York, 1963). [6] Budworth, D. W., Hoare, F. E., and Preston, J., Proc. Roy. Soc. (London), A257, 250 (1960). [7] Cheng, C. H., Wei, C. T., and Beck, P. A., Phys. Rev., 120,426 (1960): Cheng, C. H., Gupta, K. P., Van Reuth, E. C., and Beck, P. A., Phys. Rev. 126, 2030 (1962). See also review by K. P. Gupta, C. H. Cheng, and P. A. Beck, Journal de Physique et le Radium 23, 721 (1962). [8] Koster, G. F., Phys. Rev. 98,901 (1955). [9] Belding, E. F., Phil. Mag. 4, 1145 (1959). [10] Pessall, N., Gupta, K. P., Cheng, C. H., and Beck, P. A., J. Phys. Chem. Solids 25,993 (1964). [11] Bucher, E., Heiniger, F., and Muller, J., article in Low Tem- perature Physics, edited by J. G. Daunt et al. (Plenum Press, New York 1965). [12] Walling, J. C., and Bunn, P. B., Proc. Phys. Soc. 74,417 (1959). [13] Raub, E., J. Less Common Metals 1, 3 (1959). [14] Raub, E., Beeskow, H., and Manzel, D., Z. Metallkunde 50, 428 (1959). [15] Corak, W. S., Garfunkel, M. P., Satterthwaite, C. B., and Wexler, A., Phys. Rev. 98, 1699 (1955). [16] Keesom, P. H., and Pearlman, N., in Encyclopedia of Physics, Vol. 14, edited by S. Fluegge (Springer-Verlag, Berlin 1956) p. 297. [17] Schroeder, K., and Cheng, C. H., J. Appl. Phys. 31, 2154 [18] Mºll, W., Phys. Rev. I 10, 1280 (1958). [19] Stoner, E. C., Proc. Roy. Soc. (London), A154, 656 (1936). [20] Hodges, L., Ehrenreich, H., and Lang, N. D., Phys. Rev. 152, 505 (1966). Lenglart, P., Lemam, G., and Lelieur, J. P., J. Phys. Chem. Solids 27, 377 (1966). - [22] Mueller, F. M., Phys. Rev. 153,659 (1967). [23] Janak, J. R., Eastman, D. E., and Williams, A. R., these Pro- - - ceedings, p. 181. - Mueller, F. M., these Proceedings, p. 17. Heine, V., Phys. Rev. 153,673 (1967). [21] [24] [25] 585 Density of States of Transition Metal Binary Alloys in the Electron-to-Atom Ratio Range 4.0 to 6.0 E. W. Collings and J. C. Ho Battelle Memorial Institute, Columbus Laboratories, 505 King Avenue, Columbus, Ohio 4320.1 Using Ti-Mo as a prototype of binary bcc transition metal alloys for 4 ~ eſa • 6, densities-of-states at the Fermi level, n(EP), have been studied using low-temperature specific heat augmented by mag- netic susceptibility (X) measurements. A survey of the literature has revealed that the principal descrip- tors of density of states, y (the electronic specific heat coefficient), and Te (the superconducting critical temperature), generally decrease as eſa decreases below about eſa - 4.3 to 4.5. The maxima in y and Te so induced have usually been interpreted as indicating the existence of maxima in n (EF) neareſa = 4.5 for bec alloys. But since the region elas 4.5 also corresponds to that in which a submicroscopic hex- agonal-structured precipitate (a)-phase) always appears in quenched alloys, a detailed study of micros- tructure was undertaken in conjunction with the electronic property measurements. It was concluded that a steadily increasing abundance of an o-phase precipitate was responsible for the observed drops in y, T., and X below eſa = 4.5. Because of the fineness of the precipitate (70-330 Å) the physical pro- perty results themselves are indistinguishable from those usually associated with single-phase materi- als. Using magnetic susceptibility at elevated temperatures, where the prototype Ti-Mo alloy is known to be single phase bec, it has been shown that n(EF)0cc increases monotonically as eſa is reduced from 6 to 4, in agreement with deductions based on the results of recent band-structure calculations on bec 3d transition metals. Key words: Alloys; bec transition metal alloys; charging effect; electronic density of states; G. P. zone; Ginzburg-Landau coherence length: Hf-Ta; low-temperature specific heat; magnetic susceptibility; omega phase; rigid-band approximation; superconducting transition temperatures; tantalum-tungsten (Ta-W); Ti-Mo; titanium-molybdenum (Ti-Mo) alloy; tungsten (W); W-Re. l. Introduction Important theoretical contributions to our un- derstanding of the conditions for validity of, and the limitations of, the rigid-band approximation have been made by Beeby [1] and Stern [2,3]. According to the latter [3] perturbation theory is appropriate when the difference between the atomic potentials of the par- ticipating atoms is sufficiently small, regardless of solute concentration. Conversely, large differences in atomic potentials result in what is effectively a transfer of screening charge from one kind of atom to the other, accompanied by a breakdown of perturbation theory. The so-called charging effect can be parameterized by means of the quantity (|V12|)an/A, where V12 is a mea- sure of the difference between the atomic potentials of the two types of atom, and A is the zero-order energy band-width. Stern’s theory of charging thus provides a unifying interpretation of alloy behavior in terms of dif- ferences in atomic potentials. Accordingly “similar” atoms tend to form solid-solution alloys over a wide composition range, while the binary phase diagrams for pairs of atoms widely separated in the periodic table usually exhibit a plurality of intermetallic compounds. A survey of the physical and electronic properties of pure transition metals and their binary alloys leads to the conclusion that pairs of transition metal atoms chosen from adjacent columns in the periodic table probably provide optimal conditions for the validity of the rigid-band model. It follows that it should be possi- ble to transcribe the curve of y vs. eſa," in the range | y is the low-temperature electronic specific heat coefficient, and (eſa) is the ratio of the total number of valence (s-H d) electrons to the number of atoms. For a free electron gas y” (2/3) Tºkén (EF), where n (EP) refers to the density of states at the Fermi level EP for a single spin direction (i.e., one-half of the total density of states). If the units of y are m)/mole-degº, and those of n (Er) are states/ew-atom, then y"= n (EF)/0.212. 587 (i − 1 to i+ 1), into a reasonable experimental represen- tation of the density-of-states curve, for a transition metal of the ith group of the periodic table, over a small energy interval about the Fermi energy. Near the middle of a transition series, where the greatest num- ber of electrons are available for screening we expect to see the closest conformity to the rigid-band model [4]. Indeed, as McMillan has shown [5], there is re- markable agreement between the experimental rigid- band “density-of-states curve” derived from specific heat data for bec binary Hf-Ta-W-Re alloys, and the results of band structure calculations for W by Mat- thies [6]. Implicit throughout this discussion is that some degree of similarity should exist between the density-of-states curves for transition metals of adja- cent groups. The recent work of Snow and Waber [7] has enabled, for the first time, such a comparison to be made. Their calculated n(E)” curves for the bec phases of the 3d transition series exhibit a gradual change of profile on proceeding from Ti to Fe. Some similarity be- tween the hop n (E) curves for Sc and Ti has also been noted by Altmann and Bradley [8]. The extensive literature relating to calorimetric mea- surements on cubic phase (bec and foc) transition metal binary alloys has been reviewed by Heiniger, Bucher, and Muller [9]. Low temperature specific heat has not succeeded in extending the bec y curve to eſa =4 since the bec phases of Ti, Zr, and Hf (and the dilute alloys of slightly higher eſa) are not stable except at elevated temperatures. However, the limited amount of calorimetric work that has been carried out on alloys in the range elas 5 has always suggested the existence of a maximum in n (EF) near ela = 4.5 (or a maximum in n(E) near the appropriate value of E, if rigid-band con- ditions are fulfilled). Low temperature specific heat data in the range 4 × eſa • 6 are reviewed in figure 1. In agreement with these are the results of earlier stu- dies of the superconducting transition temperatures (Tc) of transition-metal binary alloys. Maxima in To have frequently been noted in Ti-base alloys near eſ a = 4.5 [10]; and for Ti-Mo in particular, Blaugher et al. [11] have suggested that the turning point observed in To might be connected with a corresponding maximum in n(EF) for that alloy system. In addition the nuclear spin relaxation data of Masuda et al. [12] in the form (TT)-1/2 vs. eſa exhibited a rather flat maximum, located near ela - 4.5, which was correlated with a * n(E) is the density of states at energy E. Many authors, including Snow and Waber [7] consider both electron spin directions when calculating n(EA). Others consider only a single spin direction resulting in an “n(EE) for one spin directi n” which is one-half of the other value. We are arbitrarily following the latter convention, cf., our figure 12 with the figure 6 of reference [7]. |2 C O O |O H. C O *: 8 H O - CNJ ON Gº. : O º, I O\ $3. CD C O 6 H. C * 5 O -> E > W- O O- \ 4 H. 3. * Ö ſ O sº 2 H O-C Sº ū O | 4.O 4.5 5.O 5.5 6.O Electron-to-Atom Ratio, eſa FIGURE 1. Plot of electronic specific heat coefficient y versus valence electron (s-H d) to atom ratio, eſa for pure transition metals and binary alloys of the 3d and 4d series. Sources of data for the (3d-3d)4 series are: (i) Ti-V [20-85 at.% (eſa=4.20-4.85)]-ref. (a); (ii) V-Cr [10-35 at.% (eja = 5. 10-5.35)]-refs. (b) and (c); (iii) V-Cr (50-95 at.% (eja = 5.50- 5.95)]-ref. (d): (iv) W-Cr (95-99 at.% (e/q = 5.95-5.99)]-ref. (e). Sources of data for the (4d-4d) series are: (i) Zr-Rh [4-8 at.% (eſ a = 4.20-4.40)]-ref. (f); (ii) Zr-Nb (50-75 at.% (eja = 4.50-4.75)]-ref. (e); (iii) Nb-Mo [10-25 at.% (eja = 5.10-5.25)]- ref (g); (iv) Nb-Mo [40–90 at.% (eſa = 5.40–5.90)]-ref. (h) (a) Cheng, C. H., et al., Phys. Rev. l 26, 2030 (1962). (b) Gupta, K. P., et al., J. Phys, Radium 23, 721 (1962). (e) Srinivasan. T. M.. and Beck, P. A., Ann. Acad. Sci. Fennicae A VI 2 10, 163 (1966). (d) (Sheng, C. H., et al., Phys. Rev. 120, 426 (1960). (e) Heiniger, G., Phys. kondens. Materie 5, 243 (1966). (f) Dummer, G., Z. Phys. 186, 249 (1965). (g) Morin, F. J., and Maita, J. P., Phys. Rev. 129, l l 15 (1963). (h) Weal, B. W., et al., Ann. Acad. Sci. Fennicae A VI 2 1 0, 108 (1966). maximum in n(EP) based on calorimetric evidence such as that of figure 1. One of the aims of the present research was to deter- mine experimentally whether the inferred density of states maximum at ela - 4.5 was in fact a property of single-phase bec alloys (i.e., a property of the rigid-band bec density of states curve). The alloy system chosen for this study was Ti-Mo, which exhibits continuous uninterrupted solid solubility in the bec field from pure Ti to pure Mo. 2. Experimental Ti-Mo alloys of nominal composition 1, 2, 3, 3.5, 4, 4.5, 5, 7, 8.5, 10, 15, 20, 25, 40, and 70 at. 9% Mo were prepared by arc-melting high-purity ingredients, the ac- tual compositions subsequently being accurately established by chemical analysis. The ingots were an- 588 FIGURE 2, Representative optical microstructures of quenched (from 1300°C). Ti-Mo alloys. A dense martensitic structure is apparent in (a) and (b), with (c) representing the end of the field. Some twinning is visible in (d) but no second phase appears to be present at this magnification. Magnification of the original photographs–50X. nealed for eight hours at 1300 °C in a titanium-gettered argon environment and rapidly quenched into iced brine. Specimens of 4 through 7 at 9%. Mo were ex- amined by conventional optical metallography (fig. 2). The quenched alloys of up to and including 4.5 at 9% Mo were apparently almost completely martensitic in structure. The martensitic field seems to terminate at approximately 5 at 9%. Mo; and the micrograph of the 7 at. 9% alloy is clear except for some slight twinning. Ti- Mo alloys in the composition range above 6 or 7 at 9% used in previously-reported physical property studies have generally been assumed to be single-phase bec. In a search for the presence of an expected second-phase precipitate in the “clear,” apparently single-phase, re- gions (cf., fig. 2) of the present alloys, specimens of 5, 7, and 10 at 9%. Mo were examined by electron diffrac- tion and electron microscopy. The results of these ob- servations will be discussed later. The principal technique used in this investigation was conventional low temperature calorimetry (1.5-6 K) which in general yields the electronic specific heat coefficient y and the Debye temperature, 60. In addi- tion for Ti-Mo a superconducting transition was ob- servable for Mo concentrations greater than 5 at 9%. Low temperature calorimetry also shows clearly the degree of sharpness of a superconducting transition, and through the relative height of the specific heat jump, the approximate abundance of the superconduct- ing component in a mixture. As a result of the metallur- gical studies we were able to show that of the quenched alloys only those of concentrations greater than about 20 at 9%. Mo could be regarded as completely single phase bec. In the low concentrated alloys, in which the bcc phase is stable only at elevated temperatures, den- sity-of-states was gauged by magnetic susceptibility (up to 1140 °C), after having established that for this alloy system at least magnetic susceptibility was, under the circumstances, a reasonably reliable substitute. 3. Results and Discussion 3.1. Low-Temperature Specific Heat The low-temperature specific heat data are sum- marized in figure 3 where they are compared with the results of previously-published calorimetric measure- Electron-to-Atom Ratio, eſq 4.O 4.2 4.4 4.6 4.8 | I | I | T | i 5 V x * 4. T C 3 >< o. O 8 H — 2 Hº V W x 7 H. — I x cº- ~y § D(Tc) 3 6 H !o * o E 5 H > uſéb) E 4. 4OO > D (Y) 3 H º 6 _^ 350 2 H. D * A QC V x — 3OO x x — 250 l | –– | –– l O IO 2O 3O 40 Atomic Percent, Mo FIGURE 3. Low-temperature specific heat data for bec and (bec + o-phase Ti-Mo alloys. or this work: A-ref [13]: V-ref [14]: x-ref [15]. The open squares refer to pure hºp Ti, the data being obtained in the present experiments excepting for T. which is due to R. H. Battſ Ph.D. Thesis, University of California. Berkeley (1964). 589 i | I | f Atomic Percent Molybdenum 2O H. X A. 5 *s-, X o 7 X © 8.5 / A |O X- X X |5 – 2 | FIGURE 4. Low temperature specific heat data for Ti-Mo alloys plotted in the usual format, C/T versus Tº. A sharp superconducting transition was found in all the alloys of concentrations greater than 7 at.% Mo. This is exemplified in the figure for Ti-Mo (7-15 at.%). ments [13-15]. There is fortunately no serious numeri- cal disagreement between the present results and those of previous authors. Figure 3 shows that the behaviors of the three properties Tc, y, and 6b are, in the absence of further evidence to the contrary, entirely consistent with a bec density of states which rises to a rather flat maximum at about ela - 4.5. To and y are seen to have their characteristically similar trends, a property which follows from the BCS-Morel [16] relationship: To/60 cc exp {-1/n (EF) V}, (1) where V is the electron pairing potential. In addition y and 60 have their frequently-observed [9] inverse rela- tionship, as do Te and 6p.” Another property usually as- sociated with homogeneous alloys is a sharp supercon- ducting transition. Figure 4 shows Tc to remain sharp * In other words, as pointed out by McMillan [5] there is a variation of coupling constant with n(EE) through a relationship between density-of-states and elastic stiffness. O O3OO | | \ _- %, Mo O.O |OOH O.OO5OH- O.OO3OH- O OO | OH O.OOO5 | O O.5 !/(0.212 y), ev-atom FIGURE 5. Plot of log (T./0p) versus (0.212 y) the slope of which according to eq (2) yields Wapp., an apparent electron-pairing potential; which is directly usable in the electron-phonon enhance- ment factor (1 + 0.212 y Vapp.) of eq (7). throughout the entire concentration range 7-15 at. 9% Mo. In the presence of electron-phonon interactions the measured y is enhanced, beyond what would be ex- pected for a free electron gas of density n(EP), by a fac- tor (1 + \) in which it is usual to assume X = n(EF) V. Equation (1) may therefore be re-written: To/60 cc exp {– 1/0.212 Yapp.) 2 (2) where Wapp. = V/(1 + \). It follows that a plot of log (T./00) vs. (0.212 y)- should be linear for any alloy system in which Wapp. is constant. As figure 5 shows such a linearity is exhibited by the entire alloy series from hop Ti through Ti-Mo (40 at. 9%) with an apparent electron interaction potential, Vapp., equal to 0.26 eV- at Orn. To summarize, on the basis of all the low-tempera- ture physical properties described above it would ap- pear that the structures of quenched Ti-Mo (> 7 at 9%) are not disturbed by the presence of a second phase, 590 and that the rigid-band bec density of states curve does in fact possess a maximum at ela - 4.5. 3.2. Microstructure Studies By now a considerable bulk of metallurgical evidence points to the conclusion that submicroscopic precipitation does occur within the bec regions of quenched Ti-M and Zr-M alloys (where M represents a transition element from groups V to VIII) in the com- position range corresponding to eſa S4.5. The occur- rence of this so-called o-phase precipitate in Ti-base al- loys has been discussed and reviewed in recent papers by Blackburn and Williams [17] and Hickman [18]. Although the detailed mechanism of its formation from the bec matrix is not yet clear, it has been pointed out by Boyd [19] that the transformation kinetics are analogous to G. P. zone formation; i.e., below some sharp solvus temperature the second phase forms rapidly as a high density of precipitates. The o-phase has a complex hexagonal structure which exhibits par- tial coherency with the parent bec lattice. According to Hickman [18] it is possible for the o-phase to be of the same composition as the matrix, in which case its rate of formation is too rapid to be suppressed even by quenching from the bec field. In quenched alloys the precipitated particles should increase in abundance as the solute concentration is reduced below the formation threshold, which for Ti- Mo alloys seems to be in the vicinity of 15 at. 9% of Mo. A correlation therefore exists between the drop in den- sity of states (as gauged by Tc and y) and the expected fraction of precipitated o-phase. To substantiate the correlation, it remains to demonstrate the presence of a)-phase in the specific heat specimens themselves. Electron microscope observations were made on samples taken from quenched specific heat ingots of compositions 4.5, 5, 7, and 10 at. 9% Mo. Typical results, those for Ti-Mo (5 and 10 at. 9%) are presented in figures 6, 7, and 8. Figure 6 is an electron diffraction photograph from Ti-Mo (5 at. 9%). Two bright spot pat- terns are superimposed: a rectangular arrangement of round spots from the bec matrix; and groups of elon- gated spots” originating from the o precipitate. The precipitate itself” may be selectively photographed against a dark background by forming an image from one of the o-phase diffraction spots. Dense precipita- tion is seen in both the 5 at. 9% (fig. 7) and 10 at. 9% (fig. 8) alloys, the particles becoming smaller in size but * According to Blackburn and Williams [17] the spot elongation is evidence of a hexagonal atomic structure and an ellipsoidal precipitate morphology. * Actually only one-quarter of the precipitate can be visualized, at one time, by this technique. more densely packed as the Mo concentration is reduced. Measurements taken from figures 7 and 8 showed the particle diameters to vary from 70-130 Å (5 at 9% Mo) to 170-330 Å (10 at 9% Mo). The structural studies have demonstrated that as ela decreases below about 4.5 the decreases in the quan- tities Tc and y correlate with an increasing proportion of a)-phase observed to be present in the same specific heat specimens. 3.3. Magnetic Susceptibility Magnetic susceptibility measurements on pure Ti have shown that the transformation from hop to bec at 883 °C is accompanied by a relatively large increase in total magnetic susceptibility [20] and presumably n(EF). It was, therefore, postulated that n(EP) for Ti-Mo alloys lay on an extrapolation of the data for the region ela - 4.5; and that the observed maximum near ela ~ 4.5 was induced by the presence of o-phase in the lower concentration alloys. If this were so, removal of the o-phase should restore n (EF) to its expected ex- trapolated value. Since the formation of the precipitate cannot be suppressed by quenching, the experiments on single-phase bec alloys would need to be performed at elevated temperatures, which eliminates low- temperature calorimetry as a technique. However, this region can be explored using magnetic susceptibility. But in order to be able to employ this technique as a valid substitute for calorimetry, it is first of all neces- sary to establish a suitable relationship between sus- ceptibility and specific heat for a series of the quenched alloys. 3.4. Magnetic Susceptibility and Specific Heat For free electrons it is well known that Xspin º 29% n (EF), (3) where pºp is the magnetic moment per spin (Bohr mag- neton); n(EF) is our conventional Fermi density-of- states referring to a single spin direction; and the con- stant of proportionality depends on the units employed. We have already shown that for free electrons: n (EF) = 0.212 y”. (4) It follows that Xspin = 13.71 y”, (5) in which, if y” is expressed as m)/mol-degº, Xspin appears as p, emuſ mole. y” may be derived from the ex- perimentally-obtained y, by allowing for electron- 591 FIGURE 6. Electron diffraction pattern from Ti-Mo (5 at.0%). Two principal types of spot patterns are superimposed: a rectangular arrangement of (twelve) bright round spots; and groupings of elongated spots originating from the w-phase precipitate. The rectangular pattern of faint round spots is due to a different orientation of the w-phase. phonon enhancement (1+A) present in a real crystal, in the following way: y"= y/(1 + \). (6) where A = n (EF) V. Using (4) and re-applying (6), A = 0.212 y V/(1 + \) = 0.212 y Vann (c.f., (2)|. The required experimentaly" is therefore y" = y/(1 + 0.212 y Vanº), with (7) Vamp - 0.26 eV - atom. The density-of-states may then be obtained by using eq (4). The chief components of the measured total mag- netic susceptibility are given by Xota = xspin + Xorn. Dark field electron micrograph showing a phase in Ti-Mo (5 at .9%). Inset is the preliminary electron diffraction pattern (shown enlarged in fig. 6), with the originating a spot indicated by the arrow. By this technique the bright patches of the photograph are specific to a phase; however, only one-quarter of the precipitate is visualized at one time. FIGURE 7. where Xoro... the orbital paramagnetism, has been discussed by Clogston et al. [21], and others. In the present work, Xorn, was derived experimentally by com- paring Xtotal (room temperature) with Xspin [calculated from the measured y using eqs (5) and (7)] for single- phase bec alloys (100-20 at 9%. Mo) and then extrapolat- ing into the two phase region." Figure 9 compares yº with Xspin for quenched Ti-Mo alloys. Agreement between the curves has of course been forced in the single-phase bec field between 100 and 20 at 9%. Mo as described above. But below 20 at 9%. Mo there is a simultaneous drop in both y” and Xºpin, which in this range is taken to be given by Xota (measured)-Xorn. The continued agreement in this latter region is sufficiently convincing to suggest that Xspin may be used, with a reasonable degree of confidence, as a substitute for y” in the high temperature measurements to be described. Magnetic susceptibility measurements were made at temperatures of up to 1140 °C. After reaching struc- * This extrapolation was practically horizontal yielding for bec Ti-Mo (0-20 at Ø xor, − 132 a emulmole. This compares favorably with xora, a 1500 emulmole for bec Ti-V alloys containing less than 70 at 9% V (and including, we assume, bec Ti) according to N. Mori [J. Phys. Soc. Japan 20, 1383 (1965)]. 592 " * wº ". º - - º Nº. º * º º º - º - º º * * sº ". . . - FIGURE 8. Dark-field electron micrograph showing al-phase in Ti-Mo (10 at.%). Inset is the preliminary electron diffraction pattern, with the originating a spot indicated by the arrow. tural equilibrium in the bec region the alloys exhibited almost independent susceptibility as shown in figure 10. Figure 11 compares the extrapo- lated room temperature total susceptibilities of single- phase bec Ti-Mo alloys with those in which al-phase precipitation has occurred. Clearly, as eſa decreases below about 4.5. Xota (bcc) continues to increase monotonically on an extrapolation of the curve for 4.5 temperature < eſa - 6: whereas the curve for Xota (bcc -- a) turns over and proceeds downwards as the proportion of al- phase increases. It is concluded as a direct result of the susceptibility work that the turning points at ela = 4.5 which ap- pear in the x, y, and Tº curves for quenched alloys are induced by the formation of an o-phase precipitate. 3.5. Analysis of the Experimental Data The essential semi-quantitative features of the ex- perimental results are immediately obvious from an in- spection of figure 11 followed by a visual extrapolation of the y' curve of figure 9. However, in order to make the best possible quantitative comparisons between the experimental X and y results and theoretical predic- tions, our data has been analyzed in a manner which is Electron-to-Atom Ratio, eſa 4.O 4.4 4.8 5.2 5.6 6.O T | i | I | I | T TI I I I I I I a 132E--- - o 100. 5 look – - E - - ~ I - S. 50k. - £ - - 8OH- ׺ 1–1–1–1–1–1–1–1 8 | O IO 20 50 IOO - - Atomic Percent Mo 4. 6 O spin - Y3—- | ol— | | | l | l | I L I O 5 to 15 20 40 60 80 ſº Atomic Percent Mo FIGURE 9. Reduced magnetic susceptibility and specific heat data for bec and (bcc-o) Ti-Mo alloys. O-reduced experimental data y' = y/(1 + \); O –Xsºn, calculated from y' in the range 100-20 at 9%. Mo: D (inset)—xor, calculated from Xota-xsºn) in the range 100-20 atº Mo, and extrapolated; E –(ximal-xon.) for (bce-w) alloys (note the un-forced agreement of the - and O. data below 20 at.” Mo); A. A –xsºn and y” for single-phase bec, respectively. best described with reference to the appropriate row numbers of table 1. From the low-temperature specific heat results, y” may first be calculated using eq (7). This is plotted (0- 100 at 9%. Mo) as the lower curve in figure 9 and is listed (bcc only, 20-100% Mo) in row 8. From y”, Xspin may next be calculated [eq (5)] and is listed (20-100% Mo) in row 9. Using the measured Xota for this concentration range, a Xorn, may be derived (row 10) and extrapolated to lower concentrations (row ll). From the results presented in figure 10 a set of values of Xota (bcc: 0-20 at. 9% Mo; 300 K) may be obtained by extrapolation. Using these values (row 5), and the extrapolated Xorn, (row ll), the corresponding Xspin is derived (row 12). These values are plotted to form the upper susceptibili- ty branch in figure 9 and appear to fall more or less on a continuation of the curve for Xspin (20-100 at 9%. Mo). y” (0-20 at 9%. Mo) is next calculated from eq (5) and is listed in row 13. These values are also plotted in figure 9. Referring to the single-phase bec data, the Xspin and y" curves of figure 9 must of course follow each other over the entire concentration range since they are con- nected throughout by eq (5). By this technique we now 593 417–156 O - 71 - 39 | | | | | | | _ – - - ooooo--o o s -- ~~ *=" §§ -5-2 rºy 2-r _- – - D E 4.6H- _* [...] —D— 3 Gl) .” S- _ — — ov-o-o-o-o- -o- 5 g44H ºr — —D-R-C-E-G-B-D-G. --- 7 T > -- T | I _ — -o-o-o-o-oo-o-o-o- -o- 84. F 42H -- _- - tra-º-º-º- -º- io - O _- T O -D 4.O H. == ‘5 _- T -o-o-o-o-o-o- -o- 5 > 3.8 – ~~ *: * . . .--------- C. Q1) O g 3.4H *= Oſ) 4— O-O Poor -o- 25 C - 3.2 H. sº Gl) 5. of . 9%. Mo # 3.0H | * * * * *-*… -- —D- 4O 2.6 | | | | | | | 3OO 500 700 900 ||OO Temperature,”K |3OO 15OO FIGURE 10. Temperature dependences of susceptibility of single- phase bec Ti-Mo alloys. Whereas the quenched 40, 25, and 20 at.% alloys are bec at room temperature, the lower temperature limit for bec Ti, for example, is 1156 K. Susceptibilities have been extrapolated back to 300 K by guessing that the temperature dependences are always similar to that for Ti-Mo (20 at.%). have a complete set of y" data with which to compute n(EP) for single-phase bec alloys using eq (4). The final results are listed in row 14 and plotted in figure 12. 4. Theoretical and Experimental n(EF) Values 4.1. n(EF) for bec and hcp Pure Titanium An immediate result of the above analysis is an ex- perimental value for the Fermi density-of-states for bec Ti, viz 1.54 (eV-atom) ". Again, applying eqs (7) and (4) respectively to the results of specific heat measure- ments on pure hop, Ti yields yhoppi"= 2.84 m.J/mole- degº; and n(EP)hoppi = 0.60 (eV-atom)−1. It follows that |n(EP) coln(EP)hop] erp. T 2.55. Band structure calculations have been carried out for both the hop and the bec phases of Ti by Altmann and Bradley [8] and Snow and Waber [7], respectively. The resulting density-of-states curves are juxtaposed in figure 13. A comparison of values at the Fermi level yields |n(EP) coln(EP)hepl theo. T 2.1. 4.2. Theoretical and Experimental Rigid-Band Density-of-States Curves An experimentally-derived curve of n(EP) vs. eſa (4 × eſa - 6) obtained from the single-phase bec Ti-Mo 24O | | | G) O X (3OO 9K) yº totd! * © *total (300 °K, extrapolated from high-temp. w bcc field). 2OO - —||8 # º E -U ^ | ity q2 5 £ S- B wº E — 16O 6 . o 5 *> × ~5 ~5 Sp Sp 5 J (ſ) C º # > |20 4. 8O | | | O 2O 4O 6O 8O |OO Atomic Percent Mo FIGURE 11. I/sing the extrapolated data from figure 10, it is seen that Xtotal (sixgle-phase, bec, - 20 at.% Mo)—O, fall on a mono- tonically-increasing continuation of the curve for Xtotal (bcc, 100-20 at.% Mo)—O; whereas the presence of a)-phase (in Ti-Mo < 15 at.% Mo) depresses the susceptibility—G). Note the remarkable agreement between the total measured molar susceptibility (OG)) and the measured (molar) electronic specific heat coefficient (A). data is presented in figure 12, for comparison with Snow and Waber’s [7]. “calculated” curve deduced from the calculated n (EF) values for Ti, V, Cr (and Mn). There is clearly a qualitative similarity between these “rigid-band density of states” curves. It may be possi- ble at least to reduce the quantitative discrepancy be- tween the experimental and calculated n (EF) curves by attacking some of the more obvious sources of error. For example on the experimental side, n (EP) for pure Mo (and possibly the Mo-rich alloys) may be too large through a possible underestimate of A, which was de- rived by assuming Vapp. = constant= 0.26 eV-atom for the entire Ti-Mo series. If instead we use for A the value quoted by McMillan [5], the experimental value of n (EP) for pure Mo is reduced from 0.367 to 0.278 (eV- atom) ', which draws the curves closer together in the range 70-100 at. 9% Mo. On the other hand, in the Ti- 594 TABLE 1. ANALYSIS OF THE EXPER!MENTAL DATA FOR SINGLE-PHASE bec Ti-Mo AL LOYS ! | Nominal At,%-M0 100 70 40 25 20 15 10 8.5 7 5 3 2 l 0 0 Comments (bcc) (hCp) 2 || Actual At.%-MO 100 71.02 || 39.81 25.36 | 19.38 || 14.92 || 10.30 8.86 || 6.96 || 5.16 2.87 | 1.9l 0.99 || 0 || 0 || Chemical analysis 3 || Moldr wt. (g) 95.95 || 82.03 || 67.03 || 60.09 57.22 55.06 52.85 52.16 || 51.24 50.38 49.28 || 48.82 || 48.37 47.90 |47.90 300°K °K 4|x. (ſem/mole, 844 322 ||904 |1997 |202.8 Measured (300 K) 300°K Extrapolated values from 5|x}. (Hemu'mole) 210.9 |216.7 |218.0 219.8 221.8 223.7 |227.0 |226.9 231.4 || – Figure 10 6|y(m/mole-degº 1.85 2.65 6.1 | 7.0 7.1 3.36 ||Measured (1.5-6°K) A = 0.212 y V, 7 A 0.102 || 0.146 || 0.336 || 0.386 0.39|| 0.185 Vºžeºn 8|y°(m/mole-degº | 1.68 2.3| || 4.66 5.05 || 5.10 2.84 ||y°-y/(1+A) -- ,0 9 Xspin (u emu/mole) || 23.0 31.7 63.9 || 69.2 69.9 Xspin ºf 13.71) 300°K 10|x. "Quemu/mole) 6|.4 || |00.5 || ||26.5 || |3|. |32.9 Xorbi XTotal TXspin 300°K Extrapolated values from 11|x. "Q1 emu/mole) 132 132 ||32 || 132 132 |132 132 |132 132 T ||Figure 9 (inset) 12 xºnºmymolº 18.9 | 84.7 |5.0 87.8 89.8 |31 |350 |949 |994 | – |xº-xrº-x, 13|y"(m/mole-degº 5.75 | 6.18 6.27 | 6.40 6.55 6.69 || 6.93 6.92 7.25 || – *-x.in/.3.7. 14|n (EF) (1/ev-atom) 0.36 || 0.49 0.99 || 1,07 | 1.08 || 1.22 | 1.31 | 1.33| 1.35 | 1.39 | 1.42 | 1.47 | 1.47 | 1.54 0.60 |n(EF) = 0.212)." rich region, the calculated n (EP) may be too small; for as Snow and Waber [7] have implied, excessively wide energy-intervals in a calculated energy histogram will reduce the apparent height of any sharp energy peak. Thus n (EP) for foc Ti which does seem to be situated at such a peak, may be underestimated in the calculations. 5. Summary In quenched single-phase bec Ti-Mo alloys, as Ti is added to Mo, the various quantities that parameterize the density-of-states viz X, y, and To increase as ela decreased from 6 to about 4.5. But, for ela - 4.5 a. second phase precipitate (a)-phase) inevitably appears in the quenched alloys. In this range n(EF) decreases as the proportion of a)-phase increases. That there is in- deed a causative relationship between these effects has been verified experimentally with the aid of magnetic susceptibility measurements at temperatures suffi- ciently high to dissolve the precipitate. For the (bec + o) alloys the observed reduced values of y would nor- mally be interpreted as the weighted averages of Yucc for the matrix, and yo, for some macroscopic precipi- tate. But this does not satisfy the requirements im- posed by the observed superconducting behavior. In the (bec-H a material the height and sharpness of each specific heat jump indicate that the superconducting transition is experienced uniformly by the entire speci- men. Were it not for the fact that the ay-phase precipi- tate can be visualized and measured by means of electron diffraction and electron microscopy, macro- scopic physical properties observations alone would lead to the conclusion that the (bec-H a material was single phase. This apparent paradox is undoubtedly related to the smallness of the precipitate size. Varying within the range of about 70-330 A, the precipitate diameter is commensurate with a Ginzburg-Landau coherence length for Ti-Mo [14]. It is postulated that detailed microstructural investigations of other Ti-M and Zr-M alloys, which have been found to exhibit ap- parent n (EF) maxima near ela - 4.5 would also reveal the presence of a)-phase below this composition. The curve of n (EF) vs. eſ a for single-phase bec Ti-Mo alloys 595 O.7H- | -: - \ |\ O.6H- | \ - | \ | | | | T. O.5H | | - 5 I W E hcp | | | / | º | T. | | LL] f | | * os| j\ ! * { \ } | f ! \, } \ / A * O.2 H. - r/ | / / | W / / O.] H. / - bc.c / | /*~~ f ~~ ~...~~~ | O -----' | | -O.6 -O.5 -O.4 -O.3 -O2 -O. | EF O. | E , ryd FIGURE 12. Calculated density-of-states curves for bec (broken line) and hcp (full line) Ti with energy referred to the Fermi level. The bec curve is due to Snow and Waber [7] and the hop curve to Altmann and Bradley [8]. Note that n (EF) in this paper arbitrarily referes to one spin direction ([7] and [8] use both spin directions). increases monotonically as ela proceeds from 6 to 4, as does a “calculated” curve based on n (EP) values for Cr, V, and bec Ti. Comparing the experimentally- derived density-of-states values for the bec and hcp allotropes of Ti we find that [n (EF)0cc/n (EF) hop]erp = 2.55, compared to a theoretical value of 2.1 based on the published results of band-structure calculations. 6. Acknowledgments We wish to acknowledge Messrs. C. J. Martin, R. D. Smith, and G. W. Waters for expert technical assistance: Dr. J. E. Enderby for stimulating discus- sions, and Dr. J. D. Boyd for his generous help with the microstructural studies. For financial assistance. We wish to acknowledge the Air Force Materials Laborato- ry, Wright-Patterson Air Force Base; and Battelle- Columbus Laboratories. 7. References [l] Beeby, J. L., Phys. Rev. 135, A130 (1964). [2] Stern, E. A., Physics 1, 255 (1965). [3] Stern, E. A., Phys. Rev. 144, 545 (1966). [4] Friedel, J., discussion in Phase Stability in Metals and Alloys, P. S. Rudman et al. Editors (McGraw-Hill, New York 1967), p. 162. [5] McMillan, W. B., Phys. Rev. 167,331 (1968). | . 6 | | | — -O- Present do to '- º- – Reference [7] == E S C I > Q) * | 2 T. Lil C C .9 *º- O Q) .*- ~5 .5 Cl Oſ) O.8 Q) C O *- O *º- (ſ) Sp C •º- U) I *}- O I > O.4 :- U) C QD -O E *- Gld Li- O | | | 4.O 4.5 4.O 55 6.O Electron- to — atom ratio , e, a FIGURE 13. “Rigid-band density-of-states curves” for 4 × eſa • 6. An “experimental” curve passes through the Fermi density-of-states data points for Ti-Mo alloys (table 1. row 14). for comparison with a “calculated” curve band on n (EP) values for Tibet – \ – Cr-\ln (after [7]), [Note: n (EF) is the single-spin-direction density-of-states.| [6] Matthies, L. F., Phys. Rev. 139. A 1893 (1965). [7] Snow, E. C., and Waber, J. T., Acta Met. 17,623 (1969). [8] Altmann, S. L., and Bradley, C. J., Proc. Phys. Soc. 92, 764 (1967). [9] Heiniger, F., Bucher, E., and Muller, J., Phys. kondens Materie 5. 243 (1966). [10] Matthais, B., Compton, V. B., Suhl, M., and Corenzwit, E., Phys. Rev. 115, 1597 (1959). Blaugher, R. D., Chandrasekhar, B. S., Hulm, J. K., Corenzwit, E., and Matthais, B. T., J. Phys. Chem. Solids 21, 2521 (1961). Masuda, Y., Nishioka, M., and Watanabe, N., J. Phys. Soc. Japan 22, 238 (1967). Hake, R. R., Phys. Rev. 123, 1986 (1961). Barnes, L. J., and Hake, R. R., Phys. Rev. 153,435 (1967). Sinha, A. K., J. Phys. Chem. Solids 29, 749 (1968). Morel, P., J. Phys. Chem. Solids 10, 277 (1959). Blackburn, M. J., and Williams, J. C., Trans. AIME 242, 2461 (1968). Hickman, B. S., Trans. AIME 245, 1329 (1969). Boyd, J. D., private communication. Kohlhaas, R., and Weiss, W. D., Z. Naturforschg. 20A, 1227 (1965). Clogston, A. M., Gossard, A. C., Jaccarino, V., and Yafet, Y., Phys. Rev. Letters 9, 262 (1962). [ll] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] 3) 96 Specific Heat of Vanddium Carbide, 1-20 K D. H. Lowndes, Jr., L. Finegold, and D. W. Bloom Department of Physics and Astrophysics, University of Colorado, Boulder, Colorado 80302 R. G. Lye Research Institute for Advanced Studies, Martin Marietta Corporation, Baltimore, Maryland 21227 Key words: Electronic density of states: specific heat: vanadium carbide. The specific heat has been measured at low tempera- tures (~ 1 to 20 K) for four large crystals of VC, with results that indicate the presence of maxima near x = 0.85 in both the Debye characteristic temperature, 60. and the electronic density of states at the Fermi level ny ($). The behavior of 60 suggests that the maximum melting temperature of VC, occurs at a composition 'close to that of the ordered compound V6V5, rather than at the composition VCo.75 proposed recently by Rudy. The variation of Ny(£) with x has been used to obtain an estimate for the density-of-states curve in the vicinity of the Fermi level. The result is discussed in terms of the behavior expected from elementary considerations of the electronic structure. Prelim- inary measurements show that the superconducting transition temperature of these crystals is lower than 50 mK. This work has been published in Phil. Mag. 21. 245 (1970). 597 Discussion on “Specific Heat of Vanddium Carbide, 1-20 K" by D. H. Lowndes, Jr., L. Finegold, D.W. Bloom (University of Colorado), and R. G. Lye (RIAS, Martin-Marietta Corporation) N. M. Wolcott (NBS): I would like to address a question to all the speakers of the group of papers deal- ing with low temperature specific heats and to the rap- porteur (John Rayne). One usually makes comparisons of the electronic density of states in terms of the number of levels per atom per eV. And in many of these comparisons, particularly where the same electron-to- atom ratio is used, I wonder if it might not be more ap- propriate to use levels per unit volume (per ce), because, at least in the free electron model, it’s the number of carriers per unit volume which determines the electronic specific heat. In many of these cases, the curve might look quite different if one included the volume change on alloying, particularly when there is a phase change in the alloy system. A. Narath (Sandia Labs.): Would anyone care to respond to that comment? (No response). J. E. Holliday (U.S. Steel): I would like to ask two questions. The first concerns the rigid band model. Our soft x-ray band spectra, the photoemission spectra presented at this conference, and the theoretical band calculations of Altman on Sc, Ti, Y and Zr show that the rigid band model is a very poor approximation. However, there has been a number of references to the rigid band model at this conference especially in the session on electronic specific heats. In light of these re- marks I would like someone to comment on the rigid band model. The second question concerns the APW calculations on TiC by Ern and Switendick and the LCAO band cal- culations on TiC by Lye and Logothetis. The APW cal- culations of Ern and Switendick show nearly complete admixture of the C-2p and the Ti-3d bands (possibly their C-2p band is at the bottom of the Ti-3d band), while the LCAO band calculations of Lye and Logothetis put the 2p band of carbon above the Ti-3d band which would result in a transfer of electrons from Ti to carbon. This direction of transfer was rather star- tling since it is opposite to the predictions of elec- tronegativity and has been quite disturbing to chemists. The rapporteur implied that the discrepancy results from differences in the APW and LCAO methods of calculations. However, is it possible that the difference is due to the original assumptions. In other words, couldn’t the APW method show a separation in the C- 2p and Ti-3d bands with the proper assumptions. I be- lieve that both Lye and Switendick are in the audience and possibly they would like to comment on this question. Our soft x-ray band spectra, and the electron spec- troscopy results at Uppsala, on TiC show a separation of the C-2p and Ti-3d bands with the C-2p band below the Ti-d band. This will result in a transfer of electrons from Ti to carbon which is opposite to that predicted by Lye and Logothetis. J. T. Waber (Northwestern Univ.); I would like to respond to the question just raised about the validity of the rigid band model. Recently Snow and I [1] published a paper giving the band structures for the bcc and foc forms of the 3d transition metals from titanium to copper. The trends in these N(E) curves were compared with the trend of the energy eigen- values of the 3d and 4s electrons in the isolated atoms. One of the unexpected results of band calculations was that the Fermi level did not steadily increase with group number but was to the first approximation con- stant when it was compared with the vacuum level (namely, compared with the zero corresponding to separation of an electron from an isolated atom). Although, of course, the Fermi level did rise con- sistently with respect to X1 (for foc) or H12 (for bec) the bottom of the d band. Note that this zero (vacuum level) can be retained when one superimposes the free atoms to form a solid. The set of foc and bec series of W(E) curves are illustrated in figures 5 and 6 of reference 1. The two peak structure is discernible for the bec metals—the structure of N(E) for the foc phases is harder to describe in a few words. However, it is clear that the bands are not rigid when such a series of ele- ments is considered. Because of the way that the Fermi level samples the N(E) curve, one will find N(EP), i.e., the density of states at the Fermi level, does display the familiar 2- peak shape which we come to expect for 3d bec alloys following the very significant deduction that Slater made thirty years ago. However, the deduction that the 598 N(E) curve is therefore fixed or universal is not correct. The distinction is that currently we believe the d-bands change both their position and shape as a function of atomic number, whereas in 1939 Slater had only one or two density-of-states curves available to him and could not be sure of a specific trend. He advanced this idea as a working hypothesis. The newer deduction has resulted from the advice and generosity of Prof. Slater, John Wood and A. C. Switendick who made their com- puter codes available to various research workers. In a discussion I made yesterday, the N(E) curves of bec titanium, vanadium, and chromium were compared with those for the bec forms of zirconium, niobium and molybdenum. In that figure, a two peak structure can be seen for the three 3d-metals whereas the N(E) curves for the three equivalent 4d-transition metals show 3 peaks. Such results reinforce the opinion that the rigid band model is not appropriate for transition metals in general. R. G. Lye (RIAS, Martin Marietta Corp.): I have two comments: (1) The rigid band model has severe limita- tions, which suggest the need for caution in its applica- tion to alloy systems in which the composition is varied over a wide range with resultant large changes in the position of the Fermi level. This limitation also poses problems in our study of VC. However, if the electronic structure near the Fermi level of VC is dominated by energy bands derived largely from d-states of the metal atom, as we believe, then our results suggest that changing the carbon concentration from VCo.76 to VCo.87 moves the Fermi level by only 0.1 eV relative to a prominent critical point in the band structure. Thus, we believe that the method we have used gives an ap- proximate description of the electronic density-of- states curve near this critical point, but we do not speci- fy how the energy of this critical point changes, nor how the band structure remote from the Fermi level changes as the carbon content is varied. These aspects of the electronic structure require separate considera- tion. (2) As Holliday mentioned, Ramqvist [2] has mea- sured the binding energies of the carbon ls levels in various materials and has determined that they are reduced in the carbides relative to carbon. Ramqvist in- fers from his studies that there is a net negative charge associated with the carbon atoms, and concludes that this contradicts our proposed band structure for TiC [3]. I believe, to the contrary, that his results provide additional support for certain features of our band structure. In particular, we found it necessary to raise the Herman and Skillman one-electron-energies of the carbon atom by 2.77 eV for the 2p state and by 4.15 eV for the 2s-state in order to make the empirical band structure agree with experimental data. This displace- ment is in the same direction and has approximately the same magnitude as that observed experimentally for the 1s state by Ramqvist ( = 3.3 eV), so we agree to this extent. The magnitude of the displacement can be related to the charge on the atoms if the spatial dis- tribution of the charge is known. Various assumptions regarding this distribution lead to a negative charge of approximately 0.2e to 0.4e on the carbon atoms. It is important to distinguish this charge transfer from the transfer of electrons between states of dif- ferent symmetry, even though the two are interdepen- dent. In particular, because the 2p-states are elevated, electrons are transferred from them into lower-lying bands derived from the 3d-states of the metal atom. This transfer compensates for some of the negative charge that otherwise may have been associated with the carbon atoms. At equilibrium, a net negative charge (0.2e to 0.4e) remains on the carbon atoms, and some 2p electrons ( = 1 1/4) have been transferred to 3d-states. This question will be discussed in more detail el- sewhere. A. Narath (Sandia Labs.): Would Dr. Switendick care to make a very brief comment? A. C. Switendick (Sandia Labs.): I really wish we could straighten this out, because we both seem to find, no matter what the experiment is, that our results agree with them. Yet the models are vastly different in the physical interpretation of what is going on in these materials. I thought that the soft x-ray results, in which the non-metal p band seems to be well below the Fermi energy and moved down as a function to say carbon to nitrogen to oxygen in these transition metal com- pounds, showed that our results were a better model, but Bob Lye says he thinks he can get the same thing out of his model. [1] Snow, E. C., and Waber, J. T., Acta Met. 17, 623 (1969). [2] Ramqvist, L., Jernkont. Ann., 153 (1969). [3] Lye, R. G., and Logothetis, E. M., Phys. Rev. 147,622 (1966). 599 Relevance of Knight Shift Measurements to the Electronic Density of States’ L. H. Bennett Institute for Materials Research, National Bureau of Standards, Washington, D.C. 20234 R. E. Watson! Brookhaven National Laboratory”, Upton, New York 1 1973 G. C. Carter Institute for Materials Research, National Bureau of Standards, Washington, D.C. 20234 The Knight shift, 7, measures the magnetic hyperfine field at the nucleus produced by the conduc- tion electrons which are polarized in a magnetic field. Knight shifts are often dominated by the Pauli term and, in its most simple form, can be written as 7% = (a)xp. Here Xn is the conduction electron Pauli spin susceptibility which depends on the density of states at the Fermi level, N(EP), and (a) is an average magnetic hyperfine coupling constant associated with the wave function character at the nucleus, ille (0)|*, for conduction electrons at the Fermi surface. The Knight shift therefore provides, through (a), insight into the wave-function character as- sociated with N (EF). Calculations of (a) involving an averaging over k-space have been attempted for a few simple metals up to the present time. For alloys and intermetallic compounds, rather different (a)'s are experimentally observed for different local environments, indicating that ºf samples the variation in local wave-function character, or a variation in local density of states. There is no unique way of separating the local variation of W(EP) from ||f|(0)|*. In this article the methods developed for relating & to the electronic properties for most of the types of cases encountered in the literature are reviewed. We discuss “simple” metals including problems of orbital magnetism and changes in 7% caused by electronic transitions such as melting. Knight shifts and their temperature dependence in metals and intermetallic compounds involving un- filled d shells, are discussed. We give estimates of atomic hyperfine fields due to single electrons, ap- propriate to those cases where problems due to electronic configurations do not make deductions from experiment too ambiguous. A density of states curve calculated for Cu is given, showing the relative im- portance of s-p, and d character for that metal. In a qualitative sense this Cu curve implies such infor- mation for other transition metals. We discuss alloy solid solutions for the cases where a “rigid” band model might be used to explain the results, and for cases where local effects have to be taken into ac- count. The charge oscillation and RKKY approaches and their limitations are reviewed for cases of dilute nonmagnetic and d- or f-type impurities. Key words: Electronic density of states; hyperfine fields; Knight shift; nuclear magnetic resonance; susceptibility; wave functions. 1. Introduction curs at about a quarter percent higher frequency in metallic copper than in a salt, CuCl. Since then, there Twenty years ago W. D. Knight [1]” discovered that have been over 500 papers reported on the theory and the nuclear magnetic resonance (NMR) of "Cu oc- observation of this effect, the “Knight shift,” in a wide *An invited paper presented at the 3d Materials Research Sym- * Work supported by the U.S. Atomic Energy Commission. posium, Electronic Density of States, November 3-6, 1969, Gaithers- * Figures in brackets indicate the literature references at the end burg, Md. of this paper. | Also Consultant, National Bureau of Standards. 601 variety of metals and alloys. The first observation of the Knight shift is shown in figure 1. This paramagnetic shift of the resonance between the diamagnetic salt, CuCl, and the diamagnetic metal, Cu, was attributed oi kops || 5 minutes || FIGURE 1. The **Cu resonance in CuCl (upper resonance) and metallic copper(lower resonance), illustrating the Knight shift [1]. [2] to the Pauli paramagnetism of the conduction elec- trons. The shift is much larger than could be explained by the average susceptibility of the conduction elec- trons. It was proposed [2] that the nuclei sampled a concentrated local susceptibility, arising from the fact that the conduction electrons in a metal have a very large probability density at the nucleus. In its simplest form, the Knight shift (7) may be written 3 = (a)xp. (1) where (a) is an appropriate sampling of the hyperfine interaction of the conduction electrons at the Fermi surface. For noninteracting electrons, Xp is proportional to N(EF), the electronic density of states at the Fermi level, Xp = pºp”W (EF) (2) where pub is the Bohr magneton. Thus in this simple ap- proximation, the Knight shift samples, via (a), local behavior of the density of states (at the Fermi level) at a particular atomic site. In this article we will inspect in detail this relation- ship of 7 with the density of states, thereby omitting several important topics on other aspects of NMR in metals. Good review articles have appeared earlier on this broader topic [3-5]. Unfortunately, as with most of the methods for study- ing the electronic density of states discussed at this symposium, untangling the factors folded in with the density of states is not an easy task. Very often, the ex- perimental Knight shift is used to measure the factor (a), Xp having been obtained from other experiments such as electronic specific heat or bulk magnetic susceptibility. The Knight shift provides a particularly complicated weighted sampling of electronic character but with these complications comes the possibility of obtaining unique information which is otherwise experi- mentally inaccessible. A more complete expression for the Knight shift would include other terms, .7% = & Paul + & dia + Zorb + higher order terms. (3) % Pauli, given by eq (1), includes isotropic and anisotropic effects, directly by contact and spin dipolar interac- tions, and indirectly via core polarization and polariza- tion of conduction band electrons below the Fermi level. The orbital paramagnetic and diamagnetic terms, 7 orb and 7 dia, are important at times. We will review in this paper the various contributions to 7%, as sum- marized in eq (3), in the light of experimental observa- tions, together with theoretical methods for relating these results to the electronic structure of metals. 2. General Observations In NMR one looks at transitions of a nucleus (with spin states m = I, I – 1, 1 – 2, . . . I – I, - I) from spin state m to m + 1, by measuring the frequency, v, of the photons involved in these transitions. The energy dif- ference between the two states, AEm-m-1 = hly, is directly proportional to the applied magnetic field, Happi. However, even for a given isotope, the propor- tionality constant is different for different solids because the electrons in the solid respond differently to Happ (paramagnetically or diamagnetically) causing an additional (positive or negative) field at the resonat- ing nucleus. This magnetization field, as seen by the nucleus, is often referred to as the “internal field,” Hint. 602 The Knight shift, º, measures the internal field at the nucleus produced by those electrons in metals which respond linearly (with one exception, noted in Sec. 6) to an applied field. Thus 7% = Hint/Happi. Specifically, this definition excludes materials with spontaneous mag- netization. For simple metals, the conduction electrons cover a broad band of energy states. Those electrons at the Fermi surface are aligned paramagnetically by an ex- ternal applied magnetic field. The resulting polarization of these electrons causes large internal fields, via the Fermi contact interaction hamiltonian, Žp, * =*uºyal sºr), (4) where y is the nuclear gyromagnetic ratio, and S(r) is the electron spin as a function of its position vector r from the nucleus. The contact (or 6-function) interac- tion samples the probability density ||4(0)|* = PA at the nucleus, for an electron in the atom. It is related to the atomic hyperfine coupling constant, aſs), by 16 a(s)=*yhup. (5) 3 The 6(r) thus restricts this effect to s-electrons and to the minor components of relativistic p-electrons. The s-effects are large, while the p-terms are almost in- variably small and will henceforth be neglected. This large hyperfine term is generally absent in nonmetallic materials, because for each s-electron of spin-up, there is an s-electron of spin-down, and these are not decou- pled by the usual applied magnetic fields. For monovalent metals, aſs) is obtained with high ac- curacy from atomic beam experiments. Values of a(s) for the alkali metals are shown in table 1. The quantity more important to Knight shift considerations, PA, is also shown. Note that PA increases monotonically with atomic number for a given group, whereas aſs) is dominated by the nuclear moment and appears ran- TABLE 1. Comparison of different ways of expressing the hyper- fine coupling of a single s-electron, for the alkali metals. a (s) (cm−1) P4 (cm−3) Hººm (k0e) "Li............ 0.0134 15.7 × 1023 122 *Na........... .0296 50.2 × 1023 390 39K............ .00770 74.6 × 1023 580 87Rb.......... : . 114 154 × 1023 1200 133Cs........... .0766 257 × 1023 2000 The data for aſs) were derived from data given by P. Kusch and H. Taub. Phys. Rev. 75, 1477 (1949); the other columns were cal- culated from these using eqs (5) and (12). dom. Except for a possible small hyperfine structure anomaly, PA is the same for all isotopes of a given ele- ment, whereas aſs) depends on the given isotope. In a metal the appropriate probability density PF is obtained by taking a suitable average over the Fermi surface, PF = (\lº (0)*), p. The Knight shift, eq (1), has shown [2] to be 8 *-*. Xplºr, (6) where X, is the Pauli spin susceptibility per atom. Sometimes an explicit volume or mass factor appears in the expression for 7%, but this depends on the ap- propriate normalization of (ilº(0))e, and on whether a mass, volume, atomic or molar susceptibility is used. If we define 8T sy. 87 p. (a) = 3 (l,(0)*)e, P. (7) then we obtain eq (1). Alternately it is convenient to introduce the effective hyperfine field Herr, which is the field measured directly in ferromagnetic materials by, for example, ferromag- netic NMR or by Mössbauer spectroscopy. Then 8 Heff F pub (a) —º pubſ’r. (8) Hence 1 *-ixºn. (9) It has been found useful to define a factor & , some- times called Knight's & factor [3], as metal - P, _ ** eff — — — — (10) t PA Hºm In the simplest cases, & has been said to express the fraction of s-character at the nucleus in the metal at EP, but as we shall see, & is more complex in its meaning for less simple cases. The Knight shift then becomes x=-1. XpéHº" (11) ALB - where - Hºm-19.65 a(s)+ º (12) ALI Here a(s) is in cm−" Hº" in Oe, and put is in nuclear magnetons. Values of H*}” are listed in table 1 for the alkali metals and in table 2 for some B-subgroup metals. The values for the monovalent metals are derived from atomic beam measurements. For polyvalent metals, Knight [3] has used measurements 603 on excited ionic states and then corrected for the degree of ionization. Knight estimated his resulting values of a(s) to be accurate to perhaps 50 percent. Rowland and Borsa [6] avoided the problem of cor- rections by using measurements on excited neutral atom states. The values for the polyvalent metals in table 2 do not depend on excited state measurements, but instead were determined by scaling, based on atomic calculations, from the known monovalent values [7]. TABLE 2. Hºff" values obtained by scaling from monovalent values [7]. Group Atom Hº" (k0e) I Cu............ 2,600 Ag............ 5,000 Au............ 20,600 II Ca............ 7,000 Hg............ 25,800 III B.............. I,000 Al............. 1,900 Ga............ 6,200 In............. 10,100 Tl.............. 34,000 IV Sn............. 12,800 Pb............ 41,400 V N.............. 3,300 P.............. 4,700 As............. 8,900 Sb............. 14,500 Bi............. 49,000 VI Te............. 17,200 X Pt............ 20,000 The § factor accounts for any deviation in hyperfine coupling from free atom behavior. It may deviate from a value of one for a variety of reasons. For example, the average conduction electron density in a metal is greater than that in the free atom (i.e., P is normalized to a Wigner-Seitz cell in the metal whereas the free atom Pº, extends over a significantly larger volume). If no other factors were present, & would then be greater than unity. A Fermi surface orbital, lip, has only partial s-character and this causes a reduction in §. In a “free electron metal,” lip is a plane wave bf (suitably orthogonalized to atomic core states) or a linear com- bination of plane waves. With increasing number of electrons in the bands the s-character of lip decreases [8]. In a metal such as Tl, Pb, or liquid Bi, this reduc- tion is quite substantial. In metals with one “free” con- duction electron (e.g., the alkali and noble metals), kr is relatively small, and the reduction could be expected to be slight. Here, other orthogonalized plane waves, dº, o (where Q is a reciprocal lattice vector) are mixed into be and the normal sign of the mixing is such that interference causes ||f(0)|* to be less than that pre- dicted by dr(0)|* alone. This, as well as d-band hybridization and core-polarization factors which will be considered shortly, tend to predominate over the normalization effect reducing & to values typically between 0.1 and 0.8 in “simple” metals. Experimental values for 7% are given in figures 2a and 2b. Using these measured values of 7%, and obtain- ing Xp as explained in the next section, the systematic trends for & seen in figure 3 are obtained.” In each period the largest & values are found for the monovalent metals, with & falling smoothly to lower values as the group valence increases, as would be expected from the wave function behavior just discussed. It is interesting that these results are obtained despite the changing crystal structures. About one-half the observed drop in & is expected from simple estimates [8] of the reduc- tion in s-character with increasing kp; the increasing atomic volumes of the polyvalent over the monovalent metals further enhances the trend. The induced conduction electron Pauli spin density may also contribute to the hyperfine coupling constant, (a), via the spin dipolar interaction Ž’sp=–2phy/Whſ . ſ: -**) • (13) rº rº where r is the vector from the nucleus to the electron. This interaction is anisotropic and contributes an orien- tation dependent Knight shift term, 7%anis, for nuclei at noncubic sites, and occasionally at cubic sites if spin- orbit coupling is present. In powders, this term results in structure and broadening of the NMR line. The induced Pauli spin density interacts directly with the nucleus only via the contact and spin dipolar interactions but it may act indirectly as well. The spin density has a spin dependent exchange interaction as- sociated with it, which arises from the Pauli exclusion principle. This may polarize the closed shells of an ion core and the paired electrons in the conduction bands below ER, producing spin densities which will then in- teract with the nucleus via the contact (and for noncu- 4 From figure 3, a & value for Zn between that of Co and Hg can be extrapolated. From this .7% for Zn is predicted to be 0.20 percent. This is in complete agreement with a prediction [9] obtained from quite different considerations. 604 I A Knight Shifts of the Melting Point ...:". H | He 2 | II A III B IV: B Ayr B AWI. B VII B Li 3| Be 4 B 5|C 6|N 7|O 8|F 9|Ne {O 453 Al 13 e e 2 loo263 932-H melting point (K) O.O.263 O.I64-H isotropic Knight shift in solid (%) - O. 164 -- . , . m º gº º f * * - No | ||Mg 12 Knight shift in liquid (%) At 13|S. 14|P |5|S |6|C 17|Ar 18 372 932 3 O. ||4 O.[64 O. |7 III. A IV. A TVIA VI. A VII A VIII A D.C. A C A I B II B O. |64 K |9|CO 20 |SC 2|Ti 22M 23|Cr 24|Mn 25|Fe 26|Co 27|Ni 28 Cu 29|Zn 3O G a 3||Ge 32|As 33|Se 34 Br 35|Kr 36 337 |356 3O3 |O92 4 o.26. O. 24 O O.155 O. 266 O.252 O. 453 O.318 Rb 37 Sr 38|Y 39 Zr 4 ONB 4 ||Mo 42 Ta 43; RU 44|Rh 45Pd 46Ag 47|CG 48 in 49|Sn 5 OSb 5||Te 52 53 |Xe 54 3|2 594 428 5O5 903 723 5 O.652 O.59 O.79 O.78 ~O.O O.66O O.80 O.789 O.76 O.7| O37 Cs 55 Bo 56 Hf 72 To 73|W 74|Re 75|Os 76||r 77|Pt 78 Au 79|Hg 80|Ti 8||Pb 82|Bi 83 Po 84|At 85|Rn 86 3O2 234 575 6OO 5.45 6 | 1.49 LG-Lu 2.7 |.. 6 |.54 |.46 2.72 |.6. |.49 |.40 Fr 87|Rd 88 7 Ac-Lw FIGURE 2a. Knight shifts in metals as compiled at the Alloy Data Center (NBS). Note: Literature references available on request. (a) Knight shifts in the solid and liquid state at the melting point. s e | N E RT I A Knight Shifts at 4 and 3OO K G A SES H | He 2 | See notes of bottom of table II A IIIB IV: B Ivº B Ivºſ. B VII B Li 3||Be 4 B 5|C 6||N 7|O 8|F 9|Ne IO 2 O.O255 if observed in nºsº if observed in liquid * FO.OO25 tº *=º- -º- qul | §§ superconducting state \ "%s", |||Mg 12 if observed at A, .#s * 14|P 15|S |6|Cl |7|Ar 18 3 O.9 high pressure .15 O. 1125 on] O. 162 ooie III. A TV. A VIA TVI A VII A VIII A D.C. A D.C. A I B II B K 19|Co. # 201S |T 2N/ 3|C 24|Mn # 25|F 26|C 27|Ni 28 9 3O 34 35 36 $ºf of o,571 || r ºf e O Ni %232 Zn k Go Se Br Kr 4 || 0.2611 || 0.3 O,254 O,69 e g g t -O. I2 | || 333" O.58O | |&#. ozsz. Jo.2 ) 37 ¥ 38 39 Zr 4 O |Nb 4 ||Nº| 2|T 43|R 44 Rh 5|Pd 46. A 7|Cd 48 Rb Sr Y r §57 C U ~O,43 º 356 3:...] - 5||Te 52|| 53|Xe 54 O.OO 5 osse os §§ O.875 oses §§ O.412" | –3.O5T | O.522| || O.4|4 O.O | - O.O26 O.695 J O,OI6 Z! Cs 5|Bo 56 Hf 2|To 73}W R 75|O | Pt A 1.59 72|To 73 Wos ſake 750s # 76||r, a f "346 hº ºft, O Po 84. At 85|Rn 86 - O.O 6 | 1.49 oº: LG-Lu |..] ] I.O6 ] –2,957 2.4 O.O __l - Fr 87|Rd 88 7 Ac-Lw |Ugº 57TCe T55Pr 59|Nd 60|Pm 6|Sm 62|Eu 63|Gd 64|Tb 65|Dy 66|Ho 67 Er 68|Tm 69|Yb 7 OILu 7| O.O 3; if | ÖğH Element No. Notes. Si | 4 Chemico shift Co 2O Predicted volue (1956) Mn 25 All data are for 3 phase Zn 3O Predicted volue Go 3 | Anisotropic shifts: K (X) = O.OI5(4) 9, ; k (Y)=-OOOI(4) 9%; K (Z)=-O.O14(4)°/o; all independent of T to 4K Sr 38 Predicted volue (1956) LO 57 F = FCC ; H = HCP OS 76 Predic fed value (1956) FIGURE 2b. Knight shifts in metals as compiled at the Alloy Data Center (NBS). Note: Literature references available on request. (b) Isotropic and anisotropic Knight shifts at room and liquid helium temperatures. 605 o solid D liquid No Cu Ag Au | n T | Sn Pb Sb Bi Te PO FIGURE 3. Behavior of Knight's & values for metallic elements as a function of their position in the periodic table. The points for each row in the table are connected using the sumbols for lines as shown outside the left lower corner of the plot. Those points shown as circles are for the metals in the solid state and those shown as squares were calculated for the metals in the liquid phase. For the former points Xp was calculated using yet values [37]; for the liquid metals Xp was taken to be 3/2(Xeror X%)using Xerpt from the Landolt-Börnstein Tables and X% from Hurd and Coodin [33]. The absolute values of § are affected by uncertainties in the estimates of Xp and Heff but the relative behavior (from element to element) along each row is probably realistic. Points for solid Ga and liquid As are omitted for lack of accurate values for 7% iso and Xp(liq), respectively. The plot provides an estimate for the Knight shift of Zn which has yet to be observed. bic sites, spin dipolar) interaction(s). These interactions arise from differences induced in the spatial behavior of spin up and spin down pairs of electrons with zero net spin induced in the electron pairs. Their existence has been established experimentally by the fact that half filled shell (p”, d”, and f") S-state atoms have nonzero hyperfine fields. While the exchange interaction is be- lieved to be the origin of this spin polarization, correla- tion effects should, in principle, be important to its quantitative behavior. These interactions have been discussed extensively elsewhere [10]. For the moment we will limit our considerations to intra-atomic contributions to the contact interaction. This necessarily involves the spin polarization of closed s-shells of the core and of the s-character in the conduc- tion bands below EF, since only these will interact directly with the nucleus. Estimates of these core polarization effects from valence electrons in the vari- ous shells are summarized in table 3. These are based on experimental data and on exchange polarization cal- culations (i.e., no correlation effects). Listed are the sign and magnitude characteristic of the core polariza- tion response to a single unpaired s-, p-, d-, or f-valence electron characteristic of various rows of the periodic table. (In the case of the open p-shell atoms, the listed response includes that associated with the closed valence s-shell.) For comparison, the direct contact in- teraction appropriate to an unpaired atomic s-valence electron is shown in table 4, for the d- and f-shell atoms. The core polarization is negative for d- and f-shells, and for np-shells, where n, the principal quantum number, is 4 or greater. The negative sign implies a core spin density at the nucleus, whose orientation is antiparallel to the unpaired spin responsible for the 606 TABLE 3. Rounded value for hyperfine fields due to the core polari- zation response to a single unpaired open valence shell electron. Comments on source(s) of core polarization values. (For fur- ther comments and details of most of the data see Ref. 10). Core polarization hyperfine field, Heft, per unpaired valence electron, k0e Open valence shell + 30 Experiment, appropriate to neutral N alone. Experiment, appropriate to neutral P alone. Experiment, (neutral As 4s.24p3 4S). Experiment (neutral Sb 5s25p3 4S). Experiment (neutral Bi 6s26p3 4S). Calculation and experiment for 3d”4s” ions. Calculation and limited experi- ment for 4d" 5s" ions. Calculation (J. W. Mallow, A. J. Freeman and P.S. Bagus, J. Appl. Phys., to be published). Inferred from hyperfine anomaly, [G. J. Perlow, W. Henning, D. Olson and G. L. Goodman, Phys. Rev. Letters 23, 680 (1969)]. Calculation and limited ex- perimental data. Estimated by calculation to make a 10–50% enhance- ment of a ns shell’s direct contact inter- action. + 15 — 50 — 150 — 300 — 125 – 350 — 750 — 600 0 to — 50 > 0 polarization. The core polarization response to an un- paired s-valence electron is positive and simply serves to enhance the contact interaction associated with the valence electron. [When using experimental atomic hyperfine data to evaluate PA, this effect is already included.] The 3d, and 4d (little is known yet for the 5d) core polarization values appear to be quite stable for their respective rows in the periodic table. The quoted values hold to within twenty percent for any member of a row and it is believed that these values are ap- propriate to the core polarization response to a d- moment in a metal. The situation is less certain and more complicated for the p-shell elements. Experimental data for which there are no competing orbital and spin dipolar terms exist only for the p" S-state atoms. It should be noted that these experimental values include the contribution coming from the polarization of the closed valence s- shells. In a metal this term is associated with the con- TABLE 4. Direct plus associated core polarization contact hyperfine fields due to a single unpaired valence s-electron for various rows in the periodic table. s shell Atoms s-contact hyperfine field, Heft, k0e 2s............ 2p1–2p3 elements 1,000– 4,000 3s............ 3p–3p3 elements 2,000– 5,000 4s............ 3d transition metals 2,000– 3,000 4s............ 4p–4p3 elements 4,000–10,000 5s............ 4d transition metals 4,000– 5,000 5s............ 5p–5p3 elements 7,000–15,000 6s............ rare earths 9,000–15,000 6s............ 5d transition metals 15,000–20,000 6s............ 6p1–6p3 elements 25,000–50,000 duction band and not the core states. There is some un- certainity as to the sign and magnitude of this core plus valence s polarization term as one goes across the 2p and 3p rows. Recent spin polarized Hartree-Fock calcu- lations of Bagus et al. [11] for these rows suggest that both the core and valence contributions are significant with the total becoming less positive (or more negative) as one goes to the lighter elements in the row. As with earlier efforts [10], these calculations do not satisfac- torily reproduce the experimental data and must be used with caution, (e.g., the wrong sign is predicted for atomic P). Bagus et al. also obtained results for the 4p row. Again the total becomes more negative by a factor of, say, two for the lighter elements but now the valence term dominates. This latter fact suggests that such atomic hyperfine constants will be of little quantitative utility when inspecting p-electron metals until one un- derstands the polarization response of s-electron character deep in a conduction band. An example in the literature of the use of a p-core polarization term larger than that shown in table 3, is Ga in AuGaº [12], where a p-term of an order of magnitude larger than that of As (table 3) was used to explain the observed negative Knight shift. One complication associated with extracting wave function and density of states information from Knight shift measurements is suggested by the numbers in table 3. Consider the 3d- and 4d-transition metals. As discussed in section 7, the Pauli Knight shift term al- most invariably has the opposite sign of temperature dependence as the Pauli susceptibility [13]. This is consistent with having d-bands at EF, with negative hyperfine constants of the sort seen in the table. Now, the s-contact densities are an order of magnitude larger than the corresponding d-core polarization hyperfine constants. Thus a few percent admixture of s-character into the Fermi surface d-states can violently affect (a). 607 While complicating matters, such interband hybridiza- tion is of considerable interest in itself and one can at- tempt to use Knight shift data to ascertain its nature and extent [14,15]. Relatively little is known of the intra-atomic hyper- fine contribution arising from the exchange polarization of conduction band states below EF, except that it probably makes a positive contribution to (a) for transition metals. Some measure of the effect can be obtained for the 3d metals by inspection of the spin polarization of the 4s” shell in the neutral 3d"4s” atoms. Experiment and exchange polarized calculations in- dicate [10,16] a 4s” hyperfine field of ~ +100 k0e per unpaired d-electron, a contribution which almost can- cels the 1s” + 2s” + 3s” core polarization. One expects a smaller effect in a metal since there are typically one, not two, electrons worth of “s” character below EP. One might expect a further reduction, in view of the fact that § values defined for Fermi level states always lie below 1. Paired Block states near the bottom and throughout an occupied “conduction” band contribute to the exchange polarization. These states have stronger hyperfine coupling than those at EF which con- tribute to the Knight shift, as is evidenced by internal conversion experiments [17]. It is probable that there is little or no reduction in this polarization term due to band effects. There may be inter-atomic as well as intra-atomic contributions to (a), since an applied magnetic field in- duces spin moments on the neighboring atoms as well as the atom in question. The two contributions are in- distinguishable for the pufe monatomic metals, but there is indirect evidence, from alloying, that the in- teratomic term is quantitatively important in some transition metals. Inter-atomic contributions will be seen to be important in transition metal alloys and intermetallics. As with the concept of a local density of states, there will frequently be ambiguity when at- tempting to divide 7% into inter- and intra-atomic terms. In addition to these contributions to the Knight shift coming from the Pauli paramagnetism of the conduc- tion electrons, there is an important contribution, espe- cially in transition metals, from the orbital magnetic moment of the conduction electrons induced by the ap- plied magnetic field. We can write, in analogy with eq (1) % orb - (b)xorb (14) where (b) is an appropriate orbital hyperfine coupling constant. In contrast to the Pauli contribution to the Knight shift, the orbital Knight shift is not proportional to N(EF). The orbital Knight shift [18-20] involves the orbital moment induced in occupied conduction electron states by an applied magnetic field, H. It is a second order term of the form 22, 1 r E-E, (iH ºf) (ſ i)6(k-k) 2 2 zº-; 1. * E (b)xorb = 2Xorb(r"). (15) Here the matrix elements are evaluated over a Wigner- Seitz cell. The occupied and unoccupied Bloch states, i and f, are admixed by the application of the field. The resulting admixture produces a moment which in- teracts with the nucleus. Except for pathological cases, where there are a substantial number of strongly ad- mixed states within kT or EF, there is little or no tem- perature dependence in this term, as is the case in the analogous Van Vleck temperature-independent para- magnetism in ionic salts. A rough estimate of the strength of the orbital Knight shift is given by [17]. | nin f (#) 2. --→º- (16) where ni and nj are the numbers of occupied and unoc- cupied Bloch states respectively and A is the conduc- tion electron bandwidth. This equation suggests that particularly strong orbital effects are expected in roughly half-filled d-band transition metals. In half- filled bands the product ninſ is a maximum, and in transition metals, A is small. Strong effects have indeed been found in W [21], Nb [20], V [19.20,22], Cr [23] and CrV alloys [24]. Although we have treated the Pauli and orbital hyperfine parameters, a and b, as multiplicative factors of the appropriate susceptibilities, it should be emphasized that (a) and (b) are not the simple averages customarily employed, but are more correctly the weighted averages = (axe) l (a) Xp (17) and, as is clear from eqs (14) and (15), (b) = (b)(orb). (18) Xorb The () denote an average over all bands. For the Pauli term, the average is over segments of Fermi surface where the contribution of a segment to Xp is multiplied 608 by the hyperfine constant, a, appropriate to that seg- ment (i.e., to its electron character). Equation 15 defines the average for the orbital term, where the in- duced orbital moments associated with initial and final states, i and f, are weighted by their hyperfine coupling constants. With this, the general expression for the Knight shift, eq (3), becomes % = (axp) + ſ^dia + (b)(orp) + higher order terms. (19) The Knight shift provides samplings of wave function and, in some senses, density of states character dif- ferent from that obtainable in other experiments. The Pauli term can in principle, and does occasionally in practice, yield considerable insight into the Fermi sur- face states contribution to N(EF). A particular case is that of alloys and intermetallic compounds with dif- ferent (a)'s at different atomic sites. This involves the variation in wave function character from site to site and, if you will, the variation in a local density of states. There is some arbitrariness as to whether one wishes to describe this in terms of (a) or a local Xp. A particularly clear example of this local nature is that of o-Mn. For this structure there are four crystallo- graphically inequivalent sites. Above the Néel tempera- ture of 95 K, four distinct resonances were observed [25,26]. Two of these had large negative shifts of –5.2 and —2.6 percent at room temperature [26], and were temperature dependent [25], with the former value in- creasing to —5.85 percent at 120 K, where X shows a maximum. The nuclei at the other two crystallographi- cally inequivalent sites showed much smaller, and tem- perature independent, Knight shifts (-0.45 and –0.15%). A number of complications have been indicated in this section. There is more than one term in the Knight shift; also the hyperfine constants (a) and (b) are sig- nificantly affected by band character and hybridization. There will in general be inter- and intra-site contribu- tions to (a). As will be seen, two important “tools” are available to aid in the identification of the different con- tributions to 7 : the Korringa relation [27] relating Knight shifts to nuclear spin-lattice relaxation times, and the temperature dependence of both 7% and X [13]. Finally one can sample Knight shift behavior at sites in- volving different atoms in an alloy or intermetallic com- pound. Matters such as these, while complicating Knight shift interpretations in terms of density of states, can supply insight into the electronic structure and local wave function character which cannot generally be obtained from other experiments. 3. Pauli Spin Susceptibility The Knight shift samples the density of states via the Pauli spin susceptibility, Xp. However, in pure metals, the density of states has usually been obtained in other ways and the Knight shift then used to explore the as- sociated wave-function character. In this section, we will review some of these methods of obtaining Xp, and their implications to our understanding of Knight shift behavior. First let us note that the expression relating the Pauli susceptibility to the density of states given in eq (2) neglects correlation and exchange effects between con- duction electrons. Electron gas estimates are frequently applied to “free” electron metals [28,29], and exchange enhancement theories to transition metals [30-32]. With a conduction-electron conduction- electron exchange interaction parameter Zeit defined in reciprocal space and taken to be constant, the ran- dom phase approximation yields [30-32] an exchange enhanced susceptibility Xp = 4–4– 1 – A eff}{} where Xp" is the unenhanced susceptibility of eq (2). The induced spin sets up an exchange field encourag- ing further polarization hence an enhanced Xp. Similar looking expressions, with N(EF) appearing in numerator and denominator are obtained from correlated electron gas theory. These as well as interband effects obviously complicate the averages taken in eq (17). In general one is forced to neglect them in (a) and assume their presence in Xp alone. Even with this simplification, there is no simple linear relation between an observed Knight shift and the density of states at the Fermi sur- face. As remarked earlier, this shortcoming is shared with the experimental data obtained by many of the other techniques reviewed in this symposium. It might appear that an adequate value of Xp could be obtained from direct measurements of magnetic susceptibilities, Xexp, especially since these would al- ready have the exchange enhancement included. We will return to the case of transition metals later, but in simple metals it turns out that bulk susceptibility results usually do not give reliable values of Xp. Con- sider, for instance, the noble metals. The bulk suscepti- bilities (Xexp) are each negative, i.e., the metals are diamagnetic. The ion core diamagnetism (Xº) plus conduction electron diamagnetism (xãº") is larger than Xp. Hartree-Fock [7] and Hartree-Fock-Slater [33] cal- culations for Xº agree to within two percent. However these calculations are for singly ionized valence states (20) 417–156 O - 71 - 40 609 which is not a totally satisfactory description in the metal. This, and interband mixing effects, raise the probable error in X; considerably. By the same token, X;" is poorly known due to electron-electron interac- tions. The net result is that Xp obtained from Xexp is probably good to no better than twenty percent for the noble metals. Various & values for the noble metals ob- tained by use of various schemes are shown in table 5. It can be seen that & values employing the traditionally quoted values for Xº are not consistent with those using more modern Hartree-Fock or Hartree-Fock- Slater estimates. The situation in the case of the alkali metals is not as bad for two reasons. First the theoreti- cal evaluation of both the core- and conduction-electron diamagnetism is on firmer ground, especially for the free electron-like metal, Na. Secondly, Xp has been ob- tained directly (i.e., without the need for the troublesome diamagnetic corrections) using conduction electron spin resonance (CESR) for the alkali metals and for Be. Combined CESR-NMR has also been used in Na and Li [34-36) to obtain Xp. For many metals it has been customary to use elec- tronic specific heat, y, measurements for information about Xp" using the one-electron relationship 0 – 3 (Bºb) X} 3(#) 'y, where k is the Boltzmann constant. There are two dif. ficulties here. One is that there may be many-body con- tributions to y (such as electron-phonon or paramagnon enhancement). The other is that the exchange enhance- ment part of Xp is missing here [see eq (20)]. The & values derived from y, given in table 5, were obtained from eq (21) and experimental y values [37], and there- fore neglect these corrections for enhancement effects. It may be that in some cases these two factors approxi- mately cancel one another. The & values plotted for the elements in figure 3 were obtained by use of uncor- (21) TABLE 5. § values for Cu, Ag and Au using various available data for Xp (see text for details). Cu Ag Au From 3/2(Xexp – X..") Hartree-Fock core........ .37 .50 .32 From Xexp-X.”-X." Hartree-Fock core..... .45 .45 .36 From 3/2(Xexp-X'...") Hartree-Fock-Slater COTC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38 | .52 .36 From 3/2 (Xexp-X.”) traditional core.......... | 45 .16 .08 From uncorrected electronic specific heat, 'y...] .53 .69 .42 From band calculations presented at this Symposium, x}* [38]............................. .57 | .68 .56 rected y values [37]. The final set of § values in table 5 utilizes a set [38] of band theory predictions for X}. Cancellations of various enhancement factors do not occur. The resultant & values are therefore larger. It is instructive [39] to compare the value of X; (obtained from y) to the value of $xp obtained by divid- ing 7% by atomic Heft values (tables 1 and 2). This is done in figure 4. The trend in our plot differs from that of Ziman [39] primarily due to our choice of atomic, rather than mass, units. The lightest alkali, Li, displays significantly less s-character than the other alkali metals [40-47]. The & values for the B-subgroup metals are all considerably lower, especially for the polyvalent elements; this might be expected, since the Fermi level lies higher in conduction bands, implying less s- character at EF and hence a smaller contact term. The associated increase in p- and d-character will generally make a negative (core-polarization) contribution to 7%, also lowering $. W I T -I I I n & =O.3 /o | & = O.9 y Cs Pb | / O / f / / 4 O H. | / / sº / = 1. | / / & = ||.O q2 / yº O | y / E }* K / / º N / 9 / -5 / / H / 2^ E 9 o / / 2. Q) oSn o Li / / 2 & = 1.5 º In f / / 2^ O CA. 2O H. / No // - - × " ... o 9/ 2 >< / Mg / / 2. (O | / / - / / C / / / 2 * / / CC%2 , p / / 2 | Aj% 2^ 2’ ! / ( 2 y^2 ! / " | 1 i l | ! 1. O 2O 4O 6O 8O 10° K/H., 9, Z koe FIGURE 4. Xp versus &l Hegfor pure metals in customary units. Lines of constant # (dashed) are shown. Due to the small & values for the heavier metals most of the data points are bunched at the left hand side of the plot. The Xp values were obtained from electronic specific heat data [37]. In the case of potassium a large uncertainty in the specific heat gives rise to an error of nearly half the size of the vertical dimension. The point shown for potassium represents the listed [37] y value. Direct measurements [239, 240] yield {(Li)=0.44 and {(Na)=0.64, rather than the values, £(Li)=0.49 and £(Na)=0.89, used in this figure for the sake of consistency. We note an interesting correlation in figure 4 in that the alkali metals, except for Li, fall on a straight line near & = 0.9. That is to say, the s-density of states ap- parently increases proportionately to the total density of states at the Fermi level for Na, K, Rb and Cs. The 610 increase in Xp from Na to Cs is attributable primarily to the large volume increase in this series. The constancy of § might seem to be surprising, since simple volume renormalization should effect š as well as Xp. It is to be recalled, that volume renormalization of § depends on the atomic volume in the metal relative to that of the free ion. It would thus appear that the alkali metal lat- tice constants faithfully reflect the sizes of the free ions, and hence & is roughly constant implying that the amount of s-character is essentially constant for the al- kali metals Na to Cs. This constancy was already noted by Pauling [47]. His calculations of s-p hybridization of bond orbitals indicated fractional s-characters of 0.72 to 0.74 for Na to Cs and a lower value (0.59) for Li, similar to the trend of figure 4. Knight shift experiments on the pressure depen- dence, as well as alloying, show that the contact density in metals is not simply an inverse function of volume [7]. This is also illustrated by recent pressure depen- dence calculations [48] on monovalent metals. The wave function effects which depress & values below 1, suppress the dependence of § on volume. In this section, we have seen the difficulty in obtain- ing a reliable value of Xp for use in obtaining & values for simple metals. Nonetheless, even in these cases, the Knight shift provides a rather unique measure of the s- contribution to the density of states in “simple” metals. 4. "Simple” Metals It is an interesting challenge to obtain the absolute value of the Knight shift, or its change with tempera- ture or pressure, from band-theoretical calculations. A number of near a priori calculations have been made with varying degrees of semiquantitative success. For example, Das and coworkers, using the orthogonalized plane wave (OPW) method, have calculated wave func- tions and densities-of-state at various points on the Fermi surface for Al [49], Be [50-53] and In [54]. A few of these papers are of particular interest in that they represent efforts to use the weighted average form of (a) as given in eq (17). In the case of the divalent metals Be, Mg and Co, the spin susceptibility has been extracted from Knight shift measurements by use of such estimates of (a) [55]. Comparison with theoreti- cal values of their bare Xp [i.e., eq (2)] permitted esti- mates of the exchange enhancement to be made. This process resulted in a reduction in Xp for Be; the authors concluded that this arose from inadequacies in the energy band estimate of Xp" and (a). The alkali metals, particularly Li and Na, have tradi- tionally attracted theoretical attention, often giving relatively close agreement with experiment [40-42]. Even in these simplest “free electron” metals, cor- rections to the hyperfine coupling constant due to non- free electron-like band structure effects, can amount to 25 percent or more [45,56,57]. It should be stressed that for the heavier simple metals (i.e., potassium and above) d-hybridization as- sociated with d-bands above or below the conduction band affects the Knight shift and other properties at (and off) the Fermi level (see, for example, Kmetko [46]). In the case of Cu and Au these d-hybridization effects are large. Such effects do not arise in the Li row, but there may be abnormal “2p” effects due to the near degeneracy of atomic 2s- and 2p-levels [58-61]. In these “simple” metals, the change in Knight shift at the melt- ing point [62] (fig. 2a) is usually not large, and its tem- perature dependence in the solid, slight (fig. 2b). As an example, consider Al. At the melting point, 7% has the same value for both liquid and solid. The temperature dependence in the solid is illustrated by the data of Feldman [63] (fig. 5). The change in Knight shift of Al is less than 2 percent of .7% over a temperature range from 4 to 300 K. The solid line in figure 5 represents a simple volume renormalization theory based on the thermal lattice expansion, which fits quite well at tem- peratures above the Debye temperature. No satisfac- tory explanation has been given for the deviation from this theory at low temperatures. Similar effects were observed in Na and Pb [63]. OTTTTT-T-I-I-I-I-I-I-I-T-I-T-I-T-I-H-I-H-I-H pº I Lºſ - O.5 H l |- P - *300 " ..., | } : - Tk. % |- 3OO ~! - - ~1 i - —T *- - ºf . ," - º | I K vs. T – Al - 2.OH- - 2.5D H–––––––––––––––––––ll O 5O |OO |50 2OO 25O 3OO T (*K) FIGURE 5. Change in the aluminum Knight shift with temperature, as taken from Feldman [63]. The solid line is theoretical. A more unusual case is that of cadmium [64-70]. In the solid the Knight shift varies considerably. Ž increases ten times more rapidly in Col than in Al, over the same temperature range (4-300 K). At 600 K, 3 in 611 Co is about 70 percent larger than its value at 4 K. This change is seen in figure 6. An additional increase in W (~ 33% of .7%) is observed upon melting (see sec. 5). The anisotropic Knight shift, Zanis, [see eq (13)] also in- creases with temperature. Col exhibits a large change - | | | | | | © THEORET CAL CAL CULATION 0.8 H A Goodrich and Khan – V Sharma and Williams o Seymour and Styles 0.7 H e Borso ond Barnes -- ~ 0.6 H – × 0.5 H …’ *-*. Vo _2~ 0.4 H -* * ...~& *—3 º — l | | | | O l OO 200 300 4 OO 500 600 700 800 o K FIGURE 6. Change in the cadmium Knight shift with temperature, as taken from Kasowski and Falicov [68]. in the shape (c/a ratio) and volume of the unit cell with temperature and it had been suspected that the changes with temperature of Żiso and 7/anis could somehow be correlated with these cell dimensions. Kasowski and Falicov [68] have explained the behavior with a different scheme: in the solid the lattice vibra- tions cause an increase in both Xp and in the s-character as the temperature is raised, thereby increasing & so. On the other hand, .7% ants arises from the non-s part of the wave-function, which of course is decreasing as the s-part is increasing. However, again invoking eq (17), we require not the average hyperfine coupling as- sociated with the non-s part, but the appropriate average over the Fermi surface. Cancellation occurs in this average at low temperatures. The reduction in the cancellation at higher temperatures more than compen- sates for the increase in s-character, thereby providing an increase in 7% anis. On the other hand, pressure de- pendence measurements of Żanis for Sn [71] was in- terpreted as due to charge redistribution rather than a change in the s-p character of the wave-function. Other data further indicate the complexity of the behavior of the Knight shift in solid Col. Borsa and Barnes [65] note that alloying Mg (in quantities of ~ 1%) with Ca will cause substantial changes in cell size and shape without affecting the Knight shift. Kushida and Rimai [72] have separated the implicit and explicit contributions to the temperature dependence of 7 by their measurements of the pressure dependence in Col. Volume renormalization was found inadequate to ex- plain the observed pressure dependence [72].” 5. Sudden Changes in 7/ The abrupt change in 7% of Cd upon melting was presumed by Ziman [39] to indicate an abrupt change in N (EP) associated with solid and liquid Ca. This con- clusion was examined in two different calculations, each employing nonlocal pseudopotentials [68,70], resulting in opposite conclusions. Shaw’s calculated [70] values for N (E) per unit volume for solid and liquid Col are shown in figure 7. N (EF) is found to be 0.7 per- cent lower in the liquid than in the solid. Shaw con- cludes that “Ziman's assertion that the strong change in the Knight shift of cadmium is a density-of-states ef. fect is not borne out by our detailed calculations.” In contrast Kasowski and Falicov [68] assume that Ca is free-electron-like in the liquid and finds that most of the change in 7% (Cd) upon melting is due to an increase in N (EP). They note that “this agrees with Ziman’s hypothesis and confirms it quantitatively.” Although these two calculations [68,70] resulted in opposite con- clusions, in part due to choice of different model-poten- tials, there is an abrupt increase in N(EF) in Shaw’s cal- culations if solid Co is compared with the free electron value, as indicated in figure 7. An interesting example of an even more abrupt in- crease in 7 upon melting is found in the behavior of the III-V compound InSb [73]. In the semiconducting solid the Knight shift of either the In or Sb in InSb is zero, but in the metallic liquid state, the Knight shifts have normal metallic magnitudes. Another example is Bi [62], in which 7% has the opposite sign in the liquid from the solid. There are various cases besides melting where a sud- den change in 7 occurs. In an alloy system, Ż usually changes smoothly with composition within a particular phase, but shows a jump across a phase boundary. An example of this is shown for the AgCd system in figure 8, taken from Drain [74]. By correlating 7% and y across a phase boundary, Drain showed that in this case the abrupt change is associated in part with changes in densities of states. * There are other examples of an observation concerning the vol- ume dependence of Ż. As noted earlier, in our discussion of figure 4, simple volume renormalization is not often useful in explaining Knight shift results. See also [7] for a discussion of this point. 612 O.O 8 H d O. O.7 H 2. H o.o.6|- LL] > M- -) — O O.O.5 H > *~ H- (ſ) LL] |-- H. O.O 4 <ſ H. - (/) LL O.O 3 H / C >~ F- / CA DM U M H / — SO L | D (ſ) go,02 ſ — — — L | Q U D # H ---- — F R E E E L E C T R O N ool || | | | | L | | ſ | | o: o, 2 O.4 O.6 O.8 ſ.0 l. 2 E / fr FIGURE 7. Calculated values of N(E) for Cal, as taken from Shaw and Smith [70]. The semiconductor-(or insulator-) to-metal transition, whether or not a Mott transition, offers a number of ex- amples in which the Knight shift changes more or less suddenly. For example, when tin changes from its metallic to its semiconducting phase, 7% changes from 0.75 percent to near zero (see fig. 2b). Also consider WO2 which is metallic above and semiconducting below a crystal structure change occurring at Tc = 68 °C [75,76]. The Knight shift is ~ —0.4 percent at 100 °C and ~ +0.2 percent below. The negative shift above Te is attributed to a metallic d-band. A case where there is an electronic transition without further structural changes is that of phosphorous-doped silicon [77-79]. At donor concentrations (na) greater than 2 × 10° crim-º, the material appears to be a metal, the Knight shift is proportional to na”, and the Korrin- ga relation holds. Below this critical concentration, 7% drops sharply. In the transition range, 3 × 10” < nd < 2 × 10”, there is a measurable.7%, but the Korringa rela- tion no longer holds. The electrons are “delocalized” in some type of impurity “band.” Other systems exhibiting nonmetal-to-metal transi- tions are the alkali-ammonia solutions. As the metal to ammonia ratio is increased, the liquid becomes gradually metallic, and the conductivity as well as the Knight shift increases substantially [80-83]. O.6 F-I-T-7 T III TTI I ſ I i I | | ! | | | i } | | | | | t | l | 1 | | | f | | | | j l I l | | | l ! I | 1 | | | | | | | | | | $ ! | | i * ! | ſ } 1 | f O.5H | | | | | | ! | - l | | | | | | I } | | | | | i | l f H | ſ ! — — — ! } | |- -# -- | | l l 9/ |--------, I k | | | | | i O ſ | | 7 O.4 H. } | | | | - | | | | | . | | | | | i } | | | ] } l Ag | ! I | | | | | | ! | | | l | | | | | ] l 1 | ! O.3H l ! | | - (q) i f | | | | | | } ! | cº |- as 3.x ºr 're e £ tº n | 1 l 1. 'i' 1 ſ 1 | 1 I —1– Ag 2O 4 O 6O 8O |OO | } H-H I | | I H l | | | f yº. k , 8% O. 4 H. } | | | | | | | | | | I | | | l ; I i CC ; : , ; ; ºs | ſ | – 93F (b) ; : *, *, C. 'a-asia, riº 8 | &+ n y) | | | | | J | - I | | 1–in–ill--- ſ Ag 20 40 6O 8O |OO ATOMIC PERCENT CADMIUM FIGURE 8. Knight shifts in AgCd for (a) the Ag resonance and (b) the Cal resonance, as taken from Drain [74]. In the two phase regions, Cd resonances for both phases are simultaneously seen, due to the rather different 7’s for the different phases. For ranges of solid solubility 7/ changes smoothly. 6. Orbital Magnetism in Simple Metals The Bardeen, Cooper and Schrieffer theory of super- conductivity [84] predicts that, as a result of spin pair- ing, Xp vanishes at T = 0. Hence it was expected [85] that 7% – 0 as T-> 0 for superconductors. This expec- tation often is not borne out. It seems certain that the residual Knight shift is predominantly of orbital origin for transition metals such as V and Nb [22]. Ferrell [86] and Anderson [87] proposed that spin-reversal scattering due to spin-orbit coupling is another possible mechanism for obtaining a residual Knight shift. This mechanism requires that 7 be a function of mean free path, i.e., particle size and impurity scattering. This 613 spin-orbit term has been shown by Wright [88] to be important in Sn. Wright also reviews earlier experi- ments on other metals. He concludes that although spin-orbit coupling is dominant in some cases, two types of orbital magnetism cannot be ruled out in sim- ple metals. These are the Van Vleck orbital paramag- netism [18-22] and a higher order mechanism in- troduced by Appel [89]. Similar higher order mechanisms were discussed earlier by Clogston et al. [20]. We have already discussed the Van Vleck term and will note its importance in the transition metals and alloys to be discussed later. The Appel mechanism in- volves spin-orbit and & H coupling to an intermediate excited conduction electron state, which contributes to the Knight shift through the contact interaction. It is of the order of \|AE times the contact Knight shift, where \ is the spin-orbit interaction energy, and AE is the energy between states connected by A. The sign of the Appel contribution to the Knight shift may be either positive or negative. Another important orbital effect is the Landau- Peierls diamagnetism [90]. The magnitude of this term is not easy to predict. In the simplest (and greatly over- simplified) free electron approximation the Knight shift, ºdia arising from Landau-Peierls diamagnetism, IS 8T / | m \? 3 (. x) (#) 2 where m” is an appropriate electron effective mass. This term has been proposed to explain a number of negative, or near zero, Knight shifts in nontransition metals. In transition metals, d-core polarization (see table 3) gives an important negative contribution to 7/, through the Pauli paramagnetism of the d-band. In non- transition elements p-core polarization has often been proposed as an alternate to 7/dia as a negative contribu- tion to 7 (see table 3). For example Das and Sond- heimer [90] first suggested the importance of the Lan- dau term to the negative Knight shift in Be. In later papers, Das and coworkers [50-52] performed detailed calculations for the contact and the p-core polarization terms in this metal. A reluctance to believe that the diamagnetic shift is as large as was originally suggested [90] is evident from these papers [50-52]. Although they do not give further quantitative estimates for 7/dia in this later work, in each case they are forced [52] to the same conclusion that the remaining negative shift is of diamagnetic origin. Using eq (22), available m” values can, in fact, give a 7/dia of -0.003% [52,53]. Yafet [91] considered the importance of .7%dia for Bi, but Williams and Hewitt [92] proposed p-core polariza- ºdia = - (22) tion as the origin of the quite substantial negative Knight shift in Bi of -1.25%. It is interesting to note that if Xp is estimated from the electronic specific heat [37], using eq (21), and if there is nos contact con- tribution to 7%, the p-core polarization necessary to ex- plain a shift of -1.25% is 800 times larger than the experimental value [93] for atomic Bi (table 3). The presence of an s term will increase this estimate of 800. Note that the experimental atomic “core” polarization term includes the polarization of the 6s-valence electrons which are part of the conduction bands in the metal. This atomic value therefore provides an estimate of the s polarization in the core and throughout the oc- cupied conduction band: the applicability of this value to the metal depends on the distribution of s-character in the occupied bands (as compared with the free atom). Granted the uncertainty in Xp and the question of relevance of the atomic hyperfine constant to this metal it would still appear that p-core polarization is at least one hundred times too small to account for the ex- perimental Knight shift. On the other hand, Bi has m|m” ratios which, using eq (22), are large enough to suggest a Landau diamagnetic shift that can approach magnitudes of the order of the observed 7' value. Other examples of negative shifts in diamagnetic, non-d materials are Tl in NaTl [9,94-96] and In in Biln at 77 K [97]. For these cases the situation is much less clear due to the lack of data for Xp, m*, and y. In addi- tion we do not have free atom experimental values for p-core polarization for Tl and In. Using some upper esti- mates for the unknown quantities, it is evident that either p-core polarization or Landau diamagnetism is hard pressed to reproduce the observed shifts. A discussion of diamagnetic Knight shifts and p-core polarization effects in these materials will be given elsewhere [98]. Das and Sondheimer [90] also indicated that oscilla- tions would be present in the diamagnetic term. These would be periodic in 1/H, similar to de Haas-van Alphen oscillations. However, when this effect was first observed by Reynolds et al. [99,100], the amplitude of the oscillations was considerably larger than that ex- pected from the diamagnetic term [101-106]. Glasser [107] explained this by proposing that the Fermi sur- face wave functions also change with 1/H, and that this introduces oscillations into the Pauli term which dominate over the diamagnetic oscillations. Goodrich et al. also observed Knight shift oscillations in Ca [108]. Their data are shown in figure 9. The importance of observing oscillations in 7 is that it is possible to obtain the Knight shift over a segment of the Fermi surface. Thus the Knight shift has become 614 O .353 H O.35 | H O. 349 H. H = |9724 G 1. | | | l O. 5 O 8 O. § 4 O H. O .338 H H = | 6 O25 G H = | 58 7.3 G O .33 S H | l | | | _l O. 625 O. 627 O. 629 (#)(o"6") FIGURE 9. Knight shift in Cal, illustrating oscillations in 7 with 1/H, as taken from Goodrich, Khan and Reynolds [108]. a potentially important tool for examining the wave- function character associated with N (EP) not only in the average sense of eq (19), but also in finer detail over Fermi surface segments. 7. Transition Metals The simple metals considered in the preceding sec- tions display, in the main, only weak orbital Knight shifts and temperature independent, usually positive, Pauli terms. Transition and noble metals with their d- bands tend to have a negative Pauli term arising from d-core polarization (see table 3). Narrow d-bands, with many states close in energy to EF, often have substan- tial orbital effects [see eq (15)]. Structure and curva- ture in W(EF) contribute a temperature dependence to the Pauli term. Given the presence of d- and non-d, or “conduction” band character, it has been normal to describe the paramagnetic transition metals in terms of a “two band” model involving discrete “s” and “d” bands. We follow common nomenclature in designating the conduction band as an s-band. (The d-bands, of course, also contribute to conduction.) The orbital Knight shift is associated with the d-band; the average taken in the Pauli term is rewritten (a)xp= (as)x} + (aa)x}(T), (23) the d-bands, a negative temperature dependent term. The latter dominates since N(EP)", hence X}, is much larger than its s-band counterpart. It is assumed in eq (23) that the temperature dependence of .7% is entirely associated with the susceptibility and not with any variation in the hyperfine coupling constants [13]. The fact that the slope, dž/dx, is a constant for these metals offers some experimental justification for this assumption. An example of this is seen [109] in figure 10, where 7% is plotted versusXiot, with temperature the implicit variable, for Pd. In this case X(T) goes through an extremum with increasing T, but is faithfully tracked by Ž(T). Pol displays the largest temperature variation in X among the paramagnetic metallic elements. As is discussed in Mott and Jones [110], such a temperature dependence arises from sampling by the Fermi func- tion of structure and curvature in the density of states in the vicinity of Ep. While the two band model has proven most useful when discussing Knight shifts and other experimental data, there is, in fact, strong hybridization of s- and d- band character and a transition metal is not constituted of discrete d and “s” bands. Some measure of this is given in figure 11, which displays the density of states obtained" for foc Cu. The results can be taken as characteristic of all transition metals. The density of states behavior is similar to that reported by Mueller for Fe [115], and to that obtained by Goodings and Harris [116], and by Cuthill et al. [117] in their estimates of soft x-ray spec- tra for Cu. The density of states has been plotted separately for the first, second and sixth bands while that for the third, fourth and fifth has been added together for the sake of legibility. In figure 11, the Cu Fermi energy is designated by EF and that appropriate to Ni by E(Ni). The high density of states peak, inter- sected by E(Ni), is due to the fifth band. Details of this 4 4 - ? ) S where the or conduction band is assumed to con- tribute a positive, temperature independent term, and band and of its Fermi surface are essential to the differ- * These results [ll]] involve a sampling of ~ 1.5 × 106 points in 1/48th of the Brillouin zone. The sampling employed a quadratic fit to a set of pseudopotential bands by Ehrenreich et al. [112, 113] involving a mesh of 28 intervals from T-X in the zone. The pseudo- potential bands were obtained from an adjusted analytic fit of some new APW calculations for Cu [114]. Spin-orbit coupling effects, though slight, have been included. The Fermi surface is in better agreement with experiment than is usual for calculations. Details of the density of states and grosser features of the wavefunction analysis are, of course, dependent on the use of a pseudopotential band description (which assumes tight binding d-bands and a set of four orthogonalized plane waves for the non-d part). These results can be considered analogous to the OPW results, obtained by Das and coworkers [49-52, 55] for the Knight shift in various “simple” metals. 615 º /ſºs O l N x;---- NS § - H. & "|x; }- N H > # -2H N Uſ) N H. N 5-3F 2 [x: (T=0) Q SP, HT. × -4 H. --~ He Xp (T-0) • -5 L ſ 1 I l i ſ - 1 OO O 1 OO 200 3 OO 40O 5 OO 6OO 7 OC 6 CO 6 #) X X 10 (# FIGURE 10. 7/(T) versus X(T) plot for Pd, as taken from Seitchik, Gossard and Jaccarino [109]. ing magnetic behavior of Ni, Pd, and Pt. The Cu Fermi level intersects the sixth band, often named the “free electron” band, which lies above the five “d” bands. The density of states associated with non-d electron character; N(E) non-a, is also shown in figure 11. It must be emphasized that details of these results depend on the scheme used to describe the bands (in this case a pseudopotential description [111] with tight binding d- functions). Hybridization effects cause a build up of non-d character at the bottom and a depletion in the middle and just above the bulk of the d-bands (i.e., in the range – 0.35 s Es –0.15 Ry). The peak seen at ~ —0.4 Ry can be important to optical and soft x-ray pro- perties. The sixth band is predominantly of d-character at the bottom and remains almost thirty percent d at the Fermi level. This particular set of results [111] yields 9.8 electrons worth of d-character out of a total of eleven electrons, in the bands below EP. [A free elec- tron parabola, holding the remaining 1.2 electrons, and with an effective mass chosen so that its Fermi level matches EP, has been drawn for comparison with the actual non-d density of states.] The lowest band is strongly free-electron like up to E - —0.45 Ry and 0.61 of the two electrons residing in the band are of non-d character. Roughly 0.35 of the remaining 0.6 non-d- 3O | T | | | T T > O ºf 25 H *: É Cu Cl- § 2O }*-* * *-* = 8* * N to (E) – # ~- Nnon-d (E) 9 2nd Orº |5 – 3rd, 4th, 5th - ſr. Lll Cl- É — º |O H. E (N) st 5 |*' BAND G 5F *m: 2 6th (a) - º----- I Xī Cl ~~~~ El- | EF| O I | | | T I <ſ 4th – e; 3 H * BAND ^ rd tº 2 F- |- c” | }= – O.4 -O.3 | –O.2 E (RELATIVE TO EF), RY FIGURE 11. (a) Total density of states and non-d density of states for the 1st, 2nd and 6th bands of Cu separately, and for the 3rd, 4th and 5th summed. The smooth, flat band is the free electron parabola containing 1.21 “conduction” electrons, as discussed in the text. This is shown for comparison with the Nnon-d(E) results. (b) Ratio of band to atomic hyperfine constants as defined by eqs (24a) and (24b). 616 electron character is associated with the one electron in the sixth band. The 3d-electron character can be expected to in- teract with the nucleus via a core polarization term of ~ – 125 k0e per pub throughout the bands. The non-d character is expected to interact predominantly via the direct contact term. Its behavior is shown at the bottom of figure 11 in the form of the ratio as(E) = (a(E)NToT (E)) (24a) Cl4 a 4Nnon-d (E ) with respect to the non-d electron density of states at E, i.e., the contact interaction normalized with respect to the non-d electron density at E, and to an atomic 4s hyperfine constant.' Omitting all core polarization con- tributions to 7%, this ratio is then related to § by NToT (E) as (E) - § (E) Wnon-d (E) gº -------' –- (24b) Cl4 A ratio of 2 to 2.5 occurs at the bottom of the bands reflecting the volume normalization enhancement of § discussed previously in connection with figure 4. Values closer to one are appropriate to the non-d character hybridized into the second, third, fourth and top of the first bands. This suggests that an asſF) set equal to a 4 can be used as a first approximation when estimating the effect of hybridization on reducing a d- band ad from a pure d-core polarization value. The ratio is higher in the fifth band but here hybridization is al- most zero. The ratio tends to fall with increasing E, as is seen in the lower part of the first and in the sixth bands. This is associated with the decrease in s- character in OPW's of increasing k. The ratio has dropped to a value of 0.78 at the Cu Fermi level. Here § (EF), defined in the manner of eq (24b), has the value of 0.57. If one adds the negative core-polarization con- tribution which can be attributed to the twenty-eight percent d-character in the bands at Ep, & (EP) becomes 0.55. The above &/EF) values agree with the upper end of the range estimated as the experimental & for Cu in table 5. Davis [118] has obtained" £(EF) = 0.67 employ- 7 An aa of 1600 k0e/pºp, omitting core polarization contributions. was used, since core polarization effects were omitted in the evalua- tion of (a(E)NTor(E)). With core polarization, correlation and relativistic effects present, the (experimental) a 4 is ~ 2600 k0e/pub. For discussion of this see [7]. *Actually Davis [118] chose to quote a 3-ratio by dividing his computed hyperfine term without core polarization by an atomic a 4 with core polarization. This yields a smaller numerical value than we quote here. ing the method of Korringa-Kohn-Rostoker. Com- parison with the experimental data may not be meaningful because core polarization terms, arising from “conduction band” spin character, have been omitted in the ſRF) estimates while being present in the quantities of tables 2 and 5. It is thus proper to make the comparison only if core polarization affects the numerator and denominator so as to leave the ratio constant. This seems unlikely since s and p character terms will contribute to the conduction electron core polarization. These two band calculations yield N(EP) values which are in good numerical agreement with the elec- tronic specific heat for Cu. Both calculations suggest that d-hybridization is a significant factor in reducing §. This is one reason for the tendency noted earlier for Ag to have a larger & than Cu or Au. The d bands are twice as far below EF in Ag, and weaker d hybridiza- tion (~ 10%) occurs at the Fermi level of Ag. Noting that the atomic s-contact interaction is typi- cally ten times larger than, and opposite in sign to, d- core polarization, figure 11 suggests that hybridization is important throughout the transition metals. Consider the case of Ni. The Fermi level intersects the high peak of the fifth band. This band has almost no hybridiza- tion, as is shown in figure 11, which was obtained with Cu bands. The sixth band has an N(EF) value which is better than an order of magnitude smaller than the fifth band at E(Ni). However the sixth band has twenty per- cent S-admixture at E(Ni) causing a large, positive ad, which compensates for this. Neglecting exchange enhancement of the susceptibility, the sixth band con- tribution can then cancel approximately one third of the Knight shift term, ad, associated with the fifth band alone. Exchange enhancement is important and an esti- mate of the exact role of the sixth band requires opinions of interband exchange effects. Scanning the lower energy parts of the plot, it appears that hybridiza- tion may affect the ad values for the lighter transition metals more severely. This hybridization trend should hold although changes occur in the crystal structures. Despite the complexities just discussed, the two band model of the Knight shift has frequently proven fruitful, with hybridization absorbed in the ad term. The various Knight shift contributions are normally disen- tangled in two ways. First, comparisons can be made between the relaxation time, T1, and Knight shift results which weight the various terms differently. The Korrin- ga relation [27] provides a test for the s-contact con- tribution. Secondly, one can employ the graphical technique of figure 10. This scheme, applied to Pt [20] 617 appears in figure 12. The experimental data are plotted for 7 versus Xtot with temperature the implicit parame- ter and, following eq (23), it is assumed that Xtotº Xdia + Xorbit Xàt Xff (25a) * = (b)xorb-H (as)x} + (aa)x}}(T). and (25b) The slope of the experimental data yields an empiri- cal value for (aa). The diamagnetic susceptibility is estimated and subtracted out, shifting the origin of the plot to point A. An estimate is then made of Xp" (usually with the free electron approximation) and of as and the S-band contributions are subtracted out shifting the origin, with respect to orbital and d-band Pauli terms, to point B. Finally, (b) is estimated, defining the slope of Worb versus the Xorb line, which is drawn until it inter- cepts the experimental 7" versus Xlot curve (at point C). The intercept defines the relative roles of orbital and d- band Pauli terms. In this case, the d-band Pauli term dominates. A value for the unenhanced Xp", estimated from specific heat data is shown in figure 12. The larger -value, deduced from the Knight shift, provides a mea- sure of the effect of exchange enhancement. While * !º ..}^{ , B2”. N { N-N- slope = {o (R) Sin” m, + 3 (R) sin 2 m.),(27) where o, (R) = (2e + 1) S (nº (ke R)-jº (keR)}(28) and 6. (R) -- (2 + 1) > (ker)n (ker). 29) The £" term in the sum is associated with the Č'" par- tial wave, m, is the phase shift of the " component, and j, and n, are spherical Bessel and Neumann func- tions respectively. To obtain the effect on a solvent metal Knight shift, the o, and 8, must be suitably averaged over R. For “simple” solvent metals and “simple” solutes the effect is presumed to be dominated by s- and p-wave scattering. Changes in the relative roles of s- and p-wave scattering are important in rationalizing the variation in A7//7′ with varying valence of the solvent, or varying valence of a solute, relative to the solvent. There is traditionally some question of how large an effect can be associated with pure S-screening. From a strictly atomic viewpoint, one might expect it to be limited to two electrons worth of charge. The recent in- vestigations of Slichter et al. [135,136] conclude that higher & scattering is very important to the screening when solute-solvent valence differences exceed two. A local spin moment will produce a spin disturbance similar to that seen in figure 13. Such a spin disturbance is obviously important to magnetically or- dered metals" but it also produces the dominant Knight shift term at some sites in certain paramagnetic alloys (see sec. 12). Consider the Knight shift of a non- magnetic site in a paramagnetic rare-earth alloy. The principal term in the Pauli susceptibility, i.e., in the spin induced by the magnetic field, is that of the open 4f-shells, and this spin will contribute to the nonmag- netic site (a) behavior via conduction electron polariza- tion. The susceptibility associated with the moment would obey a Curie-Weiss law. Examples of this are the above mentioned rare earths with their open 4f-shells, and 3d alloys, such as Fe in Cu. Sometimes the moment may arise from band-paramagnetism involving local- ized d levels which are too weakly coupled by intra- atomic exchange to produce a true local paramagnetic moment. Curie-Weiss behavior is then not followed. The 3d elements as impurities in Ag, or dilute Ni in Cu are examples of such band paramagnetism. Given an induced local spin monent of either of the above types, there will inevitably be a spin disturbance in the solvent conduction bands producing, in turn, a Knight shift term. There will be a variety of contribu- tions to this. First, and most obviously, the exchange field due to the local moment will produce a spin depen- dent scattering of conduction electrons. As described 10 We should note that in a magnetically aligned metal, cross terms will cause a magnetic imputiry to contribute a charge dis- turbance, and a charge impurity to contribute a magnetic disturbance (the charge impurity Knight shift contribution can be considered a special example of this). 620 by Ruderman, Kittel, Kasuya and Yosida (RKKY) [137. 141], the Pauli response of the conduction electrons to the diagonal exchange term, Ø (kp,kp), contributes a net spin density which is then piled up in a screening dis- tribution of the sort plotted in figure 13. Formally the theory is almost identical to the charge screening case. Exchange, rather than electrostatic Coulomb, terms are responsible for the disturbance, and details of the shape of the main peak and of the behavior of the phase and amplitude (relative to the main peak) of the Friedel oscillations should differ from the charge scattering dis- tribution. The 6 = 0 and 1 partial waves will again tend to predominate. Details [141] of the intra-atomic term in the electrostatic exchange, 2% el, are such that if the local moment is of odd (or even) & character, partial wave scattering of odd (or even) &" is enhanced (i.e., s- wave scattering is of increased importance with d- moments present while p-wave effects are amplified in 4f-moment scattering). Only s-wave spin density is non- zero at the solute’s nucleus. In the scattering picture, it describes the intra-atomic conduction electron exchange polarization term discussed in section 2. If the value of electrostatic exchange were somehow zero, the presence of a local spin moment would still cause a spin disturbance in the conduction bands [142- 144]. Resonant scattering of spin-up and spin-down conduction electrons will occur at different energies as the result of the splitting of the local virtual (or real) bound state to form the local magnetic moment. One reason the scattering differs is the different occupation of spin up and spin down orbitals on the local moment site. Consider some partial wave component, with quantum members & and m, of a scattered conduction electron at the local moment site. If the local moment had an occupied component of the same 6, m, and spin, the conduction electron component would be unaffected (except for any nonorthogonality effects which might arise); if there were a hole in that local mo- ment orbital component, the orbital could be admixed into the conduction electron function to the extent it is energetically favorable. The existence of a net spin residing in the local moment implies a difference in hybridization (and orthogonalization) effects in conduc- tion electron states of the same k and differing spin. This results in a spin density distribution similar to the core polarization effects discussed earlier. There is no net spin in the disturbance; instead there are regions of spin parallel and antiparallel to the local moment. In their original inspection of such hybridization effects, Anderson and Clogston concluded [142] that this disturbance would fall off as 1/r"; subsequent numeri- cal estimates of their model [145] are consistent with this observation. It would seem that the effect is largely concentrated at the local moment site. An effective exchange interaction arises when the next order in hybridization effects is taken [142,143]. Consider the energy shift of a Fermi surface electron. The mixing of local moment hole components into the wave function will lower the state's energy whereas orthogonality with occupied components can only raise its energy. Hybridization thus stabilizes the energy of Bloch states with spin moment antiparallel to the local moment whereas those with spin moment parallel are less favored since they undergo orthogonalization and decreased hybridization. This produces [142-144] a negative interband exchange constant Zib(kf,kr) in con- trast with Žel(kp,kr) which is always positive." A nega- tive value implies a conduction electron Pauli spin den- sity term of spin moment antiparallel to the local mo- ment. Such situations occur experimentally, implying that “interband” hybridization (and higher order ef- fects) do, on occasion, predominate over electrostatic exchange, which can only produce a net spin moment parallel to the local moments. (These effects are obvi- ously intimately related to the Kondo effect.) The earli- est evidence for negative exchange constants was ob- tained by nuclear magnetic resonance and electron paramagnetic resonance measurements for rare earths in several host metals [146-149] such as Pa. This has subsequently been borne out by magnetization and neutron diffraction studies. Interband hybridization exchange also differs from electrostatic exchange in that hybridization will only be strong between band and local moment components of common 6. The summing over individual Bloch state contributions to the spin disturbance yields partial wave scattering only from those same & components directly involved in the mixing. Thus, unlike electro- static exchange, hybridization effects with their predominant d- or f-scattering, will not contribute an ^ =0 contact spin density term to the hyperfine field at the scattering site, unless higher order (i.e., double, tri- ple, etc.) scattering processes are significant. A disturbance of the type plotted in figure 13 produces a distribution in solvent site (a)'s causing a broadening of the solvent Knight shift line. The dis- tribution in (a) will not necessarily provide a detailed, accurate mapping of the bulk conduction electron disturbance. This is due to interference effects arising from Orthogonalization of the conduction electron wave functions with the solvent site ion cores which are "Schrieffer and Wolff [143] have explored the circumstances for which Zip can be properly defined. 621 penetrated. This interference is important in that it af. fects the apparent shape of the disturbance at sites near the impurity, while providing little more than scal- ing to the result for sites at asymptotically large R. The sampling of the disturbance will inevitably cause the average solvent site (a) to increase or decrease with respect to the pure solvent value, thus causing a shift 6.7%, of the resonance line. Some sites will have values of (a) so different from the average that they will not contribute to the main resonance line, but satel- lites outside instead. This will cause a decrease in line intensity, i.e., wipe-out, upon alloying. Blandin and Daniel’s estimate [134] of one such distribution in (a) is seen in figure 14. The theoretical estimate is drawn to the same scale as an experimental [150] NMR derivative in Ag containing a small quantity of Sn. The 9°Ag |.4 Oe —- sº- <— H A Number of Otoms |* Neighbors 2nd Neighbors | || > –1 O | FIGURE 14. (Above). Rowland’s experimental [150] NMR absorption derivative curve in an alloy with 1 percent Sn in Ag. (Below). Blandin and Daniel’s calculated [134] positions and relative contributions to silver Knight shifts at silver sites in near neighbor, next near neighbor, etc. . . positions with repsect to an Sn impurity in Ag. This is plotted on the same horizontal scale as the experimental curve. near and next near neighbor (a)'s in Ag(Sn) may well be responsible for the partially resolved satellites. Another experimental example [151] of satellite struc- ture is shown in figure 15 for Pt containing small quan- tities of Mo. Here three satellites are clearly resolved. Details of the effect of alloying will depend on such Pt + O.O7%, MO (d) Pi + O. |7% MO (b) --~~ FIGURE 15. Experimental NMR absorption derivative curve in Pt-Mo alloys as taken from Weisman and Knight [151]. The resonance in (a) shows several distinct satellites and that in (b) shows satellites in the same positions but those near the central resonance begin to merge with the central line, thus causing resonance broadening for increased alloy concentrations. factors as whether or not the main peak of the dis- turbance extends out and encompasses any neighbor- ing nuclear sites. Little is known experimentally, and less from accurate calculation, concerning main peak behavior. (Most theoretical work makes the doubtful, but computationally necessary, use of asymp- totic estimates for the entire disturbance.) It is generally thought that the main peak of the Coulomb screening is largely localized at the impurity site while the spin density peak is of longer range. For Fe in Pa the latter is known to cover many lattice sites. This is due to a large 1/2kF value (which affects the main peak as well as the Friedel oscillations) and to the substantial conduction-electron conduction-electron enhancement of X, [152-154]. It has been seen that solvent data are largely limited to shifts of the main resonance line and this does not provide a unique test for any given detailed model of 622 alloy effects. Although such experiments are difficult, further observations of solvent satellite resonances in very dilute alloys would be invaluable for this purpose. Satellite lines would arise from near neighbor region (a)'s and, providing they can be disentangled, would provide a severe test for any theory. The interpretation of the alloy Knight shift data de- pends somewhat on the nature of the material in question. The change of 7%, A.7%, upon introducing a second component into a metal, will cause a change in (a)Yp. Whether one interprets A.7% as a change in Xp or in (a) [recall the latter is an average involving the product of a s and X's, over all points of the Fermi sur- face; see eq (17)] depends upon one’s preference for the particular case at hand. In a simple form one may write, in analogy with eq (1), (30) --- | 1 & alloy = (a) alloy X; oy. Permitting both (a) and Xp to vary, as in eq (30), is not as practical a viewpoint for scanning alloy data, as is hold- ing one of the two quantities constant and attributing the trend in A.7 to a variation in the other. For instance, in the case of the transition metal alloys such as the Ti- V-Cr series, the vanadium Knight shift change may be most conveniently discussed in terms of a change in density of states (i.e., Xp). We will discuss this case in more detail later and see that for this case such a description is a useful one. On the other hand, in dilute alloys, where the Friedel oscillation description may be used, the change in 7 is better described by consider- ing the different (a) values as appropriate to the dif- ferent environmental conditions of the host atoms, keeping Xp constant. A general version of eq (30), sampling the Knight shift behavior of the two types of atoms (A,B) in a bi- nary alloy is ( Cl X )älow –H C ( QX Año, (a)p in alloy B C .7%%alloy ( a)B in alloy a) A in alloy A .7%% alloy (a)4 in alloy = (1 – c) –H where 7/4 and (a)4, are the shift and averaged hyper- fine constant of atom A in the alloy. Xp is defined as the susceptibility per atom and c the concentration of B type atoms. Making the nontrivial assumptions that (a) alloy is equal to its value in the pure metal (A or B), and that there is no significant exchange enhancement (32) of X, this equation may be rewritten (using eq 17) in the form given by Drain [74], A N(E), (1–0)M(E)* .7/metal Záč –H cNB (EF) Žº (33) Thus N(EF) anoy = (1–c) Na (EF)* + cMa (EP) *, (34) which defines local densities of states NA(EF)* and Na(EF)*. The assumption of setting (a) in the alloy equal to (a) in the metal forces the whole effect of al- loying to be described in terms of these local densities of states. At times this proves useful. Drain [74] has used eq (33) and the data of figure 8 to scan the AgCd alloy system. The results are in agreement with general trends seen within and between phases obtained in a “rigid band” scan of electronic specific heat data. How- ever, using eq (33), such a scan should not rigorously reflect the variation in the density of states of Ag at and above EP for a number of reasons. These include charge screening (for discussion see ref. 8) and the fact that the hyperfine constants are held fixed. Local effects in covalent compounds, such as chal- cogenides and SiC can also be examined using eq (31). Consideration of these materials is aided by the fact that the energy bands are often more well-known in these than in intermetallic compounds. An example is n-doped silicon carbide [78,155]. The *Si Knight shift is near zero whereas a substantial Knight shift is measured for 13C. This information, together with Ti data for both sites, permitted Alexander and Holcomb [78,155] to infer important wave function symmetries. It was concluded that a zero Knight shift implied a zero wave function density at Si but that symmetry allowed a substantial shift at the carbon site. Lead telluride is another case where local effects are important and where a significant amount of experi- mental and theoretical information is available on the energy band structure. Although the results are af. fected by sample preparation, for the better samples the *7Pb Knight shift in n-type PbTe was found [156] to be temperature independent, and relatively small and positive with respect to undoped PbTe. On the other hand in p-type, PbTe .5% (Pb) was found to be large, negative and temperature dependent. This was interpreted [156] in terms of a band structure model in which the valence band possesses substantial s- 623 character with respect to the Pb atoms, whereas the conduction band lacks s-character at Pb. The small positive shifts in n-type material were assumed to be of orbital origin. The negative contact interaction is as- cribed to a negative g-value for the L-point valence band states. The same band model was used to explain the '*Te Knight shift results in these materials. 9. Correlations of .7%, X and y with Electron Concentration in Transition Metal Alloys There are many cases in the literature where the Knight shift has been observed to vary smoothly with composition in alloy systems. Where y values are available from specific heat data, or other N(EP) information is known, a direct correlation between these quantities and 7% can sometimes be found. Usually the complex nature of .7% (eq 19) causes the cor- relation to be somewhat obscured, and the fact that 7/ does not follow the N(EF) curve is not necessarily an in- dication of nonrigid band behavior. Examples are shown in figure 16a. Looking first at the 3d-alloys, there is a gradual increase in 7 with ela, with a peak at about 5.6 electrons per atom. Between ela = 5.6 to 6, there is a gradual decrease in 7%. This decrease is steepest for V-Fe alloys. The vanadium hydride results follow those of V-Cr extremely closely, as if the electron of the hydrogen is absorbed in the common conduction band, filling the band in the same manner that Cr does. Recent data by Rohy and Cotts [169,170] on V-Cr hydrides (not shown) fall on the same line. The other data, including those for the 4d alloys, all are similar to the V-Cr curve in that they show a peak in 7 at about eſa = 5.6. The Nb-Tc alloy data shown in figure 16a deviate from the general trend. The reason for discrepancy in the case of 7% measurements may be a result of a difference in N (EP), but again may be due to local effects so that no direct conclusion can be drawn from the 7 versus eſa results alone. To give a further picture of the shape of the density of states curves for these alloys we show the total susceptibility data in figure 16b. Both for the 3d and 4d series, there is a possible cusp at eſa = 5. Both X curves have quite similar behavior. The V-Tc(3d-4d) alloy system also follows this trend. From this picture we get a different impression of the density of states curve than from the curves obtained from y data as shown in figure 16c. Here there is no cusp at ela = 5 and a major peak occurs between ela = 4 and 5. Thus there is a dis- crepancy between the y curves on one hand, and the X curves on the other. Depending on which curves are used, the Knight shift data may be interpreted in a —r w W w I t I T- I | 3d alloys, K., | VCr - O.6 2*. szVTC }* ''. VRu - ſ O.4H. * l l f | | ſ l ſt 1 | - | | \, , Kre in Nb Tc – | OH 4 d alloys Y.,' * S$ (q) ge |- X2 O.8H O,6]- l 4OOF- sº 3 d alloys —Tiy | 2 O O Nº. 2º 2 O O º: 4 d d! lo y S Zr N b ( b ) / . Zr Nb Nb MO T | 8 | 9 } C e | T | > Pºm • 4d alloys/TN 4.OF - º, T. U. | 2 | e/d FIGURE 16. Variation of (a) Knight shift, (b) susceptibility and (c) density of states as measured by electronic specific heat, with ela ratio for the b.c.c. transition metals of the 3d, and 4d rows. While for 7% and X there are substantial ranges where X and 7 track one another, the N(EP), curves show less similarity. The data were taken from the following sources: (a) Ti-V [157], V-Cr [157,158], V-Tc [157], V-Fe [159], V-Ru [160], V-H [16] ], Nb-H [162], Nb-Tc [163], Zr-Nb and Nb-Mo [164]. (b) Ti-V [165], V-Cr [157,165], V-Tc [157], Zr-Nb [164], Nb-Mo [163,164], Nb-Re and Nb-Tc [163]. (c) Ti-V and V-Cr [164,166,167], V-Fe [164], Zr-Nb and Nb-Mo [168]. 624 quite different manner. In either case there is no direct correlation between 7% and the other data, and there must be an interplay of several terms as a function of eſa. A number of attempts have been made using y, X, 5%, as well as T, data and the Korringa relation [27], to derive the various contributions to 7%. For example, the results of figure 16 have been rationalized [24,163,164] by using a two-band model for the Pauli term, as in eq (23), and estimates of the Van Vleck orbital effects. Although these do explain the results, the description is not unique. An alternative explanation in terms of vary- ings-d admixture in a single band has been offered [171, 172] to explain the maximum in 7 at ela - 5.6. In the region above 5.6 both y and X are decreasing. Within this model, the decrease in 7% arises from a Pauli con- tribution which becomes less negative, and, in fact, positive with increasing ela. Only 10 to 15 percent S- character in the d-band is required to balance or over- take the negative d-core polarization term. Changes in s-character of only a few percent can produce the ob- served variations in 7%. As noted earlier for Cu (see fig. 10), admixture and variations of admixture of this mag- nitude are not unreasonable. This model is more obvi- ously appropriate to the Tife – Co., alloys, with ela from 6 to 6.5 [172], where the slope of the 7/(*Co) ver- sus X plot reverses sign across the alloy sequence. If the hybridization model is proven valid, the Knight shift can provide a useful probe of the variation of the densi- ty of s-states in “d”-bands. An example of a different type of application of a rigid band model is the proposed band structure in the lanthanum-hydrogen system by Bos and Gutowsky [173]. Lanthanum is a metal and upon adding hydrogen up to 67 percent (LaH2) the material remains metallic. At LaH2, however, the material becomes an insulator. This, together with 7% and X information, was then used [173] to propose the density of states shown in figure 17. These measurements lead to the conclu- sion that adding hydrogen means lowering the ela ratio, which can be considered equivalent to the hydrogen ab- sorbing an electron. This is in contrast to the other model in which hydrogen in the alloy gives up an elec- tron to the conduction band and remains in the lattice as a proton. This latter model has been used, for exam- ple, to describe the V-H and Nb-H results shown in figure 16a, and for the compounds Sch9 and YH2 [174]. An interstitial proton is expected to be a larger perturba- tion in the La matrix and it may bind two 1s electron states to it (as in fig. 17) whereas such electrons might not be bound in the other systems where the per- turbation is weaker. Such behavior can be anticipated from s-wave impurity scattering theory. 56 6S FIGURE 17. Proposed band structure for lanthanum dihydride [173]. For each hydrogen atom entering the metallic lattice, two electrons are assumed to be transferred to localized hydrogen ls orbitals. Formation of LaHa would correspond to complete emptying of the conduction band. 10. Solvent Knight Shifts Confidence that the Friedel oscillations (fig. 13) can be observed was given by Rowland's quadrupole wipe- out data in Cu alloys [129], and reinforced by his sol- vent Knight shift results [150]. Rowland measured the change of the Knight shift upon alloying, Aº’, for a large number of B-subgroup solutes in Ag. From these, he obtained values for T (T = %~"A?/Ac, where c is the fractional impurity concentration). These T data are plotted in figure 18. While there is a general tendency I T T I | ..[] — |.. O H. Sb — Ag host a - 2^e, Sn ~ - O Gd 2^ As T / ^ in — O.5 – …” * ./* CG ...’ _*- 2. T Au ** /..." Z." cºcº O “Ag l l | | — -O- — Cu Zn Gd Ge As . . . [] . . . Ag CC |n Sn Sb —Z\- Au Hg T Pb Bi FIGURE 18. T-1/.7 - Aºl Ac values for impurities in a Ag host versus position of the impurity in the periodic table. The dashed, dotted and solid lines connect points for impurities occurring in the Cu, Ag and Au rows, respectively. These data are taken from Rowland [150), from his table 1. As pointed out by Rowland, the T values are dependent on the range of data employed. 417–156 O - 71 – 41 625 for T to increase with solute valence, there is a slight turn back (i.e., decrease in magnitude) of T from Ag-Ge to Ag-As. For the silver row (namely Ca, In, Sn, and Sb), there is no such turn back. In figure 198 we have chosen two sets of Rowland's A.7 data, for pairs of im- purities of common valence, which clearly display the valence effect seen in figure 18. Rowland [150] noted that curved lines could be drawn to fit the datum points, as we have done for some of the data in figure 19b. Rowland points out that due to lineshape effects (for example, see fig. 14), the uncertainty of the in- dividual points is such that his representation by a straight line is all that is quantitatively reasonable. Granted this uncertainty, the possibility of nonlinearity in these plots of 7 versus c may be real, as was noted by Rowland. Using Rowland’s raw data, a T defined for low concentrations is smaller than that defined by fitting out to larger concentrations. This fact was used by Alfred and Van Ostenburg [175] in their version of the T plot which differs from figure 18, for the Sb in Ag point. By using low concentration data, this T point was reduced from the value given by Rowland, bringing it into line with their [175] predicted turn back. If the same treatment over the same concentration range is used for all of Rowland's datum points, then all the T points in figure 18 will tend to be somewhat lower but the general picture will remain as shown in our figure 18. Alfred and Van Ostenburg neither used Rowland’s choice for T, nor treated the data for all the alloys equivalently. If all of the T values are obtained con- sistently, their phase shift analysis yields neither better, nor worse, agreement with experiment than the earlier phase shift estimates of Kohn and Vosko [176], and of Blatt [177]. Similar valence effects have been seen for B- subgroup solutes in liquid copper alloys [178] and, as seen in figure 20a, in solid lead alloys [179]. The liquid copper results of Odle and Flynn [178] also display the high valence turn back. This result is more evident than in the solid Ag case, the 7 versus c plots being more linear and the turn back in T being larger, although er- rors for the points of most interest are somewhat large. Odle and Flynn [178], utilized the phase shifts of Blatt [176] and Kohn and Vosko [177] to discuss their results. In the solid Pb case, the T values are largest for the smallest valence (Hg), but a reduced effect of valence difference (i.e., the beginning of a turn back) is also evident. The raw data in figure 20a also reveal curva- ture in Ž(Pb) versus concentration for solid PbTI [179], similar to the Ag data in figure 19b. This curva- ture is not evident for the other Pb alloys. In liquid lead alloys, as seen in figure 20b, taken from Heighway and Atomic percent solute, c O 2 4 6 8 | O 12 * | | | | | Ag host - H. A * A % O O ~5 O g-ºf- zāj (d) <] Co < * ^ *s * S.O. ` - * = e • ‘D 6H- • . º &== Hg in Ag SS 8Fs, ". SO ~~. o'ss. q) 4H. ‘. . ~O – O ->< e & 2H O Cu host * - * “I (d) '.. []. Ag host [] # = O L | | l 1 I A S F- Af – 3 * . . . A # |O H. 2 Al in Au ... ' m .9. } * # Au in Au 2^ Å /* X: 8 H / 2 •- jº A. . . . . . ... • LO 2^ • ‘/\: O 2^ sº 6 H Z\ * 4 H * 2H (b) A Au host sm O L-l | I | i — — — — Cu Zn Gd Ge As * g e º ºs e º e Ag CC | n Sn Sb Au Hg T Pb Bi FIGURE 23. Impurity Knight shifts in Cu, Ag and Au hosts. Atomic effects due to the hyperfine coupling constants are divided out, using our Hoff values for the impurities, listed in table 2. The resulting trends are opposite for a gold host than for copper and silver hosts. (Data for boron in gold [195] of 15×10−5% per k0e agrees with the upward trend for the gold host. The uncertainty in this value is greater than that of the points given in the plot.) Points for impurities occurring in the same row of the periodic table are connected with the line symbols indicated in the lower left hand corner (e.g., the dotted line connecting the square datum points is for Ag, Cd, In, Sn, and Sb in a silver host). The Cu and Ag data were taken primarily from Rowland and Borsa [6]; the Au data from Bennett et al. [7]. This latter paper gives in its table 1 further references to the literature for several of the shown points. The Sb in Ag point was taken from Matzkanin et al. [194]. matter is not readily available for the Au alloys. More data of this type would be worth obtaining. 12. Magnetic Disturbances The effects of a charge impurity in a metal have been described above (sections 8, 10, 11). When a magnetic impurity is introduced into the metal, a similar Q1) I I I T I O ->< N. 8|- &E O SS # = 6– to I N 3 4. H - # & 2 - Pb host * jº o --> | | | | l e e s is tº e º e & Ag C d |n Sn Sb Au Hg T Pb Bi FIGURE 24. Impurity (B) Knight shifts in a lead host divided by the impurity hyperfine fields, H#. Data taken from Bennett et al. [193]. The dotted line connects points for impurity atoms belonging to the Ag row of the periodic table and the solid line connects those for impurity atoms belonging to the Au row. The trend is similar to that for Cu and Ag hosts and opposite that for an Au host. response occurs: spin density oscillations (rather than, or in addition to, charge oscillations) are set up around the impurity, as discussed in section 8. The behavior is similar to the oscillations shown in figure 13. The un- balanced spin at a neighboring site interacts with that nucleus via a spin-dependent interaction. Generally this interaction is rather strong compared to charge ef. fects causing correspondingly larger variations in the Knight shift and thus larger values of T. Gardner and Flynn [196] have reported susceptibili- ty and solvent Knight shift results for transition ele- ment (3d) impurities in liquid Cu. The dominant Knight shift term in these cases is associated with the 3d- magnetic moment aligned at impurity sites by the mag- netic field. The susceptibilities of alloys with Cr, Mn, Fe, and Co as impurities obey the Curie-Weiss law im- plying the existence of local paramagnetic d-moments at impurity sites. The moment values, pu, inferred from the susceptibilities are plotted in figure 25. Sc, Ti and Ni alloys do not follow a Curie-Weiss law, suggesting that local virtual d-level band paramagnetism dominates.” The Mn, Fe, and Co moments plotted in figure 25 are of some interest if one assumes that they are entirely associated with impurity site d-character, i.e., little or no moment either residing on the host lat- tice, or in conduction band character at an impurity site. The moments then equal the number of holes in the d-bands and the quantity (10-p) provides an esti- mate of the number of d-electrons at a local site. * The detailed susceptibility behavior of the Cr, Mn, Fe and Co alloys suggests the presence of a small term, of perhaps this sort, in addition to local moment paramagnetism. 630 2O - Liquid Cu host * 4 2 O O - | l | | l 1 Sc Ti V Cr Mn Fe Co Ni Cu Solute FIGURE 25. T=1/3, . Aft|Ac values for 3d transition metal impurities in liquid copper (solid line) and effective magnetic moment, p, (dashed line) plotted versus position in the periodic table. Both sets of data are taken from Gardner and Flynn [196]. The vertical scale of the p plot was arbitrarily chosen so that the height of the T and pº peaks are nearly equal. Gardner and Flynn [196] obtained values of 7.1, 6.4, and 5 for the number of Co, Fe and Mn respectively. These numbers are 0.5 to 1.0 electrons smaller than the d-electron counts believed appropriate to the pure solute transition metals and, if real, this trend offers valuable evidence as to the electronic character of these impurities. The estimate, of course, relies strongly on the assumption that the moments are en- tirely of impurity site d-character with no hybridization with the conduction bands. Electron count estimates for lighter 3d-element impurities, such as Cr, are further hampered by the question of whether or not there is any occupied d-character of spin antiparallel to the net spin of the moment. The density of states with and without such behavior is shown schematically in figure 26. It is probably reasonable to assume that a strong paramagnetic moment such as Cr in Cu has little or no d-spin moment component antiparallel to the net local moment. The T values appropriate to the various 3d-Cu alloys are also plotted in figure 25. These roughly follow the moment behavior, are negative, and are large when compared with the charge perturbation T’s of, for ex- ample, figure 17. They are large because a local 3d. susceptibility, and its associated spin density disturbance, contributes a larger Knight shift effect Cr or ; # Er FIGURE 26. Schematic density of states as a function of energy for Cr metal. In the first case the spin up (?) band and spin down (J) band do not overlap at the Fermi surface; in the second case both spin up, and spin down bands are partially filled. than the weak perturbation of charge impurities. Charge effects are undoubtedly also present in the vicinity of 3d-impurity sites, but these appear to be in- significant if a local paramagnetic moment is formed. The strength of the magnetic term, relative to other ef. fects, is a prime reason for the observed linearity in 7% versus impurity concentration. (This assumes that the magnetic term above tends to be linear.) The negative sign of the T’s might imply that the main peak of the conduction electron spin disturbance (see fig. 13) has its moment antiparallel to the local moment. This nega- tive sign is reminiscent of charge impurity effects and might instead indicate, as in the charge case, that the main peak either fails to overlap solvent nuclear sites or, if it does overlap, contributes satellite lines which are shifted out of the main resonance (e.g., see fig. 14). While the latter would be consistent with charge impu- rity experience, most workers believe that the main peak is sampled by the main resonance line, and that a negative T indicates spin moment antiparallel to the local moment. This in turn implies that hybridization and higher order effects predominate over electrostatic exchange scattering. Combined hybridization and elec- trostatic exchange terms will provide an effective exchange coupling which is not constant as one traver- ses the 3d-elements. One thus expects a crude but by no means linear relation between T and pe. This is seen to be the case in figure 25. Gardner and Flynn showed that a partial wave description involving d-wave scatter- ing crudely reproduces the trend and magnitude of the T's. Flynn and coworkers have also obtained [197] results for 3d-impurities in liquid Al and these are sum- marized in figure 27. Since band, rather than local mo- ment, paramagnetism prevails for all impurities, the added susceptibilities per mole of solute have been plotted (rather than local moment p values). The T values, except for Sc and Ti, are smaller than those ob- tained in the Cu alloys. This is largely accounted for by 631 the smaller susceptibilities (per added solute atom) in the Al alloys. The Cu and Al hyperfine fields, per effective spin moment induced on solute sites, are of the order of – 100 k0e for impurities in the middle of the 3d- series. The hyperfine fields obtained (and the é's derived from them) for the Cu alloys are plotted in figure 28. (A similar plot for 3d-impurities in liquid Al alloys results in much larger uncertainties.) One might expect a somewhat smaller value of Heft for Al relative to that of Cu, since the free atom s-contact interaction of Al is approximately half that of Cu. The fact that it has a similar value suggests that the magnetic response in the Al matrix, due to a given moment on the im- purities, is slightly larger” than in Cu. If one attributes Heft on an average solvent site to an s-moment, with its associated atomic (a), the results correspond to antiparallel spin moments of 0.05 to 0.08 pºp for Al and up to 0.05 pp for Cu for every Bohr mag. neton of moment aligned at solute sites and in the sol- vent matrix. The moment at any given solvent site is small but the total moment residing in the solvent lat- tice can become a significant fraction of that residing on the solutes thus affecting the arithmetic average of d-electron population estimates from susceptibility data. A comparison of the T behavior and the susceptibili- ties for the Al alloys (fig. 27) shows T tracking X more poorly than was the case in the Cu alloys (fig. 25). When making such a comparison it should be noted that the T for Sc, Co, Ni and Cu are of the order of charge impu- rity T's. Thus, charge as well as magnetic effects, may be contributing to T. As we have discussed, the nega- tive sign of the T’s in figures 25 and 27 would seem to indicate that hybridization exchange scattering predominates over direct exchange effects (see sec. 8). There is no reason why such hybridization effects should be constant across the 3d row and the deviation in T from the X curve in figure 27 is of a magnitude ap- propriate to such a variation in hybridization effects. Flynn and coworkers explain the trend with a particular version of such higher order effects, in which the exchange enhancement of the virtual d-level suscepti- bility (see eq. 20) plays an important role. The fact that T lies higher for the lighter 3d impurities could be due to charge effects but it would seem to imply that hybridization effects are stronger (and/or coulomb * One might be tempted to attribute this to band effects associated with the band paramagetism of the impurities in Al versus the local moment paramagnetism of Cr, Mn, Fe and Co in Cu, but note the small effective fields for the band paramagnetic impurities of V and Ni in Cu. TI I & I-I-I- 8. OH- º +1000 \ Liquid Al o \ º- \ host O 2- \ 8– YS –5 < 4.O 5OO 8– gº GD P- - - | × >< (O O o 9 | | | | | | | Sc Ti V Cr Min Fe Co Ni Cu Solute FIGURE 27. T = 1/3 A.7% |Ac values for 3d transition metal impurities in liquid aluminum (solid line) and Xi, the susceptibilities per mole of solute plotted versus position in the periodic table. Both sets of data are taken from Flynn et al. [197]. — O. O5C) Liquid Cu host }* — | OO O.O25 +.ºé – 5 O H. I l T V Cr Mn, Fe Co Ni Cu FIGURE 28. Hyperfine fields and effective & values for liquid copper sampling the effects produced by liquid transition metal impurities. Unlike the normal definition of such quantities these are defined with respect to the impurity susceptibility. This is accomplished by using composition, rather than temperature, as the implicit parameter in a 3 versus Xplot. exchange weaker) for the lighter elements in Al. The peaking of T at Cr or Mn seen in figures 25 and 27 is characteristic of the 3d elements. Quite different behavior is seen for the rare earths (e.g., see fig. 29). As already noted, the variation in Knight shift with impurity concentration is strikingly linear in both the Al and Cu alloys over the ranges of concentration studied. 632 I I I I T I - -I-I-I —18 – |5|H / A4. +|O Liquid Al ſ \z host | \ # C KSX 8 -i6 5, 6 as *— 5-4 5 C T 3 m. 4 = Ó – 5 à NZ -— 2 2 : }* Si | O IO – O | | l | l l l ſ L l 5 | l * Lo Ce Prſ JC Prm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu FIGURE 29. T=1/.7%. Aft|Ac for rare earth metal impurities in liquid aluminum (circles with error bars) and pºeff, the effective total magnetic moment (dashed line) plotted versus position in the periodic table. Both sets of data are taken from Stupian and Flynn [200]. Also plotted is the effective spin moment (S), as discussed in the text (solid lines with squares). In some cases these extended up to five and six per- cent. These are concentrations at which one magnetic impurity would have another magnetic impurity as a near neighbor roughly half the time. At such concentra- tions it is doubtful that a Friedel or RKKY type of theory should be expected to work, for they assume noninteracting impurities which are dilute enough that there are no saturation effects in the solvent. Any effec- tive local-moment local-moment exchange coupling is reduced due to the fact that the experiments were done at high temperature (above 1000 K). This may serve to reduce apparent nonadqitive effects. It may be that averaged multimoment effects are contributing to the T’s. Extremely dilute alloys were not examined; with one exception the lower ranges of alloy concentrations were one-half to one percent. Knight shift data in alloys have also been obtained by y-y, perturbed angular correlation experiments [198]. In such an experiment, a nucleus is observed which has emitted a gamma ray in some particular direction. Thus defining the nuclear orientation, one then observes that nucleus as it emits a second gamma ray in some charac- teristic multipole distribution. Application of an applied magnetic field produces a Larmor precession of the nucleus between the emission of the first and second gamma ray. The precession rate, with its associated Knight shift term, can be deduced from its effect on the second gamma ray distribution. Rao et al. [198] have recently used the technique to obtain the Knight shift of very dilute Rh in Pol over a temperature range of 4.2 to 1053 K. They then used existing susceptibility data, extrapolated to infinite Rh dilution, to obtain a 7% versus impurity site X plot, which was not linear. The 7/ versus X slope appropriate to particular temperature regimes are uncertain due to questions concerning scatter in the Knight shift data and the purity of the samples used in two different sets of susceptibility measurements. (These are strongly paramagnetic alloys and any magnetic impurities will strongly perturb the magnetic response.) The results yield a large nega- tive 7 versus X slope at high temperatures, which is of the order of 4d-core polarization effects, but a much smaller slope at low temperatures. This is consistent with a picture where the impurity contribution to the susceptibility at high temperatures is almost entirely associated with Rh sites, but, due to exchange en- hancement effects involves the entire Rh-Pol matrix at low temperatures. The low temperature 7 versus X slope is consistent with an effective magnetic moment of ~ 10pup residing largely on the solvent matrix. Such a moment was independently deduced [199] from Curie-Weiss fits for these alloys at low temperatures. Stupian and Flynn [200] studied the effect of adding rare earth impurities to liquid Al. The susceptibilities were consistent with local moments as predicted by Van Vleck [201]. With the exception of Sm(4f)” where there is strong multiplet mixing, the moments are ap- proximately - 1/2 L. : J-H2S J peft- [J (J.-H. 1)] "sun-iſºiſ; ALB > (36) where the Landé g-factor has been written out. Very substantial orbital terms contribute to p, and therefore the T’s should not, and do not, track p. The T’s are compared with (S) in figure 29 where (S) is the spin component along J, i.e. === 2S J - 2S J (S) = [J (J.-H. 1)]** *(iii; j) ; – 1 =2,...(* J ) (37) is a measure of the spin component (in pub) parallel to the aligned J. This provides a crude first order measure of effective exchange perturbations. S is antiparallel to J in the first half of the rare earth row and parallel in the second, hence the sign reversal in (S). The T’s dis- 633 play a weaker reversal which is, in part, associated with uncertainties such as the natural zero line for magnetic contributions to T. [Note that La, Yb and Lu impurities have zero-valued magnetic moments yet their T’s lie above the zero line.] The differences between T and (S) are on a similar scale to the effects seen in figure 27. Otherwise there are fundamental differences in T behavior as one transverses the rare earths in contrast to the 3d's. Negative T’s prevail suggesting a tend- ency for the conduction electron spin disturbance to be antiparallel to the spin of the rare earth moment. This is consistent with almost all experience with rare earth elements in alloys or intermetallics. This sign was also observed for rare earths as impurities in Pa [202. 204], at Al sites in REAl2 intermetallic compounds [146], and for 81P, "As and 12 Sb in PrP, PrAs, Tmp, TmAs and TmSb” [206). There is general agreement that hybridization effects are responsible for these results.” The situation with magnetic alloys is seen to be similar to the charge impurity case. Both can be described with models of the perturbations which reproduce the experimental behavior, usually crudely, although occasionally in detail. The magnetic alloy problem is complicated by the presence of several scat- tering mechanisms and by the fact that a magnetic im- purity is also a charge impurity. Solvent Knight shift ex- periments provide unique data for testing alloy models in both magnetic and charge difference systems, but as yet they have provided little unique insight into alloy behavior. Further studies of very dilute systems and of satellite lines outside the main resonance peak should prove invaluable for this purpose. 13. Intermetallic Compounds Relating the Knight shift to the electronic density of states in ordered alloys or intermetallic compounds presents some problems which we have tacitly ignored 14 Jones [205] also succeeded in observing the *Pr and *Tm Knight shifts in these paramagnetic compounds. Shifts as large as 8,900 percent were observed. Jones showed that this is consistent with theory and is due to large orbital hyperfine effects associated with the 4f-moments. He also noted that the temperature dependence of the rare earth and of the nonmagnetic site Knight shifts tracked each other quite faithfully. * But other effects may also play a role. For example, direct electrostatic exchange scattering was not added to hybridization effects in Stupian and Flynn's consideration of the rare earth-Al alloys. Reasonable estimates of the appropriate exchange integrals suggest contributions to T of the order of (and opposite sign to) the observed T behavior. Inclusion of the effect would have over- burdened the model with too many disposable parameters. when considering disordered systems. Let us review the analysis of the Knight shift results [22,207,208 for the technologically important Wax compounds [X= As, Au, Ga, Ge, Pt, Sb, Si or Sn]. It was in their now classic investigation of these intermetallic compounds that Jaccarino and Clogston developed the graphic Ż versus X analysis described earlier [13]. 7 versus X plots (with temperature an implicit parameter) are shown in figure 30 for V and Ga in V3Ga. The tempera- ture variation in X is huge. The variation in X per V atom O. 8 | | | | O.6 – sm- K v O.4 H. *= H. z º O.2 }-sº — Or ul Cl- G V., GO 2 O – 3 iſ . . . H. Li- T -O.2 }* *- Oſ) T Q - O.4 H. tºº-ºº: 2 Y Kco - O.6 }=s *=d -O.8 H *={ -1, O | | | | | O 2 4. 6 8 1O 12 x 10-6 SUSCEPTIBILITY PER GRAM FIGURE 30. Knight shift versus susceptibility for V and Ga in V3Ga, as taken from Clogston and Jaccarino [13]. as a function of temperature in V3Ga is somewhat larger than that per Pa atom in Pol metal. This strong variation requires significant structure in the density of states (e.g., see [110]) within kT of the Fermi energy. To investigate possible sources of density of states struc- ture, Weger [209] considered the role of the linear chains of V atoms which occur on the cube faces in the Vax structure. These chains impose anisotropic elec- tronic properties which, in turn, could produce strong structure in N(E) near EF. Gossard [210] has studied Knight shift and quadrupole effect changes in VaSi across the low temperature cubic-to-tetragonal phase 634 transition. He interpreted the transformation in terms of such a linear chain model. Labbé and Friedel [21]] presented an alternative linear chain model which also is in accord with the experimental situation. Strong negative .7% versus X slopes are seen in figure 30. The slope for Ga is twice as steep as that for V. The X=0 intercepts of .7 are positive and were attributed to a temperature independent Pauli term arising from a broad conduction band with s-like wave function character at both Ga and V sites. (There is probably also a significant orbital Knight shift term contributing to the V site intercept.) The temperature dependent Knight shift was attributed to a narrow V 3d-band into which Ga 4p-character is hybridized, contributing shifts of the form ºv(T)=ww.(a)»x, .5% caſT)= wga(a)cox, (38) where Xn is the d-band Pauli susceptibility per formula unit. The wi are weights per atom of V and Ga character in a formula unit in the band. They also account for any deviation in the hyperfine constants from the chosen values. With correct (a)'s chosen, then the w; are simply weights and subject to the normalization requirement” 3 wy-H waa = 1. (39) The (a)'s were assumed to arise from V 3d and Ga 4p core polarization. The free atom values of — 117 and –44 kOe/pub (consistent with table 3) were used, respectively. Given these (a)'s, the slopes of the ºf versus X plots yield wV = 0.13, and woa = 0.92. The greater waa value is in large part due to the steeper slope of the Ga plot. Testing the normalization condi- tion yields 3 wy-H wea = 1.31, (40) a sum remarkably close to one. This might suggest that the w's are essentially measures of wave function weight. Clogston and Jaccarino observed trends in Knight shift behavior of various Wax compounds which further suggest this. If the w”s are real weights, their values are surprising, for they would indicate that Gap- character, rather than transition metal d-character, dominates at the Fermi surface. Subsequent band cal- * Noting that the molar susceptibility appears in eq (38) and an atomic Xp in eqs (31) and (32), eqs (38) and (39) are equivalent to, and can be used to derive, eqs (31) and (32). culations by Mattheiss [212], yield 2 s wo/wcas 3 un- like a value of 1/7 obtained from 7% versus X plots. The Wax compounds have large N(EF)'s and their suscepti- bilities are strongly temperature dependent. Such behavior is characteristic of a d-band metal. This would suggest that the ratio obtained by Mattheiss is reasona- ble, and thus, that the wi's obtained from the Knight shifts are largely a measure of hyperfine field behavior. Assuming a value for the weight ratio, the Knight shift slopes can be used to estimate experimental (a) values for this compound. A ratio of 2 yields values of –53 and –283 k0e/pºp for V and Ga respectively. The reduced (a)V could be caused by interatomic effects, by intrasite S-band polarization, or by s-d hybridization. Three per- cent s-character admixture into the d-band at EF will ac- count for the reduction. Large negative intra-atomic ef. fects, over and above the core polarization term, are unknown. The value for the core polarization term, shown in table 3, includes the polarization of the closed valence s-shell. Wave function changes on going from a neutral atom to the metal might effect this core polarization term by a factor of two or three but not like- ly by an order of magnitude. Thus the enhanced (a) ga is most likely due to interatomic effects [(a)ca goes to –400 kOe/pub if wV/wca is taken equal to 3]. A similar situation occurs in V8Si. The (a) si is observed to be negative yet the core polarization hyperfine field ap- propriate to atomic P, and thus presumably Si, is posi- tive. The P atomic behavior might be irrelevant to Si but the result again suggests the presence of substan- tial negative interatomic terms at X sites in the Wax compounds. An X-site in these compounds has twelve nearest V neighbors. This implies the presence of a nearest neighbor spin moment which is 20 to 40 times that induced at the X site itself by the magnetic field. Conduction electron polarization effects of the order of those encountered for transition metals in either liquid Cu or Al can, given such a large neighboring moment, account for the value of (a)ca as well as the apparent sign reversal in (a)st. With such a large near neighbor moment, it is also possible that there is a substantial contribution to (a) via direct exchange polarization of the X-site ion core. Knight shift data [213,214] suggest that similar effects occur at Sn sites in the isostructural system Nb5Sn. Subsequent investigations of rare earth and transi. tion metal intermetallic compounds have often relied on 7 versus X plots to disentangle terms. Most of the data are associated with nonmagnetic atomic sites and band hybridization. Interatomic effects are featured heavily when rationalizing the behavior of the hyperfine constants. Interatomic effects are normally interpreted 635 in terms of an RKKY type of spin distribution induced by the aligned spin moments on the magnetic ion sites. A variant of the two-band description of the nonmag- netic site Pauli shift has frequently proved useful, namely 2^(T) = .7% o + 5% loc(T), (41) where 7' 0 is the Knight shift associated with the con- duction band Pauli term and 7/loc is the shift arising from the interatomic response to the aligned spin mo- ment on the magnetic atom site. Zioc, which is presumably responsible for the temperature depen- dence of 7%, has the form 2 — | f *...(t)=#| Hºx. Twº. (42) Here X, is the Pauli susceptibility of either the local moment or band type associated with the moment in- duced on the local moment site. The 209, - 1)/g, factor is included in anticipation of the rare earths, so that H'eft is the hyperfine field at the nonmagnetic site per local spin moment (per molecule) at the magnetic site. The details of the conduction electron distribution arise in the sampling Hit − X. p(R), (43) R where we have assumed that H'eft arises from the con- tact interaction and the sum spans all interatomic radii, R, connecting all magnetic sites with a nonmagnetic atom. Efforts (215, 216) have been made to relate such a sum to 7% values. These have been hampered by in- adequate knowledge of p(R). Asymptotic RKKY dis- tributions were of necessity used, although it is the near R (nonasymptotic) region which is most important to H'ett. More often the alternate approach of assuming that H'eff effectively samples the average p, i.e., the Pauli or Zener response to the local moment exchange field is used. Then H'eff=& a 2%/2pub, (44) where 2//2pub is the exchange coupling per unit local moment between the local moment and the Fermi sur- face conduction electrons. 7/a is the Pauli response of the conduction electrons to this exchange field. If one assumes that the average hyperfine coupling in the RKKY disturbance equals that associated with Fermi surface states alone, then 7/2 = .7% o and 3. (T) = .7% of 1 + (g) – 1) A Xoc(T)/g/Vup”]. (45) Knowing & 0 from an isostructural nonmagnetic com- pound, 4 can be estimated. Physically reasonable numbers for the exchange constants normally result. Even assuming that the average spin moment sampled is equal to the Pauli term in the RKKY response, it is not inevitable that 7% a should equal % o. The spin response involves states off EF and the hyperfine coupling for these states can vary radically from that at EF, as is indicated for the case of Cu in figure 11. Another possible shortcoming of the scheme is that the entire resonant scattering disturbance is not necessari- ly describable in terms of an effective exchange scatter- ing. Although A can be numerically affected by factors other than exchange coupling, tabulation of shift results in this form can prove useful when comparing results in a sequence of intermetallic compounds. For exam- ple, Jones [205] has tabulated the nonmagnetic site Knight shift results of rare earth intermetallic com- pounds in terms of A. The same results [146,217-229] are plotted in a different form in figure 31, namely in terms of § = H'eft|H. Atomic hyperfine behavior is thus normalized out, providing a crude estimate, in pub, of the spin moment residing at a nonmagnetic site due to the local moment disturbance. The resulting & s are an order of magnitude smaller than those appropriate to the transition metal alloys (compare with figure 28) implying much weaker magnetic perturbations in rare earth compounds.” The é's appear to be in three distinct groups; the Al compounds, the P. As and Sb compounds, and those involving elements in the 6s-6p- 5d row of the periodic table. (Data also exist for two hexaborides yielding £s of ~ —0.005.) We presume the grouping is associated with band and wave function character specific to the various sets of compounds. More interesting than the grouping is the variation in § across the rare earth row; & is largest at the Ce end, falling and becoming relatively constant for the heavy rare earths. The trend is very different than that seen for 3d-moments in figure 28 and appears characteristic of rare earth 4f-moment effects. This trend was first ob- served in electron spin and nuclear resonance of the REAl2 compounds [202, 146) and subsequently in ESR of rare earth impurities in Pd [203]. The negative sign of § suggests that hybridization polarization effects dominate. One contributing factor to the large & at the Ce end is the well-known tendency for the occupied 4f levels to be close to EF. The resulting small energy ” This comparison underestimates the drop in polarization be- cause the & values natural to ordered intermetallic compounds are intrinsically larger than those in alloys by the nature of the differing definition of these two & factors. For example, the & appropriate to the intermetallic compounds REAl2 and REAl3 (see fig. 31) are larger than those for the RE-Al alloys. 636 U U U T U REP R EAS RES b R E Bi R EAl2 R EAl; REP+2 — O.OO5 H. : R ESng G RE Pts (site I) RE Pts (site II) tº ^ • As © e e—º–º —º- * = *~g C *-o-o-2 1–1 – 1 – 1 – 1–1–1–1–1. O. O.O.O. " Ce NC FIGURE 31. Pm Sm EU Behavior of Knight's & factor as defined in the text, for the light metal site in rare earth intermetallic compounds. The data are Gd Tb Dy Ho Er Tm Yb from a number of sources [205,217-229), as collected in tabular form by Jones (206]. denominators tend to enhance hybridization, and hence §. An example of this 4f behavior is that a phase transi- tion occurs in metallic Ce, one phase involving no 4f electrons, and the other, one. Positive, strongly temperature dependent 7’s have been observed for nonmagnetic sites in UAl2 [215) and USng [230]. The susceptibility behavior suggests the presence of 5f band paramagnetism, rather than local moment paramagnetism. The resulting £s (~ 0.1 to 0.3) for the two compounds are opposite in sign and sub- stantially larger than the values appropriate to the iso- structural rare earth compounds (fig. 31). The authors [215,230 pointed out that the results could arise from several percent Al (or Sn) valence s-orbital hybridiza- tion into the 5f bands at EF and/or from RKKY polariza- tion with quite reasonable 2% values. The positive sign of the é's implies that electrostatic exchange then dominates. The 7 versus X plots for the two com- pounds also indicated the presence of a strong Xorb term associated with 5f character at the U sites, which makes no contribution to the Al (or Sn) site Y. Abundant data exist for a variety of transition metal compounds. In some of the more magnetic systems the results are strongly dependent on metallurgical details of the samples. For example, NiAl, CoAl and FeAl have been studied by West [231,232) and by Seitchik and Walmsley [216,233] at and off stoichiometry. West found that the Co susceptibility results in CoAl are very sensitive to the thermal history. These results sug- gested nonequilibrium magnetic clustering. The effects of thermal history on the Knight shift are less impor- tant, because the number of atoms near clusters is small and do not contribute sensibly to the observed resonance. Despite these difficulties there are several distinct features of the results which give insight into the character of these compounds. First, the Al shift in FeAl is negative and temperature dependent, suggest- ing the existence of intersite effects of the sort encoun- tered in the Al alloys and the Wax compounds. Second, while the Co shift can be strongly temperature depend- ent (depending on Co concentration), the Al shift in CoAl is small and is effectively independent of tem- perature (not depending on Co concentration). From this it was concluded that there is little Al s-character in the Fermi surface states of CoAl. The slope of a 7. (Co) versus X plot, using composition as the intrinsic parameter, is negative at room temperature and posi- tive at low temperature. Thus there are at least two par- 637 tially cancelling temperature dependent mechanisms operative at the Co site in this system. West attributed the positive slope to a temperature dependent orbital term. Finally in NiAl, the Al shift, the Al relaxation time, and the susceptibility are characteristic of an s- band metal, suggesting that Al electrons “fill” the Ni3d band. This does not imply that there are ten 3d. electrons at a Ni site in NiAl, just as there aren't at a Cu site in pure Cu (see discussion of fig. 11). Instead, charge effects have so affected the bands that there is no substantial d-band character at or within kT of EF. A similar situation appears to occur in dilute alloys of Ni in Cu (234]. Knight shift results have been obtained for both transition metal sites and nonmagnetic sites in itinerant ferromagnets [235,236,14] such as Zrzng. These systems are characterized by having ferromagnetic saturation moments, qs, which are small compared with effective moments, qe, associated with the paramag- netic susceptibility. This implies a band rather than local Heisenberg type of ferromagnetism. A plot of the qc/qs ratio for a variety of compounds is shown in figure 32. These were obtained with the Rhodes-Wohlfarth “intermediate model” [238]. There has been some un- certainty as to whether magnetic impurities drive some of the “itinerant” systems ferromagnetic. In cases, IO i T ! i W ! | l | ſ I Pdo.99 Feo.o. --A-Zr Zn2 ſº —T Fece Coo.4 Sco.76 ſno. 24 - ; FIGURE 32. The ratio qc/qs of the number of magnetic carriers deduced from the paramagnetic Curie-Weiss constant to the number deduced from the saturation magnetization. Datum points are identified by Rhodes and Wohlfarth (237] and Swartz et al. [15]. such as CrBelz (236), the NMR lines are sharp and the hyperfine fields track the magnetization, indicating that the ferromagnetism is a bulk effect, whether or not trig- gered by impurities. The slopes of the 7" versus X plots for hyperfine fields associated with magnetic atom sites, such as Zrin Zrzng, or Fe or Co in TiPezCo I-2, are generally small, ranging between 0 and + 100 kOe/pub. Similar small fields occur for Ti in paramagnetic Tibeg, and V in the V8 X compounds, suggesting the presence of band effects such as s-d hybridization. Weaker hyperfine constants occur at nontransition metal sites in the itinerant ferromagnets, implying that only weak intersite effects are present in this class of compounds. This contrasts with the X site behavior of the localized paramagnetic V3X systems which we believe is due, in large part, to substantial intersite effects. There are a number of examples where 7 versus X plots, with (a) assumed constant, have proven to be very useful. This is not always the case. For example, the Ga resonance in AuCa2 is temperature dependent [12] and while 7% follows X quite faithfully, T, data in- dicate a substantial variation with temperature in the contacts contribution to (a). This has led to a model of thermal population of an s band [238], which, however, does not explain the susceptibility behavior. As of yet, this system is not completely understood. 14. Summary In this paper, we have dealt with the Knight shift and its interpretation in terms of various models of the elec- tronic behavior in metals, emphasizing recent develop- ments. It is apparent that the relation of the Knight shift to the density of states is complicated, but there are compensations in that a large amount of closely re- lated and more intricate information may be deduced from Knight shift studies in metals, compounds and alloy systems. Information may be obtained concerning the wave functions of the electrons at the Fermi surface as probed at the resonating nuclei. Contributions to 7% can be separated into terms arising from s-electron and d-electron character, and in some instances there are indications of contributions due top-character. In addi- tion, orbital and diamagnetic contributions can be deduced at times. We have discussed most of the methods with which one obtains wavefunction insight from Knight shifts. This wavefunction information is re- lated directly to N(E) and is needed in the evaluation of (a). The relations between 7", (a), and the density of states are shown quantitatively in several equations throughout the text. These same equations display the unique relation of 3 with a local density of states due to the weighted averaging associated with (a). This becomes useful particularly in the case of intermetallic compounds and 638 less so for alloys where atoms occupy positions with a random arrangement. Often it is preferable to absorb this randomness into (a). In intermetallic compounds, the Knight shift behavior definitely suggests a descrip- tion in terms of wavefunctions and densities of states that are different for inequivalent sites. In such a situa- tion the magnetic response of one site to another is of concern. 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Letters 21, 546 (1968). [237] Rhodes, P., and Wohlfarth, E. P., Proc. Roy. Soc. (London) 273A, 247 (1963). [238] Switendick, A. C., and Narath, A., Phys. Rev. Letters 22, 1423 (1969). [239] Ryter, C., Phys. Rev. Letters 5, 10 (1960). [240] Ryter, C., Phys. Letters, 69 (1963). 642 Discussion on “Relevance of Knight Shift Measurements to the Electronic Density of States" by L. H. Bennett (NBS), R. E. Watson (Brookhaven National Laboratory), and G. C. Carter (NBS) R. V. Kasowski (DuPont and Co.); The density of states at the Fermi surface and find exact agreement states for Ca as calculated by Shaw at the Fermi sur- with McMillan’s superconductivity data and specific face does not agree with superconductivity data as is heat. shown in the W. L. McMillan paper (Phys. Rev. 167. [1] Allen, P. B., Cohen, M. L., Falicov. L. M., and Kasowski, R. V., 331 (1968)). We have calculated [1] the density of Phys. Rev. Letters 21, 1794 (1968). 643 Pauli Paramagnetism in Metals with High Densities of States” S. Foner Francis Bitter National Magnet Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts O2139 Because the Pauli paramagnetic susceptibility of a free electron gas, X} is proportional to the density of states N(0) at the Fermi energy, it is expected that mea- surements of X% of metals and alloys can yield a reasonable measure of N(0). If X} can be measured directly, then V(0) can be determined with high accura- cy. The major obstacle to this procedure is that the measured static susceptibility X, = x/(1-N(0)/)=x}D (for the Stoner model) where V is a measure of the elec- tron-electron interaction potential. Unfortunately, V is not easily determined experimentally, or theoretically, so that measurements of Xp do not yield accurate mea- sures of N(0). More detailed (realistic) models involve additional parameters which are also not accurately determined. Thus direct measurements of Xp can yield a measure of N(0), but these values of N(0) are subject to the uncertainties of various parameters in the theo- ries. In this talk we discuss the present status of experi- ments and theories of Xp in metals and alloys with low N(0) where D is generally near unity, and high V(0) where D is expected to be large. Comparisons with in- dependent low-temperature electronic specific heat measurements are discussed and attempts to derive N(0) from both susceptibility and specific heat are reviewed. Many recent investigations of metals and alloys with high densities of states have been of great interest because such systems start to approach ferromagnetic order. The properties of Pol have been investigated ex- tensively and very detailed band calculations now exist. The large exchange enhancements in this metal and its alloys have permitted considerable progress to be made. The properties of these systems are reviewed and estimates of N(0) and V are examined. Here qualita- tive and quantitative comparisons of various experi- ments can be made. The possibilities of applying large magnetic fields and thereby independently measuring N(0) and small changes of N(0) near the Fermi energy by such studies are discussed. Recent theories which examine the changes of Xp with alloying are also com- pared with experimental results. A few useful references for relevant recent papers concerning the susceptibility of nearly ferromagnetic metals and alloys are tabulated in references 1 to 4. Recent detailed band calculations in Pol have been re- ported in references 5 to 8. Several papers at this con- ference also deal with high densities of states materials. References *An invited paper presented at the 3d Materials Research Sym- posium, Electronic of States, November 3–6, 1969, Gaithersburg, Md. | Supported by the U.S. Air Force Office of Scientific Research. Density [1] [2] [3] [4] [5] [6] Foner, S., Freeman, A. J., Blum, N., Frankel, R. B., McNiff, E. J., Jr., and Praddaude, H., Phys. Rev. 181, 863 (1969). Foner, S., and McNiff, E. J., Jr., Phys. Rev. Letters 19, 1438 (1967). Foner, S., and McNiff, E. J., Jr., Phys. Letters 29A, 28 (1969). Doclo, R., Foner, S., and Narath, A., J. Appl. Phys. 40, 1206 (1969). Mueller, F. M., Freeman, A. J., Dimmock, J. O., and Furdyna, A. (to be published). Hodges, L., Ehrenreich, H., and Lang, N. D., Phys. Rev. 152, 505 (1966). [7] Anderson, O. K., and Mackintosh, A. R., Solid State Comm. 6, 285 (1968). [8] Misetich, A. A., and Watson, R. E., J. Appl. Phys. 40, 1211 (1969). 645 Discussion on “Pauli Paramagnetism in Metals with High Densities of States” by S. Foner (Massachusetts Institute of Technology) F. Ajami (Univ. of Pennsylvania): Would you care to comment on the temperature dependence of the vari- ous contributions to the susceptibility? S. Foner (MIT): Most of the contributions to the total susceptibility, XTotal, where Xtotal = Xdia + Xsp -- Xºr T Xa-H (Ximp + Xºr) + . . . are not strongly temperature de- pendent. This includes Xaia (diamagnetic core), Xsp (of the s,p bands) and Xer (the orbital or Van Vleck suscep- tibility). For the strongly paramagnetic exchange enhanced metal or alloy (which is the main topic of this talk) the largest contribution comes from Xa (the enhanced Pauli paramagnetism of the relevant d bands in say Pa) and this can be strongly temperature depen- dent. The terms in the (), Ximp (impurity) and Xsic (spin- wave contributions of ferromagnetic systems) can often be suppressed at low temperatures with a sufficiently high magnetic field. If we can use the band calculations we can in principle take into account the smearing of the Fermi distribution as we increase the temperature and hope to be able to calculate the temperature depen- dent susceptibility. There have been numerous at- tempts of this in the literature; qualitatively, they will give good results, but quantitatively, I am not sure that one can really believe them. O. K. Andersen (Univ. of Pennsylvania): About the band structure calculations, I think perhaps you have been a bit unfair to them, because I think all three or at least the two — the Mueller, Freeman et al. and the Mackintosh and Andersen agree. The Watson- Misetich-Lang results differ somewhat. The two first mentioned calculations are essentially first principles calculations, either fitted to a first principle APW cal- culation as closely as possible or done entirely with the first principles relativistic APW method. The discrepancy between different band structures calculated for the same transition metal is mainly a dis- crepancy in the width and the position of the d-bands relatively to the sp-band. Therefore, for such parts of a d-band where hybridization with the sp-band is unim- portant, the results of different calculations only differ in the energy-scale; the shape of this part of the band, which is a structural property, is unique. Now, in that narrow energy range around the Fermi level, which is relevant for the high field susceptibility of Pa, the domi- nant contribution to the density of states comes from such a purely d-like part of the bands. If, therefore, the density of states were calculated with infinitely high resolution for each of the three above mentioned Pol- bands, the derived deviations from linearity of the curve magnetic field versus magnetization would in each case have the same shape, but would differ by the scale of the d-bandwidth; If the position of the Fermi level within the d-band was the same for all those calcu- lations then they would, for instance, all predict the ex- istence of a van Hove singularity in the magnetization curve for Pd at a magnetization of about 0.06 Bohr mag- netons per atom, but the estimates for the correspond- ing magnetic field would scale with the d-bandwidths. Finally, let me mention that from our relativistic APW bands we have dared to derive the band structure contribution to the deviation from linearity of the mag- netization curve for Pol and, using the dubious rigid band model, for dilute RhB d alloys. Our results seem to agree with the measurements of Foner and McNiff, but comparison is difficult, since the effect is of the same order of magnitude as the experimental uncer- tainty. We will report on these results at the coming conference for Magnetism and Magnetic Materials. R. E. Watson (Brookhaven National Lab.): It might be best if I first reviewed the origin of the three sets of band structure results. The Andersen-Mackintosh result relied on a relativistic APW calculation: In a nar- row range around the Fermi energy the details in the density of states were calculated by tracing of constant energy surfaces and by fitting the calculated volumes to a power expansion in energy, which includes the characteristic 3/2-power van Hove term. The Freeman- Mueller, et al. results involved nonrelativistic APW bands which were then fitted with Mueller’s d tight binding-OPW interpolation scheme and spin orbit ef- fects were added. The resulting band structure was then sampled with a quadratic interpolation scheme when estimating N(E). The results of Misetich, Lange and myself involved a fit, with the Hodges, Ehrenreich and Lange tight binding-OPW scheme, to one set of the Freeman, Dimmock, Furdyna nonrelativistic APW 646 results. A spin orbit constant was chosen (different from that of Mueller et al.) and the interpolation scheme parameters were further adjusted to improve agreement with the experimental Fermi surface areas. The final set of parameters was by no means unique. There are obvious dangers in such adjustments to the prediction of other quantities such as N(EP). The bands were then sampled with a quadratic scheme to con- struct a histogram. Dr. Foner mentioned “high resolu- tion calculations.” The three sets of results are “high resolution” in only one sense: Given particular band structures, careful samplings were taken when con- structing density of states histograms. The Andersen result is, in principle, to be preferred since it did not utilize an intermediate tight binding-OPW description of the bands with attendant questions, such as those concerning subtle features of the d bands which are not reproduced by the near neighbor tight binding scheme. I would suggest that the three sets of results are in es- sential agreement, though perhaps not for Dr. Foner's purposes. Our experience with other transition metals shows it is difficult to nail down such factors as d bandwidth to the accuracy important to our estimates of N(EF) here. There may be serious shortcomings in the potentials employed to date in the “a priori” calcu- lations for the transition metals. Good predicted Fermi surfaces are inadequate tests of band results; the band structure results of Misetich, Lange, and myself are probably the poorest of the three sets due to improving the Fermi surface fit. J. Callaway (Louisiana State Univ.); It was evident from Professor Ziman’s lecture earlier in the week that d bands may be sensitive to the crystal potential in the region between atomic spheres. One should therefore be cautious in considering the APW results as valid without independent confirmation. Specifically, the density of states at the top of the d band will depend sensitively on the relative positions of the X5 and the X2 levels. These have different symmetry, and are split by crystal field effects. The relation of those levels will be affected by the non-spherical components of the crystal potential and by the potential in the interstitial region. It is thus possible that these levels may not be correctly located by APW calculations which use a spherical “muffin tin” potential. These considerations emphasize the need for self-consistency. A. J. Freeman (Northwestern Univ.); Let me point out that the calculations originally done by myself, John Dimmock and Anna Furdyna, a number of years ago, were first principles APW calculations. We considered a number of potentials; we did not just take the first potential off the shelf. These calculations are, I think, important for the following reasons: Palladium has now become perhaps the best studied metal; all these vari- ous APW calculations agree very well with each other and with experiment. I do not understand why the interpolation used by Misetich, Lange, and Watson of the bands that were calculated by the Furdyna, Dimmock, Freeman group give such different results, because in fact, the agree- ment of the relativistic calculations of Andersen and Mackintosh with our calculations is remarkable (as you saw and heard). Now as far as the question of sensitivi- ty and so forth of those levels near the point X, well, you are right, there is great sensitivity. It turns out that despite this sensitivity, we do not find differences in the results despite the differences expected, due to the use of essentially different techniques. Andersen-Mackin- tosh solved the Dirac equation. We solved the non- relativistic Schrödinger form. Finally, let me say that we have here a wealth of material to discuss, argue about, and try to understand. I think that is what Si was trying to do for us. At the moment, I would say that the two of the three groups agree—the burden of proof is on the third group, to ascertain why their results disagree. A. R. Mackintosh (Lab. for Electrophysics, Tech. Univ., Denmark): I just wanted to make a suggestion. Could you not improve the accuracy of your experimen- tal measurements by using field modulation and mea- suring the gradient of the susceptibility? I believe that this is the direction in which improvement is to be sought, rather than in attempting further refinements of the calculations which, after all, already make much more precise predictions than the experiments can test. S. Foner (MIT): The modulation technique is probably not the way to go. Estimates show that this approach lacks about a factor of 100 in sensitivity over our present magnetic moment measurements in high do fields (see S. Foner and E. J. McNiff, Jr., Rev. Sci. Instr. 39, 171 (1968)). The major limitations are the background noise in the high field magnets—even 0.01% peak to peak noise at 150 kG is a large flux change. We wish to detect a change of 0.1% in the mo- ment of Pa at 200 kG corresponding to a change of about 0.01G at this field. As improvement of a factor of 10 in the signal-to-noise is needed in order to reexamine the field dependence of X for pure Pd, where we have high resolution band calculations. I want to make another comment. One susceptibility measurement which I have not discussed is the dBiv A experiments which have been carried out so well by the Argonne group who are just starting to see parts of the pipes. If they can get more information on these major pieces of Fermi surface we would have much more confidence in 647 the various Po band calculations. Perhaps Windmiller might want to say something about this. L. Windmiller (Argonne National Labs.): We are try- ing to press our measurements further. We have seen some more of those pieces. The only other comment I wanted to make was about the g-factor. It is anisotropic and deviates from 2. When we reported measurements before, we only found the g-factor being different from 2 on the T-centered surface. Since then, in both Pol and Pt we have seen spin-splitting zeroes for the large d-like surface and in both of these the deviations from 2 are even greater on the open hole surface than they are on the T-centered surface. The relation of the susceptibili- ty to the density of states thus seems to be a question. I think we should start pushing the band structure peo- ple to see what they can do. Also, questions concerning how the g-factor varies with the Fermi surface as we start alloying things and putting in different materials so that you keep the Fermi level fixed are moot points. We just don’t know what happens right now. 648 Calculation of the Knight Shift in Beryllium” J. Gerstner cind P. H. Cutler Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802 Key words: Beryllium; Knight shift: orthogonalized plane wave (OPW); pseudopotential. A nonlocal pseudopotential theory has been used to calculate the contributions to the Knight shift in berylli- um. For the first time, the computed value has both the ( correct sign and numerical agreement with recent ex- periments. This work has been published in Physics ( ) = — 0.004.10% Letters 30A, 368 (1969) with the exception of the Calc tº following changes: diam ) = — 0.005.14% *Research sponsored in part by the Air Force Office of Scientific Research, United States Air Force, Grants No. 213-66 and 69-1704. 649 Discussion on “Calculation of the Knight Shift in Beryllium” by J. Gerstner and P. H. Cutler (Pennsylvania State University) P. L. Sagalyn (Army Matls. & Mech. Res. Cent., J. Gerstner (Operations Res. Inc.): The answer is no Mass.): I would like to ask Dr. Gerstner whether he has and no. We have essentially stopped this calculation for or possibly could, calculate the contribution of the a lack of funds and time. So I consider it a closed sub- diamagnetic field to the spin-lattice relaxation time? ject. I would be glad to discuss it with you on the side though and see what could be done. 650 Knight Shifts of the Alkali Metals A. Meyer Northern Illinois University, DeKalb, Illinois 601 15 G. M. Stocks” and W. H. Young University of Sheffield, Sheffield, England K, Rb and Cs, being leading members of transition series, have pseudoatoms with virtual bound d states; Li has a somewhat analogous p state. By contrast, Na has no such states. Evidence is offered of how the associated scattering accounts for the following observed electron transport properties: (a) Under pressure, the resistance of Li rises and, eventually, so do those of Cs, Rb and K, (b) the ther- mopower of Li is anomalous (positive) and stays so under pressure while that for Cs very quickly becomes positive when pressure is applied. The same features can now be used in the theory of Knight shifts to explain the following observa- tions. (c) The conduction electron susceptibility for Li is enhanced very significantly above that for free (and even interacting) electrons, (d) the nuclear contact density in Li is much lower than that predicted by one-OPW theory, (e) the Knight shifts of Li and Na decrease and those for Rb and Cs increase when pressure is applied, (f) the Knight shift for a given ion increases when it is successively resonated in Na, K, Rb, and Cs matrices. The key to the interpretation of (e) and (f) is the variation in the density of states (and therefore susceptibility) under conditions of pressure change and alloying. Key words: Alkali metals; alloys; cesium; Knight shift; lithium; potassium; pressure dependences; pseudopotential; resistivity ratios; rubidium; sodium; thermoelement power. 1. Introduction This paper is concerned with interpreting the ab- solute Knight shifts of the alkalis, their variations with pressure and the absolute Knight shifts in binary alkali systems. While based on precise and detailed calcula- tions [1-3], for the present conference it seems to us desirable to give an overall (and less formal) perspec- tive to the work, to highlight the main features and show how the basic underlying explanation of these is the same as for the electronic transport properties. We will use for the Knight shift, K, the basic formula [4] K= [87/3)xOP, (1) where () is the total volume of the metal, Pr is the (average) density of valence electrons evaluated at the resonant nuclear site and at the Fermi level, and X is the magnetic susceptibility per unit volume of the *Present address: Metals and Ceramics Division, Oak Ridge National Laboratory, Tennessee. valence electrons. In section 2 we indicate the develop- ment of a simple formalism that enables one to calcu- late both X and QPR in a mutually consistent manner. Guided by the Ziman [5] pseudoatom concept, the problem is reduced to the solution of radial Schrödinger equations. QPF is given, after renormalization, by the nuclear contact part, the latter being very conveniently summarized by a simple formula, as we shall see. At large distances the solutions are characterized by phase shifts, these describing X and the renormaliza- tion factors indicated above, as well as the transport properties. The phase shifts have been tabulated [6] and are not so conveniently summarized. Nevertheless, each metal has its own characteristie behavior which can be qualitatively illustrated by its transport proper- ties. It seems better to do this (as we do in sec. 3) than quote a lot of already available numbers. Turning to the Knight shift applications, in section 4, we consider the absolute Knight shifts, including those under pressure [7] and in section 5 we present an in- terpretation of recent work on alloys [8,9]. Finally, in section 6, there is a summary. 651 2. Formalism We consider an electron with Hamiltonian H=T+X, U(r–R) (2) where the U's describe the screened ion fields. (For the moment we consider a pure metal where all the U’s are the same; the generalization to alloys is straightforward and will be affected later.) Then we postulate the wave function l, ºr eikºr HX, eikº dº (r-R) (3) R which is written so as to show a plane wave together with the scattered waves demanating from the various ions. The latter interpretation is formalized by comput- ing the expectation value of (2) with respect to (3), discarding multi-center terms and using the d’s as vari- able functions. The result is (T+ U(r)) &l (r) = (hºk”/2m) &l (r) (4) where § 1. (r) = eikºr + d is (r) (5) In a similar approximation we find a mean contact density of P = (N-1 R.) | 2) = |ć, (0)|* - Wºº-Hwrºtº-III. (6) The interpretation of this is clear. If the integral were not present in the denominator, the result would cor- respond to (4), i.e., one center in an otherwise feature- less medium. The renormalizing term consists of equal contributions from each of the N sites and is an expres- sion of electron-ion affinity. The solutions (5) of (4) lead to phase shift mi(k), from which, when small (modulo T), the energy can be ob- tained by first order perturbation theory. From meſh- k”–27 (N/Q) > (2l + 1) milk, (7) one computes an effective mass at the Fermi level given by m/m” = 1 + s – or, (8) where s= (2/3T) > (2/+ 1) m (kr) (9) and O = (2/3T).X. (2l-Hl) (kömlők)r (10) and a susceptibility can then be computed from (m”/m)x where X, is the free-electron value. One must also incorporate an important valence electron correla- tion correction [10-12], but the essential features seem already to be present in the above analysis. The phase shift gradients which occur in (10) are also the parameters needed to evaluate the denominator of (6). In fact (Kittel [13]), on the mean, Jºſé, (r)|*-1}dt = (27/k”)x (21+1)(konſſøk) (11) Thus we need to compute {(0)’s for (6), and m’s and ômſók’s for (6) and (8); this can be done straightfor- wardly from (4) once U is specified. As developed above, the theory does not distinguish between solids and liquids (except by the small density change involved) and this is in agreement with experi- ment [14]. In fact the applications to pure metals under pressure given in section 4 are for solids, while those to the alloys in section 5 are for liquids. The choice of U faces us with a difficult problem. In the past, the present authors have used truncated ion potentials and guided by pseudopotential theory [5], have required (cf. (9) above) that s = 1/3 governs the choice of cut-off. On the other hand, recently, it has been suggested [15,16] that s = 0 (corresponding to muffin tins; see below) might be better. Indeed Lee [17] has fitted the observed Fermi surface anisotropies in the solid alkalis to phase shifts which roughly satisfy this criterion. It seems to us [18] that the resolution of this difficul- ty lies in remembering that pseudopotentials overlap somewhat and are rigidly located on the various ions. Thus, as the ions move about, fluctuations in the net potential in the interstitial regions are described [19]. This physical aspect of the pseudopotentials is missed by the muffin tins which correspond only to the mean minimal nonoverlapping potential. If fluctuations are unimportant, then the use of s = 0 should be better for formulas derived on the basis of perturbation theory, since the corresponding phase shifts are smaller. On the evidence of computations, fluctuations are important in electron transport theory for which s = 1/3 seems more appropriate. (Computations based on the latter criterion are fairly successful [20-24]; similar cal- culations using a mean muffin tin are not.) It may well be that s=0 is more appropriate for cal- culating m” via (8). Certainly, in the low temperature 652 solids, fluctuations should be negligible; also the ex- perimental evidence [25] suggests that X for Li is little changed across the melting point. However, at the present time, we have not made enough systematic cal- culations to report here fully in this approximation, though it does appear that both criteria lead to the same general conclusions, both explaining semi-quantitative- ly the broad features of the Knight shifts in the alkalis and their alloys. Below, then, we rely on pseudopotential screening. Ion core potentials are truncated to form U(r) and solu- tions of (4) are found with the large r behavior (kr)-1 sin (kr – 1/2 litt-H mi(k)). In general, those phase shifts for k = k will not satisfy s = 1/3, but the cut-off radius in U(r) is adjusted until the criterion is satisfied. Once U(r) is fixed, mi(k) can be found for any k. Phase shifts found in this way have been successful in explaining the electron transport properties [20-24]. On evaluating the s-wave functions at the origin, we obtain values of the contact term. On analyzing these, we find they are all well represented by the formula |éºr (0)|*=5.3Z(krao)-lºanº (12) when Z is the atomic number and ao- h”/me”, the first Bohr radius. 2. 3. Phase Shifts and Electron Transport For very general reasons [26], the low energy phase shifts for a screened ion are described by m ºr k”; kóm/0k = (2 + 1)n, (13) and, in fact, this is quite a reasonable representation of the situation up to quite near the Fermi level (fig. 1). The higher energy behavior then depends on the specific ion. Most stay small over the ranges of energy of interest, but certain exceptional m’s (two of which are shown in fig. 1) grow large. The latter behavior may be interpreted using the con- cept of the virtual bound state [27]. The lowest unoccu- pied energy level of the Cs atom is the 5d level. When the atoms join to form the metal, this level is not lost, though it is no longer sharply defined. It may be de- tected by the strong scattering it produces on electrons of appropriate energy. Thus me for Cs rises to high values characteristic of resonant scattering (m2 - Tl2 for maximum cross section; 6°m2/ök*= 0 for maximum delay time [26]). Similar behavior is found in the d waves of Rb and K, though the onset of resonance oc- curs at higher energies than for Cs. In Li, the 2p level plays a role somewhat akin to the 5d level in Cs, 2.0- 1.5- +d 0.5– O 02 0. 06 Öğ fo k (a.u.) FIGURE 1. Energy dependence of two pseudoatom phase shifts (m2 for Cs and v, for Li). Rb and K have similar m's to Cs, though the resonance occurs at higher energies. All other phase shifts for the alkalis are small over the energy range indicated. This seems to be the common feature correlating the experimental results of figures 2–5 and of table 1. though, as is usual for a p level, the curve is much broader. If we squeeze or alloy a metal, the scattering charac- teristics of the ions have to be worked out afresh. How- ever it is a useful first approximation (and excellent for illustrative purposes) to think of such curves as those shown in figure 1 to stay the same, with only the Fermi level moving. To indicate that we have a basically correct picture we quote a number of pieces of experimental evidence and we emphasize that out comments on them have been backed up by detailed computations [20-24]. Resistance measurements are, of course, good for revealing strong scattering at the Fermi level; the 'stronger the scattering, the higher the resistance. Figure 2 shows the variation of the resistivities of the al- kalis under pressure [28.22]. Normally, the resistance of a metal drops under pressure, the explanation being [29] that the stiffer lattice is more favorable for coherent diffraction by conduction electrons. Figure 2 indicates, however, that under pressure all the alkalis behave abnormally; to explain this, we turn to the other effect involved, namely, the strength of conduction electron scattering at single sites. Returning to figure 1, we see that for Li, even at zero pressure, the Fermi level occurs well up the mi curve, the application of pressure causing it to move further up. The scattering is thus increased in intensity and this is reflected in the increased resistance. In Cs, the stiffening of the lattice wins to begin with, but very soon the growing d-wave scattering dominates and one ob- 653 2.0 Cs /l Rb 2% Z 5 s 8 0-7 0-6 0-5 ſh/no FIGURE 2, Experimental resistivity ratios versus fractional volume changes [28]. The rising curve for Li reflects kr in figure 1 rising up the mi curve. The subsequent rise in Cs is due to the same effect for the m2 curve in figure 1. tains a sharp increase in resistance. To a much lesser extent, Rb and K exhibit the same characteristics as Cs, the resistivity increases being more delayed in these cases because of the higher-energy positions of the virtual bound states. This picture is corroborated by the thermopower measurements [30]. The thermoelectric power of a metal depends on the change in the cross section ex- perienced by electrons as the Fermi level is traversed. Clearly the presence of a virtual bound state not far above the Fermi level gives rise to a rapidly increasing cross section at the Fermi level. The (positive) ther- mopower thus produced is anomalous in that, contrary to the simplest free-electron arguments, electrons dif- fuse from cold to hot regions in the metal. Figure 3 shows the experimental situation. Li already shows an anomalous thermopower, even at zero pressure, in- dicating the already dominant mi, while the ther- mopower of Cs becomes positive on applying only modest pressures. That for Rb is beginning to turn in the expected direction and at higher pressures the same should happen to K. (Note that the compression range is smaller in fig. 3 than in fig. 2.) - With the above general picture in mind, we now turn to the Knight shifts. 4. Knight Shifts in Pure Metals 4.1. Absolute Knight Shifts in Li and Na Usually, susceptibility measurements refer to a sam- ple as a whole. Actually, only for Li [31] and for Na : 85 ſh/ſo FIGURE 3. Experimental thermoelectric powers versus fractional volume changes [30|. The positive and constant value for Li arises from the shape of mi in figure 1, while the shape of m2 explains the positive value for Cs under pressure. [32] has X for the conduction electrons alone been ob- served (conduction electron spin resonance being the technique used). Since the Knight shifts are also known [7,14], then, assuming (1), both X and QPR are known independently for these metals. The experimental situation and our computed numbers are shown in table 1. To understand the experimental effective masses, we need, first, to remember that valence electron correla- tion plays an important role [10-12]. In fact, this effect can be “subtracted out” and, according to Rice's calcu- lations [12] one needs an mº/m, from eq (8), slightly greater than unity for Na but considerably greater for Li (see table 1). Such behavior is easily explained by the present ap- proach. Na has no nearby virtual bound state (cf. fig. 2). All its phase shifts are rather small and also (cf. eq (13) TABLE 1. Susceptibility and nuclear contact data m*/m | | Target "... 1 -QPr ſº QPs M l t. :}; 1 81CUl- C81CUI- eta (expt.) | mº/m lated) (expt.) lated) or W) Li.....….. 2.6 1.6 1.8 15 2] 54 Na.................... 1.7 1.1 — 1.2 1.0 119 120 176 * mºlm (expt.) is the ratio of the observed to the free-electron susceptibilities. According to theory [10–12], this contains a substantial contribution from conduction electron correla- tion effects; when these are removed, one obtains a target m”/m which a one-electron theory must aim at for eventual agreement with experiment. (There is a little uncertainty here due to our imprecise knowledge of the susceptibility of an interacting electron gas. The numbers quoted rely on the calculations due to Rice, the lower value for Na arising from the Hubbard and the higher from the Silverstein interpolation formula. Both schemes give the same num- ber, to the accuracy shown, for Li). The target m”/m's may thus be compared with those calculated from eq (8). The experimental QPF's are obtained from eq (1) using the observed K’s and X's and directly following them are the values obtained by the present techniques. The one-OPW results quoted in the final column give a measure of the improvement wrought by the present method of calculation. 654 so are its phase shift gradients. It follows, therefore, that mºlm, as given by eq (8) will be near to unity. But for Li, the dominant mi corresponds (cf. eq (13) and fig. 1) to an even more dominant gradient and so eq (8) leads to an elevated value of mºlm (table 1). Turning to the contact term, we see that the present approach improves on the single-OPW method, espe- cially for Li (table 1). To explain the numbers, note that eq (6) gives 35 and 170 for Li and Na respectively and in fact the corresponding OPW numerators [33] are not radically different from these. The difference oc- curs in the denominator of eq (6). In the OPW approxi- mation this is always less than () [33] (corresponding to a displacement of the free electrons into the intersti- tial regions) and is responsible for the enhanced OPW results shown in table 1. But remembering the domi- nant p wave in Li, on the basis of eq (11) we expect, and find, a renormalized volume considerably in excess of Q in eq (6). This leads to the substantially reduced (and very satisfactory) calculated value shown in table 1. The strong p character lowers the contact density which, of course, represents the s-wave component only. The result for Na, though less drastically affected, is in the same direction and agrees well with experi- ment. 4.2. Volume Dependence of Knight Shifts We now provide an explanation of the experimental results of Benedek and Kushida [7] shown in figure 4. From (6), (8), (10) and (11), we may write (1) in the form léke(0)| | + O. 8T. X, 3 1 + S – Or K= (14) This omits the important effect of correlation on the ab- solute magnitude of X (cf. 4.1. above) but is quite adequate for explaining the qualitative variations of K exhibited in figure 4. In view of (14), one might consider as normal those cases where the volume dependence is adequately described by Xºlºr (0)|*. Recalling (12) and remembering that Xf or kr, we see that normal behavior (as thus defined) corresponds to a slight decrease of Knight shift with increasing pressure. To understand departures from this behavior, one must examine the two denominators of (14) which, un- like the numerators, are peculiar to the metal under consideration. The important variable is or for, in the present pseudopotential screening approximation, s = 1/3 and even for muffin tins eq (13) applies. As has been emphasized, all phase shifts and gradients for Na remain small, even under pressure (fig. 2). It follows, therefore, that Na will be normal as indeed it is (fig. 4). Our calculations indicate that a 1. A — Experiment. ----Theory. K/Ko 1-3- 12 1-1H 07 FIGURE 4. Knight shift ratio versus fractional volume changes. The experimental values are taken from ref [7]; the theoretical values are computed using eq (12) and the data of ref. [6]. The use of muffin tins (cf. sec. II) should improve agreement with experiment. The rise shown for Cs and the fall shown for Li are explained by the curves of figure 1. similar explanation seems appropriate for Cu (eq (12) being an adequate, if rather less accurate, representa- tion of the contact term in this metal). The normal behavior of Li follows since, while mi is significant at the Fermi level (and therefore giving, for example, as we saw in 4.1., an enhanced m”), its gradient (cf. fig. 1) is rather constant. There is a complete parallel with the thermopower situation (fig. 3). The high value of the gradient gives the anomalous negative result, while the insignificant variation with pressure of the gradient im- plies the constant thermopower shown. Finally, to explain the observed abnormal response of Rb and Cs we note the nearby virtual bound states in these metals (figs. 1-3), these giving rise to sharply in- creasing or’s when pressure is applied. The variation of 1 + s — or in (14) is more important than that of 1 + or since the effect of the zero in the former (or, cf. (14), the pole in X) can be felt even at the densities of interest. (The eventual divergence in X is, of course, a purely mathematical consequence of the terminated perturba- tion expansion (7); physically a greatly enhanced value should be anticipated.) Thus, one concludes that the rising Knight shifts shown in figure 4 primarily arise from the sharply increasing effective masses. Con- sistent with figures 2 and 3, the curve for Cs rises more rapidly than that for Rb. 5. Knight Shifts in Alloys The observed Knight shifts of binary alkali systems vary quite linearly with concentration across the com- 655 — Experiment. 025 * = * = Theory. 0: 20 0.15 0.10 × 2 º No. K Rb Cs FIGURE 5. Experimental Knight shift of a resonant ion, scaled by its atomic number for various matrices. The pure metal results are taken from Drain's review [14] and the alloy results from van der Molen et al. [8]. (There is little difference between the latter's results and those of Kaeck [9]). The present theory can explain (using figure l; see also the caption) the trend from matrix to matrix, but not the variations for a given matrix. The theoretical results labelled MNY were computed using the data of ref. [6]. The results for Na, K, and Rb matrices include correlation in the calculation of X but not that for Cs which would, in that case, be divergent (recall the limitations of eq (7) discussed in sec. II). We believe the slight drop thus calculated in moving from K to Rb matrices arises only from our inability to calculate with adequate precision. To illustrate this, we show also theoretical results (labeled SY) using the data of Stocks and Young [34] which are, in principle, an improvement on those of ref. [6]. position diagrams [8,9]. The experimental situation may, therefore, be summarized as in figure 5 where we have plotted the Knight shifts for very low concentra- tion solute ions as well as those for the pure metals. To explain figure 5 we need to generalize section 2 to the case of alloys. This is formally quite easy; one adds appropriate subscripts to the U’s and d’s of eqs (2) and (3) and finds a Schrödinger equation of type (4) for each type of ion, the appropriate kr being that of the alloy. Specializing to the case of solute shifts, the numera- tor of (6), the unrenormalized contact density is for the solute since this is the resonant ion but this must be evaluated for kr defined by the solvent. Also, since the denominator of (6) arises from the matrix ions (see the comment following that equation), this, like X, will be given by the solvent. Thus, from (1), (6), (11) and (12), we have — 1 .45 A, -3 8T 5.3%resonator (k; matrix a0) 'do 3 Xmatrix K= (15) | + O. matrix We can look upon (15) as a generalization of the pure metal formula, though in the latter case every ion is par- ticipating in the resonance and the signal is much Stronger. On scaling these Knight shifts by the atomic number on the resonant ion, we expect, therefore, a number de- pending only on the matrix. As figure 5 shows, this is a fair first approximation, though there are systematic trends for various ions in a given matrix which the present theory cannot account for. The increase as one moves to heavier matrices is, however, easily ex- plained. Just as in section 4.2., the susceptibility dominates the renormalization term and, in view of figures 2-4, this should increase as one moves from Na to heavier matrices. This conclusion is consistent with the observed electronic specific heat data [35-37] though the correlation is not clear-cut because of the significant electron-phonon contributions to the effec- tive masses in the latter cases [38]. The realization that the susceptibility dominates this problem was due to Kaeck [9]. Note that we have not discussed the rate of change of Knight shift as the alloy concentration is varied. This would have involved us in different and, as yet, un- resolved problems [39,40]. 6. Summary The scale and shape of the calculated mi curve for Li shown in figure 1 are corroborated by (a) the rise in re- sistance with increasing pressure (fig. 2), and (b) the sign and lack of variation of thermopower with pressure (fig. 3). When applied to the theory of Knight shifts, it explains also (c) the enhanced susceptibility (table 1), (d) the failure of the single-OPW method to predict the contact term, and (e) the fall of Knight shift under pres- sure (fig. 4). Similarly, the virtual bound state shown in the m2 curve for Cs in figure 1 is corroborated by (a) the even- tual rise in resistance shown in figure 2, and (b) the change of sign of the thermopower shown in figure 3. In the theory of Knight shifts, it explains also (c) the rise of Knight shift with increasing pressure (fig. 4) and (d) the enhanced Knight shifts exhibited by solute ions in a Cs matrix. Virtual bound states associated with Rb and K account for the corresponding properties of these metals in a similar way. Na has no strong phase shifts and so has unremarka- ble transport properties: (a) a resistance which falls and stays small when pressure is applied (fig. 2), and (b) a negative thermopower which does not change much under pressure. In the theory of Knight shifts (c) its contact density and susceptibility can be understood by free-electron arguments (table 1) though preferably, in a correlated framework for X, (d) it has a normal (cf. sec. 4.2.) falling shift (fig. 4) when pressurized, and (e) the decreased shifts exhibited by solute ions dissolved in Na is thus explained. 656 7. Acknowledgment G. M. Stocks acknowledges a United Kingdom Science Research Council award. 8. References [1] Micah, E. T., Stocks, G. M., and Young, W. H., J. Phys. C. 2, [19] Ziman, J. M., Proc. Phys. Soc. (London) 88,387 (1966). [20] Young, W. H., Meyer, A., and Kilby, G. E., Phys. Rev. 160,482 (1967). [21] Dickey, J. M., Meyer, A., and Young, W. H., Phys. Rev. 160, 490 (1967). [22] Dickey, J. M., Meyer, A., and Young, W. H., Proc. Phys. Soc. (London) 92,460 (1967). [23] Thornton, D. E., Young, W. H., and Meyer, A., Phys. Rev. 166, 746 (1968). [24] Meyer, A., and Young, W. H., Phys. Rev. 184, 1003 (1969). [25] Hahn, C. E. W., and Enderly, J. E., Proc. Phys. Soc. (London) 92,418 (1967). [26] Mott, N. F., and Massey, H. S. W., The Theory of Atomic Colli- sions (Clarendon Press, Oxford, 1965), 3rd ed. [27] Friedel, J., Nuovo Cim. (Suppl.) 7,287 (1958). [28] Dugdale, J. S., Science 134, 77 (1961). [29] Mott, N. F., and Jones, H., The Theory of the Properties of Metals and Alloys (Clarendon Press, Oxford, 1936). [30] Dugdale, J. S., and Mundy, J. N., Phil. Mag. 6, 1463 (1961). [31] Schumacher, R. T., and Slichter, C. P., Phys. Rev. 101, 58 (1956). [32] Schumacher, R. T., and Vehse, W. E., J. Phys. Chem. Solids 24, 297 (1963). [33] Bennett, L. H., Mebs, R. W., and Watson, R. E., Phys. Rev. 171,611 (1968). [34] Stocks, G. M., and Young, W. H., J. Phys. C. 2,680 (1969). [35] Martin, D. L., Proc. Roy. Soc. (London) A 263, 378 (1961); Phys. Rev. 124,438 (1961). [36] Lien, W. H., and Phillips, N. E., Phys. Rev. A 133, 1370 (1964). [37] Martin, B. D., Zych, D. A., and Heer, C. V., Phys. Rev. A 135, 671 (1964). [38] Ashcroft, N. W., and Wilkins, J. W., Physics Letters 14, 285 (1965). [39]. Watson, R. E., Bennett, L. H., and Freeman, A. J., Phys. Rev. Letters 20,653, 1221 (1968); Phys. Rev. 179, 590 (1969). [40] Van Ostenburg, D. O., and Alfred, L. C. R., Phys. Rev. 161, 569 (1967); Phys. Rev. Letters 20, 1484 (1968). 1653 (1969). [2] Micah, E. T., Stocks, G. M., and Young, W. H., J. Phys. C. 2, 1661 (1969). [3] Stocks, G. M., Young, W. H., and Meyer, A., J. Phys. C. (to be published). [4] Knight, W. D., Solid State Phys. 2, edited by F. Seitz and D. Turnbull (Academic Press, New York, 1956), p. 93. [5] Ziman, J. M., Advanc. Phys. I 3,89 (1964). [6] Meyer, A., Nestor, C. W., and Young, W. H., Proc. Phys. Soc. (London) 92,446 (1967). [7] Benedek, G. B., and Kushida, T., J. Phys. Chem. Solids 5, 241 (1958). [8] vander Molen, S. B., vander Lugt, W., Draisma, G. G., and Smit, W., Physics 38,275 (1968); 40, 1 (1968). [9] Kaeck, J. A., Phys. Rev. 175,897 (1968). [10] Pines, D., in Solid State Phys. 1, edited by F. Seitz and D. Turnbull (Academic Press, New York, 1955), p. 367. [11] Silverstein, S. D., Phys. Rev. 130,912 (1963). [12] Rice, M., Ann. Phys. 31, 100 (1965). [13] Kittel, C., Quantum Theory of Solids, (John Wiley and Sons, New York, 1963). [14] Drain, L. E., in Metallurgical Reviews 12, edited by J. S. Bristow (Institute of Metals, London), p. 195. [15] Ziman, J. M., Proc. Phys. Soc. (London) 91, 701 (1967). [16] Ball, M. A., J. Phys. C. 2, 1248 (1969). [17] Lee, M. J. G., Phys. Rev. 178,953 (1969). [18] Meyer, A., and Young, W. H., Phys. Rev. Letters 23, 973 (1969). 417–156 O - 71 - 43 657 Discussion on “Knight Shifts of the Alkali Metals” by A. Meyer (Northern Illinois University) and G. M. Stocks and W. H. Young (University of Sheffield, England) M. Natapoff (Newark College of Engineering): Do the phase shifts mentioned here differ from those previ- ously published by you? A. Meyer (Northern Illinois Univ.); This is just the comment I would make. The original sets of phase shifts that were published were obtained using repul- sive pseudopotentials. They are the ones that we primarily relied on. However, we indicated that we have others in which we do not use repulsive pseu- dopotentials but use instead the full atomic potential truncated to simulate screening. We simply subtract our multiples of T from the phase shifts so obtained and we get the phase shifts modulo TT which are close to the original set, and are far easier to obtain. Presumably we would use this method in other cases for which doing a full pseudopotential treatment would be just too hard. A. Narath (Sandia Labs.): I would like to apologize to the authors of the post deadline papers. In view of the hour, I think we should skip them and I therefore declare this session closed. 658 Role of Exchange Effects on the Relationship Between Spin Susceptibility and Density of States of Divalent Metals P. Jend cind T. P. Dds Department of Physics, University of Utah, Salt Lake City, Utah 841 12 S. D. Mdhanti Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey O7974 G. D. Gaspari Department of Physics, University of California, Santa Cruz, California 95060 The influence of exchange and correlation of conduction electrons on the spin susceptibility, Xs, is well understood in alkali metals. There is reasonable agreement between theoretical and experimental results where the latter are available. No such comparison has been reported for the divalent metals, mainly because of a lack of experimental information. Only for Be is X, known (from spin resonance measurements). We have made semi-empirical estimates of Xs for Mg and Ca by adjusting the measured Knight shifts with theoretical values of the core polarization and (lº(0)). The values of X, so deduced are compared to theoretical estimates made by the method of Silverstein, who treated the exchange enhancement by an interpolation procedure analogous to that of Nozières and Pines for calculating cor- relation energies, and included the effects of band structure through the calculated thermal (band) ef- fective mass. Agreement is good for Mg, which behaves more like a free electron metal, and poor for Be and Col. Possible sources of the discrepancies are discussed. Key words: Be; Cd; density of states; divalent metals; exchange enchancemnt; Mg; spin suscepti- bility. 1. Motivation for this Work One of the major properties related to the density of states, g(EF) at the Fermi surface (F.S.) in a metal is the spin susceptibility, Xs. In the one-electron approxima- tion the relation between Xs and g(EF) is well-known, [1] namely Xs= pºig (EP) (1) where pºp is the Bohr magneton. However, when one in- cludes the effects of exchange and correlation among electrons, the response of the electrons to the magnetic field is expected to alter in the direction of increased susceptibility. Theories developed for quantitative treatment of these effects are limited in applicability to free (uniform) electron systems [2,3] and only intuitive extensions [3] have been made to take into account the Bloch character of electrons in the metal. Such theories are therefore expected to be reasonably valid for alkali metals which are nearly free electron like in their prop- erties and this is indeed found to be true [3] in the cases where Xs has been obtained directly from electron spin resonance (esr) measurements. To subject the theory to a more severe test for Bloch electrons, we need to analyze the cases where there are strong depar- tures from the free electron behavior. The divalent metals, beryllium, magnesium, zinc, and cadmium are particularly suitable examples in this respect. These metals have four electrons per unit cell which fill up a volume in k-space equal to two Brillouin Zones (B.Z.). As a consequence, there are substantial intersections between the Fermi surface and B.Z. boundary. The electrons on the Fermi surface are therefore expected 659 to depart rather markedly from free-electron character. Among these metals, direct measurement of X, from esr experiment has been made only for beryllium [4]. However, one can extract experimental values of X, for the other metals through analysis of Knight shift (K.) data from nuclear magnetic resonance (nmr) measure- ments. For this purpose, one needs to evaluate the con- duction electron spin density at the nucleus and make use of the well known expression [5] 8T K=#. X,0( |ll. (0) |*)ay (2) Q being the volume of the unit cell over which the wave function, l,(r) is normalized. The average in eq (2) is taken over the Fermi surface. Unfortunately, the Knight shift of zinc is not currently available due to dif- ficulties in observing nmr signals in this metal. Careful determinations of the Knight shift have, however, been made in magnesium [6] and cadmium [7] and concur- rently, band structure studies are available for these metals [8,9] to permit determination of the spin densi- ties (ºl. (0)|*)ay. We have analyzed the direct and exchange core polarization (ECP) contributions to the spin density in both of these metals [10,12,14] in order to obtain X, and study the trend of the agreement between theory and experiment for X, in the series formed by these two metals and beryllium. 2. Extraction of Spin Susceptibility of Mg and Col From Knight Shift Data The direct spin density, (lili, (0)|*)ay requires a detailed scanning of lil, (0)|* over the F.S. with ap- propriate weighting according to the local density of states [10,15]. The ECP contribution was calculated using the moment perturbation (MP) procedure [16] which has been applied in the past to the study of this effect in several metals [17]. The averaging of the ECP effect was carried out in the same manner as for the direct effect. In the case of magnesium, there are four segments of the F.S. [8] described respectively as lens, cigars, but- terflies, and monster. Of these segments, the lens and the butterflies were found [10] to have predominantly s-character whiles the other two had substantial p- character. In reference 10, the spin densities and local density of states were calculated by using the orthogonalized plane wave [OPW) procedure and a potential constructed from first principle by Falicov [11]. A more extensive scanning [12] of the Fermi sur- face has been carried out using the nonlocal pseu- dopotential of Stark and Mueller [13] and the spin den- sities have been evaluated at 110 nonequivalent points on the Fermi surface. The results of this calculation in- dicate that the s-character of electrons occupying the cigar and the monster have been underestimated in the earlier OPW calculation [10]. This may be due to (1) in- accuracies in the potential, (2) use of pseudo-core wave functions instead of crystal-core functions for the pur- poses of orthogonalization, and (3) insufficient scanning of general points on the Fermi surface. Lens has the strongest s-character, but in view of the large surface areas associated with the butterflies and monster, the latter two make the dominating contribution to the spin density. The s-part of the ECP contribution from vari- ous segments was found to follow the same relative trend as the direct spin density. The p-type ECP effect was contributed to mainly by the monster. ECP con- tributions from higher l-components of the wave func- tions were found to be small. The s-type ECP contribu- tion to the spin density was found to be positive and about 25% of the direct ECP effect. The p-type ECP contribution was also positive and about 13% of the direct. On including the small negative contributions to the spin density from higher l-components, the net ECP effect was found to be about 38% of the direct. In cadmium, the F.S. has only two segments, lens and monster. Both of these segments were found to have substantial s-character, with the monster making the major contribution to the spin density both due to its larger s-character and larger surface area. The ECP contribution was a smaller fraction (9%) of the direct spin density. This smallness of the fractional ECP spin density was a consequence of two factors. First, the larger direct spin density compared to magnesium, and secondly, the reduced exchange interaction between the conduction electrons and s-cores due to the screen- ing effect of the intervening 4d-core electrons. The p- type ECP effect was again found to be positive but much smaller in relative importance, namely 1.3% of the direct spin-density. Substituting the total calculated spin density including direct and ECP effects in eq (2) we have obtained the experimental Xs listed in table 1. TABLE 1. Susceptibility enhancement factors and other pertinent parameters in Be, Mg, and Ca [Extrapolated to 0 K. Xs eXpt. Xs band. Metal c/a miſm (cgs. vol. (cgs. vol. nexpt |ntheo. units) units) Be........... 1.567 || 0.45 0.20 × 10-6 || 0.64 × 10-6 || 0.3] 1.09 Mg.......... 1.624 .95 | 1. 14 X 10-6 .93 X 10-6 | 1.23 | 1.32 Col........... 1.862 .54 | 1.03 × 10−6 .54 X 10-6 || 1.90 | 1.17 660 It is appropriate to remark here about the possible in- accuracies of these results for lººp due to the neglect of the orbital contribution [18] to Ks. This latter effect (only Landau diamagnetism) was found to be of deter. mining importance [19] in explaining the negative sign of Ks in beryllium. However, beryllium was somewhat special in that the p-type ECP effect was substantial and negative [15] and cancelled out the major part of the direct spin density contribution to Ks. In contrast, the p-type ECP effect in magnesium [10,12] and cad- mium [14] is positive and the absolute values of the total spin densities are orders-of-magnitude larger, due to a larger fractional s-character. Thus, any contribu- tion from either the Landau [20] or Van Vleck-Ramsey type orbital effect [21] to Ks is expected to be a small fraction of the spin-contribution to Ks. 3. Comparison of Theoretical and Experimen- tal Spin-Susceptibilities — Discussion The commonly used procedure for studying the ex- change enhancement is the one based on the random phase approximation [22] (RPA) of many-body theory. This procedure involves a self-consistent treatment of the wave functions for the electron gas in a magnetic field. The difficulty with this procedure is that the RPA approximation is strictly valid for high densities, that is, for electron sphere radii rs' less than 1, and takes ap- propriate account of the long range interactions (small momentum transfer). Silverstein [3] has improved upon the RPA by utilizing a momentum transfer inter- polation procedure analogous to that developed by Nozières and Pines [23] for correlation energy calcula- tions. The interactions in the presence of a magnetic field involving small momentum transfers (long range effect) are handled by the RPA procedure while the large momentum encounters (short range effect) are handled by second-order perturbation theory, an inter- polation procedure being applied for intermediate mo- mentum transfers. Additionally, Silverstein [3] has at- tempted to include the effects of band structure through the thermal (band) mass, m, related to the den- sity of states by the relation miſmo = g(EF)/go (EP) (3) where go (EF) is the density of states for free-electrons (mass mo). Silverstein’s expression may be expressed in the compact form n=1|–º (4) where 8= molm, (5) o = x*x." Xs” being the susceptibility obtained by Silverstein in the absence of the band effect and band 'm =Xºlx. (6) with X,” given by eq (1). In table l are listed the values of cla, 1/8, X,**P*, and m"he" from eq (4). For beryllium [15] and cadmium [24], the density of states at the F.S. g(EF) has been calculated with great accuracy and we have used these values to calculate I/3. For magnesium, no such detailed calculation of g(EP) is available. We have therefore utilized [10] the density of state obtained from the specific heat mea- surement after making appropriate corrections for the phonon enhancement of g(EP). The results in the last two columns show that agreement between me"P" and m” for beryllium and cadmium is quite poor where as for magnesium, the agreement is reasonably good. Amongst the three, magnesium behaves more like a free electron system and it is not surprising that an ef- fect mass approximation for treating the exchange enhancement of X is reasonably good for this metal. The small theoretical value of m for cadmium is primari- ly due to the small value of milm, which indicates that there are not enough electrons at the Fermi surface to take part in the exchange enhancement process. It should be pointed out that our derived experimental value of m for cadmium is 1.9 compared to 1.5 obtained by Kasowski and Falicov [24]. This discrepancy may be due to (1) the choice of Hartree-Fock-Slater core states in [24] and (2) the use of smaller number of OPW basis functions to construct the conduction elec- tron wave function which overestimates the theoretical spin so density and hence underestimates me"P". There seems to be an interesting trend in the value of me” in going from beryllium to cadmium, although there is no trend with respect to the parameter 8. A part of the difference between experiment and theory could perhaps be due to the neglect of some additional contributions to Knight shift in deriving Xs. For beryllium, the theoretical situation is rather poor since m seems to have a de-enhancement from the band susceptibility while theory predicts the reverse. As a matter of fact, under no circumstances can eq (4) lead to m” less than unity. While the Knight shift involves the susceptibility Xs in a uniform field, the nuclear spin lattice relaxation 661 rate, T-" requires an average [25,26] involving the mo- mentum dependent susceptibility, X (q) over the range q = 0 to 2kr. We have utilized the calculated direct and ECP spin densities for beryllium and cadmium to cal- culate T-1 in these two metals where (TT)-1 has been studied experimentally [27,7]. In keeping with the trend for X, the theoretical (TT)-1 in beryllium is found to require a de-enhancement factor of .60 to agree with the experiment [27] while cadmium requires an enhancement factor of 3.10. Evidently, two types of improvements are necessary in the theory of the response of the electrons to a mag- netic field. First one needs to handle actual electronic densities (rs21) more rigorously and second, to incor- porate Bloch effects which lead to nonuniformity in the electron density distribution. It appears that the latter is the more pressing problem for the case of divalent metals. It is hoped that present self-consistent procedures [2] for the free-electron gas can be ex- tended without too much complexity to the study of Bloch electrons. 4. Acknowledgments The portion of this investigation carried out at the University of Utah was supported by the National Science Foundation. 5. References [1] Pauli, W., Z. Physik, 41, 81 (1927). [2] Hamann, D. R., and Overhauser, A. W., Phys. Rev. 143, 183 (1966). Gell-Mann, M., and Brueckner, Phys. Rev. 106, (1957). [3] Silverstein, S. D., Phys. Rev. 130,912 (1963). [4] Feher, G., and Kip, A. F., Phys. Rev. 98,337 (1955). [5] Townes, C. H., Herring, C., and Knight, W. D., Phys. Rev. 77, 852 (1950). [6] Rowland, T. J., in Progress in Materials Science, Bruce Chal- mers, ed. (Pergamon Press, New York) Vol. IX, p. 14, (1961). Dougan, P. C., Sharma, S. N., and Williams, D. L. (to be published in Canadian Journal of Physics). [7] Seymour, E. F. W., and Styles, G. A., Phys. Letters 10, 269 (1964). Borsa, F., and Barnes, R. G., J. Chem. Phys. Solids 27, 567 (1966). Sharma, S. N., and Williams, D. L., Phys. Letters 25A, 738 (1967). Dickson, E. M., Ph. D. Thesis (unpublished), University of California, Berkeley, California (1968). [8] Falicov, L. M., Phil. Trans. Roy. Soc., A255, 55 (1962). Kim- ball, J. C., Stark, R. W., and Mueller, F. M., Phys. Rev. 162, 600 (1967). [9] Stark, R. W., and Falicov. L. M., Phys. Rev. Letters 19, 795 (1967). [10] Jena, P., Das, T. P., and Mahanti, S. D., Phys. Rev. (in press). [ll] Falicov, L. M., Ref. 8. [12] Das, T. P., Jena, P., and Mahanti, S. D., Addendum to Ref. 10, (to be published). [13] Kimball, J. C., Stark, R. W., and Mueller, F. M., Ref. 8. [14] Jena, P., Das, T. P., Gaspari, G. D., and Mahanti, S. D., Phys. Rev. (in press). [15] Jena, P., Mahanti, S. D., and Das, T. P., Phys. Rev. Letters 20, 544 (1968), and Ref. 10. [16] Gaspari, G. D., Shyu, W. M., and Das, T. P., Phys. Rev. 134, A852 (1964). [17] Shyu, W. M., Das, T. P., and Gaspari, G. D., Phys. Rev. 152, 270 (1966). Also see Refs. 10, 11, and 12. [18] Kubo, R., and Obata, Y., J. Phys. Soc. (Japan) 11, 547 (1963); Obata, Y., J. Phys. Soc. (Japan) 18, 1020 (1963). [19] Gerstner, J., Ph. D. Thesis (unpublished), Pennsylvania State University, University Park, Pennsylvania (1969). [20] Landau, L, Z. Physik 64,629 (1930). [21] Ramsey, N. F., Phys. Rev. 78,699 (1950). Ibid., 86, 243 (1952). [22] Pines, D., Elementary Excitations in Solids, (W. A. Benjamin, Inc., New York, 1964). [23] Nozières, P., and Pines, D., Phys. Rev. 109, 762 (1958). [24] Kasowski, P. V., and Falicov, L. M., Phys. Rev. Letters 22, 1001 (1969). [25] Moriya, T., J. Phys. Soc. (Japan) 18, 516 (1963). [26] Narath, A., and Weaver, H. T., Phys. Rev. 175, 373 (1968); Ma- hanti, S. D., and Das, T. P., Phys. Rev. 183, 674 (1969). [27] Barnaal, D. E., Barnes, R. G., McCart, B. R., Mohn, L. W., and Torgeson, D. R., Phys. Rev. 157, 510 (1967), T represents the absolute temperature. 662 Discussion on “Role of Exchange Effects on the Relationship Between Spin Susceptibility and Density of States of Divalent Metals” by P. Jena and T. P. Das (University of Utah), S. D. Mahanti (Bell Telephone Laboratories) and G. D. Gaspari (University of California, Santa Cruz) R. V. Kasowski (DuPont and Co.); I have calculated the Knight shift in cadmium and also the spin-suscepti- bility. I find that the enhancement due to, say, many- body effects, is about .54 (this was published in Phys. Rev. Letters in March). I think in the units that the author, Professor Das, uses, it is about 1.54. So I would like to know what potential he used and how the density of states at the Fermi level was calculated? We used the potential that was fitted to data by Stark and Fal- icov and it has been quite successful in a number of Fermi surface properties for cadmium. S. D. Mahanti (Bell Telephone Labs.): I believe that for cadmium, the value of the exchange enhancement factor for Xs (the spin susceptibility) as calculated by Dr. Kasowski is 1.54 compared to our value of 1.90. The potential utilized in both the calculations are the same — that of Stark and Falicov. The density of states used in our band susceptibility was that of Kasowski and Falicov (your Phys. Rev. Letters paper of March 1969). However, the difference is in the calculated spin density at the nucleus. Two plausible reasons that come to mind are: (1) We have utilized Hartree-Fock cores for constructing the OPW functions whereas Dr. Kasowski uses Hartree-Fock-Slater cores. We have found that this may lead to 15-20% difference in the spin density. (2) The second reason is the difference in the number of OPW's utilized to construct the conduc- tion electron wavefunction. We have utilized nearly 25 OPW's, and if Dr. Kasowski has utilized a fewer num- ber of OPW's, then this may explain the differences in the calculated spin densities and the exchange en- hancement factor. 663 Correlation of Changes in Knight Shift and Soft X-Ray Emission Edge Height Upon Alloying L. H. Bennett, A. J. McAlister, J. R. Cuthill, and R. C. Dobbyn Institute for Materials Research, National Bureau of Standards, Gaithersburg, Maryland 20760 In simple metals and alloys—those having no significant local d-character at or near the Fermi level—the Knight shift provides a measure of the local s-electron density of states. If the particular atom under study has a p-like core level, then soft x-ray emission arising from transitions of electrons at the Fermi level into p-like core vacancies should provide a similar measure. Using Al as the example, we compare results of these two techniques by studying fractional changes, relative to Al metal, of the Knight shift and L2.3 soft x-ray emission edge height of Al in NiAl, AuſAl2, Mg2Al3, Mg17 Alig, and Al2O3. A distinct correlation is observed. Key words: Aluminum oxide (Al2O3); electronic density of states; gold aluminide (AuſAl2); Knight shift; magnesium-aluminum alloy phases; nickel aluminide (NiAl); soft x-ray emis- Sion. 1. Introduction Among the various experimental techniques yielding information on the occupied density of electronic states in solids, we can distinguish two classes or families: the Fermi level probes (Knight shift, specific heat, suscep- tibility), and the deep band probes (photoemission, ion neutralization, soft x-ray emission). There have been many comparisons of results of techniques within a family. However, so far as we are aware, there has been no systematic attempt to correlate results of techniques belonging to different families. In this paper, we exhibit such a correlation, focusing attention on the Knight shift and soft x-ray emission. Analysis, given in the next section, shows that such a correlation is predicted by the one electron model for simple materials—those having no significant local d- character at or near the Fermi level. An experimental search for the correlation can serve two useful pur- poses. If the one electron model is valid, these two techniques should serve as sensitive tests of band theory, particularly for alloys, since alone in each fami- ly, they are capable of yielding information on the con- tribution of a specific orbital component of the Bloch waves to the state density, at a specific ion site. On the other hand, if many body effects, such as exchange enhancement of the Knight shift [1] or resonant behavior of the soft x-ray emission rate at the Fermi level [2], are important, we would expect to observe systematic deviations from the predicted correlation. As our example, we have chosen to study Al, in Al metal, in alloys with Au, Ni, and Mg, and in the oxide Al2O3. For the cases considered so far, a distinct cor- relation is observed. No systematic deviations are noted. Thus, many body effects are small, constant upon alloying, or compensating. 2. Andlysis In simple materials as defined above, the Knight shift provides some measure of the local density of s-electron states at the Fermi level. If the ion under study has a p- like core level, the soft x-ray emission arising from transitions of Fermi surface electrons into p-like core vacancies, should provide a similar measure. For the Knight shift, we assume a simple contact in- teraction from which we write k & X ||,(0)|*6(E(k)—Er), (1) k, n - which, aside from constants, is the first term in the usual expression, k = asn's (EF) + adna (EF) + . . . . . . 665 The second term in the latter expression is an in- direct contact term, arising from a net spin density at the nucleus produced by polarization of cores-levels via exchange interaction with unpaired d-states. The ratio of hyperfine coupling constants, aaſas, is typically nega- tive and of the order of 0.1. However, in our assumed simple materials, na(EP) is assumed to be negligible, and we omit this and higher terms from consideration. For the soft x-ray emission process, the spontaneous L-emission rate from states near the Fermi level may be written RL (ha)) OC X. (Er – Ec)* < lººp|ril, X- |*6(hay–Ep-HEc) k, n (2) Both sums are carried out over the Brillouin Zone. n is a band index, lik is a band wave function; ill.(0) its value at the nuclear site. lººp is the soft x-ray core state, and Ec its excitation energy. The suggested correlation can be clearly exhibited by introducing orbital state weights, wi', which are con- veniently defined for an augmented plane wave (APW) calculation employing a muffin tin potential by Sºftwº-ſ ll lſ” (k, E) iſ (k, E) dºx = 1 l, v w? (k, E) = 4T (21+ 1) X. cc;ekºr, Pi (cos 65).ji (ki/Ay).ji (k;A,) i,j R}(r #) (3) Av X r2 2 | dr (# E) 1Upw = X. coſmo, — 47 X. A}eikºr, |kij i,j Here v denotes the v" ion in the unit cell. A, is the APW sphere radius for the ion under consideration. gi is a reciprocal lattice vector, and ki = k + gi. The ci’s are coefficients of the normalized band eigenfunction expanded in APW’s.j(ki/A) is an lºh order spherical Bes- sel function. kg =ki-ky. 95 is the angle between two vectors ki and kj. Pi is the lº" order Legendre polynomi- al. Q is the primitive cell volume. These weights can be introduced into a sum over states to yield local orbital state densities, n° (E), q denoting ion type. It is then tedious, but straightfor- ward to show that the sums (1) and (2) may be rewritten R{(0. #) n}(Ep) Ka Oº (#} Er)/ I'm (4) and - (14,0)* 2 (14, 2)” R# (ho) or * nº (E.) 45 * ng (EP) (5) where --- A Q •) R; (r, EF) ) 1,–ſ dr r (###,(Aq, EP) IQ -- Aq d 3 Rq R (r, EP) ºrſ r r 2p (r) (i. (Aq., #) The Ri’s in these expressions are the orbital com- ponents of the radial wave functions with the APW sphere. . For pure Al and the alloys under consideration, it is reasonable to assume that n2^*(EF) makes a negligible contribution to Riſhop) [3,4]. We may therefore rewrite eq (5) as R#! (0, #) ná' (Er .4 l * R! woe (#) * ) ſéAl Al | dr rº R; (r) (; (r, #) (6) R; (0. EP) Only slight variation upon alloying is anticipated for the integral on the right hand side of eq (6). Thus the essen- tial difference in kai and RLA'(hop) should be only a con- stant. Both should yield measures of no"(EP). Further, a plot of Rallow/RA versus Kallowſka should yield a straight line of unit slope. 3. Corrections to Measured Soft X-Ray Emission Rote Soft x-ray emission rates measured at constant volt- age and current density differ from true emission rates, and must be corrected for differences in absorption coefficient, p, and the density of radiating atoms, pl.(3), as a function of depth & below the sample surface. 666 These quantities enter the emission rate through a fac- tor ŚL - 1–ſ" dio, (e) cº- (7) &L is a maximum depth, beyond which excitation of L shell vacancies is no longer energetically possible. It is a function of tube voltage, angle of incidence of the electron beam, the rate of electron energy loss, and the probability of inner level excitation. l is the x-ray takeoff angle. These quantities are schematically illus- trated in figure 1. By taking the measured emission rate, subtracting the background continuum from it, and then dividing the difference by a reasonable ap- proximation to the factor I, an estimate of the true emis- sion rate can be obtained. We calculate I in the following approximate way. First, straight, average paths are assumed for the elec- trons in the exciting beam, with electron energy varying along the mean trajectory according to the prescription V(£) = Vo- (£18 sin d) la (8) This expression is consistent with Feldman’s empirical range formula for 1 to 10 keV electrons in solids [5]. r= 8V;. Vo is the initial electron energy in keV, and B=250A/pZaſº, o: = 1.2/ (1 – 0.29 log10 Z), with A the atomic weight, Z the atomic number, and p the density. For alloys, appropriate averages of A and Z are used. We now write pl (£) = n.a. PL(£) where nAi is the density of Al sites in the material stu- died, and PL(#) is the probability of direct 2p shell ionization of an Alion. For the L shell ionization probability, we use Bethe’s nonrelativistic expression, as modified by Worthington and Tomlin [6], Pl(s) or ln U/V}U, (9) where U = V(£)/VL, and VI is the 2p excitation energy. hy º \ } eT * : I 4 ZZZZZZZZZZZZZZZZ z, d 8. i FIGURE 1. The geometry of soft x-ray generation. Electrons incident at angle (b to the sample surface excite inner level vacancies at various depths. §, below the sample surface. To be collected by the spectrometer, an emitted photon must traverse a distance {|sin ill of the sample, and may be absorbed along the way. The absorption coefficients, p.(ha)), are not known for the alloys. We therefore make the estimate |MAI, B = O'Ain Al-F Ohn B from the absorption cross sections ori of the pure metals, and the appropriate ion densities in the solids. Table 1 lists our estimates of ori and the data from which they are evaluated. TABLE 1. Numerical values for atomic density, linear mass absorption coefficient, and absorption cross section Atoms per cm" | Mass absorption Absorption Metal Ili coefficient cross section pi (cm−") ori = pºi/ni (cm”) A]16 6.07 × 1022 0.1 × 105 (). 16 X 10-18 Mg|7 4.30 2.2 5.1] Aulº 5.93 7.5 12.64 Njić 9.17 8.6 9.37 From the expressions above, we write the factor I as #ſ. sin q (Vo – V.)” VI, Jo {In [Vo- (ºlº sin (b)"]- ln VI), {Vo- (£13 sin (b)'''} dć Hº: | exp ––– sin il, (10) and evaluate it numerically. Appropriate data and results are listed in table 2. 667 TABLE 2. Numerical values for atomic densities, linear mass absorption coefficient, and excitation electron range parameters, O., 9 Mass ab- Alloy || Atoms per | Atoms per sorption O. B cm” nA cm” nE coefficient (cm−1) Al 6.07 X 1022 () X 1022 (). 1 × 105 | 1.77 2.57 × 102 Al; Mg2 || 3.1 2.08 1.1 | 1.76 || 3.11 Alſº Mg17 | 2.05 2.90 1.51 1.75 | 3.32 Al, Au 3.70 1.85 2.40 2.17 |0.85 AlNi 4.14 4.14 4.0 1.93 || 0.97 We have also estimated, by extrapolation of Green's [6a] experimental results, the change in backscatter loss on alloying. For the experimental conditions used in the NiAl and AuſAl2, we expect fractional losses relative to Al metal of .04 and .09 respectively. For the Mg alloys, the effect should be nil, owing to the negligi- ble change in average atomic number. While fairly realistic, these approximate values of I are generally overestimates, because of our straight electron path assumption [7]. A further complication, in the case of AuſAl2, is the possibility of enhancement of the Al Lemission rate via fluorescent excitation of Al cores by Au N4.5 radiation. The experimental soft x-ray data employed here are from three sources. In our own laboratory, we have made measurements on Al, NiAl, and AuAl2. The fol- lowing experimental parameters were employed: l = 90° q = 20°, Vo- 2.5 keV. We have also included in this analysis the data of Appleton and Curry [8], who em- ployed l = 90°, b = 30°, and applied a 4.0 keV peak sinusoidal voltage to their x-ray tube, and corrected the observed spectra for variations in nAi. Since I is not a linear function of V, we have calculated a time average of it for application to their data. Also included in Fomichev’s [9] measurement of the L-spectrum of Al in Al2O3. There remains one important consideration in the reduction of the soft x-ray data. At what point along the emission edge should one choose an intensity value characteristic of emission at the Fermi energy? There is some arbitrariness involved in answering such a question. Aside from possible singular structure men- tioned above, the emission edge is always broadened by the instrument, by the finite temperature at which mea- surements are made and by the finite lifetime of the ini- tial state. All of these factors combine to render an exact answer an arduous, if not impossible, task. We therefore assume these factors constant with alloying and use the intensity, above background, of the first definite structural feature at or below the Fermi energy. The emission edge of aluminum, in the metal and in the alloys considered here, exhibits the sharp cut-off at the Fermi energy characteristic of free electron metals, and consequently, the location of this intensity value is unambiguous. The Al spectrum of the oxide shows no emission edge. The Fermi level falls within the band gap, and the Fermi level emission rate is zero. The soft x-ray measurements made in our laboratory were performed on 99.999% pure aluminum and on the single-phase alloys AuſAl2 and NiAl. Details of sample characterization and instrumentation are reported else- where [4,10]. Results for Ala Mg2 and Al12Mg17 are based on measurements reported by Appleton and Cur- ry, to which we have applied only the correction nai/I. Knight shift measurements on pure aluminum [11], AuſAl2 [12], and NiAl [13] have been previously re- ported. Those on Ala Mg2 and Al12Mg17 were taken at both 77 K and room temperature, on powders crushed from arc-melted ingots. Measurements at 77 Kindicate a 0.01% reduction in k from room temperature values in each case. The diamagnetic reference is AlCl3. X-ray diffraction and metallographic analyses showed them to be primarily single-phase alloys, with minute traces of second phase at the grain boundaries (<1%). Table 3 (a,b) gives the measured mean values of emission rate TABLE 3. Numerical values of: measured Al L2, 3 emission edge heights, at constant voltage, per scan, per unit current density, AR; density of Al sites, per A*, na; the partial range-absorption factor, I'-Iſnai; fractional backscatter loss, y; corrected emission edge height, R(hor); Al Knight shift, k; and values relative to pure Al of emission edge height and Knight shift, R' = Ralloy/RA and k" = kalloyſkal. Mate- || AR Il Al I' 'y R(hor)| K R" | K' rial Al 458.4 || 0.0606 | 1.000 || 0.00 || 7564. || 0.164 | 1.00 |1.00 NiAl 23.1 | .04.14 | .311 | .04 | 1866. .061 | .25 | .38 AuſAl2 42.9 || 0370 | .391 | .09 || 3232. .061 | .43 | .38 Material AR I' R(hor) K R’ k' Al............... 22.0 1.000 22.0 .164 1.00 1.00 AlaMg2......... 13.3 .616 21.6 .135 .98 .82 Alig Mg17...... 9.7 .509 19.0 .138 .86 .84 above background, AR, the correction I and nai, and measured Knight shifts, k. The data on Ala Mg2 and 668 |.O |- O Al2WGſ, º .5 wº- Knight Shift Ratio, Kalloy / KA FIGURE 2: L2, 3 soft x-ray emission edge height versus Knight shift of Al in a number of compounds. Each is expressed as a fraction of the pure metal value. The solid straight line of unit slope is the one electron prediction for simple materials. Alış Mg17 are shown separately (table 3b) since changes in emission rate from these alloys are calculated with respect to the pure aluminum data of Appleton and Curry, and they have already been corrected for changes in nAi. Also listed are the ratios Railouſ RAI and Kalow/KAI, plotted in figure 2. 4. Discussion The correlation between the soft x-ray intensity at the Fermi edge and the Knight shift is illustrated in figure 2. The solid line of unit slope passing through the origin is the one electron prediction, and not an attempt to fit the data. The point for Al metal falls on the line by definition, and is not a verification of the correlation. The point for Al2O3 at the origin is a valid data point, and presumably is typical of nonmetallic compounds. Here the Knight shift is expected to be zero because there is no Pauli paramagnetism. All electrons are paired in spin. Actually, quadrupole effects have prevented a simple determination of a Knight shift. The soft x-ray emission edge height is zero because the Fermi level falls within the band gap, where the state density is zero. We would expect semi-metals to have near-zero Knight shifts and Fermi edge emission heights as well. The principal uncertainty in the Knight shift involves such factors as chemical shifts, orbital shifts (paramag- netic or diamagnetic), p-electron core polarization. All must be considered carefully, or systems chosen in which they are known to be small. Uncertainty in the soft x-ray values comes mainly from our estimates of the factor I, defined above. The values of I used can reasonably be taken as upper limits. Thus, the points for NiAl, AuAl2, Mg2Al3, and Mg17Alſº lie no higher on the plot than shown. The extreme, and definitely incor- rect, procedure of correcting only for changes in the density of Al sites would reduce the corrected emission edge heights by factors ranging from .3 to .6, and move the points substantially below the line of unit slope. Thus, there exists the possibility of a systematic devia- tion from the expected correlation which, if real, could be interpreted as due either to enhanced Knight shifts or to reduced emission edge height enhancement in the alloys, with respect to Al metal. A more exact estimate of I can be made by replacing the assumption of average straight line electron trajectories with a diffuse scattering model. Considerably more labor would be in- volved. The calculations of I can be experimentally tested by studying emission edge height as a function of exciting potential at constant current density. We hope to make these better estimates in the future. How- ever, our present feeling is that no gross error has been made, and that the indicated correlation is real. Within the approximations and uncertainties out- lined above, we have shown that Knight shift and soft x-ray estimates of local s-wave state densities are simply related. Although consideration has been given to Al alloys, we expect that other simple metal alloy systems will show similar correlation. We also expect that phenomena such as melting, which produce changes in the Knight shift in certain metals, will produce comparable changes in soft x-ray emission edge heights. Lastly, we would suggest that investiga- tion of the soft x-ray emission edge heights can shed light on the question of whether the small Knight shifts observed in a number of metals and alloys [14] are due to dominant p-character (i.e., little s-character) at EF or to cancellation of s-character by core polarization. 5. Acknowledgment We thank R. L. Parke for his able technical assistance with the NMR measurements of the Al-Mg alloys. 6. References [1] Pines, D., Solid State Physics, edited by F. Seitz and D. Turn- bull (Academic Press, Inc., New York, 1955), Vol. 1; Narath, A., Phys. Rev. 163,232 (1967). 669 [2] Mahan, G. D., Phys. Rev. 163, 612 (1967); Nozières, P.. and DeL)ominicis, C. T., Phys. Rev. 178, 1097 (1969); Ausman, G., and Glick, A.J., Phys. Rev. 183,687 (1969). [3] Rooke, G. A., J. Phys. C1, 767 (1968). [4] Williams, M. L., Dobbyn, R. C., Cuthill, J. R., and McAlister, A. J., these Proceedings, p. 303. [5] Feldman, C., Phys. Rev. 117,455 (1960). [6] Worthington, C. R., and Tomlin, S. G., Proc. Phys. Soc. A69, 401 (1956). [6a] Green, M., Ph. D. Thesis, Univ. of Cambridge, 1962, un- published. [7] Archard, G. D., in X-Ray Microscopy and Microanalysis, edited by A. Engstrom, W. Coslett, and H. Pattee (Elsevier Publishing Company, New York, 1960) p. 331. [8] Appleton, A., and Curry, C., Phil. Mag. 12, 245 (1965). [9] Fomichev, V. A., Soviet Physics, Solid State 8, 2312 (1967). [10] Cuthill, J. R., McAlister, A. J., and Williams, M. L., J. Appl. Phys. 39,2204 (1968). [ll] Knight, W. D., Solid State Physics, edited by F. Seitz and D. Turnbull (Academic Press, Inc., New York, 1956), Vol. 2. [12] Jaccarino, W., Weger, W., Wernick, J. H., and Menth, A., Phys. Rev. Letters 21, 1811 (1968). [13] Miyatani, K., Ehara, S., Sato, T., and Tomono, Y., J. Phys. Soc. Japan 18, 1345 (1963); West, G., Phil. Mag. 9,979 (1964). [14] Schone, H. E., and Knight, W. D., Acta Met. 11, 179 (1963); Bennett, L. H., Phys. Rev. 150, 418 (1966). [15] Haensel, R., private communication. [16] Townsend, J. R., Phys. Rev. 92,556 (1953). [17] Haensel, R., Kunz, C., and Sonntag, B., Phys. Letters 25A, 205 (1967). 670 TUNNELING; SUPERCONDUCTORS; TRANSPORT PROPERTIES CHAIRMEN. J. R. Leibowitz J. W. Gadzuk Tunneling Medsurements of Superconducting Quasi-Particle Density of States and Calculation of Phonon Spectra’ J. M. Rowell Bell Telephone Laboratories, Inc., Murray Hill, New Jersey It is an unfortunate fact that the tunneling technique, which has proved incredibly successful in the study of superconductivity, has given little information about the normal state properties of metals and semiconductors. It will be shown that, in the determination of the superconducting quasi-particle densi- ty of states, it is the change in density induced by the onset of superconductivity which is measured rather than the total density. Returning to the problem of normal materials, a review of the limited achievements and failures of tunneling will be presented. This will include the influence of band edges on tunneling in p-n diodes and metal-semiconductor contacts, the structures observed in tunneling into bismuth and the negative results obtained in nickel and palladium. The dominant effect ºf the change in barrier shape in most of these tunneling characteristics will be illustrated. Key words: Density of states; phonons; semiconductors; superconductivity; tunneling. Many of the talks heard this week have outlined ex- perimental techniques which determine the density of electron states in metals and semiconductors over ener- gy ranges which are typically 1-10 volts. I would like to discuss a technique which is much happier in the range of 1-10 millivolts, is the only method which can measure the change in density of electron states induced by su- perconductivity, but which to date must be classed as a failure in the determination of band properties of nor- mal metals. For semiconductors and semimetals, because of smaller energies and larger fractional changes in electron density at band edges, the situation is a little better. Even in these materials, however, we can only say that density-of-states effects are observed and cannot generally deduce a density measurement from the experimental results. After those opening remarks, it will be obvious that most of you attending this conference should not be familiar with the tunneling technique so I will briefly review the experimental method. Two structures are commonly used, the metal-insulator-metal (M-I-M) junction, which relies on the oxide of the first metal to form the insulator, and the metal-semiconductor (M-S) contact which uses the depletion layer (Schottky barri- er) at the surface of the semiconductor. In the M-I-M case it is generally hard to prepare sufficiently smooth clean surfaces of bulk metal that oxidize in a controlled way so films have been used in all but a few cases. After evaporation of the first metal film (Al, Pb, Sn, Mg, for example), oxidation takes place by exposure to air, oxygen, or a glow discharge in oxygen, and the second film is then cross evaporated to complete the junction. Typical oxide thicknesses are 15–20 Å. To make the M-S contact a semiconductor is cleaved in vacuum and covered with the metal very rapidly. The barrier is lower and ~ 100 Å thick. In some materials, where a suitable Schottky barrier does not form at the surface (e.g. InAs), a M-I-S structure can be made by oxidizing the semiconductor before evaporation of the metal. These three tunneling structures are shown in figure l, with the circuit used to measure the current-voltage characteristic or, if greater detail is required, the *An invited paper presented at the 3d Materials Research Sym- posium, Electronic Density of States, November 3–6, 1969, Gaithers- burg, Md. 417–156 O - 71 – 44 673 dynamic resistance (d/)/(d1) versus voltage. The figure also shows schematically the potential barrier (with typ- ical energies indicated) which separates the electrodes. - ACTAl- Eziº Onº. Woº. PAET PA- ~ 2V NSS-metau. |SEratonboºroºl A, 2V NS-Merau Pºzzzzºz.2222222221 - ox be $ENA\Cot' buctoſ. | º SI —ſ V **, FIGURE 1. Schematics of three widely studied tunneling structures and the circuit used to measure their I-V characteristics. Considering the M-I-M junction as the most favorable case for studying metals, I would like to make a number of points regarding the theoretical possibilities of ob- serving band structure effects and the probable experi- mental difficulties. The tunneling current through a barrier of the type shown in figure 2 is given by [1], j-ºx'ſ depºp.(E) P(E) Iſº k —f(E – eſ/)], (1) where pa(E) and pºſB) are the density of states for a given transverse momentum k, and total energy E, f(E) $6. N) º E. f's (e) <-4- The potential in an ideal trapezoidal barrier between metal electrodes. FIGURE 2. is the usual Fermi distribution and Er the total electron energy perpendicular to the barrier. The tunneling probability P(Ex) has the form P(E)-4 exp(-|| 2nººn–E)]"dº (2) where d is the barrier thickness and [d(x, V)-Er] is the barrier potential at position x when a voltage V is ap- plied. The pre-exponential factor A describes the frequency with which an electron arrives at the barrier interface and its exact form determines whether one ex- pects to observe density of states effects at all. In the W.K.B. approximation, which makes the metal-barrier interface properties vary slowly compared to the electron wavelength [1], A is proportional to I||pa(E)pºſB)] so that the current in (1) is independent of electron density of states. The other extreme limit, with the metal-barrier interface absolutely sharp, gives a complicated prefactor A which does not exactly can- cel the density terms in eq (1) [1,2] so some density of states effect in tunneling might be expected. However, as the interface properties can never be known in this detail, a serious interpretation of any such observation would be impossible. Rather than looking for results of the slow energy variation of p(E) within a band, a more likely experiment is to look for effects of band edges, where the number of final available states for the tun- neling electron changes more abruptly. It is generally felt that band edges can be observed in tunneling as long as an appreciable fractional change in total elec- tron density is produced by the new band. As we will see later, this usually happens at convenient energies only in semiconductors. Duke [3] has also pointed out that an increased tunneling probability into the new band will enhance the magnitude of the current onset near the band edge. Apart from the theoretical question of the exact na- ture of the metal-insulator interface and its role in producing density of states effects, there are a number of serious experimental problems which lead me to question whether these effects could be observed in metals. These problems can be illustrated by reference to figure 3. First, typical barrier heights in oxides are ~ 2 volts, which is a small energy compared to inter- esting band structure in most metals. For an applied bias - dº, electrons from the Fermi level (at A for example) tunnel into the conduction band of the insula- tor rather than into the second electrode. Second, the tunneling probability is much greater for electrons at B than for those at C, which enter the second metal just above the Fermi level. In figure 3b, for example, where 674 we assume dR = db = 2W, the transmission probability at B is ~ 10-9, whereas at C it is ~ 10-1°. Thus the chance of probing band edges far below the Fermi level in the left metal seems remote, as practically none of the ^ %L f —H-\--A 2V ſ —H--- r — 3 t Ør. 2V J - - - - -- C. (a) (b) FIGURE 3. Figure (a) shows the possible injection of electrons into the conduction band of the insulator (Fowler Nordheim tunneling). Figure (b) shows two possible tunneling paths into the second electrode. current, flows from these levels. The possibility of probing band edges above the Fermi level (at B in the right metal for example) raises the third problem, namely that the current-voltage characteristic of the junction is dominated very strongly by the changes in barrier shape. Typical results obtained by Fisher and Giaever [4] are shown in figure 4. At low voltages the junctions are ohmic (small deviations are observable only in detailed conductance plots) but from 0.2 to 1.4 V the current depends almost exponentially on voltage. This strong current dependence is purely the result of the barrier becoming lower (for electrons leaving the Fermi level) as the voltage increases. Thus any small changes in current (or conductance) due to band edges, superimposed on this exponential behavior, will be ex- tremely hard to observe. There are also other difficulties which one has to consider in many tunneling experi- ments. If evaporated metal films are used it is unusual for these to have the properties of single crystals or even clean polycrystals. Even more critical are the surface properties of the films, as tunneling between normal metals is sensitive to the material only within a screening length of the surface. As this surface is in contact with (or diffused into) the insulating oxide the chance of it having bulk properties seems remote. After this very pessimistic survey of why tunneling is not the right technique for the study of density of states effect in metals, let me review a number of ex- periments where band edges at least are observed, or affect the tunneling characteristic noticeably. ; & 2 O |O 20 tº ILLIVOUYS (a) º 1. Ç0 ; i /2 3, O} I ſº º | ! f _ gº ..? & .6 .8 1,0) Łº Iº, VOLTS FIGURE 4. I-V characteristics reported by Fisher and Giaever [4]. (a) At low voltages, the current through thin oxide films is proportional to voltage and to film area. Curves shown are for five films with areas in the proportions 5:4:3:2:1 as indicated. (b) At higher voltages, the current increases exponentially with voltage. 1. Metal-Semiconductor Contacts In a heavily doped n-type semiconductor the Fermi energy is relatively small (< 100 mV.) and the depletion layer formed at the surface is thin enough to permit tun- neling. Unfortunately the barrier height is rather low and barrier thickness varies with bias, leading to a strong dependence of conductance on voltage. Nevertheless, the calculation of Conley et al. [5] pre- dicts that the minimum conductance occurs when the bottom of the band crosses the Fermi level of the metal (fig. 5). This has been observed in the experiments of 675 V Fog Musitauta ConDurvatAct FIGURE 5. Metal-semiconductor contact with bais applied to produce a minimum in conductance. Steinrisser et al. [6], as shown in figure 6. The barrier parameters were determined independently and the agreement between calculated and experimental con- ductances is orders of magnitude better than in most 50 T-I-I-I | H---|--|--|--|--|--|-- }- (d) - Pb on Ge: Sb z (b) n = 7.5 × 1019/cm3 T = 4.2 °K 2O H ſ i 5 (a) High Extreme - (b) Theory (c) Typical Junction . (d) Low Extreme H–––––––––––––––––. - 50 O 5O 1OO Bios Voltage (mv) l - 1CO FIGURE 6. Calculation and measurement of conductance in M-S contacts by Steinrisser et al. [6]. Comparison between three experimentally measured conductance curves on n = 7.5 × 10*/cm” Sb- doped Ge [solid lines (a), (c), and (d).] at 4.2 K and the calculated conductance [dashed line (b)] for a barrier height Vn = 0.63 eV obtained from capacitance measurements. The most commonly observed conductance curves were similar to (c), whereas (a) and (d) represent the high- and the low-conductance extremes. The contact metal is Pb and the contact area is 2.5+0.5×10−1 cmº. Structure associated with the superconducting energy gap has been omitted. The Fermi degeneracy pºp = 25 m/ has been indicated. tunneling systems. The density of states in the semiconductor band was assumed constant in this cal- culation. The general aim of this type of experiment has been to obtain such a satisfactory agreement with the calculated conductance rather than to determine a den- sity of states variation. Uncertainty in the energy de- pendence of the barrier parameters would make such a determination exceedingly difficult. A second band structure effect, observed in Au-Ge surface barrier contacts by Conley and Tieman [7], is the onset of tunneling into the k=0 conduction band minimum, roughly 150 mV above the Fermi level. The influence of the band edge is rather dramatic in this case, as shown in figure 7, and a decrease of resistance by at least a factor of 10 is observed within - 10 mW of the edge. Note that the band edge is marked by a decrease in resistance, or increase in conductance. 200 I -I T u | I | I | º *—.154 W. —: : |50H- mº – SAMPLE IO84 S. Au ON Ge/Sb “g {} § |OOH- T= 4.2°K. == s: 92 º/) # ă g ºf - 95 2: O —l 1– | A. _l | | f | -.25 O +.25 Wo APPLIED BIAS (VOLTS) FIGURE 7. Tunneling resistance of a Au-Gesurface barrier contact, measured by Conley and Tiemann [7]. The onset of transitions to the zone-centered (T2') conduction band in Ge observed in the incremental resistance dv/di at an applied bias Wa = –0.124 V. Note that that threshold is 0.154 Vfrom the maximum, a value which corresponds to the interband L – T2' separation. 2. Metal-Insulator-Semiconductor Junctions In order to investigate tunneling into semiconductors to higher voltages, or study those materials which do not form surface barrier contacts, an oxide can be grown on the surface before evaporation of the metal. Alternatively, for semiconductors which can be evaporated, junctions of the type aluminum-aluminum oxide-semiconductor have been fabricated. The current calculated by Chang et al. [8] for such a structure, on a degenerate p-type semiconductor, is shown in figure 8. The current is a maximum at the top of the valence 676 band and a minimum at the bottom of the conduction band. At higher voltages it is interesting to see that pro- perties of the insulator, namely the increased tunneling probability at voltages corresponding to the heights br and bl, dominate the characteristic. Experiments on a number of different III-V and II-VI semiconductors have confirmed this type of behavior except that the O |- Sº IOT" | -> - FORWARD CURRENT IO-8– (J1-J2) H. REVERSE CURRENT (J2-JI) FP IO-10 | Eð. O | VOLTS current minimum is not so sharp as the calculated one, as shown in figure 8 for SnTe. The observation of interface states and impurity bands has also been reported from tunneling studies. The left-hand plots of figure 9 show the conductance results, obtained by Gray [9], for two M-I-S structures with different boron doping levels in silicon. IOT” E H IO-5 H 10 *E. (ſ) sº Lil º H # IOTY = s: |- H 10-8 T H — IOT * – O V (VOLTS) FIGURE 8. Calculation and measurement of current in metal-insulator-semiconductor junctions by Chang et al. [8]. (a) Theoretical current-voltage characteristics for a M-I-S junction. The semiconductor is degenerate p type and the conduction band of the insulator provides the tunneling barrier. (b) Current-voltage characteristics for Al-Al2O3-SnTe junctions at 4.2 K. Three sets of curves are shown corresponding to samples with various oxide thickness. 3. p-n Diodes The I-V characteristic of the Esaki diode [10] is a clear observation of the influence of band edges on tun- neling but recently this system has received little atten- tion, probably because the barrier profile is difficult to measure. It is also difficult in this case to decide how much of the junction current is due to tunneling. An interesting effect which has been observed in p-n diodes is the influence of Landau levels on the tunnel- ing conductance. This was first reported by Chynoweth et al. [11] for InSb diodes and extensive studies of Ge diodes have been made by Bernard et al. [12]. Re- cently Landau levels in InAs have been measured by Tsui [13] in an M-I-S structure. Because only one elec- trode is a semiconductor the latter results are simpler to interpret. 4. Metal-Insulator-Metal Junctions This type of tunneling structure is usually comprised of aluminum-aluminum oxide-second electrode of semimetal or metal. Considerable effort by various groups has gone into trying to observe the band edges in bismuth. Tunneling structures at low energies (<50 mV) have been reported by Esaki et al. [14] but these results have not been reproduced elsewhere [15]. At higher voltages the characteristics for polycrystalline films were first reported by Hauser and Testardi [16]. In figure 10 we show the good agreement between their results and those of Sawatari and Arai [17]. However, the exact position of the bands which give rise to this structure is difficult to determine from the tunneling characteristic. -- 677 H-I-T-I-T- SAMPLE CL-11 (I) 24H- 77°K * to \ – 60; a sm. P type £ e 20 \ At 0.54 mm? / > \ SAMPLE CL-20 (2) / * "c, \ 77 °K M 16H- \ || || ----| 0.014 n.cm. P type f * § \||t (REDUCED 1 100) 2… C As 0.38 mm.” 2: 12 H. * H. £ = 27 º c & C / }* ^e º 4 `-- 2’ ol—4->=#=4–4– -0.6 -04 -0.2 0 0.2 0.4 0.6 0.8 8|AS IN WOLTS | | | .008 — CL= 20 (3) 0.063 () cm (p) 770 K ºmme A= 0.38 mm. emº .004 H. m= () | | | 0.] 0.2 0,3 B|AS IN WOLTS FIGURE 9. Observation of impurity levels in tunneling by Gray [9]. (a) de conductance versus bias showing the effect of a boron-impurity band at two different doping levels. The conductance of the more heavily doped sample is reduced X100 for comparison. (b) Detailed dc con- ductance near an impurity band (0.19 W). This sample was cleaved in air before mounting in the vacuum jar. BiAS VOLTAGE (rnw) !.O 2.O ºf OO y —w Wr t- 5 X- (a) +8 % tºº. i N- an § 46 º * > Absorption (3OO°K) S. vo |." 4 42 `--- l A $. I ſº I {OO 2OO 3OO 4 OO 5CO 6OO 7OO PHOTON ENERGY (mv) {OO (b) | _go a Bof § 2 J40 .3 2 * or 60H > g ‘o É - O 3 4OH > s tº e * 20- plus bios to Al plus bids to Bi l_ ſl { ſº ſt ſi ſl —l -1.5 - I.O -O5 O O5 IO 15 20 25 V Vbl? Fää0 5T at 6-565-7:55– BIAS VOLTAGE (mv) FIzöð ‘FE65 FIGURE 10. Tunneling conductance for Al-I-polycrystalline Bi, as determined by Hauser and Testardi [16] (left figure) and Sawatari and Arai [17] (right figure). (a) Insert: dV/dI vs V for an Al-Al2O3-Bijunction. Bottom curve, adsorption as a function of energy. The dashed line shows the contribution from free carriers. (b) di/dV vs V (solid line) and background curve (dashed line); curve of (dL/dV) (dI/dV), ºn vs V. (+V corresponds to Bi positive.) (c) Conductance-voltage curve in the voltage range - 1.4 - +2.3 V at 77 K. The sample is different from that of (a) and (b). 678 The conductance characteristics of M-I-M junctions generally show considerable structure at voltages 0 K, Sen(T) decreases linearly in T while SesſT) falls much more rapidly. As first noted by Daunt, Horseman, and Men- delssohn [3], the implication of this circumstance is that, for sufficiently low T. lim AS(T) = yT, (2) T— () where y is the temperature coefficient of the normal electronic specific heat. Early efforts to apply (2) were limited by the technical difficulties of making measure- ments below 1 K and the resulting y values were rela- tively imprecise [4]. The analytic form of SesſT) can be estimated using the BCS theory [5] from which it can be shown that Ses 681 becomes negligible for temperatures lower than about 0.25 Tc. For temperatures below this value it follows that the shape of the critical field curve is wholly deter- mined by the normal electronic specific heat. Equating the right sides of (1) and (2) and integrating yields H}(T) = Hà– (4try”) Tº (3) where y” = y/V. It should be noted that this differs from the parabolic dependence of Ho on T which is com- monly assumed. Three considerations make (3) an especially useful and reliable means for the experimental determination of y: (a) The validity of (3) is a general thermodynamic consequence of two well established experimental facts, the linear T dependence of Sen, and the fact that Ses vanishes more rapidly than Son as T approaches 0 K. (b) The experimental parameters of (3) are quantities which can be measured under conditions of static equilibrium with very high precision. (c) The degree of linearity exhibited by experimental HoſT) data when plotted according to (3) provides a test of the validity of the analysis under the prevailing conditions of mea- surement. For sufficiently low temperatures, this linearity has been well confirmed by measurements on several different superconducting elements [2]. For most Type I superconductors the range of validi- ty of (3) lies below 1 K, but new techniques have made temperatures down to about 0.03 K relatively accessi- ble in recent years [6]. The experiment may be done as a difference measurement between the He values of two specimens held in isothermal equilibrium. This reduces the effect of experimental uncertainty in the absolute value of T and puts the main burden of accuracy on dc magnetic field measurements which can be made with high precision using conventional measuring methods. The resulting values of the ratio, y” = (y/V), from the present measurements have a relative accuracy of about 1:3000 which is substantially better than the typi- cal precision of low temperature calorimetry. The principles just described have been applied to study the change in y” under hydrostatic pressures up to 1000 atm for In and Tl. In each case the measure. ments consist of comparisons of He(T,p) for pressurized and unpressurized specimens mounted in thermal equilibrium in a He” cryostat. The experimental details will be published separately but qualitatively similar techniques have already been reported [2,7]. The critical field is displaced by pressure and, using (3), the observed difference may be written as AH}(T, p) = AHä(p) – 4TAy” (p) Tº (4) | | | | | 24OO F 950 Cim — 2300 H T]. c. 17OOH T % 690 Otm O - - SR © 16OOH Tl H. ,-fº 2. cJö || 5O - -: I 4|O Otm º - * IO5OH TL 35OH 125 dim T - * I T I - r l * t 25OH | | | | | - O 2 4. .6 8 |O |.2 Tººk?) FIGURE 1. Isobaric shifts in the critical field of In near 0 K plotted according to (4). where AH}(T, p) = H. (T, p) – H. (T, 0), AHä(p) = Hä (p) –H3 (0), and Ay”(p) = y^(p) – yº(0). Data for In are plotted in figure 1 for measurements at four different pressures. The experimental points show a satisfactory linear dependence on T” as expected from (4) and the slope is directly proportional to the change in y” with pressure. The critical field constants for In are such that a change of 100 units on the AH.2 scale corresponds to a shift in Ho of approximately 0.18 G within the temperature range of figure 1. The variation of y” calculated from the data of figure 1 is shown by the points in figure 2. The solid curve represents the parabolic relation Ay” (p)=–6.5 × 10−7p + 2 × 10−10p”. Using the pressure dependent compressibility for In [8] yields y(p) = y(0) – 1.4 × 10–5p +3.4 × 10-ºp? (5) where p is in atm and y in miſmol. deg. For comparison with other measurements (5) may be used to calculate the logarithmic derivative, d lny/d ln V. the nonlinearity of y(p) results in substantial varia- tion of d lny/d ln V within the pressure range of the present work. Its value drops from 3.40 + 0.1 at p = 0 to 1.80+ 0.05 at p = 1000 atm [9]. 682 O &T C) Qt) -U G E - 2 > E * > - Re 1.0 NS 0.5H O 500 FIGURE 1. model from the experimental data of the low tempera- ture specific heat coefficient yo without including cor- rections due to the many-body effects of electrons, phonons and paramagnons, were made use of. From the experimental and calculated results of R, k, y and Xs, it is easily seen that there are certain cor- relations among their temperature variations for paramagnetic transition metals. The object of this paper is to explain the correlations among the tempera- ture dependences of thermoelectric power S, R, k, y and X, and the sign of S in connection with the shape of w(e) and the position of er for typical transition metals of the plus and minus groups. Our recent calculated results [6] of the temperature variations of R, k, and S at high temperatures for molybdenum, tungsten, rhodium, iridium, palladium and platinum metals by the usual theory of conduction in a model similar to the Mott model of s-d scattering [1] are shown in section 3. By using simple approxima- tions R, k and S are expressed in terms of v(e) and parameters of electronic band structures. These calcu- 2.5]- b – Cr Zr * H f O * -- * 1.5 – Tj | MO º % U W Rh 1.0 F = Ta Pł 0.5H V Pd Nb | | L 0 500 1000 1500 T ok (a) Experimental and (b) calculated results of the reduced electronic specific heat coefficient y” = y/yo. lated results are compared with the experimental ones. In section 4 it is concluded that the temperature varia. tions of R, k, and S and the sign of S at high tempera- tures are strongly dependent on the shape of v(e) and the position of ep. 2. Temperature Dependences of Electronic Specific Heat and Magnetic Susceptibility The part of the electronic specific heat Ce= yT is evaluated by subtracting the Debye specific heat and dilatation correction from the observed specific heat at constant pressure in the usual way [2]. The tempera- ture variations of the ratio yeap” = y/yo obtained from the experimental values of y=CE/T and yo and of the ratio of the observed value of X at T to the one at 300 K, i.e., Xerp” =X/X300 K, for various paramagnetic transition metals are shown in figures 1a and 2a, respectively (cf. the references 2 and 5 for details of the experimental data). From figures la and 2a, it is easily seen that for the plus group metals, where the ep occurs in the 686 a _2- 1.5 1.5H -- . J iſ §º Pd Rh Ti Hf 8 R Aeº Zr º Pt (Y) e 27–ºf >< Pt 24- º N e= W M & >< 2- * S ; 1.0 * *- :- > 1.0 "..., Cr *E=ºss V Tj *{- à Ta L >< Nb x & >< Pt Pt O.5H ---. 0.5H PC/ PC/ I | | | | f O 500 1000 15OO O 500 1000 1500 T ok ok FIGURE 2: (a) Experimental and (b) calculated results of the reduced susceptibility X*=XX300 K. TABLE 1. Estimation of the molecular field coefficient O. and the constant part of the magnetic susceptibility Xe o: (104 moleſ emu) Xe (10−4 emuſ mole) Ti V Cr Ti V Cr 0.82 () () 0.73 1.8 1.32 Zr Nb Mo Rh Pol Zr Nb Mo Rh PG 0 0.24 0 1.33 0.72 0.83 0.82 0.55 — 3.2 () Hf Ta W Re Pt Hf Ta W Re Pt 0.88 () 3.06 () 0.74 0.14 0.71 0.31 0.42 — 0.15 neighborhood of a minimum of v(e) and hence the values of yo and Xs are small, Yerp” and Xeºp” increase from one with increasing temperatures and for the minus group, where the ep occurs in the neighborhood of a maximum of v(e) and hence the values of yo and Xs are large, they decrease from one with increasing tem- peratures. Rhodium and rhenium metals are inter- mediate between the plus and minus groups. The CE and Xs can be calculated in the one-particle approximation by the usual electron theory of metals, if the shape of v(e) is given. The curves of v(e) for paramagnetic transition metals of 3d-, 4d- and 5d- groups with bec and foc crystal structures were ob- tained in the rigid band model from the experimental data of yo for various metals and their alloys, as shown in figure 3, where the positions of eF for various metals are also shown. It is easily seen that the width of the 4d- band is broader than that of the 3d-band and narrower than that of the 5a-band. From the v(e)'s shown in figure 3, the temperature variations of y = CE/T and Xs are calculated in the usual electron theory of metals. The total magnetic suscepti- bility X is calculated from the calculated results of Xs by using X=Xs/(1 – Oxs) + Xe, where 0 is a molecular field coefficient and Xe is the sum of the constant orbital- paramagnetic and core-diamagnetic susceptibilities, Xe =Xorb-H Xa. The calculated results of the ratios, year * = y/yo and Xcal” = x/x300 K, are shown in figures lb and 2b, respectively. The values of o and Xe are deter- mined so as to get the best fit of the calculated values of X to the observed ones for respective metals. These values are shown in table 1. It seems that these values of O. are reasonable except rhodium and tungsten metals, where the values of O. are anomalously large, and the values of Xe are consistent with the calcula- tion of the Xorn by Place and Rhodes [7], except rhodium metal. The agreement between the calculated and experimental values of y” in rhodium is very 687 's 3– a . b C C | – 3 - b. f C C – Q. .S. o S. S Uſ) Sº S O s s PC/ i. * S- S- 2 !--- --- 3 § § J Uſ) Uſ) *- ‘S O > > ; 1 - Cl Cl Pt 5C Rh Ir al- -1 O 1 Energy (eV) FIGURE 3. (a) Electronic densities of states for transition metals of 3d-, 4d-, and 5d-groups with bec structure and (b) those with foc structure. satisfactory, nevertheless the variation of X for this metal is very large and it is very difficult to explain the T628 SOIl. - From figures 1 and 2 we can see that there are qualitative agreements between the calculated and ex- perimental results of y = CE/T and X for various paramagnetic transition metals of the plus and minus grOupS. 3. Electrical Resistance, Thermal Conductivity and Thermoelectric Power By using a model similar to the Mott model of s-d scattering [1] and by the usual theory of conduction [8], the temperature variations of R, k and S at high temperatures are numerically calculated for molyb- denum, rhodium, palladium, tungsten, iridium, and platinum metals. The main scattering of electrons at high temperatures is due to electron-phonon interac- tions, and for the sake of simplicity their matrix ele- ments are averaged over the Fermi surfaces and as- sumed to be a constant denoted by 8. Because of the ap- proximation for the matrix elements of electron-phonon interactions and of the effective mass approximation in the simple model of electronic structure, R, k and S can be expressed in terms of the v(e) and the parameter of electronic band structures as follows: R = (e”B05/£)-1 (T/C), (1) k = (B05/£) (GA – G*/G1)/T", (2) ** S=– (G2/G1)/ (eT), (3) 6.--ſ de” (e-o'-vo-ºx mº, () where B is a constant which depends on the lattice con- stant, density, etc., 6)d the Debye temperature, f(e) the Fermi-Dirac distribution function, & the chemical potential, and mi the effective mass ratio of electrons in the ith band. Further, Ei in (4) is given by Ei = e – ee or en–e for an electron or hole band, where ee or en is the energy at the bottom or top of the respective bands. The values of the parameters of electronic structure mi, ee and en in (4) are determined in the following way. For palladium and platinum the two band model of s- electrons and d-holes, for rhodium and iridium the four band model of two hole surfaces around the X point and two electron surfaces around the T point, and for molybdenum and tungsten the four band model of two hole surfaces around the N and H points and two elec- tron surfaces around the T point and centered on the A axis are made use of. These closed Fermi surfaces are replaced by spheres of equal volume so as to be con- sistent with the calculated results of electronic bands and the observed results of yo and the de Haas-van Alphen oscillations. 688 0.8 H £- 0. 6 3000 FIGURE 4. Calculated results (solid curves) of the reduced electrical resistivity divided by T. Broken curves are obtained for constant (90's. 2.0 PC/ Pł º O 3 1.5 H _2~ × _2~ N Pt(exp) --- > | / 2 -----T | | / > --~~~T * 22--" > 1.0% Ir W MO 0.5 | | | 300 500 1000 1500 2000 T ok FIGURE 5. Calculated (solid curves) and observed (broken curve [9]) results of reduced thermal conductivity k” = k|Kano k. By using the values of mi, en and ee determined in this way and the v(e) shown in figure 3 and by taking ac- count of the temperature variation of 0D due to the thermal expansion [1] in (1) and (2), the temperature variations of R, k and S are calculated by (1-4). The value of £ in (1) and (2) for each metal is determined in such a way that the calculated value of R agrees with its observed value at the highest temperature, and the values of § obtained are shown in table 2. It is noted that the values of § for superconducting metals are larger than those for normal metals. The calculated results of R at high temperatures agree almost completely with the observed results [9] 30H MO __2=-->====fºl.---------> *W__ --~~~~ IFT------ O *. == ,2T) MO S——º: ss. ---------- El Q-- 7 SSJ T--~~~ º, Rh ssss T------- ~ T *** - - SS-> TTT--~~~ Pt ~ | N \ --~ TTT-------' >< | N \, S-- o | N S >>s > -50F N N Tºss Poſ - –3 | N \, ºss ^_^ Uſ) Pt – 100 H. - PC/ | | | O 500 1000 1500 2000 T ok FIGURE 6. Calculated (solid curves) and observed (broken curves [9]) results of thermoelectric power S. TABLE 2. Values of electron-phonon coupling con- stant & in (e/)? Metals Mo Rh PG W Ir Pt Ś................ 10.5 1.92 1.63 13.3 4.56 2.46 for all metals except palladium metal (cf. [6]). The cal- culated results of the ratio of R/T to its value at 300 K are shown by solid curves in figure 4, where the broken curves are obtained without taking account of the tem- perature variation of 69p. The calculated results of the ratio of k to its value at 300 K and of S are shown in figures 5 and 6, respectively, where broken curves are the observed results [9]. It is seen that there are qualitative agreements between the calculated and ob- served results of k and S. 4. Discussion and Conclusion At lower temperatures, R, k and S in (1-4) can be ex- panded as power series of T as [6] (RT)/(RIT) -- (mºt) (3,4-º), 5 K/(k)T-0= 1+. (TkT)? (; vi-É. w) (6) = - 2 \ 2 I 2 37 2 42 7 l/3 S 3e (Tk)*Tv, {1+. (TkT) (. vi-g tº Fä #). (7) 417–156 O - 71 – 45 689 where un- (d"v(er)/dep")/v(ef) and the contributions from Ei in (4) are all neglected. The expression (5) was given before by Jones [3]. The low temperature expan- sion for X, and y were given by [4,2] | Xs=2p;|v (€F) | 6 (nºw-vo). (8) •) | | • (Tk)*v (ef) | 6 (T/T)? (*-*. º). (9) From (5-9) it is expected that X, y, R/T, l/k and Tl|S| at lower temperatures increase with increasing tempera- tures for the plus group metals where ep occurs in the neighborhood of a minimum of v(e) or v2 > vſ’ = 0, and they decrease for the minus group metals where ef occurs in the neighborhood of a maximum of v(e) or vº < 0. Moreover, from (5), (8) and (9), the temperature variations of R/T are expected to be smaller for the plus group metals and larger for the minus group metals, respectively, than those Xs and y. Accord- ingly, as shown by broken curves in figure 4, the deviations of the values of R/T from unity at high tem- peratures are relatively small for the plus group metals and large for the minus group metals, respectively. As the temperature variations of R/T are very small for rhodium and iridium of the plus group it seems that v2 is nearly equal to 311” in these metals. In the plus group metals the temperature variations of Xs and y show a maximum at a certain temperature because X, and y become zero at infinite temperature. It has been shown [2,10,11] that even in the minus group metals, e.g., pal- ladium, X, and hence y may show a maximum if viº — v2 < 0 as seen from (8) and (9). As the sign of S at low tem- peratures is determined by the sign of v1 as seen from (7), from the results in figure 6 it is concluded that the positions of eF in v(e) shown in figure 3 are all con- sistent with the experimental results of S. From the comparisons between the calculated and experimental results shown in figures 1, 2, 4, 5 and 6, it is concluded that the temperature variations of R, K and S and the sign of S at high temperatures are strongly de- pendent on the shape of v(e) and the positions of ep, and the Mott model for the s-d scattering is a satisfactory approximation to calculate the temperature variations of R, K and S at high temperatures for paramagnetic transition metals. By our calculations of R, k and S, it is confirmed that for the transition metals of the plus group, where the Fermi level occurs in the neighbor. hood of a minimum of the v(e) curve and the values of yo and X, are small, R/T and 1/k, as well as X, and y, in- crease with increasing temperatures and for the minus group metals, where the Fermi level occurs in the neighborhood of a maximum of the v(e) and the values of yo and X, are large, they decrease with increasing temperatures. 5. Acknowledgments The author wishes to thank T. Takahashi, A. Katsu- ki, H. Yamada, T. Aisaka and K. Ohmori for their cooperation and assistance. 6. References [1] Mott, N. F., and Jones, H., The Theory of the Properties of Metals and Alloys (Oxford Univ. Press, London, 1936) p. 268; Mott, N. F., Advances in Phys. 13, 325 (1964); Proc. Roy. Soc. A153,699 (1936). [2] Shimizu, M., Takahashi, T., and Katsuki, A., J. Phys. Soc. Japan 17, 1740 (1962): 18, 240 (1963). [3] Jones, H., Encycl. of Physics, S. Flügge, Editor (Springer, Ber- lin, 1956) XIX, p.265. [4] Stoner, E. C., Proc. Roy. Soc. A 154,656 (1936); Acta Metallur- gica 2, 259 (1954). [5] Shimizu, M., and Katsuki, A., J. Phys. Soc. Japan 19, 1135, 1856 (1964); Katsuki, A., and Shimizu, M., ibid. 21, 279 (1966); Shimizu, M., Katsuki, A., and Ohmori, K., ibid. 21, 1922 (1966). [6] The details of this calculation will be published by T. Aisaka and M. Shimizu elsewhere. [7] Place, C. M., and Rhodes, P., J. Appl. Phys. 39, 1282 (1968). [8] Wilson, A. H., The Theory of Metals, 2nd ed., (Cambridge Univ. Press, London, 1953) p. 193. [9] Touloukin, Y. S., Thermophysical Properties of High Tempera- ture Solid Materials l (Macmillan Co., New York, 1967). [10] Elcock, E. W., Rhodes, P., and Teviotdale, A., Proc. Roy. Soc. A221, 53 (1954). [11] Allan, G., Leman, G., and Lenglart, P., Journ. de Phys. 29, 885 (1968). 690 Discussion on “Temperature Dependence in Transport Phenomena and Electronic Density of States for Transition Metals” by M. Shimizu (Nagoya University, Japan) J. F. Goff (NOL): I have treated k and p for chromium (which Prof. Shimizu does not treat) and find compati- ble results. I would recommend that people use this method of moments formulation due to Klemous for analysis. Since one would expect dense d-bands not to conduct, do your results for the elements to the right of the transition series imply the contrary? M. Shimizu (Nagoya University, Japan): In our model of transition metals, both s- and d-electrons in the neighborhood of the Fermi level contribute to conduction. 691 Metal-Semiconductor Barrier Junction Tunneling Study of the Heavily Doped N-Type Silicon Density of States Function* Y. Hsid” cind T. F. Tao University of California, Los Angeles, California 90024 Experimental and analytic techniques and procedure used in the study are described. Experimen- tal data showing the dependency of the Fermi level on the dopant types of the heavily doped n-type sil- icon are reported. A dopant type dependent density of states effective mass is postulated to describe the effect of different dopants on the Fermi level. The deviation of the experimental data curve from the cal- culated curve is ascribed to the effect of degenerate semiconductor band tailing. In addition, through in- terpretation of incremental conductance versus applied bias characteristic curves of the different tun- nel junction evaluated, a consistent description of the density of states function of the heavily doped sil- icon is obtained. The density of states function, dependent on the dopant type and dopant concentra- tion, is generally parabolic above the band edge, but towards the band edge, band tailing can be severe. Key words: Antimony-doped silicon; arsenic-doped silicon (As doped Si); band absorption; band tailing; depletion layer barrier tunneling; electronic density of states; Esaki diode; gallium-arsenide (GaAs); heavy doping with As, Ga, P, Sb; luminescence experi- ments; P doped Si; Schottky-barrier; silicon; transport properties; tunneling. 1. Introduction The study of the transport properties across a tunnel junction has been an active field of research for a decade. There are three types of tunnel junctions that have been studied in various degrees: (1) p-n junction, (2) insulated layer tunnel junction, and (3) depletion layer metal-semiconductor tunnel junction. In this paper, we are presenting some of our experimental results on the metal-semiconductor barrier junction tunneling study of the heavily doped n-type silicon den- sity of states function. It was Gechwend et al. [1] who first indicated the possibility of a metal-semiconductor depletion layer tunnel junction. Since then, Conley, Duke, Mahan, and Tiemann [2] calculated the tunneling current through the depletion layer in a metal to heavily doped semicon- ductor junction and found that d//dl has a maximum at a bias potential equal to the Fermi degeneracy calcu- lated by the parabolic band model. Conley and *This paper is based on part of a dissertation submitted by Y. Hsia in partial fulfillment of the requirements for a Ph. D. degree in the School of Engineering and Applied Science, University of California, Los Angeles, 1969. **Mailing address: Litton Systems, Inc., Guidance and Control Division, Woodland Hills, California 91364. Tiemann [3] reported reasonable verifications of the calculations based on their measurements of metal to n-type Ge Schottky barrier diodes. Conley and Mahan |4,5], also studied n- and p-type GaAs. For p-GaAs, they found that the location of d//dl maximum was af- fected by the band tailing in heavily doped semiconduc- tor. For n-type GaAs, it occurred at a bias much smaller than the calculated Fermi degeneracy. The deviation was accounted for by the effect of band mixing in the forbidden gap. Using a thin oxide layer as a tunneling barrier, Chang, Esaki, and Jona [6] had earlier re- ported their tunneling study of metal to both n- and p- type InSb junctions, and suggested that the d1/dV minimum observed in the case of p-InSb occurred at Fermi degeneracy. The depletion layer barrier tunneling appears to be an amenable tool for the study of the properties of heavily doped semiconductor near and about the band edge, including the determination of the Fermi degeneracy, the amount of band tailing, and generally the effect of impurity band on the overall band struc- ture of the heavily doped semiconductor; because the tunnel current measurement is referenced to the zero bias which is located at the Fermi level. This is an im- 693 provement over the band absorption [7–10] and the lu- minescence experiments [11-13] commonly used in the study of the heavily doped semiconductors. In all of these optical measurements, they are referenced to the energy band gap, whereas the band edge phenomena to be evaluated are only a small proportion of the reference energy level. In addition, when measure- ments are made as a function of dopant concentrations, for example, in the study of the Fermi degeneracy due to impurity dopant, data interpretation becomes ex- tremely hazardous because of the existence of the band gap narrowing phenomenon introduced by heavy impu- rity doping || 14-16]. 2. Andlyses of Tunneling Transport 2.1. Physical Model When a metal-semiconductor junction is formed, there is an exchange of carriers, such that the Fermi level is continuous across the junction. In most of group IV (Si included) and zinc blende III-V semiconductors, Mead and Spitzer found that the Fermi level at the junction interface is fixed very close to one-third from the valence edge, and thus the potential barrier at the interface which results from the redistribution of charge depends not on the metallic element or the dop- ing of the semiconductor [17]. The potential barrier is maintained by the electric dipole layer at the contact. + Met etal eVs The positive charge at the semiconductor side of the junction is resultant from the fixed ionized donors hav- ing a density much less than the ionized lattice metallic atoms in the metal. Therefore, the ionized donors at the semiconductor contribute a distributed charge layer near the junction interface, quite different from the sur- face charge layer at the metal element. The distributed charge layer in the semiconductor is commonly called the depletion region of the junction as the region has been depleted of charge carriers, leaving behind the fixed ionized impurities locked in the lattice. Figure 1 is a diagram showing the potential distribution at a metal-semiconductor junction. - The I-V characteristics of a metal-semiconductor tunnel junction is shown in figure 2. At zero bias, elec- trons tunneling through the depletion layer barrier from the metal to the n-semiconductor are in equilibrium with electrons tunneling in the reverse direction, and we have zero current flow. With forward bias, the elec- trons in the conduction band of the n-semiconductor are brought to opposite to the empty states above the Fermi level of the metal, thus increasing the tunnel cur- rent which is proportional to the joint probability of the occupancy of the conduction band of the n- semiconductor and the availability of empty states in the metal. At forward bias above V, where (a ) Zero Bias (b) Reverse Bias FIGURE 1. e/p + Er – Ec (1) Semiconductor EF Fo N -— d —- \ Ec eVA ––––––5 N (c) Forward Bias The idealized metal-semiconductor tunnel junction. 694 80 - 60 — 40 — 2 O Vp –60 º º f 40 6 O 80 if l2O | | | — -20 H -40 Voltage in my FIGURE 2, The idealized I-V characteristics of a metal-semicon- ductor tunnel junction. no additional filled electron states in the n-semi- conductor are brought to opposite empty states in the metal with increasing applied bias, and the tunnel current remains approximately constant. The slight in- crease in the tunneling current at bias above /p is at- tributed to the modification of the barrier transparency by the applied bias field. 2.2. Mathematical Model The generalized tunneling equation that we are using to study the analytic behavior of the metal-semiconduc- tor barrier tunneling is where fin (E) = the Fermi function describing the occupancy of the density of states function of the metal f, (E) = the Fermi function describing the occupancy of the density of states function of the semi- conductor * gº (E) = the density of states function of the metal/ superconductor g; (E) = the density of states function of the semi- conductor P(E) = the tunneling probability A = the constant of proportionality This generalized tunneling equation is very similar to the original tunnel diode equation proposed by L. Esaki except that the tunneling probability is P(E) instead of I = A | [f, (E)-f(E)]g,(E)g,(E)P(E) dB (2) an assumed constant [18]. It is also similar to the basic equation used in the interpretation of superconductor tunneling experiments [19]. Quantitative data on su- perconductivity based on the interpretations of the tun- neling experiments have been obtained on the assump- tion of the tunneling current being directly proportional to the density of states of the quasi-particles in the su- perconductors [20,21]. To determine the metal-semiconductor barrier tun- nel current according to the generalized eqs (2-48) we first derive the potential distribution of the tunnel barri- er and then arrive at the tunneling probability function P(E) using the WKBS approximation solution of the Schrödinger's equation describing the electronic wave function across a barrier potential. The potential distribution at the metal-semiconduc- tor junction can be readily derived from Poisson’s equa- tion. The parabolic potential in the depletion region results from the assumption of uniform distribution of the impurity charge density No. The one-dimension Poisson’s equation is given by * Vo-" 6 x* € (3) where } (x) = potential distribution as function of x q = electronic charge e = dielectric constant of material 695 Assume width of depletion layer = d, then solution to eqs (3) is given by N V(x)="; (d-o-º-V, (4) where d=|*-ū, W,-Vol" =|# B-H / I' o (5) and VA = applied voltage bias VF = Fermi voltage Vn = barrier height of the junction in volts For the Schottky barrier of heavily doped Si, in the volt- age bias range of interest (<|+100 mv), the width of the depletion layer d is of the order of 100 A. Experiments to determine the barrier height of metal-semiconductor junctions were carried out by C. A. Mead and W. G. Spitzer on Si, Ge, and many group III-V compound semiconductors [17]. The barrier height was found to be nearly independent of the metal- lic element work function or the doping of the semicon- ductor. It was shown that one can estimate the barrier height to a fair degree of accuracy by tº , lip = e/p -#s (6) A detailed study of the interface surface states at the metal-semiconductor junction by J. Bardeen [22] and recently by V. Heine [23] led to the conclusion that the barrier height is independent of the metal used pro- vided the density of surface states is sufficiently high (> 1018 cm−2), a condition adequately met for the group IV and III-V semiconductors. In terms of our diode fabrication technique as will be described in a later section, it is expected that the description of the barrier height can be given by eq (6) [24]. In the development of the barrier potential function (eq. (4)), we neglected the effect of the image force ex- perienced by an electron when it approaches the metal. C. R. Crowell and S. M. Sze [25] have shown that even at room temperature, with low doping impurity materi- als and at large bias, i.e., high field, the effect of image force barrier potential lowering does not affect ap- preciably the current voltage characteristics of a Schottky barrier. Therefore, image force consideration is not included in our development of the barrier poten- tial function. Also assumed in deriving the barrier potential func- tion is the uniform distribution of the doping impurities in the semiconductor. The effect of random distribution of impurities in the depletion region on the Schottky barrier potential function has been studied by J. W. Conley and G. D. Mahan [3]. They showed that inclu- sion of the fluctuation in barrier height due to nonu- niform impurity doping distribution changes little the theoretical prediction of the tunneling current through the barrier potential. Therefore, the assumption of uniform distribution of impurity doping distribution in the development of the barrier potential function can be made. The WKBJ approximation solution [26-28] of the Schrödinger's equation describing the electronic wave function across the depletion layer barrier potential U(x) = e/(x) (7) yields the tunneling probability P(E) = e-W(P, U) (8) W(E, U) = 2 ſ |# (E-UGo"a, I (9) The integral function W(E,L) is the WKBJ integral. We calculate the tunneling probability for the metal- semiconductor barrier by eq (8), with W(E,L) obtained from eq (9) where W(x) is as given in eq (4). Then Vºßd FKF-dvk-É { Voy-E Bal-Ekapº — 02 p(p, u(,) (2nd B) \ot Bdt & –Bºltºp ln 2V1. (10) 4k 4k 8k Voy-- B at 5 vº where o-º-º-º-º-o-º: 2e qVo 8-d _q\o K-5, 696 It is noted that the independent-particle theory of barri- er tunneling using the WKBJ method [29] assumes a tunneling velocity in k-space and thus the tunnel velocity must be included in the integrand as v(k) in the generalized theory. And the general form for the tunnel- ing current so obtained is independent of the density of states function because the tunneling velocity in the k- space is taken to be reciprocal of the density of states. J. W. Conley et al. [2] have calculated an exact solu- tion of the one-dimensional Schrödinger equation for a parabolic depletion layer potential and determined the transparency of the barrier in the generalized tunneling equation originally derived by Fredkin and Wannier [30]. —2. c • dºko, •) I ſº * j-–2 ſ; vsz|T, ..]*[f, (E) —f, (E)] (11) The barrier transparency |T m—s is found to contain the factor E.1/* describing the parabolic density of states function of the semiconductor. To obtain analytic and numerical correspondence between the exact barrier transparency calculation containing the density of states term E}/*, represented to be PCDMT, and the ap- proximate transmission coefficient obtained from the independent-particle WKBJ tunneling calculation, Pºrkh), J. W. Conley and G. D. Mahan arrived at the fol- lowing expression [5]: 12, I = 3 | dEE'ſ Pirka, (12) * 1 where Pºw-exº-2 ſ dzk (z, E) an expression equivalent to the generalized tunneling eq (2) we are using in this study. The validity of the eqs (2) or (13) are shown empirically with the study of Con- ley et al. on the Schottky barrier tunneling studies on GaAs [6,7]. 2K.,.]!” Pirkby = exº-ſº | * b }}l () Then, from (11) and (12), according to the development given by Mahan and Conley, the expression can be ob- tained: D * PopMT gº E*Prºpy (16) And in the case of GaAs, through their numerical cal- culations, Padovani and Stratton show that expression – 2K? - bno 1/2 2K. 1/2 | -- - : * * N.J.' }}l J" I] |# | + ( bno ) | Comparing eq (12) with (13), R. Stratton and F. A. Padovani [31] obtain: 4. PopMT Im. — | T mm F. | 17ls (Erm - Ers + V-H E) bmokh, + 4 exp (— bn0) | I?l in b mob, ()() | I?ls (Erm - Er's + V + E) - (1_1 is \ll 12 - (*_l kº ) *||--|-}ki)+; K. G-: ki. where bno= (Ep -- Erm - V)/E00 K? -- E/2E00 mir Eoo =#qh (N/mme)" And the WKBJ approximation of P is determined to be Pirkby = (bno)"ina (e/K%, V2) ºr eXp (— bno) (14) assuming E = EF, that is, all participating tunneling electrons are to originate at the Fermi level. Stratton and Padovani then determine from both the exact and approximate tunnel current calculations of incremental junction resistance, the occurrence of sharp peaks in p- type GaAs; n- and p-type Ge and Si are at the voltage corresponding to the semiconductor Fermi energy and in the case of n-type GaAs, at voltage lower than the corresponding Fermi energy. They also obtain very ex- cellent agreement for the WKBJ calculations with the experimental incremental resistance data of the Ge Schottky barrier by Conley and Tiemann [3]. More recently, F. A. Padovani and R. Stratton [32] questioned the general applicability of the eqs (2) and (12) through their detailed numerical analysis on Pepyſ and on Pwkuſ derived with the same identical assump- tions as those for the exact Popiſt. The WKBJ expres- Sion is given as (15) 2 is valid to an accuracy of about 6 percent only if E → 2.1 meV, a very narrow range. Assuming a parabolic density of states function for the heavily doped semiconductor, the tunneling current of the metal-semiconductor is obtained as a function of the voltage bias with either the WKBJ in the exact method. Figure 3 compares the calculated 4.2 K tunnel 697 l O Calculated with *CMDT Calculated with *WKBJ 2NP2 ll Diode 3NP2l O Dio de 4NP22 Dio de amams am = * * * - - - * * *-* * - - - - * _- - - - ** º- am-e ºs -" amme - = -- - * - - - * amme - = <- --------- E-L –50 -40 – 30 –20 —l O O 10 J 20 30 Cl Vo2 Wo 3 Applied Voltage in my FIGURE 3. junction incremental conductance characteristics of several diodes using both methods. Both methods yield the same characteristic features, especially the ex- istence of an incremental conductance minimum cor- responding to the Fermi voltage of the semiconductor. Figure 3 also reveals that the parabolic density of states function of the heavily doped semiconductor is reflected in the incremental conductance of the junc- tion diode. Study of eqs (2) and (13) indicate that at heli- um temperatures, the equations can be approximated by the incremental tunneling conductance equation: dI/dVoſ’ſ’khs (V)2, () ) (17) giving a mathematical designation of the dependence of the incremental junction conductance of the semiconductor density of state function. Comparison of calculated tunnel junction incremental conductance characteristics, using Pºkby and PcMDr. To assure that variations of barrier heights due to dif- ferent surface properties on different samples, image force and nonuniformity of doping impurity distribution do not significantly alter the interpretability of the data generated by the tunneling current measurements, Con- ductance curves of different barrier heights (Vn = 1/2 Vy, 2/3 V, 3/4 V.) are compared (fig. 4). It is noted that even though the absolute magnitude of the tunnel cur- rent differs almost in orders of magnitude with the dif- ferent barrier height assumed, the normalized con- ductance curves do not differ widely with each other (less than+ 10 percent within the range of experimental interest). With the validity and accuracy of the function P(E) and the mathematical model established, we can con- clude that the calculated normalized conductance curves based on eq (2) and typified by those given in 698 l OO 8O 7 O 60 5 O 40 3O | | | | | | 6 7 8 9 l O 2 O 3 O |→l—1– l 2 3 4 5 Incremental Conductance in Arbitrary Unit FIGURE 4. Study of the effect of barrier height on incremental conductance. figure 5 can be utilized to evaluate the experimental tunneling curves with regard to the density of states function of the heavily doped semiconductor. 3. Experimental Techniques and Procedures 3.1. Diode Preparation Silicon wafers heavily doped with different dopants and uncompensated, cut from an (111) pulled ingot are purchased from Wacker Chemical Corporation, Los Angeles, California. The dopant types chosen are As, P, and Sb for n-type Si and B for p-type Si, because of their relatively high solid solubility in Si so that heavy doping concentrations are more readily available for study. The highest concentration chosen for this study in each dopant type is more than an order of magnitude less than the maximum theoretical possible concentra- tion so that we will not encounter experimental results complicated by precipitation effects in the extremely heavily doped materials [32]. These wafers are generally 3/4 inch to 1-1/4 inch in diameter, 10 to 15 mils thick, and mechanically lapped on both sides with all traces of saw marks removed. Resistivity measure- ments are made on the wafers with a four-point probe [33]. Several measurements are made on each wafer for a measure of the uniformity of electrical properties of the wafer. With few exceptions at extremely low re- sistivities, with measurements sensitive to electrical statics due to low output signal voltages, variations in resistivities measured for any wafer are within + 10 per- cent. Resistivity variations for the majority of the wafers are within + 5 percent. Carrier concentrations for the n-type materials are interpreted from the re- sistivity data interpreted from the published data by Irvin [34], Logan, Gilbert and Trumbore [35], and Fu- rukawa [36]. One of the early procedures used was a mechanical polish step after completing the resistivity measure- ments. Starting with a large (~ 15 micron) particle size abrasive, and progressing downward to abrasive size of 0.3 micron, the wafers were mechanically polished to a mirror finish. This polishing step was eliminated after it was found that the chemical etch that followed was effective in etching down the wafer by a few mils and polished it to mirror finish at the same time. The chemical etch used is a modified CP-4 solution [37], the etchant composition is 2 parts hydrofluoric acid, 9 parts nitric acid and 4 parts acetic acid. At room temperature, with a tumbling agitator, the etchant is ef- fective in etching down uniformly the Si wafers to a depth of 2-5 mils in about 30 minutes while at the same time polishes the wafer to a mirror finish. The etch depth is more than sufficient to eliminate all surface damages introduced when the wafer is cut from the ingot [38]. If the surface damages are not removed from the wafer, dislocations in the damaged semicon- ductor crystal will distort the measured semiconductor density of states function significantly by contributing localized imperfection energy levels in the energy gap. The electrons no longer need to tunnel completely through the depletion layer barrier, but can make use of these localized levels as trapping centers resulting in a smearing of the tunneling current whereby the volt- age bias can no longer be considered as a good measure of the energy of the electron involved in the current transport [39]. The etch polished wafer is then diced into small dice of 75 mils square with a string saw. The string saw is used so that saw damage to the die edges will be minimal. The dice are then mounted on a transistor TO- 5 header by a Si-Au eutectic alloy technique to result in an ohmic contact. We use a gold plated base TO-5 transistor header. A 98 percent Au-2 percent Si perform is placed between the die to be mounted and the transistor header. Using 699 G). ll l2 l O (Y) (U G): 7 G)G): T 4. 2° K VB 2/3 Vg m*/m O. 33 O X 104°em" 3 o N1 ~ * 9 * N l. O X l O Cm 2 | 9 - ? N., 3.0 x 10 cm -) º l OO l ºme V2, \ – L- 1–1–1–1– | C +-I- | l—— | | | – 5 O – 4 O – 3 O – 2 O —l O O l O 2 O 3 O 4 O 5 O 6 O 7 O 8 O 9 O Voltage in mv FIGURE 5. Computed incremental conductance curves of metal-semiconductor tunnel junctions. a molybdenum strip heater set at 450 °C, with a slight pressure applied on the die to break the thin oxide layer on the underside of the die, the Si-Au eutectic bound can be obtained. The Si-Au eutectic is at 370°C. The die mounting operation is performed under flowing nitrogen gas to minimize oxide formation on the top sur- face of the mounted die. The mounted die is then subjected to ultrasonic cleaning. This cleaning serves two purposes. First, it provides a check on the die bounding process. If a Si- Au eutectic has not been obtained, the mechanical bound of the die to the header is weak and the die will break loose under ultrasonic vibration. Secondly, the cleaning solvent (deionized distilled water) and the ul- trasonic action will remove the last traces of organic im- purities on the Si surface that have not been burnt away in the die bounding process. After this cleaning process, it is assumed that the Si surface is entirely free of organic contaminants, residual abrasive particles and dust particles, etc. From this point onward, ex- treme care is taken to prevent introduction of contami- nants on the Si surface. Between the processing steps to follow, the mounted dice are stored under deionized distilled water in chemically inert containers which have been carefully cleaned. To handle the transistor header on which the die is mounted, cleaned teflon- coated tweezers are used. The Si die is then masked for evaporation of the metal-superconductor contact. The masking material used is the black wax which does not outgas in a vacuum. Enough area of the Si die is covered such that damaged areas on the edge of the die which resulted from the cutting action of the string saw during dicing of the wafer are masked and eliminated from the junc- 700 tion area of the tunnel barrier diode to be formed. The masking procedure is performed under flowing nitrogen gas on the molybdenum strip heater at the temperature under which the wax will have the right viscosity for ease of masking. This temperature is esti- mated to be less than 75 °C. Immediately prior to metal evaporation, the Si die is etched in a solution of 1 part hydrofluoric acid and one part H2O to remove the surface oxide formed during the different preceding preparation steps. The evaporation is done in a bell jar vacuum system at 2 × 10−6mm Hg and lower; the pressure is measured by an ion guage. The metal shots used for evaporation (Pb, Sn, and In) are guaranteed 99.9999 percent chemi- cally pure. No attempt is made to pre-etch the metal shots before evaporation to remove the surface oxide. The various forms of the metallic oxides either dis- sociate before reaching the boiling point of the metal or sublime in vacuum at relatively low temperatures. Or- ganic surface contaminants on the shots are burnt off at even lower temperatures. To prevent contamination of the junction surface, a shutter is used. The shutter is opened to permit evaporation onto the Si dice only dur- ing the mid-period of metal evaporation, preventing the deposition of low boiling impurities, as well as eliminat- ing the possibility of the deposition of high boiling point impurities when the metal source is depleted. A very high rate of evaporation, approximately 1000 A per minute, is used since it is found that it gives the best results in terms of good evaporated films of highly metallic appearance and subsequent good tunnel diodes. It is suspected that a slow evaporation rate may cause some oxidation of the evaporated films due to the prolonged thermal radiation from the tungsten boat, resulting in thin, dull appearance. The thickness of the films are monitored during the evaporation by a Sloan quartz crystal film thickness monitor. The film thickness is made to be at least 5000 A, so that later on when measurements are made at temperatures below the superconductor transition temperature of the metal electrode, the superconductors will behave as a bulk superconductor, without the uncertainties introduced by thin film superconductor behavior [40-42]. After the metal contact electrode evaporation, the wax mask is cut to remove all electrical shorts produced by the evaporated film. Then the electrode is connected to the TO-5 header emitter binding post with a 2-mil gold wire. The electrode contact solder used is a liquid Hg-In-Tl alloy [43]. A liquid solder contact is used so that there will be eliminated the possibility of mechanical damages to the Si crystal or to the metal- semiconductor junction using the commonly available wire bounders. The liquid alloy is found to be the best among the several contact materials tried: the conduc- tive paint has a tendency to open up when subject to thermal stress, and the different low melting indium alloy solders do not wet the metal surface as readily and are difficult to use with manual tools. However, because of the excellent wetting properties of the Hg- In-Tl alloy solder, care has been taken to avoid possi- bility of either destroying the evaporated film or form- ing complicated superconducting alloy films [44,45] right at the metal-semiconductor junction by applying the alloy solder contact to the evaporated film on top of the masking wax immediately adjacent to the junction area (fig. 6). After completion of the liquid solder contact to the metal electrode, the diode fabrication is completed and the device is ready for study. Figure 7 is a schematic drawing of the metal-semiconductor barrier tunnel diode thus made. Several of the P doped n-type Si dice are prepared slightly differently than reported above in that after chemical etch of the wafer, the wafer is processed by M. Weiss of TRW Semiconductors, into Si dice with steam grown Si dioxide mask ready for mounting, and metal electrode evaporation, after a ten second etch in a 10 percent HF solution for removal of surface oxide introduced in the junction area by transit and die mounting and with further black wax masking needed only for ease in the removal of electrical shorts caused by the evaporated films. Diodes made from the masked dice so obtained exhibit no difference in electronic pro- perties when compared with the others, and are not treated separately. 3.2. Electronic Medsurements For each junction, two types of curves are obtained: the current versus voltage bias curve (I-V) and the in- cremental conductance versus voltage bias curve (dI/dV-V). Two separate ranges of voltage bias are used for both types of curves: In the study of superconduc- tivity structures about zero bias, a small voltage bias range centered at zero volts and extended in both for- ward and reverse bias directions over only a few mil- livolts is used. For the study of Fermi degeneracy in heavily doped Si, the voltage bias range used extends from approximately 80 mV negative to over 120 mV positive. To study the effects of superconductivity on the barrier junction, the I-V and di/dV-V curves are evaluated over a temperature range extending from 4.2 to 1.1 K. In the case of Fermi degeneracy studies, the experiments are performed in liquid helium under at- mospheric pressure (4.2 K). 701 Silicon Wafer 4-Point |Resistivity NO Black Maski Probe Measured Determined' Wax as king Modified Chemical surface Chemical SiO2 CP-4 Etch Strain HF Etch Rémoval Removed - g Pb, Sn, In sºng Dicing iºn Electro de y Evaporation Wax Mask ! ! Evaporated Metal Electrode l Si Elimina- Silicon rºtta, tion of Dices Removal Electrical Shorts N2 Atm. & Contacting = 4000 C c: *:::::::: Evaporated AU : Si Electrode Metal- Semicon- ductor Diode FIGURE 6. Fabrication procedure of the metal-semiconductor FIGURE 7. Schematic of the metal-semiconductor tunnel diode. tunnel diode. Voltage Source 32 : 1200 3 AA A www | la R1 = Ro?--w P 500 Hertz --- Q s scillator -E- R., 3 Test [Y] 2. R4 O | * tº º 2 S Diode - T wº A/\/\, R5 tº- L ) ©— t + Moseley *HL l 2 Keiºsy > x 7030A y | -- 3 3 R5 XY gº l Preamp 2 Ho Recorder o-Ho- * = - 2 wº | Input § l: 2 - - G. g EMC-RBJ § 3 Lock-In Amp º' Il ºr FIGURE 8. Measurement circuits; position 1, current versus voltage, position 2, incremental conductance versus voltage. A schematic of the measuring electronics is shown in figure 8. The applied voltage bias is directly measured across the tunnel diode, the tunnel current is measured over a current sensing resistor. Using an X-Y recorder and a variable voltage supply which can be swept auto- matically with a timing motor drive, the measurements of the I-V characteristics of the tunnel diode are easily made. To avoid loading of the circuit, the input im- pedance of both the X and Y channels must be high with respect to the resistance of the diode and the cur- rent sensing resistor. The X-Y recorder used has input impedance = 1 megohm, a value much higher than the usual kilo-ohm range of the diode, and the 1 to 100 ohm resistance value of the current sensing resistor. To measure the incremental conductance of the tun- nel diode, we bias the tunnel diode at the de level about which the incremental value is to be measured, and ob- tain the incremental conductance by measuring the ac conductance of the diode under a small ac voltage signal. A 500 Hertz signal is introduced into the diode 702 circuit via an isolating, impedance matched trans- former. The magnitude of the ac signal is adjustable via the voltage divider R5RA. For good definition of the zero bias tunneling structure, as well as the determination of the incremental conductance minimum, the ac signal is set to be š 50 p.V. The incremental conductance value is picked up by the current sensing resistor because the ac signal voltage is coupled into the diode circuit at a constant magnitude. Gac – bac (18) 1) (to K's late T- Rs (19) where Gac = the diode incremental conductance Vs = signal voltage at Rs The signal voltage is first amplified by the Keithley AC amplifier (model 104) which is an ultra-low noise preamplifier with high gain (100 to 1000) and high input impedance (100 megohm). In addition, it provides high and low frequency cut-off filter to reduce the effective band width and improve signal to noise ratio, a feature most suited for our application because our ac signal is narrow band (500 Hertz sinusoidal). The signal from the preamplifier is then further amplified and converted to a do output which is proportional to the diode incre- mental conductance, using the EMC model RJB, Lock- In Amplifier employing the standard phase lock-in technique of derivative measurements [46,47]. The do output from the Lock-In Amplifier can be applied directly to the Y input of the X-Y recorder for the mechanized plotting of the incremental conductance versus voltage bias curve of the tunnel diode. It is important to note that the variable do voltage bias source is in series with the tunnel diode in both the I-V and di/dV-V measurements. In order that the ob- served bias dependent structures in the I-V and di/dV- V measurements are solely the results of the bias de- pendent properties of the tunnel diode, the variable voltage source must be such that its internal impedance is independent of output voltage. In addition, its inter- nal impedance must be low compared with the tunnel diode, otherwise the measured I-V characteristics reflect the internal impedance of the voltage source, with the tunnel diode providing a second order pertur- bation effect on the I-V curve. A detailed analysis of the measurement circuits shows that under no case will the voltage source constitute more than 0.5 percent error to the measured tunneling characteristics of the diode under study [48]. Finally, it should be noted that the ac signal pickup at the current sensing resistor Rs is of the order of one microvolt. Because of the low signal level, and the long lead length needed for diode measurements in the cryo- stat, ground shields and low noise shielded cables must be incorporated in the measurement set-up. Some basic considerations such as avoidance of ground loops, use of electrostatic shielding guards in the diode circuit, minimization of lead length, etc., are included in the packaging design of the measurement electronics and the sample holders for use in the low temperature cryo- stat and in the helium dewer for measurements at 4.2 K. 4. Experimental Data and Andlyses There are two independent variables that can be ex- perimentally varied in the study and evaluation of heavily doped n-type Si. They are: (1) impurity dopant concentration and (2) the type of impurity dopant used, be it Sh, P, or As. We expect that the density of states function of the semiconductor will be modified to some extent according to the impurity doping concentration and to the type of dopant used, and this modification of the density of states function will show up in the mea- sured tunneling current and more so in the measured incremental conductance of the depletion layer tunnel diode. Figure 9 presents a current-voltage and incremental conductance-voltage x-y recorder plot of a typical metal-semiconductor tunnel diode. There is a zero bias conductance structure similar to many reported in literature. We found that by assuming the metal elec- trode is superconducting and using the BCS theory in describing the superconductivity, the zero bias con- ductance structure can be matched very accurately over the temperature range by the generalized tunnel- ing eq (2) [48]. In addition to the zero bias features due to supercon- ductor tunneling, we observe in the figure the presence of an incremental conductance minimum which occurs at a voltage V min in the forward bias direction. Accord- ing to the analytic study given earlier, the incremental conductance minimum is a result of the sharp change in the density of states function at the bottom of the conduction band, and qVnin is equivalent the Fermi energy of the semiconductor. With increasing carrier concentration, the density of states function is occupied at higher and higher energy levels, hence a corresponding increase in the Fermi 703 6 O —H 3. O 40k- (ſ) I —V 3 – 2.5 # 4NP22 = 2 OH- P doped Si º Temperature at 4.2 °K º Ç. •r- T 2.0 a 5 : 4–) +) OH- O 5 # g - 1.5 5 O O r— º –2 O H. 5 GI – - L. O = # - v º O C H V min –60 | | | – 6 O –4 O —2 O O 2O 40 6 O 8 O l OO | 2 O Voltage in my FIGURE 9. Transport properties of a superconducting Pb-Si tunnel junction. energy of the semiconductor. Figure 10 compares the conductance curves of two diodes showing the shift of the conductance minimum with impurity dopant con- centration. In addition to the dependence on concentra- tion, the Fermi level is found to be strongly dependent on the type of impurity dopant in the semiconductor. Figure 11 is a log-log plot of EF versus ND showing the dependence of the Fermi level on impurity dopant con- centration and on the type of impurity dopant. The boundary of the rectangle about each point is the esti- mate of possible error due to accuracy of measure- mentS. A brief study of the figure will reveal several impor- tant features: (1) the Fermi level of the heavily doped Si at an impurity concentration is strongly dependent on the type of impurity dopant in the semiconductor; (2) the variation of the Fermi level as a function of impurity dopant concentration approximates the same power de- pendence of EF and ND irrespective of the dopant type; (3) the separations in the EP versus ND curves for Sb, P, and As doped Si are more than that can be ac- counted for by the differences between their ionization energies in Si which are 39, 45, and 49 meV, respective- ly [49]. The separation increases with dopant concen- tration. For example, in the case of P doped Si versus As doped Si, it varies from 3 meV at the dopant concen- tration of 2 X 1018 cm-3 to 10 meV at 4 X 10 meV, com- pared to a difference of 6 meV between the ionization energies of P and As in Si. It is noted that the Er versus No curves for the three different types of dopants tend to approach the slope as given by the parabolic dependence of EF and ND. This observation is further explored in figure 12, where as- tride each of the EP versus ND experimental curve is an assumed curve which corresponds to the parabolic de- pendence of EF and No. The offset of the three curves can be interpreted to be the result of the dopant type dependent density of states effective mass. A study of the experimental curve and its adjacent assumed parabolic curve reveals that they are actually separated approximately by a small constant 6 throughout the en- tire range. That is (Er – 6) or NH3 (20) 704 l. 8 —H 2.5 l. 6 T- (ſ) (ſ) 3 3 à l'. 4 Temperature at 4.2°K .E r – P doped Si – 2 ... O I r— º r— 'E 'E l. 2 a *- H Q) Q) 3 1. o H — 1.5 3 Qū Qū 4–9 4–) O O # 0.8 H £ C C O O C — 1. O 9 ... o. 6 H 'º 4–) 4–) C G Q) (l) § 0.4 – # ; — O. 5 : C C H H O. 2 H O | | | | | | | O G) –2 O —l O O l O 3 O 40 5 O 6 O – 4 O –2 O O 2 O 6 O 3O l OO l2O Voltage in my FIGURE 10. The effect of dopant concentration in Si on the incremental conductance of the Pb-Si tunnel junction. where TABLE 1. Function Dependence of 6 on N = 1 meV for P-doped Si p N., cm -3 º 8(P), meV º 1.6 meV for Sb-doped Si II, 6. — 1.5 meV for As-doped Si 5 × 10 18 1.6 1.2 — 1.6 3 × 10 18 1.6 I. 1 — 1.3 - tº - 2 × 10 18 1.6 1.0 — 1. Therefore, according to the band-filling model of | X 10 18 1.4 0.7 1.1 electron occupancy where the dependence of the Fermi * - Ie e s e º 'º e e level on the carrier concentration is taken to reflect directly the density of states function of the semicon- ductor, we can conclude from the empirical curves of the Fermi dependence on impurity dopant concentra- tion that allowing for some deviations at the lower range of the density of states function, the parabolic model of the density of states function accurately describes the heavily doped Si, provided we can assume a density of states effective mass which is dependent on the impuri- ty dopant type. At the lower energy range where E is of the same order of magnitude as 6, the parabolic model of the density of states function breaks down in that the 2/3 power dependence of Er and ND is no longer correct even in the sense of eq (20) because 6 is only approxi- mately a constant, as is shown in table 1. Following the above discussion, according to eq (20), we can approximate the density of states function of the heavily doped semiconductor, in the region where E > ô as follows: /2\3 / no k\ 3/2 (, (E)=87, V2 (*) (#) (E-8) ſº (21) 7710 in which m” is the dopant type dependent density of states effective mass. It is 0.178 mo, 0.330 mo, and 0.418 mo for Sb, P, and As doped Si, respectively, ac- cording to our experimental data as given in figure 12. Reference can be made to figure 13 which summarizes the resistivity vs carrier density measurements of several investigators. The dopant-type dependence of the resistivity is studied as an impurity effect upon electron mobility in the heavily doped Si: O = p = nep, (22) 417–156 O – 71 – 46 705 l OO - 1 O H. – A Sb doped Si |- e P doped Si º- I. As doped Si l 1–1–1–1–1–1 || || | | | | | | | | | | | |-|--|--|-- 1017 1018 101.9 1020 Dopant Carrier Density in cmTº FIGURE ll. At a given carrier concentration n, the resistivity in- creases in the order p(Sb) < p(P) < p(As), so that mo- bility decreases in the order p(Sb) > p(P) > p.(As). The mobility differences are not due to long range impurity scattering which considers the effect of the Coulombic forces acting at large distances proportional to the elec- tronic charges involved and give rise to a mobility which depends only on the impurity dopant density. In- stead, the mobility increases in the same order as the ionization energy decreases, suggesting that there ex- ists a scattering mechanism which interacts at close range between the carrier electron and the impurity atom. Description of the scattering mechanism must take into account the nature of the potential well of the impurity atom in the semiconductor. The same con- siderations given for the differences in mobility of the differently doped Si can be given for the differences in the density of states functions as manifested in the in- equality m”(Sb) < m” (P) < m”(As). And, if we are to use the simple hydrogenic model to describe the ionization of the impurity atom in a Measured Vmin Dependence on dopant type and on dopant carrier concentration. semiconductor, we obtain for Si the following: E;= H = 99.3 (...) meV (23) 1710 From which we can obtain the impurity dopant depen- dent effective mass which takes into account the effect of the close range interaction between the conduction electron and the impurity atom affecting its ionization energy. They are m”(Sb) = 0.393 mo, m*(P) = 0.444 mo and m”(As) = 0.494 mo. The order and magnitude of the differences are in general agreement with that sug- gested for eq (21). A detailed discussion on the significance of the do- pant dependent density of states effective masses, and their correlation with other known experimental data available in literature will be presented elsewhere by the same authors. Thus far we have directed our attention to the posi- tion of the incremental conductance minimum and its functional dependence on impurity dopant and dopant concentration. Additional qualitative and quantitative 706 l O à – E C •r- ſº [...] r– 10 l- () |- :- (l) He’ H •r- – F. mºs $– (l) [L. º IIl " IIl Experimental Data Curve Calculated FF CK N2/3 ºr emº º º mº l | | | | | | | | | | | | | | | | | | | | | | | | | lol 7 1018 lol? lo20 Dopant carrier Density in cm * FIGURE 12. Dopant dependency of fermi level location in heavily doped silicon. information on the properties of the heavily doped Si can be obtained from further analyses of the incremen- tal conductance versus voltage bias curve based upon the analytic treatment given earlier. In figure 14, we compare the experimental incremen- tal conductance curves of P-doped Si with the normal- ized calculated incremental conductance curves based on Pwkby and PCMDT and the corresponding tunneling equations, assuming for the heavily doped semiconduc- tor, the parabolic density of states function. Near the band edge, i.e., below approximately 12 meV, the experimental conductance curves drop off at a much lower rate than that predicted by the calcula- tion using Pwkby or PCMDT. Furthermore, the direction from the parabolic density of state function is observed to be less for the very heavily doped material. The den- sity of states tail has been interpreted as the result of strong electron interaction with clustered impurity atoms, assuming no correlation in the distribution of the dopant impurity atoms. At high impurity concentra- tions, the distribution of the impurity atom is strongly correlated, resulting in minimizing the density of states tail. And, hence, less deviation from the parabolic den- sity of states function at higher dopant concentration. Above the band edge, the experimental curve follows closely the normalized calculated curve, demonstrating the applicability of the parabolic form of the density of states function for the heavily doped Si, in agreement with the physical model for heavily doped semiconduc- tor [48,50,51]. At high energies the experimental curve increases more rapidly than the calculated curve, indicating the density of states function increases at a more rapid rate than parabolic. A review of the conduction band E(k) function of Si shows that at the high energy range, using the band filling model, we may be into the double degeneracy at the X point (on the 100 axis of the diamond structure). Reference can be made to figure 12 and the cor- responding discussion on it, where we observe that the As doped Si is found to depart more from the parabolic dependence of EF and Np than P doped Si. From the 707 —l l O º-s, – — 5 lotº| Furukawa ‘s l M-me É t O }* SQ C * •r- |- > ºms +) -r- S. º – (ſ) º – 3 () l O |-mm. ſr. – C T Logan, Gilbert and Trumbore – Hº- 10-4 | | | | | | | | | | | | | | | || | | | | | || || | | | | | | | || | | | | | | 1016 101.7 1018 1019 1020 1021 Carrier Density in cm * FIGURE 13. Summary of resistivity versus carrier density data curves for heavily doped n-silicon. study of the incremental conductance curves, a similar difference between the P doped Si and the As doped Si is found. Figure 15 compares the experimental incre- mental conductance curves of the As doped Si with the corresponding calculated curve. In comparison with figure 14, it is seen that in the low energy range, band tailing is more severe in magnitude and extent in the case of As doped Si than P doped Si. And the energy range in which the density of states functions follows the parabolic form is relatively narrow in As doped Si. On the other hand, it appears that the X point double degeneracy for As doped Si occurs at a much higher energy level for the impurity dopant concentration. The incremental conductance of the metal-semicon- ductor junctions is directly proportional to the products of the tunneling probability Pwkeſ(E) and the density of states function gs(E) according to eq (17). In the log-log plot of the calculated incremental conductance curve, as in figures 14 and 15, we can determine the power de- pendence of Pwkbj(E) on E from the slope of the incre- mental conductance curve, given the parabolic density of states function assumed in the calculation of the in- cremental conductance. Then, with the power depen- dence of Pwkh (E) on E known, we can determine the power dependence of g(E) on E from the slopes of the experimental incremental conductance log-log plot for all E. Based on the above procedure, we determine from figure 14, the power dependency of g(E) on E for P doped Si at E = 5 meV as follows: N = 8.9 × 1018 cm-3, g,(E) or E=0.01 N = 4.7 × 1019 cm−3. gs (E) oc E0.48 N = 5.6 × 1019 cm−3, g,(E) or E0:38 We note from figure 14 that the slope of the log-log plot of the conductance curve is a monotonically decreasing function of E near the band edge. We ob- serve also that Vmin exists for N = 8.9 × 1018cm−3 implying a sharp drop of the density of states tail at eV min, even as the corresponding gaſe) for Si of this do- pant concentration at 5 meV above the band edge has a negative power dependence on E, implying increasing magnitude of the density of states function at that ener- gy level. Therefore, we conclude that near the band edge, for the less heavily doped Si, the impurity band produces a small hump on the density of states function of the semiconductor. At higher dopant concentrations, 708 l OO Z 9 OH- |- 2l O lºm 4NP22 3OH- 2NP2ll — 3NP / lºmº Q — 70| N = 8.9 x 104°om 3 N = 4.7 x 101°om 3 n = 5.6 x 10**cm−3 6 O |- 5 O – 4 O lºmas 3 O }* 2 O — S. £ 5 lo > (l) 9 lººsas / Un 8 H. / º / + 7 ſº / | 9 6 – / 5 *ms / / 4 — ſ / / 3 – / / | Experimental 2 / — — Calculated Mº- | G (E) or El/2 p WKBJ | º- «-» - 4- - Calculated G (E) or *CMDT | | | - | | | | | | l 2 3 4 5 6 7 S l 2 3 4 5 6 7 8 l 2 3 4 5 6 7 8 Incremental Conductance in Arbitrary Units FIGURE 14. The effect of the density of states band tail in P doped Si on the incremental conductance of the metal-semiconductor tunnel junction. by the evaluation of figure 14 and eq (21), based on similar analysis, it can be concluded that the impurity band near band edge becomes fully merged into the density of states function by its complete disap- pearance. Figure 16 summarizes the estimated density of states functions for different levels of dopant concen- trations of P doped Si. Included in the figure is also the summary curves of the estimated density of states func- tion of As doped Si based on similar analysis of the in- crement conductance curves of the corresponding metal-semiconductor junctions. It is observed from the figure that for the As doped Si, the density of states function as well as the band tail near the bottom of the conduction band is distorted from the parabolic dependency on energy much more severely when compared with P doped Si, in agreement with earlier observations in the study of figures 12-15. The hump in the Si density of states function due to the impurity band obtained experimentally by us is in general agreement with the work by G. D. Mahan and J. W. Conley [4] on their study of the heavily doped p- type GaAs density of states function, except that the Si impurity hump is confined to a narrower energy range near the bottom of the conduction band than that ob- tained for p-GaAs. 5. Conclusion In this study, we have developed the necessary mathematical analyses of the metal-semiconductor junction for the evaluation of the experimental data. Both the WKBJ approximation and the exact transmis- sion coefficient calculations of the tunnel diode equa- tion were used. For the case of silicon diode, both methods yielded comparable solutions which predict the characteristic features of the metal-semiconductor tunneling current experimental curves. Through the study of both the incremental resistance and conductance curves of the metal-semiconductor 709 l OO 9 O 2NAll 6 19 T 4NAll 2 / ; N = l. O X l O cmT3 T N = 5. 2 x lolº'em-3 60 / º- / 50 / º- 40 H. / wº- / 30 — / |- / 20 k- |- / S. E C / • H (l) g ſ +) l 0 ||— Pºº- # E / T / :- 8 – / º / 7 – / º- / / 5 – }* ſ / 4 I- / — / / // / / 3 H / - / / / / / Experimental 2 # / / — — Calculated l/2 G (E) or E *WKBJ - - - - - - Calculated | G (E) or *CMDT | | | | I I | | | | | | l 2 3 4 5 6 7 8 l 2 3 4 5 6 7 8 Incremental Conductance in Abitrary Units FIGURE 15. The effect of the density of states band tail in As doped Si on the incremental conductance of the metal-semiconductor tunnel junction. diodes, it is concluded that the incremental con- ductance curve describes to a greater degree of clarity some of the important properties of the bulk material under study than that possible with the incremental conductance curve. Based on this evaluation, we mea- sure directly the incremental conductance curves of the diodes under study. In addition, the measurement circuits have been analyzed to assure the accuracy of the measurements made, as well as to determine the magnitude of the inclusion of extraneous impedance ef- fects from the measurement circuits in the diode con- ductance curves. Our tunneling data are interpreted by two different procedures: (1) evaluation of the Vmin versus N curves, the curves being the compilation of the most prominent features of all the diode conductance curves; (2) the evaluations of the individual diode conductance curve. Using the Vmin curves, we discover the dopant type de- pendence of the Fermi level of the semiconductor, lead- ing us to postulate a dopant type dependent density of states effective mass for Si. This density of states effec- tive mass also can be considered to be a measure of the effect of different dopants on the density of states func- tion of the heavily doped Si. Using the individual diode conductance curves, we can determine the overall characteristic features of the density of states function of the heavily doped Si at particular dopant concentra- tions and with specific dopants. And with either procedure, we arrive at the same consistent interpreta- tion that the density of states of the heavily doped semiconductor is generally parabolic above the band edge, but near the band edge, band tailing can be severe, depending on the dopant type and on the do- pant concentration. - Quantitative data on the density of states function of the heavily doped Si are obtained through the com- parison between the calculated curves, using known functions for the density of the functions of the 710 60 H. 50 H 4 O H. à à E. £ : º ă. à º 3 O H. X- C g [r] [r] 2O H. P doped Si l O H. N = 9 x 10*om * l 9 — 3 — — — N = 4 x l O’ cm º -> *-* -º º ºx! N = 6 x 10**om * O *— N (E) FIGURE 16. Density of states of the heavily doped n-silicon. semiconductor, and the experimentally derived curves. The data so obtained include the dopant dependent density of states effective masses and the power depen- dence of the density of states functions on energy in the band tail region. Density of states functions of the heavily doped Si, at different dopant levels and dif- ferent dopant types, can also be derived from the ex- perimental conductance curves. It is hoped that our ex- perimental data on heavily doped Si will be of interest to those active in the theoretical modeling of the heavily doped semiconductors. 6. References [1] Gschwend, W. F., Kleinknecht, H. P., Neft, W., and Steiler, K., - Z. für Naturforschung 18a, 1366 (1963). [2] Conley, J. W., Duke, C. B., Mahan, G. D., and Tiemann, J. J., Phys. Rev. 150,466 (1966). - [3] Conley, J. W., and Tiemann, J. J., J. Appl. Phys. 38, 2880 (1967). [4] Mahan, G. D., and Conley, J. W., Appl. Phys. Letters 11, 29 (1967). [5] Conley, J. W., and Mahan, G. D., Phys. Rev. 161,681 (1967). [6] Chang, L. L., Esaki, L., and Jona, F., Appl. Phys. Letters 9, 21 (1960). - l / / 60 — f / / / / / / / / 40 k- 30 — 20 H As doped Si 10 N-ºxiºn. * * * * * N = 5 x 10" cm * O 2-N (E) [21] [7] Burnstein, E., Phys. Rev. 93, 632 (1954). [8] Hill, D. E., Phys. Rev. 133, A866 (1964). [9] Lucovsky, G., Solid State Commun. 3, 105 (1965). [10] Pankove, J. I., Phys. Rev. 140, A2059 (1965). [11] Curie, D., Luminescence in Crystals (Wiley, New York, 1963). [12] Lucovsky, G., Varga, A. J., and Schwarz, R. F., Solid State Commun. 3, 9 (1965). [13] Morgan, T. N., Phys. Rev. 139, A343 (1965). [14] Haas, C., Phys. Rev. 125, 1965 (1962). [15] Fowler, A. B., Howard, W. E., and Brock, G. E., Phys. Rev. 128, 1664 (1962). [16] Pankove, J. I., and Aigrain, P., Phys. Rev. 126,956 (1962). [17] Mead, C. A., Spitzer, W. G., Phys. Rev. 134, A713 (1964). [18] Esaki, L., Phys. Rev. 190, 603 (1958). [19] Giaever, I., Phys. Rev. Letters 5, 464 (1960). [20] Giaever, I., and Megerle, K., Phys. Rev. 122, 1101 (1961). Shapiro, S., Smith, P. H., Nicol, J., Miles, J. L., and Strong, P. F., IBM Journal 6, 34 (1962). Bardeen, J., Phys. Rev. 71,717 (1947). Heine, V., Phys. Rev. 138, A1689 (1965). Geppert, D. V., Cowley, A. M., and Dore, B. V., J. Appl. Phys. 37,2458 (1966). Crowell, C. R., and Sze, S. M., Solid State Electronics 9, 1035 (1966). Mathews, J., and Walker, R. L., Mathematical Methods of Physics (W. A. Benjamin Inc., New York, 1964) pp. 26-37. Morse, P. M., and Feshbach, H., Methods of Theoretical Physics (McGraw-Hill, New York, 1953) pp. 1092-1106. [22] [23] [24] [25] [26] [27] 711 [28] Kohn, W., Shallow Impurity States in Silicon and Germanium, Solid State Physics 5, F. Seitz and D. Turnbull, Editors (Academic Press Inc., New York, 1957). [29] Harrison, W. A., Phys. Rev. 123, 85 (1961). [30] Fredkin, D. R., and Wanniev, G. H., Phys. Rev. 128, 2054 (1962). [31] Stratton, R., and Padowani, F. A., Solid State Electronics 10, 813 (1967). [32] Runyan, W. R., Silicon Semiconductor Technology (McGraw- Hill, New York, 1965). [33] Buehler, M. G., and Pearson, G. L., Solid State Electronics 9, 395 (1962). [34] Irvin, J. C., The Bell System Technical Journal 41, 387 (1962). [35] Logan, R. A., Gilbert, T. F., and Trumbore, F. A., J. Appl. Phys. 32, 131 (1961). [36] Furukawa, Y., J. Phys. Society of Japan 16, 577 (1961). [37] Holmer, P. J., IEEE Proceedings 106, B861 (1959). [38] Gatos, H. C., and Lavine, M. C., Chemical Behavior of Semiconductors: Etching Characteristics, Semiconductors, A. F. Gibson and R. E. Burgess, Editors (Temple Press Books Ltd., London, 1965), pp. 1-45. [39] Chynoweth, A. G., Feldmann, W. L., and Logan, R. A., Phys. Rev. 121,684 (1961). Progress in [40] Vogel, H. E., and Garland, M. M., J. Appl. Phys. 38, 5116 (1967). [41] Vogel, H. E., Ph. D. Thesis, University of North Carolina (1962). [42] Blumberg, R. H., and Seraphim, D. P., J. Appl. Phys. 33, 163 (1962). [43] King, W.J., Rev. Sci. Instr. 32, 1407 (1961). [44] Claeson, T., Phys. Rev. 147,340 (1966). [45] Nemback, E., Phys. Rev. 172,425 (1968). [46] Rowell, J. M., Anderson, R. W., and Thomas, D. E., Phys. Rev. Letters 10,334 (1963). [47] Schrieffer, J. R., Scalapino, D. J., and Wilkins, J. W., Phys. Rev. Letters 10,336 (1963). [48] Hsia, Y., Ph. D. Thesis, University of California, Los Angeles (1969). [49] Tao,T. F., and Hsia, Y., Appl. Phys. Letters 13, 291 (1968). [50] Bonch-Bruyevich, W. L., The Electronic Theory of Heavily Doped Semiconductors (American Elsevier, New York, 1966). [51] Bonch-Bruyevich, W. L., Effect of Heavily Doping on the Semiconductor Band Structure, Semiconductors and Semimetals, Vol. 1, R. K. Willardson and A. C. Beer, Editors (Academic Press, New York, 1966), pp. 101-142. 712 Discussion on "Metal-Semiconductor Barrier Junction Tunneling Study of the Heavily Doped N-Type Silicon Density of States Function” by Y. Hsia and T. F. Tao (University of California, Los Angeles) G. D. Mahan (Univ. of Oregon): In silicon metal- semiconductor junctions it is very easy to get a signifi- cant oxide layer unless proper precautions are taken. This would seem to be the case here. Thus you should be cautious about over-interpreting your data. Y. Hsia (Litton Systems, Inc.): There is probably a residual layer of oxide about 30 A to 50 Å at the junction. The effect of the oxide on the tunnel current should be dopant independent. The main problem is its effect on the assumed tunnel barrier and consequently the theoretical calculation of the tunnel current. (Fig. 4 of the text compares the effect of the tunnel barrier varia- tion on the normalized incremental conductance fea- tures which are utilized in comparison with our experi- mental data.) The determination of the Fermi level by interpreting Vmin measurements on oxide junctions has been shown experimentally by the work of Esaki et al. [1] on many materials. M. Cardona (Brown Univ.); Have you compared your results with those obtained from infrared reflectivity measurements? Y. Hsia (Litton Systems, Inc.): The dopant dependence of the effective mass in heavily doped semiconductors has actually been previously obtained by Spitzer et al. [2] in their reflectivity measurements on As, P, and Sb doped Ge to determine carrier density dependent den- sity of states effective mass (though interpretation of their data presented in fig. 8 of the cited reference). The dependence is similar but differs in having smaller differences of values. We are planning to do free carrier reflectivity measurements in the infrared region on the variously doped silicon to obtain density of states effec- tive mass data for comparison with this tunneling work. [1] Esaki, L., and Stiles, P. J., Phys. Rev. Letters 14,902 (1965); Es- aki, L., and Stiles, P. J., Phys. Rev. Letters 16, 574 (1966); Chang, L. L., Esaki, L., and Jona, F., Appl. Phys. Letters 9, 21 (1966); Chang, L. L., Stiles, P. J., and Esaki, L., J. Appl. Phys. 38, 4440 (1967). [2] Spitzer, W. G., Trumbore, F. A., and Logan, R. A., J. Appl. Phys. 32, 1822 (1961). 713 TRANSPORT PROPERTIES: APPLICATIONS CHAIRMEN: A. l. Schindler A. Feldman RAPPORTEUR. R. J. Higgins The Effect of Hydrostatic Pressure on the Galvano- Magnetic Properties of Graphite I. L. Spain Institute for Molecular Physics, University of Maryland, College Park, Maryland 20742 Measurements of the Hall Effect and magneto-resistance in crystals of graphite with current flow in the basal planes at high pressures are described. Galvanomagnetic measurements enable the mag- neto conductivity tensor components Orr and Orry to be obtained, and from them the electron and hole densities and mobilities. The results are compared with the band model for the semi-metal graphite proposed by Slonczewski and Weiss. Of particular interest at the present time is the information that these results give about the assignment of carriers at the symmetry point K in the Brillouin zone of gra- phite and the properties of mobile minority carriers. Key words: Electronic density of states; galvano-magnetic properties; graphite; Hall Effect; mag- neto-resistance; pressure effects. 1. Introduction The determination of the effect of pressure on the electronic properties of graphite is of interest for several reasons. The highest occupied valence band overlaps the lowest unoccupied conduction band by about 32 meV [1], so that graphite behaves like a semi- metal when the current flows along the layer planes. As a result, changes in the lattice spacing induced by hydrostatic pressure are expected to produce quite large effects in the electronic properties. The large elastic anisotropy enables the change in lattice parame- ter in the plane (ao) to be neglected compared to the change in c-axis spacing (co) ((Ac/co) - 29(Aalao)) [2] so that changes in electronic properties are expected to originate from corresponding changes in the overlap in- tegrals between planes. The generalised band model proposed by Slonc- zewski and Weiss [3], with adjustable parameters re- lated to overlap integrals, has been successful in In- terpreting many of the experiments on graphite, and some experiments have attempted to directly relate pressure effects to changes in these parameters [4-8]. However, recent studies of Schroeder, Dresselhaus, and Javan [9], have proposed several fundamental changes in the band structure that have important con- sequences on the interpretation of galvanomagnetic data. A program to study the effect of pressure on the gal- vanomagnetic properties of graphite, with current flow both along and perpendicular to the planes, is being carried out. So-called “c-axis effects” are very interest- ing because of the possibility of observing the role of lo- calised “electron-hole” or “exciton” states on the pro- perties of solids [10] but this problem is not discussed here. Measurements of pressure on the Hall Effect and magneto-resistance of highly oriented graphite are re- ported at two temperatures and the information that they give about the band parameters and the density of states discussed. 2. The Band Model of Graphite The Slonczewski-Weiss band model [3] concerns itself only with those states of interest to free-electron phenomena near the vertical Brillouin zone edges (HKH'). The variation of energy for electron states along (HKH') is shown in figure la, being given by: e1 = A + 2y cos (b e2 = A – 2 yi cos (b e3 = 2)2 cos” (b where *- k-Co b== and y1)2.4 are related to out-of-plane overlap integrals. The theory obtains the energy of those states near the zone boundary by a perturbation calculation. In the event that the effect of (other overlap) parameters 717 G l G TK | | HOLES < | A -14' 2^ K-1 O |H (E Cº- F Ty- K 2x2 | | N ELECTRONs | f*— * K | A-2x - m /Co O m/Co Cº- kz |KF x 0°cm' |.5 H. I.O. H. 0.5 H HOLES ELECTRONSW / HOLES O º — m/Co O m /Co Kz lvexio'cm/sec |OH- 8 6 jºs 4. ºns *F Holes HOLES O ELECTRONS Jº — — m/Co O 7/Co Kz FIGURE 1(a). along the vertical zone edge HKH' of the Brillouin zone of graphite. The assignment of electrons at point K is used in this diagram in agreement with the work of Dresselhaus et al. The variation of energy with wave-ratio component perpendicular to HKH' is indicated. FIGURE 1(b). Sketch of the variation of electron and hole wave- vector KF along the Brillouin zone edge of graphite. FIGURE 1(c). The basal velocity (vr) of carriers with the Fermi energy. Yºyºy, appearing in the theory may be neglected (ys accounts for trigonal warping effects), the dispersion relation may be written: e1,2 + e3 e1 , 2 - €3 3yśašk” 1/2 2 +|| 2 ) + 4. €12 = €1 Or €2 € where a0 is the lattice constant in the plane = 2.46 Å yo is an in-plane overlap integral k= (k} + k})* For the “four-parameter band model,” the basal com- ponent of the effective mass tensor along the zone edge is given by mºr” (k=0)=4|3h”(y – yº)/3a.0°yo” -- 0.061 mo cos (b. The wave-vector KF is plotted as a function of q in figure 1b, and the basal velocity (VP) for electrons Sketch of the variation of electron energy for states on the Fermi surface in figure le. The following values of the band parameters were used [9,11]: yo = 3 eV y = 0.395 eV y2 = - 0.016 eV A = +0.02 eV er = –0.0208 eV at T= 0 K. Until recently, the value of ye was taken to be posi- tive, indicating that holes occupied that volume of the Brillouin zone around point K. From magneto-reflection data, Schroeder et al. [9] concluded that ye was nega- tive, implying electron occupancy at this point. The as- signment is discussed later in relation to galvano-mag- netic data. Dresselhaus and Dresselhaus [12] also suggested that A was positive, implying that the tips of the hole re- gions of the Fermi surface protrude into the next zone. As a result of spin orbit splitting, a pocket of holes (with a positive value of y2, the pocket would be occupied by electrons) is located at the zone corner, having very small effective mass. - Dresselhaus has also suggested [13] that the value of ya ( ~ 0.3 eV) is much larger than previously thought, so that a perturbation calculation of the electron energy for states near the zone edge is no longer applicable. A large value for ya implies that the electron energy sur- faces are trigonally warped, possibly to the extent of producing pockets of electrons split off from the main body. 3. Galvanomagnetic Effects In the experiments to be described, measurements of the Hall Constant (RH), zero field resistivity po and re- sistivity in a magnetic field p(H) are reported. In general it is convenient to compare theoretical models with the conductivity tensor components. In the case of current flow along the planes, magnetic field in the c- axis direction (H2) and no preferred direction along the planes, the relationships between measured effects and conductivity components are [14]: Rº-4–*– H TH (orº, + O.,) p = OT.cº. (oft, + O., ) with inverse relationships: 718 =–4– OT.cº. | + (Riſor H)* OTry = OTrr (Riſor H) or = p ' = measured conductivity The conductivity components may then be expressed as integrals over the volume of reciprocal space in the following way [15,16]: — e.” v:T (k) ôfo — I — , — c13 4Tº J (1 + ay”Tº (k)) 0e dºk. Orr (H2) F _-e H. vºt” (k) ºf 0 3 or (H,)=-1. ſ mi, (1 + ay” (k)) de where _ eHz 0) = I. In ºr C and T(k) = relation time for scattering electrons. These equations hold to high magnetic fields in the approximtion that the basal plane effective mass tensor mºr” = myſ” does not vary appreciably with energy in a plane in k-space perpendicular to k2. The approxima- tion is not strictly valid for graphite, since the disper- sion relation in such a plane is hyperbolic, not parabol- ic. In addition, the strong variation of carrier properties along the zone edge make the conductivity integrals dif- ficult to calculate. Even if the details of the bands are known, a detailed knowledge of the relaxation time as a function of wave vector is required. Since this is determined by the scattering of electrons by lattice im- perfections it cannot be calculated precisely, and the fine details of the bands become hidden by gross as- sumptions made in the function T(k). A simple solution presents itself, since the condition of charge neutrality in the crystal implies an equal number of free carriers in the valence and conduction bands. Soule [14] then suggested the following multi- band formula: n;= number of carriers per unit volume in band i N= total number of bands ei=+|e for holes, - |e for electrons O'oi = measured conductivity of band i in zero magnetic field Hi– characteristic field of carriers in band i cmº c eTi Li pi = carrier mobility O'Oi = niepti. The underlying assumption in this theory is that the properties of the carriers can be approximated by averaging effects from groups of carriers with different properties as though they all had the same properties. In addition, the relaxation time appearing in the denominator of the integrals is assumed to be indepen- dent of energy. Galvanomagnetic data has been fitted using these formulae by Soule for natural crystals and Spain, Ubbe- lohde and Young [17], for synthetic material. In both cases one majority band of holes and electrons was as- sumed, with a minority carrier to explain marked devia- tions from two-band behaviour in the Hall Constant at low fields. In the approximation of two majority carriers a sim- plification may be made, enabling a mean mobility to be calculated. If a =# ~ | suffix (1) refers to majority electrons 1 º-ſ, ~ | suffix (2) refers to majority holes , , (Ap ſh 2 ~ a2 ( | en pu- ~ c połłº - In summary, this simplified analysis enables the gal- vanomagnetic data to be fitted approximately with averaged parameters for the electron and hole bands, yielding values for the average mobility pi and number of carriers ni. The method enables the properties of minority carriers to be explored from low-field Hall data. From the number of carriers of each type, a direct comparison may be made with the theoretical predic- tl On: n–ſ d i n(e) fo(e) de It must be remembered, however, that the number of carriers calculated from galvanomagnetic data must first be corrected by a numerical factor (~ unity) which comes from the integration procedure. 719 The measurement of the effect of pressure on the gal- vanomagnetic coefficients gives information about the changes occurring in the constant energy surfaces brought about by changes in overlap parameters. This is reflected in the change in the number of carriers with pressure and in the mobility, since the relaxation time depends on the energy wave-vector relationship. In the case of scattering by lattice vibrations, the relaxation time is inversely proportional to the density of states into which scattering takes place [18], while for scat- tering at crystallite boundaries it is inversely propor- tional to the velocity of the carriers (i.e., proportional to e-1/* for both scattering mechanisms for ellipsoidal bands of standard form). 4. Experimental Details of the experimental apparatus used for the present work are to be reported elsewhere. Using heli- um fluid as the pressure-transmitting medium, mea- surements could be made to 10 kbar in magnetic fields up to 15 kG, with the temperature controlled to +0.02 °C. Conventional D.C. techniques were used to measure the galvanomagnetic coefficients. Techniques for fabricating bridge specimens and at- taching leads to them have been described elsewhere [17]. One unexpected problem was encountered that has still not been solved. After compression to above about 3 kbar, irreversible changes occurred in the measured galvanomagnetic coefficients. This probably arose from changes in the current and potential con- tacts to the specimen. With a normal isotropic metal or semiconductor, small changes in the probe configura- tion do not affect the isopotential lines in the body of the specimen, provided that the length to width or thickness ratio is greater than about 5. However, in gra- phite, the high anisotropy ratio (> 10°) ensures that equal sharing of current between the planes is extreme- ly difficult to achieve. Small changes in contacts could easily produce the changes observed. Experiments described in this paper are limited to the pressure range below about 3 kbar for this reason. Attempts are being made to improve contacts using different plating and soldering techniques. Experimental results are presented for two specimens cut from different types of synthetic materi- al. Material for specimen A-2 was obtained from Union Carbide Corporation, hot pressed at 2500 °C and an- nealed above 3000 °C. Stress recrystallised material was used for specimen A-6 obtained from Pennsylvania State University. Values of measured parameters for these specimens are given in table 1, comparison being TABLE 1. Galvanomagnetic coefficients of specimens A-2 and A-6 compared with data from other sources Ratio of Mobility at” Value of Hall Room Sample resistance 77.5 K COnStant at temperature p295| (cm”/volt-s) || maximum value | resistivity 77.5 K. A-2: ........ 1.30 || 4.87 × 104 – 0.005 cm3/C 3.54 × 10-5 A-6f......... 1.59 5.3X 104 +0.060 cm3/C 4.12 × 10-5 EP-14f...... ~ 1.82 | 6.8× 104 **-i- 0.18 cm3/C 4.42 × 10-5 SA-20+...... 1.4] | 4.85 × 104 — 0.005 cm3|C 4.42 X 10-5 SA-26+...... 1.74 5.75 × 104 .07 cm3/C 4.43 × 10-5 fpresent measurements. #Soule, reference [14]. #Spain, Ubbelohde and Young, reference [17]. *All data taken at 3.00 kG for exact comparison. *A plateau rather than a maximum. made with data from Soule’s crystal EP-14. It can be seen that A-6 compares quite closely with his crystal and probably has a grain size greater than 10 microns. A-2 is definitely an inferior specimen with a smaller grain size. For comparison purposes, data is also in- cluded from previous work on similar synthetic materi- al [17], close similarities being observed between sam- ples SA-20 [17] and A-2; and SA-26 [17] and A-6. 5. Experimental Results at Room Temperature Graphs of the variation of the Hall Effect and mag- neto-resistance coefficient with pressure at room tem- perature are shown in figures 2 to 5. It can be seen that the trend in the Hall Constant is to more positive values (dR/dp = 0.0025 cm3/C.kbar for A-2, dR/dp = 0.0032 cm°/C.kbar for A-6). This trend is in agreement with the results of Arkhipov et al. [6], but in numerical dis- agreement. Their zero pressure value at 300 K at un- specified magnetic field being - – 0.03 cm3/C com- pared to ~ 0.05 cm3/C for A-2 and A-6, while their value of dB/dp -- 0.001 cm”/C.kbar. This may be due in part to differences in specimen perfection, Arkhipov et al. [6], working with natural crystals of unspecified per- fection. The largest error in the measurements reported here arises from the c-axis dimension (S2) of the crystals (+ 5%). Assuming a correct value for S2, the Hall Con- stant was then measured with a precision and repro- ducibility of ~ + 0.0002 cm3/C at 15 kG, reducing to approximately + 0.0005 cm”/C at 1 kG. The trend in the magneto-resistance coefficient (Ap/poliz) is to lower values as the pressure increases, indicating that the average mobility (p = (Ap/poliz)''” X 10° cra”/volt-sec) also decreases. However pº varies 720 | | H-04 |85O bors /T. bors /– 6 bors / | bor – O 4 H. Somple A-2 — — O5 - O 5 — — .O6 R 3O || 4 bors R H 27O7 bors H (cm3/c) –7 (cm/c) Sample A-6 () – O6 – Ø —- O7 1996 bors |395 bors 87 | bors | bor – O7 H. – O8 | | | - - O 5 |O 15 Magnetic Field (H) (kilogauss) FIGURE 2: The effect of pressure on the Hall Constant (Rn) plotted as a function of magnetic field for two samples at room temperature. Note that the ordinate corresponding to sample A-6 is on the left-hand side, that cor- responding to A-2 on the right-hand side. linearly with pressure rather than p.". Values of p. were taken at 14 kG. Since the resistance does not vary as the square of the magnetic field, a consistent criteria for calculating p is required. The magnetic field was correspondingly chosen to be that value for which Ap/po ~ 2.00. At this condition, minority carrier effects should be unimportant, and the data susceptible to accurate analysis. |400 I I I I T I I I I ( ( { |3OO -: | bor { - 87 bors { |,395 bars |2OO H. amº A P 47 2 Połº x 10" goussº |||OO H. > Ö D } |,995 bors 1000 - 2,707 bars St 3,OI4 bors } > 900 | l | l l l | l l 5 6 7 8 9 |O || |2 |3 |4 |5 H (kilogauss) FIGURE 4. The value of the magnetoresistance coefficient (Ap/poR") for specimens A-2 and A-6 as a function of pressure at room temperature. The change in basal plane resistance was extremely small [(1/podpoldps 10 °/kbar)] indicating that an in- crease in the number of carriers produced by swelling of the energy surfaces was compensated very closely by the decrease in mobility. This result agrees with that of Spain [19] and Yeoman and Young [20]. The large decrease in (Ap/poh”) with pressure reported here (~ – 3.2%/kbar for A-2, - 4.3%/kbar for A-6) is larger — O7 | | | O | 2 3 Pressure (kilobar) FIGURE 3. The value of the Hall Constant at 10 k(, for specimens A-2 and A-6 plotted as a function of pressure at room temperature. FIGURE 5. I I I |||OO xlo' gauss’ |OOO Sample A-6 at 298 K l | 900; | 2 3 Pressure (kilobar) The variation with pressure of the parameter (Ap/poh”) obtained for sample A-6 at 14 kG and 298 K. 417–156 O – 71 – 47 721 TABLE 2. Experimental coefficients for A-2 and A-6 Temp. K Parameter A-2 A-6 Ap 1/2 298 pl = º #) X 10° (cm”/volt s) (0.90+0.01) × 104 (1.05+0.01) × 104 () [1" p0 (Qcm) (3.45+ 0.15) × 10-5 (4.12+ 0.15) × 10-5 71. tou-º (cm−3) (17.8+0.8) × 1018 (14.5 + 0.7) × 1018 RH at 10k.g. (cm°/C) — 0.058] -- 0.0002 — 0.0656 -- 0.0002 dRH , , dp (cm”/C.k.bar) + 0.0025 + 0.0002 + 0.0032 + 0.0002 d (or (#) / dp (k.bart' )) — 0.032 + 0.003 — 0.043 + 0.003 ()I1" d (log po)/dp (k.bart') – (0.0004+ 0.0003) — (0.0005+0.0003) Ap \!!” - 77.5 p = º #) X 10° (cm”/volt-s) (4.87+0.01) × 104 (5.35+0.01) × 104 () po (Qcm) (2.72 + 0.10) × 10-5 (2.61 + 0.10) × 10-5 n total= |* (4.7+ 0.25) × 1018 (4.45+ 0.25) × 1018 ep. a=!“ 0.99 -- 0.005 0.99 + 0.005* Ill b=#. 1.00 + 0.01 0.91 + 0.0]* *A three band model does not fit the data for SA-6 beyond about 8 k bar. attributed to a minority electron, while at higher fields, the trend is produced through a small excess of majori- than that reported by Arkhipov et al. [16], (~ – 1.8%| kbar) who also reported a linear variation of (Ap/p011%) with pressure. Results are summarised in table 2. Computing the value of the number of carriers in all e bands from the simple formula: O 5 H (kilogauss) |O |5 Sample A-6 | bor gives the value (17.8+ 0.8) × 1018 for A-2, (14.5 + 0.7) × 1018 for A-6. This compares with the value computed for the four-parameter band model, with parameter - e Rh values given earlier. (cm3/c) — .2 - - 18 n = 13.3 × 10 Sample A-2 | bor 6. Experimental Results at 77.5 K 755 bors |O90 bars The variation of Hall Constant with magnetic field for — .3 H - sample A-2 at 77.5 K is shown in figure 6. The curve at 1 atmosphere is typical for synthetic material, with a | | ! º *-* - 3 maximum ( 0.0050 cm3/c) at about 2 kG. Below FIGURE 6. The effect of pressure on the Hall Constant of Sample this, the downward trend in the Hall Constant may be A–2 at 77.5 K. 722 | I I 2^s | *-*~~ | bor + .O5 H * --- 7OO bars — e |e A-6 º Sample – 1850 bars tº º & Rºs & E * * * g º a |94O bors RH 3 ‘. (cm’/C) & Sample A-2 — | bor O } | |O 15 H (kilogauss) — O5 – * —. O L | l FIGURE 7. The effect of pressure on the Hall Constant of sample A–6 at 77.5 K. ty electrons (a - 0.99). The effect of pressure is to move the curves to more negative values (dB H/dp - – .001 cm3/C.kbar) — a smaller effect than at room tempera- ture, and in the opposite direction. The curve for specimen A-6 is shown in figures 7 and 8. Several different features are observed. Firstly, the maximum in the room pressure curve occurs at a posi- tive value of the Hall Constant (~ + 0.050 cm3/c) in- dicating a smaller value for the ratio b = p/pº (b - 0.91 for A-6; b - 1.00 for A-2). Secondly, a minimum in the Hall curve at approximately 13 kG (RH -- 0.084 cm3/c) can be seen — a feature not previously reported in the literature. The cause of the minimum is not known at present. This feature invalidates the simple three band model, and awaits detailed interpretation until mea- surements at high field are made. Thirdly, the Hall Constant does not change in a simple way with pres- sure, decreasing below about 3 kG and increasing between - 3kG – 15kG. For both specimens the small changes in Hall Con- stant and magneto-resistance coefficient (fig. 9) are to be noted, and were too small to effectively determine whether the pressure variation was linear or not. The number of carriers calculated for this tempera- ture is (4.7+ 0.25) × 1018 for A-2; (4.45+ 0.25) × 1018 for A-6. This compares with a value n = 4.2 × 101° based on the four-parameter band model. 7. Galvanomagnetic Data and the Assignment of Holes at the Point K in the Brillouin Zone of Graphite For the interpretation of any electronic property of graphite, the carrier assignment at point K is of the ut- + O.O5 H. -: | bor 7OO bors + O.O25 H |350 bor S *4 |94O bors Sample A-6 RH (cm3/c) O i | —H t 3 H (kilogauss) – O.O25 H --- — O.O5 º I ! FIGURE 8. The effect of pressure on the Hall Constant of sample A-6 at 77.5 K for low values of magnetic field only (0-3 k(;). most importance. From galvanomagnetic data it is con- cluded that electrons are located at point K, in agree- 29 w | I I 28H *: 27 H * 26 H. * Af. 25– sm: PoHº xio" 24 H. * gauss’ | bor 23 H. 7OO bars. * |350 bars 22 H. 1940 bars *sº 2|H * 2O H. * |9 H. * 18 H- - | | l O 5 |O 15 H (kilogauss) FIGURE 9. The effect of pressure on the magnetoresistance coeffi- (Ap/poh”) for sample A-6 at 77.5 K. 723 ment with the assignment of Schroeder et al. [9]. The strongest reasons for this are as follows: (1) For the most perfect specimens, the ratio b = pºi/p.2 is less than unity for the temperature range - 50 to 200 K. Since the electrons are predominantly scattered by phonons in this temperature range, this fact is explained only by a heavier effective mass for electrons than holes. (2) At a given temperature in the temperature range below about 150 K, increase of specimen perfection decreases the ratio b. For poorer specimens, boundary scattering is increased. If the average hole velocity is greater than that of electrons, then the observed behavior can be explained (see fig. lo). (3) As the temperature is reduced below about 50 K, the ratio b is observed to increase. This is consistent with an increase in the importance of boundary scattering as the phonon-electron scattering probability decreases, and is con- sistent with the new assignment. Above about 200 K, the observed rise in the ratio b above unity for even the best specimens can be at- tributed to the following effects: (1) A shift in the value of the mean chemical poten- tial to more negative values (see fig. la) as the temperature is raised. (2) Effects arising from the excitation of carriers to levels within about 2kT of the mean chemical potential, thereby averaging the properties of electrons and holes. (3) The increased importance of intervalley scatter- ing which acts to average the properties of the carriers. g (4) The change in overlap parameters arising from thermal expansion of the crystal. This last effect may be estimated from the gal- vanomagnetic data reported here. A pressure of about 2kbar reduces the c-axis spacing by approximately the same amount that a 100 K change in temperature in- creases the spacing. Since the pressure coefficient of the Hall Effect is ~ 0.0025 cm”/C.kbar, a temperature change from 200-300 K probably increases the ratio b by about 4%, through the expansion of the lattice only. 8. Minority Carrier Behaviour With the new assignment of electrons at point K, and with a positive value of A demanded by de Haas-van Alphen measurements [21], the pocket of minority car- riers at points H and H' can only contain holes. The Hall Constant at low fields for synthetic materials clearly indicates the presence of a mobile minority elec- tron, since the sign of the contribution to the tensor component Ory is only determined by the sign of the charge of the carrier. The de Haas-van Alphen period A (1/H) = 2.24 X 10-4 gauss observed by Williamson et al. [21] cor- responded to an extremal cross sectional area of minority carriers in the basal plane (S) equal to 4.25 × 10" cm−2. This compares with the value predicted by the Slonczewski Weiss model [3]: — 37tep (e. - A) S :=====2.16 × 101 cm−3 4añy; with parameters defined before. A period anisotropy of only 1.9 in synthetic material corresponds to a spheroidal Fermi surface of the same axial ratio containing 3.2 × 101° available electronic states per cm3 of material. For the specimens used in the present measurements, the minority carrier density is ten times greater than this (~ 3 × 10" cm^*). This points to the conclusion that the minority carriers ob- served in the de Haas-van Alphen effect (not identified as holes or electrons by this measurement) do not originate from the same volume of the Brillouin zone as: the minority electrons observed in the Hall Effect. An attempt was made to observe minority hold behavior in the Hall Effect. Assuming that the minority holes are more mobile than the observed minority elec- trons, an upward trend in the Hall Constant should be seen at very low magnetic fields. Data was obtained down to 10 gauss (fig. 8) where some indication was ob- tained for this effect. However, the Hall voltage was ex- tremely small at this field (0.12p.v.) and even with re- peated measurements and data averaging, the best that can be said at the moment is that the possibility of such minority hole behavior is not ruled out. Shortly, mea- surements will be made utilizing a two frequency Hall technique with a PAR Lock-In-Amplifier. With this system measurements can be made in principle to 0.1 gau.S.S. Another indication that the minority electrons do not originate from the corners of the Brillouin zone is given by the effect of pressure on the Hall Constant at low magnetic field values (fig. 8). Application of pressure does not distort the curves, but only moves them downwards. This implies that the number of minority carriers and their mean mobility does not change with pressure within the precision of the measurement. However, measurements by Anderson et al. [5] on the change of de Haas-van Alphen periods for minority electrons at point H indicate that the parameter A, 724 which largely controls the properties of these carriers, changes by approximately 9%/kbar. The mobile minority electrons could originate in the feet of the energy surfaces, connecting electron-like with hole-like surfaces. Dresselhaus [22] has sug. gested that if the trigonal warping is large enough, aris- ing from a value of yº ~ 0.3 eV, outrigger electron sur- faces may be formed there. Such electrons might have relatively low values of the effective mass, and have properties relatively independent of pressure. 9. Conclusions Measurements of the Hall Effect toresonance are reported for two synthetic specimens of graphite at 298 and 77.5 K at hydrostatic pressures up to 3 kbar. The data supports the assignment of elec- trons at point K in the Brillouin zone of graphite as sug- gested by magneto-reflection experiments of Schroeder et al. [9]. Minority electron behavior observed in the Hall Effect at low fields at 77.5 K cannot be accounted for using the “four-parameter band model.” When a calculation is made for the dispersion relation of elec- trons at points near the zone edges with yº ~ 0.3 eV it is possible that pockets of electrons near the “feet” of the energy surfaces (outrigger pieces) may account for the observed properties. The number of carriers cal- culated from the four-parameter band model agrees quite well with the number obtained from the experi- mentS. and magne- 10. Acknowledgments The author wishes to thank the United States Atomic Energy Commission for a grant supporting this work, also, Dr. A. W. Moore of Union Carbide Corporation and Dr. C. Roscoe of Pennsylvania State University for supplying the graphite used in this work. Thanks are also due for technical and secretarial assistance in the Institute for Molecular Physics and to Mr. Raymond Crafton III for helping with numerical calculations. ll. References [1] McClure, J. W., Phys. Rev. 108,612 (1957). [2] Vereschagin, L. F., and Kabalkina, S. S., DAN SSR 131, 300 (1960). English translation Doklady 5, 373 (1960). [3] Slonczewski, J. C., and Weiss, P. R., Phys. Rev. 109. 272 (1958). [4] Itskevich, F. S., and Fischer, L. M., JETP 5, 141 (1967). English translation JETP Letters 5, 114 (1967). [5] Anderson, J. R., O'Sullivan, W. J., and Schirber, J. E., Phys. Rev. 164, 1038 (1967). [6] Arkhipov, R. G., Kechin, V. V., Likhter, A. I., and Pospelov, Yu. A., JETP 44, 1964 (1963). English translation JETP 17, 1321 (1964). - [7] Likhter, A. I., Kechin, V. V., Fizika Tverdoga Tela. 5, 3066 (1963). English translation in Soviet Physics Solid State 5, 2246 (1964). [8] Kechin, V. V., Likhter, A. I., and Stepanov, G. N., Fizika Tver. doga Tela. 10, 1242 (1968). English translation in Soviet Physics Solid State 10,987 (1968). [9] Schroeder, P. R., Dresselhaus, M. S., and Javan, A., Phys. Rev. Letters 20, 1292 (1968). Spain, I. L., reported at the Ninth Biennial Conference on Car- bon, Boston (1969) to be submitted to J. Chem. Phys. Dresselhaus, M. S., and Mavroides, J. G., Carbon 3,465 (1966). Dresselhaus, G. and Dresselhaus, M. S., Phys. Rev. 140, A401 (1965). Dresselhaus, M. S., reported at the Ninth Biennial Conference on Carbon, Boston (1969). Soule, D. E., Phys. Rev. 112,698 (1958). Blochinzev, D., and Nordheim, L., Z. Physik 84, 168 (1933). Shibuya, M., Phys. Rev. 95. 1385 (1954). Spain, I. L., Ubbelohde, A. R., and Young, D. A., Phil. Trans. Roy. Soc. 262, 345 (1967). Radcliffe, J. M., Proc. Phys. Soc. A68, 675 (1955) see also F. J. Blatt “Theory of Mobility of Electrons in Solids,” Solid State Physics, Vol. 4, F. Seitz and D. Turnbull, editors. Spain, I. L., Ph. D. Thesis, London University (1964). Yeoman, H. L., and Young, D. A., reported at the Ninth Bienni- al Conference on Carbon, Boston (1969). Williamson, S.J., Foner, S., and Dresselhaus, M. S., Phys. Rev. 140, A1429 (1965) and Carbon 4, 29 (1966). Dresselhaus, M. S., private communication, September (1969). [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] 725 Electrical Resistivity as a Function of Hydrogen Concentration in a Series of Palladium-Gold Alloys A. J. Mdeland Department of Chemistry, Worcester Polytechnic Institute, Worcester, Massachusetts 01609 The changes occurring in the electrical resistance of a series of gold-palladium alloys during hydrogen absorption have been measured. The results are presented for each alloy in the form of rela- tive resistance, R/Ro vs. hydrogen concentration H|M; R is the resistance of a particular gold-palladium alloy containing a certain amount of hydrogen, given by H/M, the atomic ratio of hydrogen to metal, and Ro is the resistance of the same hydrogen free alloy. For pure palladium the relative resistance increases as a function of hydrogen concentration to a maximum value of ~ 1.80 at HIM =0.75; further hydrogen absorption results in a decrease in R/Ro. Similar maxima are found in some gold-palladium alloys; how- ever, the maxima occurs at decreasing R/Ro values and also shifts to lower H/M values with increasing gold concentrations. At sufficiently high gold contents the maximum disappears and a continuous decrease in resistance with increasing hydrogen content occurs. The results are evaluated in terms of the band model. Key words: Electron donation model; electronic density of states; gold-palladium alloys; hydrogen absorption; hydrogen in palladium-gold alloys; palladium-gold alloys; rigid-band approximation. 1. Introduction The earliest application of the band theory of solids to a process of chemical interest [1] appears to be that of Mott and Jones [2] who proposed the electron dona- tion model for hydrogen absorption by pure palladium. According to this model, hydrogen is considered to be absorbed as protons plus electrons. The electron from hydrogen is assumed to be donated primarily to the par- tially empty d band of palladium. The protons enter in- terstitial sites in the palladium lattice where d-band electrons pile up to screen them. Subsequent work has shown these sites to be the octahedral interstices in the face-centered cubic palladium lattice [3]. The behavior of the magnetic susceptibility of H-Po alloys [4] is usually cited, perhaps unjustifiably so, as strong evidence in favor of a simple rigid-band model which in- dicates that pure palladium has roughly 0.6 holes per atom in the d band. However, recent work [5] on the band structure seems to point to a much smaller number, namely 0.36. Thus the rigid-band approxima- tion appears to be invalid. The electron donation model, however, is not invalidated by these results. It has, in fact, been substantially strengthened by some very recent low-temperature electronic heat capacity mea- surements on the hydrogen-palladium system by Mackleit and Schindler [6] who obtained direct evidence that the density of states in the d band of pal- ladium decreases with hydrogen absorption. The utility of the electron donation model may be tested by applying it to alloys of palladium. Gold-pal- ladium alloys are of particular interest in this respect because gold is believed to donate a 6s electron to the d band of palladium. Thus in the gold-palladium- hydrogen system both hydrogen and gold act as elec- tron donors to the d band of palladium. We have previ- ously reported x-ray and thermodynamic results on the gold-palladium-hydrogen system [7]. The data, we feel, was successfully interpreted in terms of this model, thus contributing further evidence in its favor. In the present experiments we have measured the electrical resistivity of a number of gold-palladium alloys as a function of hydrogen concentration. The results may be rationalized in terms of the electron donation model referred to above. 2. Experimental The gold-palladium specimens were in the form of wires 0.012 and 0.025 cm in diameter and were supplied 727 |.2O |..]O + O Crº N |OO - 0. O.90+ O.80 h # } } h { O.O5 O.IO O.15 O.2O O.25 O.3O O.35 FIGURE 1. Relationship between relative resistance and hydrogen COPllent. []. 18.80 percent Au, A. 26.47 percent Au: ©. 35.07 percent Auf à. 44.76 percent Aux L., 55.76 percent Au. by Engelhard Industries, Inc., who also performed the analysis. The samples had been prepared from 99.99% purity gold and palladium melted under argon. X-ray powder patterns revealed that the alloys were face-cen- tered cubic and showed no evidence of long-range order. Hydrogen was introduced by direct absorption from hydrogen-stirred dilute HCl solutions (< 0.04 N) under slow absorption conditions; i.e., the hydrogen pressure was reduced to below 1 atm. by dilution with helium. Details of the technique as well as the sample holder are available in the literature [7,8]. The re- sistance was determined potentiometrically; the volt- age across the specimen was matched with that across a standard resistor. All measurements were made at 25.0 + 0.1 °C. Hydrogen contents were established by vacuum degassing in a calibrated volume apparatus. The procedure was to remove the specimen from the reaction vessel after charging it to a certain hydrogen concentration and submerging it in cold acetone con- taining sulfur (catalytic poison). The sample holder was then attached to the vacuum line which was sub- sequently evacuated. Degassing the sample was accom- plished by passing a current directly through the specimen until it reached a dull red color. This heating was sufficient to remove all of the hydrogen from the |..] 5 |..[O + 3 105+ / * IOO. Cr. O95; O90} 8 O85 { { º | { O OO2 OO4 OO6 OO8 O.I.O H/M FIGURE 2, Relationship between relative resistance and hydrogen content in the O-phase. 1. Pure Pd: 2, 5.66 percent Au; 3, 11.90 percent Au; 4, 15.26 percent Au, 5, 18.80 percent Aug 6, 26.48 percent Au, 7, 35.07 percent Au; 8, 44.76 percent Aux 9, 55.77 percent Au. specimens. Any loss of hydrogen occurring between the time of measuring the resistance and the time of degassing was shown to be minimal [9]. The two sizes of wires used gave essentially identical results indicat- ing again no hydrogen loss during the transfer and evacuating stage. Any such loss would have affected the smaller diameter specimens proportionally more than the larger because hydrogen loss would be propor- tional to the surface/volume ratio. 3. Results and Discussion The relative resistance, R/R0, as a function of hydrogen content, H/M, was determined for a number of gold-palladium alloys. Some of the results are shown in figure l; here only those gold-palladium-hydrogen al- loys which exhibit one phase behavior of 25 °C are in- cluded; i.e., alloys having more than 17 atomic percent gold [7a). Alloys having less than 17 atomic percent gold behave like the palladium-hydrogen system in the sense of having a two phase region at room temperature [7a). The two phases are commonly referred to as cº- phase (hydrogen poor) and 3-phase (hydrogen rich). Figure 2 shows in detail the changes in resistance oc- curring at low hydrogen contents; here we have in- cluded some alloys which do have a two phase region (curves 1-4). As in the palladium-hydrogen system there is a discontinuous change in the slope, d(R|R0)/d(HIM), at the point where the second phase begins to form (not shown in fig. 2). The change in slope may, in fact, be used to determine the phase boundary [7a]. 728 It may be noted that some of the curves in figure l show maxima in R/R0 vs. H|M. A maximum has also been observed in the Pd-H system; this maximum oc- curs at R/Ro ~ 1.80 and H/Pd = 0.75 [10]. The maxima move toward lower values of R/Ro and H/M with in- creasing gold content. At sufficiently high gold concen- tration the maximum disappears and a continuous decrease in resistance with increasing hydrogen con- tent occurs (e.g., the 44.76% Au alloy). A qualitative rationalization based on electron dona- tion to the partially filled d band of palladium may be made as follows: The electrical resistance in transition metals is believed to be mainly due to scattering processes in which the electron makes a transition from the s to the d band; the current is carried by s electrons. The probability of an s-d transition is proportional to the density of states in the d band; the density of states is in turn proportional to the cube root of the number holes in the d band. Therefore the resistance, R, may be written [2] Ro" (1) Ils where na stands for the number of holes in the d band and ns denotes the number of s electrons. On one hand the introduction of hydrogen into the palladium lattice destroys the periodicity of the poten- tial field of the lattice and leads to an increase in the re- sistance. Furthermore, the lattice expands causing ad- ditional increase in resistance [ll] (d ln R/dln V = 4.0 at room temperature for pure Pol). On the other hand, hydrogen reduces the s-d scattering if its electron en- ters the d band of palladium, thereby decreasing the number of holes in the band. This leads to a decrease in the resistance which can be estimated from (1). dR #(º) (2) cy –- —- dna 3ns n; 3 We have assumed here than n, is constant. This is ap- proximately true since the density of states of the d band in pure palladium is about 10 times that of the s band. Consequently the electrons from hydrogen enter primarily the d band without increasing the number of s electrons appreciably. The first two effects initially dominate the resistance behavior as hydrogen is ab- sorbed. However, as the d band gradually fills, the s-d scattering becomes less probable and the resist- ance begins to decrease. Furthermore, as the d band fills, its density of states in relation to the density of states of the s band decreases and the electrons begin to enter the s band, resulting in further decrease of the resistance. The decrease in slope of R/R0 vs. H/M, figure 2, with increasing gold content may be explained similarly. The addition of gold also decreases the number of holes in the d band of palladium. From eq (2) it can be seen that dF/dna decreases more as na becomes less (by ad- dition of gold, e.g.). If the two other effects, i.e., the in- crease in resistance due to alteration of the periodicity of the lattice and volume increase, are assumed to be essentially independent of gold content, then the net ef- fect should be a smaller increase in R/R0 vs. H/M with increasing gold content. Also when a substantial por- tion of the d band holes in Pol has been filled by the ad- dition of Au, dB/dna becomes large enough to dominate the resistance behavior. In addition relatively more electrons enter the s band providing more conduction electrons. As a result, when enough gold has been added, the addition of hydrogen causes a decrease in the resistance. 4. Acknowledgments Financial support by U.S. Atomic Energy Commis- sion is gratefully acknowledged. We wish to thank the Engelhard Industries Inc., for the gold-palladium alloys used in this work. 5. References [1] Flanagan, T. B., Techn. Bull. Engelhard Ind. Inc. 7, 9 (1966). [2] Mott, N. F., and Jones, H., The Theory of Metals and Alloys, (Oxford Univ. Press, Oxford, 1936). [3] Worsham, J. E., Jr., Wilkinson, M. K., and Schull, C. G., J. Phys. Chem. Solids 3, 303 (1957); Maeland, A. J., Can. J. Phys. 46, 121 (1968). [4] Svensson, B., Ann. Physik 18, 299 (1933); Biggs, H. F., Phil. Mag. 32, 131 (1916). [5] Vuillemin, J. J., and Priestly, M. G., Phys. Rev. Letters 14, 307 (1965); Kimura, H., Katsuki, A., and Shimizu, M., J. Phys. Soc. (Japan) 21, 307 (1966). [6] Mackleit, C. A., and Schindler, A. J., Phys. Rev. 146, A463 (1966). [7] (a) Maeland, A. J., and Flanagan, T. B., J. Phys. Chem., 69, 3575 (1965); (b) Allard, K., Maeland, A. J., Simons, J. W., and Flanagan, T. B., ibid., 72, 136 (1968). [8] Simons, J. W., and Flanagan, T. B., ibid., 6.9, 3581 (1965). [9] Simons, J. W., and Flanagan, T. B., J. Chem. Phys. 44, 3486 (1966). [10] Barton, J. C., and Lewis, F. A., Z. Physik. Chem., Nem Folge 33, 99 (1962). [11] Bridgmann, P. W., Proc. Acad. Sci. 79, 125 (1951). 729 The Volume Dependence of the Electronic Density of States in Superconductors R. I. Boughton,” J. L. Olsen, and C. Palmy Laboratorium für Festokörperphysik, Swiss Federal Institute of Technology, Zürich, Switzerland The volume dependence of the electronic specific heat coefficient y can be obtained from measure- ments of the low temperature thermal expansion, from observations on the pressure dependence of the superconducting threshold curve, and from the volume change occurring at transition. We make use of recent theoretical results to obtain values of the change in the density of states with volume from the ex- isting experimental data for a number of metals. Key words: Aluminum (Al); electronic density of states; electronic specific heat; gallium; Gru- neisen parameter, electronic; superconductivity; thermal expansion; thorium; volume dependence of density-of-states. 1. Introduction The volume dependence of the electronic specific heat y is of interest since it is a function of the volume dependence of the electronic density of states, and of the electron-phonon interaction and its volume depen- dence. It is a quantity that is difficult to measure directly, but a number of simple thermodynamic rela- tionships exist which allow it to be determined in- directly by several methods. The logarithmic volume derivative of y, commonly referred to as the electronic Gruneisen parameter Te, can be obtained from the elec- tronic thermal expansion coefficient, from a knowledge of the change of the superconducting critical field curve under pressure, and from measurements of the difference in volume between the normal and supercon- ducting states. It is the purpose of the present note to present data from measurements of the critical fields of aluminum, gallium, and thorium under pressure and to make a brief summary of available results on other supercon- ductors. 2. Thermodynamic Expressions The relations connecting the measured properties with the electronic Gruneisen parameter Te in the three methods referred to above are the following: *Present address: Northeastern University, Department of Physics, Boston, Mas- sachusetts 02.115. Te is related [1] to the electronic volume expansion Be by |W Te - kyT Be (1) where T is the absolute temperature, and k is the isothermal compressibility. Due to the fact that this type of experiment is easily carried out only at zero pressure, it can only give the inital slope in the depend- ence of y on volume. The second method, which is ap- plicable only to superconductors, involves the measure- ment of the superconducting threshold curve as a func- tion of pressure. In general, the threshold relation for any superconductor can be written where Ho is the critical field at 0 K.; t = TIT, is the reduced temperature; and, f(t) is the normalized threshold function. The electronic specific heat coeffi- cient is related to the threshold parameters by the well- known relation [2] – V Hà , 7~ 1. Tif (0) (3) where f"(0) is the second derivative of the threshold function at 0 K. Here, y(V) is determined directly from measurements of Ho and To as functions of the volume. The threshold function, f(t), is usually assumed to be in- dependent of pressure (the similarity principle) and most of the existing experimental evidence indicates that this is indeed so [3,4]. 731 Finally, we arrive at the third method which involves the measurement of the change in volume at the normal-to-superconducting transition as a function of temperature. This quantity is related [5] to the deriva- tive of the critical field with respect to pressure by the following equation *-*-*(*) e y T V, 4T ôp I (4) The derivatives of Ho at T = 0 K and T = T, can be re- lated to Te by making use of (2) and (3), and by assuming f(t) to be unaffected by pressure. We find.[6] — 2 | |) | = — A — | – || + +} + 1 Teº ii. |Hº (; T. dp where f' (1) is the slope of the threshold function at T= Tc. The experimental techniques required to make such measurements are quite similar to those used in mea- suring thermal expansion, and unfortunately suffer from the same limitation, namely that only a zero pres- sure value of Te can be obtained. (5) 3. Theory In order to relate the electronic specific heat coeffi- cient to the band-structure density of states, it is neces- sary to include the effect of electron-phonon enhance- ment [7,8] to give y = }Tºk},N(0) (1 + \) (6) where A = N (0) Vep, and Vep is the electron-phonon in- teraction. From this expression it follows that the band- structure Gruneisen parameter TN is given by _0. In N(0) r A .0 in A Ts-º-To-Tººij, (7) The second term on the right-hand side is thus seen to have an influence which tends to reduce the total value of TN below the observed Te if Ó In A/6 ln V is a positive quantity as it usually is. Several authors [9-13] have obtained expressions for A which differ from each other somewhat. It has been demonstrated elsewhere [14] that by using an expres- sion due to Baryakhtar and Makarov [12], a reasonably good fit to the existing data on the volume dependence of T. is obtained for several elements. On this basis, the following expression results ô ln \ r *) – _r 2 6 ln }=(2r. Ty })=(1+x)(2r. Te #) (8) where Ta is the lattice Gruneisen parameter. Twice To is usually larger than (To + 2/3), and therefore A is in general a decreasing function of pressure. Substitution of (8) into (7) yields Tv = T, (1 + \) – A (2To –%). (9) Although no theoretical calculations have yet been made to determine TN for any real metals except copper, the nearly-free-electron (nfe) model predicts a value of 2/3 for an isotropic metal [15]. This is simply the result of the fact that in this approximation the Fermi surface scales with the reciprocal lattice as the volume is changed. Such a result would, of course, be most likely in the case of a cubic metal like aluminum under hydrostatic pressure. However, for an anisotrop- ic material, an applied hydrostatic pressure gives rise to a nonuniform strain field and may change the band structure considerably, even in the nfe model. It is therefore reasonable to assume that somewhat dif- ferent values of TN will result from measurements on anisotropic metals. It should also be pointed out that measurements on anisotropic materials using uniaxial stress or change in length at transition should give a more detailed idea of how the band structure is af. fected. 4. Results Experiments on the change in the superconducting threshold curves under pressure for aluminum, galli- um, and thorium have been carried out and the results are presented in figure 1, where y, as determined from (3), is plotted against the reduced volume, using published values of the volume dependence of the com- pressibility at room temperature [16]. It is apparent that within the present range of experimental accuracy \ Y, m Joule K2 mol ſº THORIUM 45- © ALUMINIUM l,OF GALLIUM O5 —l w – t | 99 OO 98.00 97.OO 96.OO looxy. FIGURE 1. Electronic specific heat as a function of relative change in volume for gallium, aluminum, and thorium. TABLE 1. Experimental values of Te at zero pres- sure and calculated values of TN Element | Method” To (p = 0) \oq | Top |TX(p = 0) Ref. Al............... Te.......... 1.8 + 0.1 || 0.38 2.18 1.1 ((1) PE 1.9 + 0.7 1.2 (b) Ca.............. TE......... 0.7+ 1.5 | .38 2.30 | – 0.5 (c) PE......... 5.4 +2 6.0 (d) Ga.............. PE......... — 0.8-H 1 .40 | 1.45 | – 1.4 (b) Hg a ........... PE......... 7.3 + 0.3 | 1.0 | 3.00 9.3 (e) VC......... 10.2 = 2 15.1 (f) In............... TE......... 3.2 + 0.4 |0.69 || 2.48 2.4 (ſſ) PE......... 3.4 + 0.1 2.8 (r) VC......... 1.0 + 0.2 — 1.3 (f) Ir............... TE......... 2.7-E 0.3 | .34 || 2.49 2.2 (h) La.............. TE......... — 2.0 + 0.2 | .84 || 0.74 || – 4.4 (i) VC......... — 3.4 + 1.0 — 6.9 (f) Mo............. TE......... 1.6 + 0.3 | .41 | 1.65 1.2 (c) Nb.............. TE......... 1.5 + 0.2 | .82 | 1.74 0.4 ((1) Pb.............. TE......... 1.7 -- 0.5 | 1.12 || 2.84 || – 2.0 ((1) PE......... 6.0 -- 0.9 7. I (j) VC......... 1.8 + 0.4 — 1.8 (f) Re.............. TE......... 4.5 + 0.5 || 0.46 || 2.66 4.4 ((‘) Sn.............. TE......... = 1.0 .60 2.27 | – 0.7 (), ) PE......... 2.0 + 0.3 ().9 (e) US......... 1.7 -H 0.3 0.4 (()) Ta.............. TE......... 1.3 + 0.1 | .65 | 1.82 0.2 ((1) PE......... 4.0 + 0.5 4.7 (l) Th.............. PE......... 0.6 -H 1 .53 | 1.41 | – 0.2 (l)) Ti............... TE......... = 2.0 .38 | 1.33 2.0 (m) Tl............... PE......... — 4.0 + 1 .71 2.3 — 9.6 (m) V... | TE......... 1.7+0.1 | 60 | 1.55] 1.2 | tº VC......... – 0.8 + 0.6 — 2.7 (f) Zn.............. PE......... 4.7 -H 2 .38 || 2.1 5.2 (ſl) Zr.............. TE......... 0.0 -H 0.2 .41 0.82 0.4 (m) *TE – Thermal expansion: PE— Pressure effect; VC – Volume change: US–Uniaxial StreSS. “J. G. Collins and G. K. White, Progress in Low Temp. Physics, Vol. IV, Ed. C. J. Gorter (North Holland Publ. Co., Amsterdam, 1964). b Present work. * K. Andres, Phys. kondens. Mat. 2, 294 (1964). d I. V. Berman, N. B. Brandt and N. I. Ginzburg, Zh. Eksp. i Teor. Fiz. 53, 124 (1967): Sov. Phys. JETP 26, 86 (1968). “J. E. Schirber and C. A. Swenson, Phys. Rev. 123, 1115 (1961). J H. Rohrer, Helv. Phys. Acta 33,675 (1960). ! J. G. Collins, J. A. Cowan and G. K. White, Cryogenics 7, 219 (1967). h E. Fawcett and G. K. White, Jour. Appl. Phys. 39, 576 (1968). i K. Andres, Phys. Rev. 168, 708 (1968). j M. Garfinkel and D. E. Mapother, Phys. Rev. 122, 459 (1961). k G. K. White, Phys. Letters 8, 294 (1964). ! C. H. Hinrichs and C. A. Swenson, Phys. Rev. 123, 1106 (1961). * J. A. Cowan, A. T. Pawlowics and G. K. White, Cryogenics 8, 155 (1968). * M. D. Fiske, J. Phys. Chem. Solids 2, 191 (1958). ° C. Grenier, Compt. Rend. 241, 862 (1955). P See Ref. 16. Q See Ref. 6. r G. Dummer and D. E. Mapother, Proc. N.B.S. Electronic Density of States Symposium, Nov. 3–6, 1969. no definite conclusions as to the detailed behavior of y as a function of volume can be made, other than that the slope appears to be positive (y decreases with decreasing volume) for Al and Th while being nearly zero for Ga. Earlier pressure effect data on Al yielded values of Te several times as large as those obtained from thermal expansion data. The present results show that the two methods are capable of yielding consistent results. For Th and Ga our measurements provide only an order of magnitude estimate of the magnitude of the ef- fect. In order to summarize the present situation in this field, we compare the existing data for a number of ele- ments in table 1. As is evident from the table, there ap- pear to be rather large discrepancies between the su- perconducting and thermal expansion results for Te. At present, this lack of agreement is still unexplained and will certainly require further experimental work to be resolved. The most reliable results appear to be those obtained from thermal expansion measurements, because of the small errors reported in most cases. In particular, the Ty values for the nontransition elements are of the predicted order of magnitude and, for exam- ple, the experimental value of 1.1 for Al is not very far from the nfe model’s prediction. 5. Conclusion It is readily apparent that more accurate techniques for the measurement of the threshold parameters are necessary to determine the volume dependence of N(0) with any accuracy. The present experiments only serve to give an order of magnitude estimate of the effect and therefore any detailed comparison with theory cannot be attempted. It should be realized, however, that theoretical information will be required on both the band-structure density of states and on the volume de- pendence of the electron-phonon coupling in order to completely resolve the question. 6. References [1] Collins, J. G., and White, G. K., Progress in Low Temp. Physics, Vol. IV, C. J. Gorter, Editor (North Holland Publ. Co., Amsterdam, 1964). [2] Lynton, E. A. Superconductivity, (Methuen, London, 1962), p. 19 for example. [3] Berman, I. V., Brandt, N. V., and Ginzburg, N. I., Zh. Eksp. i Teor. Fiz. 53, 124 (1967); Sov. Phys. JETP 26, 86 (1968). 733 [4] Harris, E. P., and Mapother, D. E., Phys. Rev. 165, 522 (1968). [5] Shoenberg, D., Superconductivity, (Cambridge Univ. Press., London, 1952), p. 74. [6] Boughton, R. I., Olsen, J. L., and Palmy, C., Progress in Low Temp. Physics, Vol. VI, C. J. Gorter, Editor, in press. [7] Buckingham, M. J., and Schafroth, M. R., Proc. Phys. Soc. 67A, 828 (1954). [8] Migdal, A. B., Zh. Eksp. i Teor. Fiz. 34, 1438 (1958); Sov. Phys. JETP 7, 996 (1958). [9] Ziman, J. M., Phys. Rev. Letters 8, 272 (1962). [10] McMillan, W. L., Phys. Rev. 167,331 (1968). [ll] Seiden, P. E., Phys. Rev. 179,458 (1969). [12] Baryakhtar, W. G., and Makarov, V. I., Zh. Eksp. i Teor. Fiz. 49, 1934 (1965); Sov. Phys. JETP 22, 1320 (1966). [13] Fröhlich, H., and Mitra, T.K., Proc. Phys. Soc. 1,544 (1968). [14] Boughton, R. I., Brändli, G., Olsen, J. L., and Palmy, C., Helv. Phys. Acta 42, 587 (1969). [15] Harrison, W. A., Physics of Solids at High Pressures, C. T. Tomizuka and R. M. Emrick, Editors (Academic Press, New York, 1965), p. 3ff. [16] Gschneidner, K. A., Solid State Physics, Vol. 16, F. Seitz and D. Turnbull, Editors (Academic Press, New York, 1964). [17] Ziman, J. M., Principles of the Theory of Solids (Cambridge Univ. Press, London, 1964), p. 116. 734 Alloy Fermi Surface Topology Information from Superconductivity Medsurements * H. D. Kaehn and R. J. Higgins Department of Physics, University of Oregon, Eugene, Oregon 97.403 Key words: Electronic density of states; Fermi surface; In-Co alloys; pressure; strain; supercon- ductivity. A great deal of progress has been made recently in calculating the energy band structure of random alloys [1]. In searching for an experimental test of these theo- ries, we have found that the rapid variation with con- centration of the pressure and strain derivatives of the superconducting transition temperature (dTe/dp and dTeſde) observed [2,3] in the In-Co alloy system has a direct connection with the alloy density of states n(E) at the Fermi level EF. These measured derivatives are proportional to the energy derivative of n(E) at EF, hence show strong structure near van Hove singulari- ties where the Fermi surface (FS) topology changes. The amount of broadening of the singularities is a mea- sure of the Bloch state lifetime, and the sign and shape of the observed structure is a direct indicator of the type of FS topology change. The basic formalism for the dTe/dp calculation and its application to In-Col has been presented elsewhere [4]. We shall concentrate here on important cor- rections to that calculation which were found in the course of interpreting the dTe/de data on whiskers which has recently been reported [3]. Since pressure and uniaxial stress are just special cases of a general- ized stress, the analysis is similar. As in [4] we take into account the rapid variation in n(E) near a singular point Ec by separating n(E) into a rapidly varying part ôn (EF-Ec) plus a monotonic background no (EF). It follows that the varying part of the derivative may be written: dTe. To dº dil dX 2no (EF) dm dX X = p, e where m is (EF – Ec)/T, T is h|T, and F(m) is 6nſBF — Ec, T) convoluted with a broadening function of width kTc. The calculated impurity dependence of dTe/dp is shown in figure 1 compared with experiment assuming electron saddle point and pocket singularities in n(E) near 0.8% and 1.9% Col, respectively. These features are consistent with available information on the pure In FS. It is convenient to compare the calculated ratio of the derivatives at the peak values dTe /dTe dp/ de with the experimental value of –2 × 10−4 (% e)/(kg/cm3). We find that by neglecting the derivatives of Ec, as in [4], the calculated value is an order of mag- nitude too large. We associate E, with a feature of the p kg/cm3 CALC. 2,O – | |O 2.O 3.O at. 7% CO in In FIGURE 1. Computed dTeſdp for In doped with Cd [4] compared *Research supported by the National Science Foundation. This is a portion of the Ph. D. thesis work of Mr. H. D. Kaehn, the full details of which will be published shortly elsewhere: H. D. Kaehn and R. J. Higgins (to be published). with experiment [2]. 735 third zone Fermi surface near the corners T of the Bril- louin zone and write Ec = ET + Eps where Eps is the band structure contribution to the free electron value of ET for the bottom of the band near T. The calculation of R including the derivatives of ET and Ebs leads to a value in good agreement with the experi- mental data. It is interesting to note, however, that the pressure derivative of ET nearly cancels the pressure derivative of Er so that the smaller term from Ebs is sig- nificant. References [1] e.g., Soven, P., Phys. Rev. 151,539 (1966). [2] Makarov, V. I., and Volynskii, I. Ya., JETP Letters 4, 249 (1966). [3] Overcash, D. R., Skove, M. J., and Stillwell, E. P., Bull. Am. Phys. Soc. 14, 129 (1969). [4] Higgins, R. J., and Kaehn, H. D., Phys. Rev. 182 (1969). 736 Hydrogencition Effects on Palladium Tunnel Junctions W. N. Grant, R. C. Barker, cind A. Yelon Yale University, New Haven, Connecticut 06520 The effects of hydrogenation on the electron tunneling characteristics of Al-oxide-Pa junctions have been investigated. It is found that the impedance of the junctions increases from 1 to 5 percent with increasing hydrogenation, with the greatest increase occurring at large positive bias on the Pd. We attribtue this effect to the introduction of electrons from hydrogen into the d bands of palladium. Key words: Al-oxide-Pa; electronic density of states; hydrogen in palladium; rigid-band approxima- tion; tunnel junctions. 1. Introduction We have measured the tunneling characteristics of the Al-insulator-Pa junctions, ranging in zero-bias im- pedance from 50 to 12,000 ohms, at 4.2 and 120 K. The experiment was designed to investigate the effect of hydrogenation upon the level and shape of the dif- ferential impedance curves of the junctions. Hydrogen dissolves in sizable quantities in palladi- um, and dissociates, leaving an increased number of electrons per Pol atom, plus interstitial protons [1,2,3]. Measurements with increasing hydrogen show a decrease in the specific heat [4], paramagnetic suscep- tibility [4], temperature dependent resistivity [5], and thermoelectric power [6]. All of these measurements, combined with de Haas-van Alphen experiments [7], galvanomagnetic experiments [8], and, more recently, band calculations [9] indicate that Pd has a closed electron Fermi surface, centered about the T point of the Brillouin zone, and two hole surfaces, one of which is open, and both of which are centered about the X point of the zone [7,9]. The Fermi level of Pol is near the top of the d band of the copper-like band structure, resulting in a rapidly varying density of states with sharp peaks in the vicinity of the Fermi surface. Hydrogenation of the Pd raises the Fermi level in the approximately rigid d band by filling some of the d states. This reduces the energy density of states at the Fermi surface, and moves the peaks with respect to it. The tunneling current is given by a sum of the con- tributions from the portions of k-space falling within the bands in question. The energy density of states does not appear explicitly in these contributions. However, the departures from simple one-band spherical energy surfaces in k-space and the degeneracy of the bands, which account for the variations in the energy density of states, also determine the E(k) relations in the in- tegral and the limits of the integrals [10]. Thus, if any metal were to show any reflection of its electron struc- ture in its tunneling behavior, Pd, with its rapidly vary- ing density of states near the Fermi surface, might be expected to do so. 2. Experimental Results Hydrogenation experiments have been performed with a large number of samples. On each substrate there are three similar junctions, 6.3 × 10−4 mm” in area, of Al-oxide-Pol, and a control junction with a counter electrode usually of Al, Pb, or Ag. The oxide is produced by glow discharge in dry oxygen, and all layers are formed before the evaporator is opened. Dif- ferent junctions with the same counter electrode on the same substrate have zero-bias resistances typically within 20 percent of each other. However, their dif. ferential resistance curves normalized to the same zero- bias level are usually within 2 percent of each other over the range —0.5 to +0.5 volts. The resistance level for different junctions is determined by time in the glow discharge. The maximum in the differential resistance of the Al- oxide-Pa junctions is shifted toward positive bias on the Pol, by about 100 mV. A similar shift occurs in junc- tions with control electrodes, and is characteristic of our method of oxidizing. It suggests a barrier height at the counter-electrode oxide interface about twice that at the Al-oxide interface [11]. 417–156 O – 71 – 48 737 |.OOO H. 3 .95O H. C d lº, § Cº. .900 H. ~5 Q) .N # 5 ,850 H. 2. C .8OO LC , A B |→l—l | | | | | \ A 2OO |OO O |OO 2OO 3OO 4OO 500 mV + PC Applied Bids FIGURE 1. Differential resistance versus voltage for an Al-oxide-Pai junction at 120 K for different levels of hydrogenation. A is the unhydrogenated sample. B and C have increasing levels of hydrogen. An example of the effect of hydrogenation upon the differential resistance curves of Al-oxide-Pol junction is shown in figure 1. The junction was hydrogenated at 120 K, and measurements at this temperature are shown. The curves A, B, and C were taken before hydrogenation, after 1/2 hr. of hydrogenation, and after 4 1/2 hrs. of hydrogenation at 300p. pressure, respec- tively. The zero-bias differential resistances were 22200, 2240(), and 22900 respectively. The curves have been normalized at zero-bias for comparison. After the completion of these measurements, the sam- ple was brought to room temperature in dry nitrogen, to drive off the hydrogen. A fourth curve, which falls directly upon curve A, was then run at 120K, indicating a reversible hydrogenation process, and reversible response in the tunneling resistance. In order to clarify the shift of resistance with hydrogenation, we show in figure 2 the percentage change in resistance, AR, as a function of voltage for the sample of figure 1, for a second Al-oxide-Pa sample, and a control junction with an Al counter electrode, all on the same substrate. The zero bias differential re- sistance for each of the junctions is shown in the figure. The variation of R with V for the control is due to a com- bination of data plotting error and instrument zero shift totalling about 1/4 percent. The downward shift of 1 percent represents a reduction in R0 from 18100 to 1795() upon hydrogenation. Ro remained at 1795Q. after dehydrogenation. Ro returned to its original value for both Pol they were dehydrogenated and measured, 5 days after these ex- periments. In figure 2, we observe a minimum in the percent resistance rise at applied voltages close to the junctions, when 5 H- Pd 4 H. 3. Pd C- 3 O - § 2 H ſº E | H § O H. Ç 2 - |- $ -2 H Re-1810 | | | | | | | | –2OO – |OO O |OO 2OO 3OO 4OO 5OO mV. + counterelectrode Applied Bias FIGURE 2, Percent change in differential resistance versus voltage due to hydrogenation for Al-oxide-Pó and Al-oxide-Al junc- tions on the same substrate. |- ſ 28OO |.OO8 H * º |- F- E JC 9 2600 H. |.OO4 H. g § ºw- C F- 5 |.2°k : tº 2400 H. : I.OOO H. § 3 OY O }* O § .996 H. l l 5 |- O |Omv E 5 .992 H. * 2 988 — | l | l | | –3O mv – 20 -|O O |O 2O 3O mV + Applied Bids PG FIGURE 3. Differential conductance near zero-bias at 4.2 K, for an Al-I-Pa junction, compared with data of Rowell [12], and resistance structure due to the superconducting energy gap of the aluminum electrode, at 1.2 K. resistance maximum at about 65 mV, followed by a general rise to the right, reaching 4 or 5 percent at +500 mV. We have also performed such measurements at 4.2 K. For low-impedance junctions at the higher tempera- ture, application of the relatively large biases in mea- surement warms the junction enough to drive out the hydrogen. The results of measurements at 4.2 K are not significantly different from those obtained at 120 K, but the use of the low temperature reduces the effects of heating. As an indication that the observed effects are in fact due to tunneling, we show in figure 3 the zero-bias structure at 4.2 K, compared with the data of Rowell [12], and the aluminum gap at 1.2 K. Calculation from 738 this gap yields a transition temperature of 1.4 K for the aluminum. 3. Conclusions We have found a reversible tunnel resistance in- crease of up to 5 percent at positive Pd bias, in Al-I-Pol junctions. This effect increases with increasing applied voltage with that polarity. No evidence for such an ef. fect is found with Al, Pb, or Ag counter electrodes. The change in tunneling characteristics seems to be due to the introduction of electrons into the Pol electrode. It may arise directly from a change in density of tunneling states near the Fermi surface of the Pol, or from a change in the effective barrier height at the Pa-oxide in- terface, either one produced by the introduction of the electrons. We are conducting further studies, in order to clarify this point, and are also attempting to deter- mine quantitatively by other measurements, the amount of hydrogen introduced into the samples. [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] 4. References Ebisuzaki, Y., and O’Keefe, M., Progress in Solid State Chemistry 4, (Pergamon Press, N.Y., 1967) p. 187. Mott, N. F., Advances in Phys. 13, 325 (1964). Tsuchida, T., J. Phys. Soc. Japan 18, 1016 (1963). Hoare, F. E., in Electronic Structure and Alloy Chemistry of the Transition Elements, P. A. Beck, ed. (Wiley, N.Y., 1963); Hoare, F. E., and Yates, B., Proc. Roy. Soc. A240, 42 (1957); Shimizu, M., Takahashi, T., and Katsuki, A., J. Phys. Soc. Japan 18, 240 (1963); Froidevaux, C., Launois, H., and Gauti- er, F., J. Appl. Phys. 39,557 (1968). Coles, B. R., and Taylor, J. C., Proc. Roy. Soc. A267. 139 (1962); Burger, J. P., Am. Phys. (France) 9,345 (1964). Fletcher, R., and Greig, D., Phil. Mag. 17, 21 (1968). Vuillemin, J. J., Phys. Rev. 144, 396 (1966). Alekseevskii, N. E., Karstens, G. E., and Mozhaev, V. V., Soviet Phys. JETP 19. 1333 (1963). Mueller, F. M., Freeman, A. J., Dimmock, J. O., and Furdyna, A. M., to be published. Duke, C. B., Tunneling in Solids 10. (Academic Press, N.Y., 1969) p. 52; Friedkin, D. R., and Wannier, G. H., Phys. Rev. 128, 2054 (1962). Barker, R. C., and Gruodis, A. J., to be published. Rowell, J. M., J. Appl. Phys. 40, 1211 (1969). 739 Calculation of Thermodynamic Information Based on the Density of States Curves of Two Allotropes of Iron & D. Koskimaki and J. T. Waber** Northwestern University, Materials Science Department, Evanston, Illinois 60201 The use of density of states curves to obtain thermodynamic information associated with the al- lotropic phase transitions in iron is discussed. The density of states curves for both body-centered cubic and face-centered cubic iron are determined using a program which randomly interpolates between previously calculated eigenvalues to generate a large number of new energy solutions. These new eigen- values enable a more accurate determination of density of states curves than is possible by plotting and averaging the original eigenvalues themselves. The density of states curves determined for each phase are used to obtain the energy sum of the eigenvalues of the valence electrons, the shift in the Fermi potential with temperature for the two curves, and the electronic specific heat versus temperature curves for both phases of iron over the temperature range from 0 K to the melting point. Key words: Electronic density of states; iron; phase transitions; thermodynamic information. 1. Introduction 1.1. Phase Changes in Iron Pure iron experiences two allotropic phase changes as it is heated from 0 K to its melting point. Iron is body- centered cubic (o-iron) from 0 to 1183 K, face-centered cubic (y-iron) from 1183 to 1673 K, and once again body- centered cubic (6-iron) from 1673 to 1812 K, the melting point. In addition, Q-iron experiences a ferromagnetic Curie point at 1042 K. From a thermodynamic view- point, these phase changes have been explained from experimental specific heat and enthalpy values ad- justed for thermodynamic consistency. Very little suc- cessful work has been done, however, to derive these thermodynamic relations entirely from the basic elec- tronic structure of the phases, mostly because energy band calculations at least initially lacked the accuracy needed for these calculations. The present work is a calculation of the information which is directly obtaina- ble from density of states curves for both crystal struc- tures, that is, the summation of the energy eigenvalues of the valence electrons, and the electronic specific *This research was supported by the Advanced Research Projects Agency of the Depart- ment of Defense, through the Northwestern University Materials Research Center. **Graduate student and Professor, respectively, Department of Materials Science and Materials Research Center, The Technological Institute, Northwestern University, Evan- ston, Illinois 60.201. heat versus temperature curves for the entire tempera- ture range of solid iron. To our knowledge, a true elec- tronic specific heat versus temperature curve based on an actual density of states has never been calculated for any element. 1.2. Previous Work In 1932, Austin [1] first discussed the allotropic phase changes in iron in light of the temperature depen- dence of the specific heats and free energies of both crystal structures. He indicated the necessary ther- modynamic equations and used a mixture of observed and hypothetical specific heats and enthalpy dif- ferences to obtain a consistent set of free energies for the two phases. As Austin indicates, the most stable phase at any temperature must have the lowest value for the Gibbs free energy in relation to other possible phases at this temperature. The Gibbs free energy is given by G = H –TS OT f T T cº-hºrſ car-Tſ % dT (1) () 0 T where Ho is the enthalpy at zero degrees, and C, is the specific heat at constant pressure for the phase. Since 741 the entropy term or third term on the right side of eq (1) has a greater magnitude than the second term, the free energy will decrease with increasing tempera- ture. The temperature T in front of the entropy term in- tegral gives the term an increasingly large magnitude at higher temperatures, while the temperature T' in the denominator of the integrand causes the specific heat at higher temperatures to become increasingly less im- portant in relation to the specific heat at lower tempera- tures. For this reason differences in the specific heat at low temperatures have a more profound effect on the free energy and hence the phase stability at higher tem- peratures than do the specific heat differences at these higher temperatures. Obviously, since the stable phase at zero degrees is body-centered cubic, the free energy at zero degrees is lower for the bec phase than for the foc phase. At low temperatures the specific heat of fec iron must exceed the specific heat of bcc iron, if the free energy of fec iron is to be lowered relative to bec iron as the temperature is raised, causing the first phase transition. However, if the bec phase is to reoc- cur above 1673 K, the specific heat versus temperature curves must intersect at some intermediate tempera- ture far enough below 1673 K to allow the specific heat difference to have an appreciable effect in lowering the free energy of bcc iron relative to foc iron as the temperature is raised. Obviously, to predict this behavior, one needs the specific heat curves for both phases at all temperatures between zero degrees and the melting point, along with the difference in enthal- pies between the two phases at 0 K. Since each phase exists only in a certain range of temperatures, the specific heat curves and enthalpy difference at zero degrees must be derived theoretically for these phases outside their observed range of existence. Austin obtained the specific heat relations for y-iron in the temperature range from 0 to 1183 K by ex- trapolating the specific heat values for face-centered cubic iron alloys of 19.4 and 30% manganese con- tent. Using these values for fec iron and experimen- tally observed values for bec iron, he obtained curves for the temperature dependence of the free energy. Then, by adjusting these curves to coincide at the first transition temperature, he found an enthalpy difference of 960 calories per gram-mole at 0 K. In 1942, Seitz [2] showed that the necessary behavior of the specific heat versus temperature curves for the two crystal structures could be explained if bec iron had a higher electronic specific heat, while foc iron had a higher lattice specific heat in relation to the other phase. Since the electronic specific heat is a significant part of the total specific heat only at high and very low temperatures, the higher electronic specific heat of bcc iron would allow the total specific heat of this phase to exceed that for fcc iron at high temperatures, while the higher lattice specific heat of fec iron would enable the total specific heat of this phase to dominate at lower temperatures. The total specific heat curves of the phases would then be con- sistent with the behavior necessary for the phase transi. tions as suggested by Austin. In 1943, Manning [3] and Greene and Manning [4] calculated the electronic band structures of both bec and fec iron, using a Wigner-Seitz method. They were able to show that the electronic specific heat coefficient of bec iron does indeed exceed the electronic specific heat for fec iron. At low temperatures the electronic specific heat is proportional to the density of states at the Fermi level and can be found approximately by using a formula derived by E. C. Stoner [5]: Coi = 0.209 × 10−4 n (Epº) RT (2) where n(EP) is the density of states at the Fermi level Ep. in states/Rydberg K, and R is the gas constant. Manning and Greene obtained Fermi level density of states values of 17 and 11.4 states/atom Rydberg for bcc and foc respectively. However, due to the ap- proximate nature of their density of states curves, they did not attempt to derive any further thermodynamic information from these curves. By extrapolating and adjusting existing specific heat curves to conform to the observed phase changes in iron, Johanson [6] and Darken and Smith [7] have ob- tained values for the total specific heat of both crystal structures of iron at temperatures outside the range of their observed existence. Zener [8] and Weiss and Tauer [9] have divided the total specific heat of cy-iron into magnetic and nonmag- netic contributions. Using their nonmagnetic specific heat curves and Manning and Greene's values for the electronic specific heat coefficients, Weiss and Tauer concluded that fec iron has a Debye temperature of 335 K as compared to 420 K for bec iron, and that foc iron would be stable by 130 calories/mole at 0 K. in the absence of magnetic effects. Kaufman, Clougherty and Weiss [10] have analyzed thermal expansion data, the effects of pressure on elec- trical resistance, and extrapolated results from foc al- loys of iron to conclude that fec iron has two spin states. At very low temperatures fec iron is antifer- romagnetic with a moment of less than one Bohr mag- neton, while at high temperatures it has a high moment, greater than two Bohr magnetons, and is not ferromag- 742 netic. As the temperature is raised, the transition from the low spin to the high spin state imparts an extra en- tropy to the foc structure, lowering the free energy for fec iron and causing the stabilization of this phase above 1183 K. As the temperature continues to rise the entropy resulting from the magnetic disordering above the Curie point for bec iron is multiplied by an in- creasing value for temperature and eventually over- rides the entropy of the foc structure, causing the bcc phase to once more become stable above 1673 K. Based on their calculations, Kaufman and coworkers found that the values of the Debye temperature and the electronic specific heat coefficients are approximately the same for both phases (0) = 432 K, ype = 12 X 10-4 cal/mole K). In addition to the work of Manning and Greene, ener- gy bands have been calculated by Wood [11], Stern [12], Abate and Asdente [13] and Snow and Waber [14]. The present work is based primarily on the calcu- lations of Snow and Waber. 2. Theory 2.1. The Enthalpy Difference at 0 K The electrons in a metal are generally divided into core electrons and valence electrons. In iron and other 3d transition elements, the core electrons exist in the argon configuration, that is, 1s”2s22p*3s°3p°. Between two phases, the core electrons are altered only slightly and, as such, their effect on the enthalpy difference for two phases is ignored in this paper. The enthalpy difference due to the valence electrons consists of a magnetic part and an electronic part. The magnetic enthalpy difference results from the fer- romagnetic ordering of unpaired electron spins in ad- jacent atoms of bcc iron, which has been calculated by Weiss and Tauer [9] to lower the energy of bcc iron by approximately 2086 cal/mole at 0 K. To calculate the electronic enthalpy difference, the eigenvalues of the energy band calculation are con- sidered. In most calculations, the eigenvalues are given by e = (T + V) + X V, HX, W, (3) j where T is the kinetic energy of the ith electron with respect to the nucleus, Vi is the electrostatic potential of the ith electron with respect to the nucleus, Vij is the electrostatic potential of the ith electron with respect to the jth electron, and Wij is the exchange potential energy of the ith electron with respect to the jth elec- tron. The summation j is over all the occupied orbitals including j = i. The total energy, which is used to find the electronic enthalpy difference, is given by Er-X. e-A X Wu-, X. Wiſ (4) i ij ij where the second and third terms on the right side of the equation are correction terms due to the fact that each electron pair is counted twice in eq (3). The first term on the right is the sum of the eigenvalues, which is easily evaluated at 0 K by integrating the density of states times the energy, or EE, er-X. e-ſ En (E) dE. (5) i () The electrostatic and exchange energy corrections can be approximated using an equation given by Snow, Canfield and Waber [15]: | S. W. H. S. Wu- ſ p (r) [V(r) + W(r)] dºr (6) where p(r) is the charge density which is found by summing the square of the radial electron wave func- tions, or p(r)=> 1 () (). (7) V(r) is the self-consistent crystal potential used in the energy band calculation and W(r) is the exchange potential which can be approximated using Slater’s p1/8 method [16]. Thus º)" (8) wo--6 (; where W(r) is in Rydbergs when p is given in elec- trons/(atomic unit)*. 2.2. The Shift in Fermi Potential with Temperature At any temperature above zero degrees, eq (5) must include the Fermi-Dirac distribution, or e-ſ Eſte, T, on(e)at (9) where E2 and E are the upper and lower limits respec- tively of the density of states curve, and f(E,T, () is the Fermi-Dirac distribution given by f(E., T. () = H º (10) l + exp ( |T 743 The function f(E,T, () gives the probability that an elec- tron will occupy a state at a given energy and tempera- ture. The Fermi potential ((T), which is equal to the Fermi level EE at absolute zero, is defined as the energy level at which the Fermi-Dirac function f(E,T, () equals 1/2. The Fermi level EP is defined as the energy at which the number of vacated levels below this level equals the number of excited electrons above. As the temperature is raised above 0 K, the Fermi potential shifts in energy from its value of Er at 0 K, the Direction and magnitude of this shift depending upon the tem- perature and on the shape of the density of states curve near the Fermi level. This behavior can be understood by studying figure 1. At any temperature, the electron dis- tribution is given by the product of the Fermi-Dirac dis- tribution curve and the density of states curve, as shown in figure 1(c). However, if the Fermi potential is prevented from shifting, the number of states vacated from below the Fermi level will not necessarily equal the number of electrons excited to above the Fermi level, due to the asymmetry of the density of states curve about the Fermi level. In order that the number of valence electrons remain conserved as the tempera- ture is raised, the Fermi potential must shift, as shown in figure 1(d). Mathematically, this shift can be un- derstood from the following argument. The total number of valence electrons n is given by /l F ſºn(E)/(e. T, () dE E1 (11) At absolute zero the Fermi-Dirac distribution is unity for energies less than the Fermi level and zero above, as shown in figure 1(a). Therefore, at 0 K eq (11) can be reduced to E I, In F | H n (E) dB (12) * 1 At any temperature, the integral in eq (11) can also be split into two integrals at some constant energy which can be chosen as the Fermi level Ep., EH' 77 - E1 Substituting eq (12) for n, one obtains n(E)f(E. T., ()dE. (13) n (E)f(E., T. () dE +| E2 EF 1 (14) The left side of this equation represents the number of empty states below the Fermi level Ep, as can be seen ſº n (E) [l – f'(E. T. (..) |dE = |. n (E)f(E. T. () dB. | T = 0 °K ... T f(E,T) ~ T > 0°K (d) O EF n (E) (b) STATES VACATED BY \\\\\\ Exºd'ºïsos ſº STATES FILLED BY n(E) f(E,T) 2 Excit ED ELECTRoNs (c) EF n(E) f(E.T.) | | | | (d) EF º E N E RG Y FIGURE la. The Fermi-Dirac distribution, which gives the proba- bility that a state will be occupied at a given energy and temperature. In this case, the Fermi potential is assumed to equal the Fermi level. FIGURE lb. A hypothetical density of states curve. FIGURE le. The Fermi-Dirac distribution for some temperature than zero multiplied by the density of states curve of figure l (b) to give the distribution of electrons. The Fermi potential (, is assumed equal to the Fermi level EP. The cross-hatch regions represent states vacated by and filled by thermally excited electrons. Note that the area representing the states vacated by these electrons is unequal to the area of the states filled by º electrons, violating the fact that electrons must be conserved as the temperature 1S Tal Sé Cl. FIGURE 16. The Fermi-Dirac distribution for some temperature greater than zero multiplied by the density of states curve. In this case the correct Fermi potential is used. equating the areas representing states vacated and filled by thermally excited electrons. by the fact that if f(E,T, () is the probability that a state at a certain energy and temperature is occupied, then |l-f(E,T, ()) is the probability that this state is unoccu- pied. The right side of the equation gives the number of electrons excited to states above the Fermi level. When a certain temperature and density of states curve are given, then eq (14) determines the position of the Fermi potential ((T), contained in the expression f(E,T, (), since the Fermi potential is the only quantity in this equation which remains a variable. It is important to note that at all temperatures the definite integrals of eq 744 (14) are integrated over the limits of E! to EF and EF to E2 rather than from E to Ç(T) and ((T) to E2, as this condition is necessary to conserve the number of elec- trons occurring at 0 K. If the latter limits were taken, then eq (14) would no longer uniquely determine the position of Ç(T). 2.3. The Electronic Specific Heat The electronic specific heat is found by taking the derivative with respect to temperature of the energy et given by eq (9). Ideally, the derivative of the total elec- tronic energy ET given by eq (4) should be used instead, but the temperature dependence of the electrostatic and exchange potential energy corrections is assumed small enough to be neglected. Values for the electronic specific heat must be obtained numerically, since the energy integral of eq (9) must be obtained numerically. The Fermi potential shift with temperature complicates this energy integral even further. Ziman [17] has given an expression for the elec- tronic specific heat which is usually quite accurate at low temperatures: 2 - T2/. Cel - Tn (EF) + [n (() |# | Tºkºſ an (E) | (15) The second term accounts for the Fermi potential shift with temperature, which can be found using the ap- proximate formula r Tě (KT): |º] {(T) = Er 6 n (EF) || |E |p = Er (16) Eq (16) is only approximate since 0n (EF)/6E gives the slope at the Fermi level, but other than this, no account is made of the shape of the density of states curve near the Fermi level. 3. Procedure The starting point in the calculation was a set of eigenvalues for both bec and fec iron calculated by Snow and Waber using the augmented plane wave method. They used a reciprocal lattice grid spacing of 2T(m,n,p)/4a where a is the unit vector of the real lat- tice. For bec and foc lattices, this lattice spacing amounts to 128 and 256 points respectively, within the Brillouin zone of each structure. Snow and Waber calculated these eigenvalues for two electron configurations, 3d 74s, and 3d64s”, and con- cluded on the basis of calculated occupancies for the bands, that 3d'4S' is the preferred configuration for both phases. Stern [12] also found previously that the 3d'4S' configuration was the most compatible with his self-consistent field calculations for bec iron. Since there are substantial changes in charge distribution and energy with changes in configuration, the 3d 74s' configuration is used for both phases, although we recognize that the Engel-Brewer theory [18] predicts the configuration of fec iron to be 3d%3p”. Although the density of states curves calculated by Snow and Waber have a resolution of roughly 0.03 Ryd- bergs, which is better than any previously published curves for iron, this resolution is not small enough for an accurate calculation of the temperature dependence of the Fermi potential and the density of states at the Fermi potential. The energy range of the electrons af. fected by the Fermi-Dirac function is about 1 or 2 kT on each side of the Fermi level, and at 1000 K this energy span amounts to only 0.02 Rydberg. Thus, any fine peaks or other detailed structure in the density of states curve which might affect the Fermi-Dirac distribution, would be partially or wholly smoothed out if such detail occurred over a span less than the resolution of 0.03 Rydberg. Fortunately, an ingenious program written by F. M. Mueller, et al. [18], was made available to the authors, which takes the set of original reciprocal lattice points and interpolates quadratically for points between them to obtain new eigenvalues. In order to prevent false structure in the density of states curve from the regular choice of these points, the new points are chosen ran- domly. In this way 20,000 new points and their eigen- values were obtained on a CDC 6400 computer in just five minutes of machine time. The reliability of this QUAD method is discussed by Kennard et al. in another paper of this Conference (p. 795). Using the density of states curves obtained in this way, the Fermi levels for both phases were found by using eq (12) to fill the density of states curves to con- tain eight electrons. 3.1. The Shift in Fermi Potentical The Fermi potential shift was found by solving eq (14) for the Fermi potential ((T). Since both sides of the equation had to be numerically integrated, the Fermi potential shift could not be obtained directly. Instead, a trial and error method was used in which the Fermi potential was first chosen randomly, and then interpo- lated between successive solutions for the right and left sides of the equation, until the two sides were equal. 745 2.5 – - FACE – CENTERED CUB |C 2.O |- O.5 H. z 2.5 H. – BODY - CENTERED CUB] C 2.O. H. O.5. T O —l | | | FF —|.3 — 1.2 –I, — 1. O —O.9 –O.8 —O.7 — O.6 ENERGY |N R Y DBERGS —O, 5 FIGURE 2, The density of states curves for 3d 4s' iron. These curves were obtained by using eigenvalues calculated by Snow and Waber [14] as input for the QUAD interpolation program of Mueller et al. [19]. Twenty thousand points in the Brillouin zone were used. For each temperature, eq (14) had to be solved for about five iterations to obtain a consistent value for ((T). 3.2. Total Energy and Specific Heat The energy summation of the eigenvalues was found for a range of temperatures by using the shifted value for the Fermi potential and numerically integrating eq (9). The electronic specific heat was found by plotting the energies obtained from eq (9) for a range of tem- peratures and determining the slope of this curve at each temperature. A smooth curve was constructed from these graphical values. 4. Results and Discussion 4.1. Density of States Curves The density of states curves obtained for both crystal structures are presented in figure 2. The energy scale is chosen to give the same zero energy as that used in the self-consistent crystal potential of the APW calcula- tion by Snow and Waber. This energy scale gives the I | | -T T- | | 2.5H - FACE – CENTERED CUB IC 2.0 – - > O H. <ſ | > Lil N (ſ) Lu H. <[ H. (/) z - 2.5 H. -4 BODY – CENTERED CUB |C tº 20– - H. <ſ H. C/) 1.5 H. - Li- O f |.O H. - C/) 2. Lal ° 0.5| - O | | | | | EF | – i. 3 -1.2 —|, | —I.O —O.9 —O.8 —O.7 —O.6 —O.5 ENERGY |N RYDBERGS FIGURE 3. The smoothed density of states curves for 3d 4s' iron as - calculated by Snow and Waber. One hundred twenty-eight and 256 points in the Brillouin zone were used for body- centered cubic and face-centered cubic iron respectively. Note that these curves lack much of the structure of the curves of figure 2 which were obtained using the QUAD interpolation program. energy of an electron state in the density of states curve relative to the energy of an electron just free of the solid. For the purposes of comparison, the density of states curves calculated by Snow and Waber are given in figure 3. The curves of the present calculation con- tain much more structure than the curves of Snow and Waber. However, one should note that the Fermi levels and the density of states at the Fermi levels agree sur- prisingly well. 4.2. The Summation of the Eigenvalues At 0 K the summation of the energy eigenvalues was found to be - 7.1868 Rydbergs/atom for bec iron and - 7.0257 Rydbergs for fec iron. To obtain a value for the enthalpy difference of the valence electrons, these values must be corrected for the electrostatic and exchange potential energies given in eq (6). This calcu- lation is presently being carried out by the authors. If one can assume that the largest term in the enthalpy difference between two phases is that given by the dif- ference in the sum of the eigenvalue energies, then a 746 2.5 (ſ) | T- | I I | H I O Dr. Lil CO 2 O - e “F FACE – CENTERED CUB | C Or: º Sº I. 5 || z 1. 1.0 || – | SJa 0.5 – 4 OO 800 1200 1600 Uſ) O 6 — -: Or # BODY – CENTERED CUB | C C >- k== -: 3: 4 º z 2 T *- ~. Lil | O SJº ~ – 2 F- - –4 | | | | | | | O 400 8OO |2OO | 6OO TEMPERATURE * K The shift in the Fermi potential with temperature for both phases of iron. FIGURE 4. rough indication of the stability of one phase with respect to another is possible by comparing these sums. As such, the difference in the eigenvalue sums for both phases is 0.1611 Rydberg/atom at 0 K with the b.c.c. phase having the lower energy. This value, about 50,400 cal/mole, is large compared to the approximately 1000 calories/mole accepted by most workers to be the enthalpy difference at 0 K. However, when compared to the sums of the eigenvalues for both phases, the value represents only about 2% of these sums. 4.3. Electronic Specific Heat Figure 3 gives the result of calculating the Fermi potential shift with temperature for a range of tempera- tures from 0 to 2000 K. Figure 4 gives the correspond- ing change in the density of states curve at the Fermi potential. From figure 1, one could expect the Fermi potential to shift along the density of states curve from regions of higher density toward regions of lower den- sity. As can be seen from figures 4 and 5, this is indeed the case. Face-centered cubic iron shifts down the den- sity of states curve over the entire temperature range. Body-centered cubic iron shifts down the slope initially, but as the temperature is raised the direction of the !. Ol I | | | | I- T | FACE – CENTE RED CUB | C |. OO H. O.99 H. O.97 H. l– + t 4 —!. 4 OO 8 OO |2OO |6OO |.85 H. - BODY – CENTERED CUB | C | 8 || H. * | 4 O O 8 OO |2OO TEMPERATURE °K The change in the density of states at the Fermi potential with temperature. FIGURE 5. shift is reversed. The reason is that the Fermi-Dirac function extends over an increased energy range to in- clude the whole peak rather than just part of the double peak in the immediate vicinity of the Fermi level. At this higher temperature the tendency of the Fermi potential is to move away from the center of the entire double peak and to do this the Fermi potential must shift temporarily up the slope of one part of the double peak. The density of states at the Fermi level obtained by various authors are compared with the present results in table 1. The experimental electronic specific heat coefficient of bcc iron is slightly higher than that pre- dicted by eq (2), since the electron-phonon enhance- ment has not been included in the present work. On the basis of their energy band calculations, Snow and Waber have estimated this enhancement to increase the value of the electronic specific heat for 3d 74s' iron by factors of 1.09 and 2.09 for bec and fec iron respectively. The electronic specific heat versus temperature curves are presented in figure 6 with numerical values given in table 2. The values extrapolated from low tem- perature are plotted as dotted lines. It can be seen that 747 TABLE 1. Density of states at the fermi level and elec- tronic specific heats found by several authors States/ew-atomſ|10−4calories/mole K n(EF)* | n(Ep.)” ya yy Present work........................ 1.823 | 1.005 10.3 5.7 Greene and Manning [3,4] ..... 1.25 0.85 7. I 4.8 Snow and Waber [14] ............ 1.862 .978 10.5 5.5 Kaufman et al. [10]...............] ...... . ...... 12 12 Cheng, Wei, and Beck [26].....] ...... ...... *11.9 | ...... Deegan [27]......................... 1.67 1.12 9.4 6.3 *Obtained experimentally. TABLE 2. The electronic specific heat and the elec- tronic components of the Gibbs Free Energy for bec and fec iron between 0 K and 1812 K. Values for the Specific heat are in calories/K mole and values for the free energy are in calories/mole. T, K C à C. —(Go-GT)* —(Go-Gº)" G#- G} 100 0.102 0.056 0 0 O 200 .208 . 112 10.3 5.6 — 4.7 300 .312 . 169 30.9 16.9 — 14.0 400 .421 .230 61.5 33.9 – 27.6 500 .532 .295 103 56.4 — 47 600 .645 .363 155 84.8 — 70 700 .762 .433 218 122 –96 800 .865 .507 292 162 — 130 900 .943 .586 376 208 — 168 1000 .990 .664 47] 262 — 209 1100 1.024 .737 575 323 — 252 | 183 1.043 .800 668 380 – 288 1200 1.045 .812 690 390 — 300 1300 1.06] .890 812 463 — 349 1400 1.07] .966 94.2 542 — 400 1500 1.077 1.043 1082 631 — 45] 1600 1.085 1. 112 1224 726 — 498 1700 1.096 1. 180 1376 822 — 554 1800 1. 102 1.240 1536 929 — 607 1812 1.104 1.246 1555 94.2 — 613 the electronic specific heat deviates substantially from the linear behavior usually assumed by most authors. With increasing temperature, the bec curve deviates first positively and then negatively from linear behavior while the foc curve deviates positively. This behavior can be only partly predicted by eqs (15) and (16). Calcu- lations performed using these equations correctly give the initial change in Fermi potential along with the resulting change in the electronic specific heat. How- ever, eq (16) is an approximation which gives the change in Fermi potential as a function of the slope of the density of states at the Fermi level. No provision is made for the fact that the slope of the density of states curve may change substantially on both sides of the Fermi level. Perhaps eq (16) could be improved using terms of a higher order with respect to T, but it would be simpler and more accurate to obtain the electronic 5. O | | I | | 7 | / / / 4.5 H. * ; / H- / º / 4.01– ;: / *- / B / > / or 3.5 – / * (O / * 'e / 2^ B. C. C. / / / 2 / / * / 2^ 2.5 – / -4 H. / 2^ <ſ / / # / 2^ 2.0 – '/ 2’ == O / 2^ L. / 2 *- / / ... 1.5 H / 2^ -: % %2 / O |.0 — /*~ F. C.C - Z a * * * , O Cº. 5 Nº. 0.5– * —l Lid O | | | | | | | | O 400 8OO |2OO |6OO TEMPERATURE °K FIGURE 6. The electronic specific heat versus temperature for both phases of iron. The dotted lines are extrapolated from the low temperature behavior. The two curves intersect at 1560 K. specific heat directly by finding the change with respect to temperature of the energy given by eq (9). The unusual behavior of the electronic specific heat of bcc iron is due to the fact that the Fermi level coin- cides with a large double peak in the density of states curve. The Fermi-Dirac function changes from 0.99 to 0.01 over an energy span of about 13 kT. The number of electrons which are thermally excited is a rough func- tion of the average density of states in this energy span. At 1000 K this energy span amounts to about 0.08 Ryd- berg, while the total width of the double peak is only about 0.06 Rydberg. As the temperature is increased above 1000 K, the number of electrons thermally excited is a function of a decreasing average density of states, and this behavior is reflected by the decreasing slope in the electronic specific heat curve above 1000 K. Similarly, around 600 K the energy span of the major change in the Fermi-Dirac function occurs over about 0.05 Rydberg, which is less than the total width of the double peak. Since the Fermi level occurs in the dip between the peaks, the average density of states in this region is increased, causing the positive deviation from linear behavior of the electronic specific heat curve around 600 K. The behavior of the electronic specific heat curve for fec iron can be explained in a similar manner. 748 4.4. Comparison of the Contributions to the Total Specific Heat To determine what effect the electronic specific heat has in relation to the phase transitions in iron, a rough calculation was performed to resolve the total specific heat for each phase into its various components. These are compared in tables 3 and 4. The free energy dif- ference resulting from each of these components was then calculated using eq (1) and compared in table 5. This calculation is described in the following para- graphs. The total specific heat is given by CMT)=c, (%) [1+ ayſ c. (T) + cºt), (ii) where C, (6.0/T) is the lattice specific heat at constant volume; 60 is the Debye temperature; or is the volume coefficient of thermal expansion; Ce(T) is the elec- tronic specific heat; and C#(T) is the magnetic specific heat. The Gruneisen constant is given by O'ºk' Cog (18) where 8 is the compressibility or – 1/V(ÖV/ðP).T. The first term, C, (6.0/T) [1 + opyT], is the lattice specific heat at constant pressure, Cp. When T * 0D, C, is pro- portional to the cube of the temperature, and when T => 0D, C, is approximately constant and equal to the Du- long-Petit value of 3R or 5.96 cal/K mole. Basinsky, Sutton, and Hume-Rothery [19] have mea- sured on in the range 300 K s T s 1812 K, thus deter- mining or for both crystal phases of iron. For O-iron, the value of the Gruneisen constant y has been determined by Kittel [20] to be 1.6 and by Slater [21] to be 1.4, em- ploying a correction to Slater's value given by Dugdale and MacDonald [22]. However, the present authors could not find an experimentally determined value for the Gruneisen constant for y-iron, nor a value of the compressibility or bulk modulus of y-iron measured at high temperatures. Referring to eq (18), one would ex- pect the Gruneisen constant for y-iron to be slightly higher than the constant for o-iron, since oº is higher for y-iron and 8 could be expected to be less for the more densely packed foc structure. A rough estimate of the Gruneisen constant for fec iron is possible from the temperature dependence of Young's modulus as determined by Koster [23]. Since Young's modulus is approximately equal to the bulk modulus (assuming Poisson’s ratio = 0.33), then the bulk modulus of fec iron is about 12 × 10° Kg/cm * at 1200 K, giving a com- TABLE 3. Components of the Total Specific for bec iron. All values are in calories/K mole. The elec- tronic specific heat includes an electron-phonon enhancement factor of 1.15 T. K C, Cp–C, Cel Cp Cp Cp (total) magnetic | unknown 100 2.72 0.01 0.11 0.02 ............... 2.86 200 || 4.77 .05 .24 .10 l.............. 5.06 300 5.38 .07 .36 .19 |.............. 6.00 400 || 5.63 .10 .48 28 l.............. 6.49 500 || 5.76 .15 .6] .53 0.02 7.07 600 || 5.83 .20 .74. .79 .05 7.61 700 || 5.87 .25 .88 1.20 .10 8.30 800 5.88 .31 1.00 1.80 .22 9.2] 900 || 5.89 .37 1.10 2.90 .32 10.58 1000 || 5.90 .43 1.14 5.70 .46 13.63 1100 || 5.91 .50 1.18 2.10 .73 10.42 1183 || 5.91 .56 1.20 0.90 .75 9.32 1200 5.92 .57 1.21 .80 .76 9.26 1300 || 5.92 .65 1.22 .45 .90 9.14. 1400 5.92 .74 1.23 .25 1.06 9.20 1500 || 5.93 .83 1.24 .12 1.20 9.32 1600 || 5.93 .92 1.25 .06 1.37 9.53 1673 || 5.93 .99 1.26 .03 1.50 9.71 1700 || 5.93 1.02 1.26 .02 1.53 9.76 1800 || 5.94 1.12 1.27 .0] 1.68 |0.02 1812 || 5.94 1.13 1.27 .01 1.70 10.05 TABLE 4. Components of the total specific heat for foc iron. All values are in calories/K mole. The elec- tronic specific heat includes an electron-phonon enhancement factor of 1.11 T. K Cr Cp–Ce Cel Cp (total) 100 3.65 0.07 0.06 3.78 200 5.2.2 .20 .12 5.54 300 5.6] .32 .18 6.11 400 5.76 .44 .26 6.46 500 5.82 .54 .33 6.69 600 5.88 .65 .40 6.93 700 5.90 .75 .48 7. 13 800 5.9] .85 .56 7.32. 900 5.92 .96 .65 7.53 1000 5.92 1.06 .74 7.72 1100 5.93 1.16 .82 7.91 1183 5.93 1.24 .89 8.06 1200 5.93 1.26 .90 8.09 1300 5.94 1.36 .99 8.29 1400 5.94 1.46 1.07 8.47 1500 5.94 1.56 1.16 8.66 1600 5.95 1.65 1.24 8.84. 1673 5.95 1.72 1.29 8.96 1700 5.96 1.74 1.3] 9.00 1800 5.95 1.83 1.38 9. 16 1812 5.95 1.84 1.38 9.17 749 TABLE 5. Free energy differences between foc and bec iron due to the various components of the total specific heat for each phase. All values are in calories/mole. Note that the total free energy dif. ference refers both phases to zero energy at 0 K. To get the actual free energy difference, 1131 calories/ mole must be added to the free energy of the foc phase (Gº-Gº') (Gº-Gº') (Gº-Gº') —Gº ||—Gº |(Gº-Gº') T, K due to AC, due to elec- unknown mag- | total - A (Cp–Cr) tronic netic 100 - 29 ......... . . . . . . . . . . . . . ...... — 29 200 – 146 — 6* 5 | ..... 2 – 145 300 — 287 – 20 15 ..... 8 – 284 400 — 438 – 41 31 ..... 2] — 427 500 — 592 — 7] 53 | ..... 41 — 569 600 — 747 – 108 81 | ..... 7] | – 703 700 — 904 — 153 114 2 113 —828 800 – 106] — 205 152 4. 165 –945 900 | — 1218 – 265 187 10 242 | – 1043 I000 | — 1375 — 330 247 18 350 | – 1090 1100 | – 1534 — 403 300 32 481 – 1124 1183 || – 1664 — 467 347 49 604 || – 1131 |200| — 1692 — 4.81 356 52 626 – 1139 1300|| — 1849 – 565 415 79 778 — 1.146 |400| — 2008 — 665 475 | 12 942 | – 1144 1500 | – 2166 — 749 538 153 1083 || – 114] 1600 — 2324. –848 600 202 1235 | – 1135 1673 || – 244] –924 646 243 1345 || – 113] 1700 | – 2483 — 952 663 260 1388 || – 1124 1800 — 2642 — 106] 726 326 1539 || – || 12 1812 — 2661 — 1074. 733 335 1557 | – 1110 pressibility of 8.33 × 10−7 cm */Kg. Using this value and the value of or determined by Basinsky, et al., the Gru- neisen constant of fec iron was found to be 2.48. Using the thermal expansion data of Basinsky, et al., and Gruneisen constants of 2.48 and 1.6 for fec and bcc iron respectively, Co-Co was calculated for each phase. The lattice specific heat was calculated for each phase using Debye temperatures of 330 and 432 K for foc and bec iron respectively. - Assuming that there is no magnetic contribution to the specific heat for fec iron, an experimental value for the electronic specific heat can be found by sub- tracting the calculated lattice specific heat Cp at con- stant pressure from the experimental specific heat measured for fec iron over its stable range of tem- perature. Using the experimental results of Anderson and Hultgren [24], it was found that the electronic specific heat determined in the present investigation should be multiplied by an electron-phonon enhance- ment of approximately 1.11 to be consistent with ex- perimental data. This value is very close to the electron-phonon enhancement of 1.15 which results for bcc iron when the low temperature experimental heat coefficient determined by Cheng, Wei and Beck [25] is compared with the results of this investigation. The magnetic specific heat due to the second-order ferromagnetic transition in bec iron was found graphi- cally by assuming that this contribution is almost zero below 200K and above 1700K. The nonmagnetic specific heat was then interpolated between these temperatures and subtracted from the total specific heat, leaving the magnetic specific heat. It was found, however, that when the contributions from the lattice specific heat and the enhanced electronic specific heat are added and compared to the experimental specific heat above 1700 K, a large part of the total specific heat was unac- counted for. To account for this unknown contribution, which amounts to 1.7 cal/K mole at 1800 K, an electron- phonon enhancement of about 2.6 at 1800 K would be necessary for bec iron, in contrast to the value of 1.15. at low temperatures. It was decided to treat this con- tribution as a separate component and simply call it Cp (unknown). When the free energies resulting from each specific heat contribution are calculated and compared in table 5, it is apparent that the largest free energy differences are due to the ferromagnetism of bcc iron, and to the difference in C, between fec and bec iron. Since all values in table 5 are determined by subtracting the free energy of bcc iron from the free energy of fec iron, all negative values for the free energy difference tend to stabilize fec iron and all positive values tend to sta- bilize bec iron. Thus the negative values are responsi- ble for the bec zºº foc transition at 1182 K, and the positive values are responsible for the foc & bec transition at 1673 K. It is apparent that the major cause of the first transition is the free energy difference due to the larger C, for fcc iron, while a minor cause is the free energy difference due to the larger rate of thermal expansion for fec iron, which is determined from A(Cp — Co). The largest component responsible for the second phase transition is the magnetic free energy due to the ferromagnetic specific heat of bcc iron. The contribution to the free energy due to the larger elec- tronic specific heat for bec iron is about one-half the contribution from ferromagnetism. In this calculation, any effect due to the low spin & high spin transition described by Kaufman et al., was ignored. This was done as a matter of convenience, as we were only interested in showing the effect of the electronic specific heat in relation to the phase transi- tions. Presumably, the free energy difference which 750 results from differences in C, for the two phases could be accounted for instead by a low spin º high spin transition as Kaufman et al., have done. It should also be noted that Kaufman et al., assumed that CŞ, as C3, and that C = C, in disagreement with the results of this investigation. However, when taken together, the contributions to the free energy difference due to ACe, A(Cp–C) and C, (unknown) approximately cancel each other out, unaffecting the rest of the investigation by Kaufman et al. 4.5. Factors Needed for a More Complete Andlysis The authors clearly recognize that a number of im- portant factors have been neglected in this paper. These are briefly summarized here. (a) As previously mentioned, the electrostatic and exchange potential corrections have not been taken into account to calculate the difference in enthalpy at zero degrees. The summation of the eigenvalues gives only a rough comparison as to which phase is stable at zero degrees. (b) Although the core electrons can normally be neglected, for the purpose of calculating the enthalpy difference between two phases they must be taken into account. Since the enthalpy difference is a small quan- tity which is found by subtracting two comparatively large sums, any small factor such as the difference in energy of the core electrons may have a substantial ef- fect on the total enthalpy difference. (c) The effect of the thermal expansion of the lattices on the energy bands, and hence on the density of states curves has not been considered. In fact, the fec lat- tice parameter used by Snow and Waber is for 910 °C and the bec lattice parameter is for 25 °C. For this reason, the energy summation of the eigenvalues may be slightly in error, and there will be a change in this energy summation as the temperature is changed. (d) The bands for up and down spins will be slightly different, giving a different density of states curve for each spin, with a slightly different density of states at the Fermi level for each spin. 5. Conclusion The density of states for the 3d'4s' configuration of both crystal structures of iron have been determined using a set of eigenvalues previously determined by Snow and Waber and an interpolation program which randomly generates new eigenvalues. The density of states curves established by these new eigenvalues were found to contain more sharp peaks and dips than the original density of states curves determined by Snow and Waber; however, the Fermi levels and densi- ty of states at the Fermi levels were very similar. The density of states curves times the energy were integrated for both crystal structures to establish a rough indication of the enthalpy difference at 0 K and the phase stability. The difference in energy was found to be 50,400 calories with bec iron having the lower energy. This is quite large when compared to the ex- perimental enthalpy difference of approximately 1000 cal/mole. However, a large number of factors were neglected in the calculation, the most important of which are the electrostatic and exchange potential cor- rections. The electronic specific heat versus temperature curves were determined for the two phases. These curves were found to deviate substantially from the linear behavior given by Cel–Yeſ" where yel is the elec- tronic specific heat coefficient. Although at lower tem- peratures the electronic specific heat of bcc iron is greater than that for fec iron, the two curves intersect at 1560 K. When the values obtained for the electronic specific heat were added to calculated lattice specific heat values, an electron-phonon enhancement factor of 1.11 for fec iron was needed to adjust the theoretical values to the observed values for the total specific heat curves. For bec iron, the low temperature electron- phonon enhancement was found to be 1.15. At high temperatures, there is an increasingly large unknown component for the specific heat of bec iron. The high temperature phase transition at 1673 K is due to: (1) the magnetic specific heat resulting from the ferromagnetic phase transition in bec iron; (2) the electronic specific heat difference due to the higher electronic specific heat for bec iron; and (3) an un- known component in the specific heat of bcc iron. The phase transition at 1182 K is due to: (1) either a low spin -e high spin transition in fec iron, or a dif. ference in C, for the two phases with 0p = 330 K for foc iron and 00 = 432 K for bec iron; and (2) a higher (Cp – Co) value forfec iron. 6. Acknowledgments The authors would like to take this opportunity to thank Dr. F. M. Mueller for making available his QUAD interpolation program, and Mrs. David Kennard for converting this program for use by the CDC 6400 com- puter. 751 [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] 7. References Austin, J. B., Ind. and Eng. Chem. 24, 1225 (1932). Seitz, F., “The Modern Theory of Solids,” (McGraw Hill, New York and London, 1940) p. 487. Manning, M. F., Phys. Rev. 63, 190 (1943). Greene, J. B., and Manning, M. F., ibid, p. 203. Stoner, E. C., Proc. Roy. Soc. (London) A154, 656 (1936). Johanson, C. H., Arch. Eisenhuttenw. 11, 241 (1937). Darken, L. S., and Smith, R. P., Ind. and Eng. Chem. 43, 1815 (1951). Zener, C., Trans. AIME 167, 513 (1946). Weiss, R. J., and Tauer, K. J., Phys. Rev. 102, 1490 (1956). Kaufman, L., Clougherty, E. V., and Weiss, R. J., Acta Met. 11, 323 (1963). Wood, J. H., Phys. Rev. 126, 517 (1962). Stern, F., Phys. Rev. 116, 1399 (1959). Abate, E., and Asdente, M., Phys. Rev. 140, A1303 (1956). Snow, E. C., and Waber, J. T., Acta Met. 17, 623 (1969). Snow, E. C., Canfield, J. M., and Waber, J. T., Phys. Rev. 135, A969 (1964). [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] Slater, J. C., Phys. Rev. 81,385 (1951). Ziman, J. M., “Principles of the Theory of Solids,” (Cambridge University Press, London, 1964) p. 124. Hume-Rothery, W., Progress in Materials Science 13, 229 (1967). Mueller, F. M., Cohen, N. H., Garland, J., and Bennemann, K., to be published. Basinski, Z. S., Hume-Rothery, W., and Sutton, A. L., Proc. Roy. Soc. (London) A229,459 (1955). Kittel, C., “Introduction to Solid State Physics,” 3rd ed., (John Wiley & Sons, New York and London, 1966) p. 184. Slater, J. C., Phys. Rev. 57, 744 (1940). Dugdale, J. S., and MacDonald, D. K. C., Phys. Rev. 89, 832 (1953). Koster, W., Z. Metallk. 39, 1 (1948). Anderson, P. D., and Hutgren, R., Trans. AIME 224, 842 (1962). Cheng, C. H., Wei, C. T., and Beck, P. A., Phys. Rev. 120, 426 (1960). Deegan, R. A., submitted to J. Phys. Chem. 752 Discussion on “Calculation of Thermodynamic Information Based on the Density of States Curves of two Allotropes of Iron" by D. Koskimaki and J. T. Waber (Northwestern University) O. K. Andersen (Univ. of Pennsylvania): In calcula- tions like this, where you have calculated the shift in the Fermi energy as function of temperature, you seem only to be interested in a small energy region of about 10 mRy. Why do you in this case use a histogram technique which puts equal weight on every point in the Brillouin zone? Actually the information used is con- tained in a tiny fraction of the zone, and even your 20,000 points|zone probably does not give very good statistics. One would think that tracing of constant energy contours is a more appropriate technique, and it has the additional advantage of yielding the Van Hove singularities. J. T. Waber (Northwestern Univ.); I think your point is well taken and we would be well-advised to look at the actual region of the Brillouin zone from which the electrons are thermally excited and into which they go. The nature of the electronic contributions to the specific heat would be revealed. We have not done that. But I think that spikes from electron-phonon interac- tions and Van Hove singularities might very well con- tribute more to the specific heat, etc., than has been in- corporated in detail in either our work or literature on iron in the past. The point I wish to make is that we have established that it was not necessary to invoke a magnetic transformation to be able to get the BCC → FCC → BCC phase transformation in Fe. I will not go into the details here. M. B. McNeil (Mississippi State Univ.): How does the A E calculated on Fe due to Fermi level shift compare, at high temperatures, to TAS due to differences in the lattice specific heats? J. T. Waber (Northwestern Univ.); Considering just the body-centered cubic gamma phase at high tempera- tures, TAS due to lattice specific heat is approximately 20,000 cal/mole. This is offset by an enthalpy of about 10,000 calories/mole, to give a net change in free energy of around 10,000 calories/mole. The change in the elec- tronic free energy is about 1,500 calories per mole, which is quite a bit smaller. When the two phases are compared, the free energy difference due to lattice specific heats is about 3,600 calories/mole and the free energy difference due to electronic specific heats is around 700 calories/mole. It is important to realize that AE for the electronic energy is not due to the Fermi level shift, but is primari- ly due to the redistribution of the electrons having ener- gies nearly equal to the Fermi level. The Fermi level shift results from this redistribution. K. J. Duff (Ford Motor Co.): Just a comment to per- haps dampen your enthusiasm for the prospect of doing the calculation sufficiently well when magnetic interac- tions are included. I would refer you to a paper presented on Monday by Prof. Das and myself. It is our conclusion that the density of states at the Fermi sur- face is probably the least reliable number that can come out of even a careful band calculation. We found that satisfactory agreement with all other aspects could be obtained with a variation of that number by a factor of something like 4. J. T. Waber (Northwestern Univ.): Of how much? K. J. Duff (Ford Motor Co.): A factor of 4 would be the range of theoretical predictions of the density of states at the Fermi surface. J. T. Waber (Northwestern Univ.); I believe that those properties which involve integration over the entire density of states, such as magnetism does, may be in- sensitive to N(EP). But when you attempt to calculate the temperature dependence of that property, the slope and height of the N(E) curve, both above and below Er, become significant. They sensitively influence the answer. Well, I will not try to quote numbers here about how reliable our density of state curves are, but will refer you to our post-deadline paper [1]. We are trying to make an effort to find better ways of obtaining relia- ble curves. We recognize that peaks may exist which lead to very high values of N(E) which we might not ob- serve or see in our “smoothing” method. It is less clear at this time, whether “false” peaks might be developed. when using either the specific sampling method of Snow and Waber, or the QUAD interpolation scheme; I am inclined to think not. However, if there are large narrow peaks, then a small error in determining EP could have a profound effect on the value of N(EP). The 417–156 O - 71 - 49 753 specific Question of how well we can perform the in- interval E; + AE, will improve OUIT reliability of both the tegration of N(E) to locate Ep precisely, has not been in- N(E) curve and the Fermi level. vestigated in our post-deadline paper and will be a topic for future work. Nevertheless, it is clear that by in [1] Kennard, E. B., Koskimaki, D., Waber, J.T., and Mueller, F.M., creasing the number of E(k) values which lie in each these Proceedings, p. 795. 754 Potential-Independent Features of Crystal Band-Structure * M. M. Sdffren Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91 103 Key words: Band structure; crystal potential; electronic density of states; pseudopotential. Using theorems in addition to those previously developed [1] it has been shown that a portion of the band structure of a crystal is independent of crystal potential depending solely on the space-group to which the crystal belongs. As a result, the band structure of a crystal can be analyzed in terms of its potential-depen- dent and potential-independent parts. The latter, which we call the “invariant” band structure, is predicted as soon as the symmetry of a crystal is known. The invariant band structure of the face-centered, body-centered, and tetragonal lattices have been deduced. One of the more interesting results is the in- variant structure associated with the 3-d, and 4:f bands. This structure remains the same regardless of the width these bands may have. The existence of an invariant band structure can be applied to the problem of the determination of crystal pseudopotentials. In fact, it can be shown that any local pseudopotential for a 3d-transition metal must nearly be as strong as the actual crystal-potential itself. In subsequent work, it has been shown that the band structure of two or more phases of the same material can be precisely related. In particular the correlation between band structure of a material in its body-cen- tered and face-centered phases have been deduced. The correlations then allow the band structure of one phase of a material to be deduced from that of any one of its other phases. Reference [1] Saffren, M. M., Phys. Rev. 165, 870 (1968). *This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract No. NAS7-100, sponsored by the National Aeronautics and Space Administration. 417–156 O - 71 - 50 755 Discussion on “Potential-Independent Features of Crystal Band Structure” by M. M. Saffren (California Institute of Technology) J. F. Goff (NOL): In the first long period it is observed that the transport properties of the transition metals and their compounds show a remarkable similarity when they are compared in terms of their electron num- bers [1]. It would seem that the potential independent portions of the Fermi surface must be responsible for these similarities. Thus, the transport properties of these solids can serve as a sort of test of the effect of core potential on the higher energy portions of the band Stru Cture. M. M. Saffren (California Inst. of Tech.): I don’t quite see the connection really. J. F. Goff (NOL): You have the band structure as inde- pendent of the core potential essentially. And transport properties seem to be dependent upon that indepen- dent portion of the band structure. Did I understand you? M. M. Saffren (California Inst. of Tech.): Well per- haps we can talk about it privately. J. T. Waber (Northwestern Univ.): If I understood you correctly, you said that the ns level is always below np level. My recollection is that for heavy elements, where spin-orbit splitting is important, the order can be 2p 1/2 < 2 s 1/2 < 2. p 3/2. Have you considered relativistic ef- fects such as spin-orbit splitting and the relabelling of your states? M. M. Saffren (California Inst. of Tech.): No I did not say that. In fact np and ns cannot be ordered indepen- dent of potential. As for relativistic levels, I have not found a way so far to order relativistic states. The results stand as results for non-relativistic bands. [1] Goff, J. F., J. Appl. Phys. 39, 2208 (1968). 756 Nonlineqr Optical Susceptibility of Semiconductors with Zincblende Structure * M. I. Bell” Physics Department, Brown University, Providence, Rhode Island 02912 A simple model for the band structure and electronic density of states in zincblende semiconduc- tors has been used to calculate the dispersion of the nonlinear susceptibility responsible for second-har- monic generation. The calculation requires no adjustable parameters, and results have been obtained for GaAs, InAs, ZnTe, and InSb in substantial agreement with experiment in the energy range 0-2.0 eV. Key words: Electronic density of states; gallium arsenide (GaAs); indium antimony (InSb); indium arsenide (InAs); joint density of states; nonlinear optical susceptibility; optical pro- perties; semiconductors; zincblende structure; ZnTe. 1. Introduction The dispersion of the nonlinear optical susceptibility X%)(a)) which is responsible for the second harmonic generation of light (SHG) has been calculated for four semiconductors of the zincblende type (GaAs, InAs, InSb, and ZnTe) in the energy range 0-2 eV. A highly simplified band-structure model has been used, and as- sumptions introduced which reduce the calculation to a form similar to that of density-of-states calculations of the linear dielectric constant. SHG is by now a well-known phenomenon, and its quantum-mechanical description has been developed fully, both for localized (atomic) systems [1] and in the context of band theory [2-6]. The process is described by a third-rank tensor X%) which relates the applied macroscopic field e to a polarization which is quadratic in the field: P. (20) = x}(o)ej (o)ek (a)). (1) The profusion of variables and indices surrounding X” has led to a variety of conventions and notations, most of which are described by Robinson [7]. The definition of eq (1) will be used here. 2. Theory 2.1. General In materials with zincblende structure (point group 43m) the tensor X%) is particularly simple. The only non- *Supported in part by ARPA contract SD 86. **Corning Glass Works Foundation Fellow. zero components Xiji") are those in which no two of i, j, and k are equal. In addition, all such nonvanishing com- ponents are equal. Hence X”) is completely specified by the single component X125°) which will be calculated here. The results of Butcher and McLean [3], evaluated in the one-electron approximation, give for the suscepti- bility i € (2) as –- –- 3 3 X% (a)) 32h2 (i. ) X. |. d°kfi, rSt (**** + P., (k)P3,(k)P}, (k) [ors (k) +20) [ort (k) + ol | P:(k)P}(k)P.(k) + Pºſk)P,(k)P#(k) [ors (k) – all ſort (k) + ol P},(k)P3,(k)P}, (k) + P},{k)P., (k)P *] (2) [ors (k) - o] [ort (k) *-ºs 20] Here the electrons are assumed to occupy Bloch states l,(r) = u,(r)e”, (3) specified by the band label n and the wavevector k. The frequencies are complex and restricted to the upper half-plane; frk is the Fermi-Dirac function for the state lirk; and Prs(k) is the matrix element of the linear momentum operator between states lirk and list. Two useful observations can be made concerning eq (2). First, the result of applying the time reversal opera- 757 tor T = —io-yſ (where or is the Pauli spin operator and K is the complex conjugation operator) to a Bloch func- tion ill, is another Bloch function iſ k which is degenerate with and orthogonal to ill. Since TPT-1 =-P one obtains (r, kPs, k) = – (r, – kP|s, – k)*. (4) Thus in the integration over the Brillouin zone a product of the form Prº (k) Ps/ (k Pir’ (k) can be replaced by its imaginary part. This result is not altered when the spin-orbit interaction is taken into account, since this requires only that P be replaced [3] in eq (2) by P’ = P + X Tº " VV(r), (5) and TP'T' =– P'. The second observation is that if all the momentum matrix elements in (2) are replaced by components in the [111] direction - | Pºn - – * (p +P.+P) (6) one obtains a quantity X(11.1%) which is given by xº-ºxº (7) Thus X123°) can be written in a form which depends only upon the [111] components on the momenta. Both of these results have been used elsewhere [8] to obtain simplified expressions equivalent to (2). In the summation over bands in (2) three types of terms can be distinguished [5]. The labels r, s, t may refer to states in one, two, or three bands. In the first case it is easy to see (by writing (2) in terms of Pijl”]) that the contribution to X125%) vanishes. For two bands one may use the symmetry requirement X128 = X213 = X512 and the first observation above to show that these contributions must also vanish. As Kelley [5] has pointed out, the two-band contributions may in general be nonzero; the present zero result is due to the par- ticular point-group symmetry of the zincblende struc- ture. Thus only three-band terms need be included in the summation in (2). 2.2. Three-Band Model Even with these simplifications eq (2) is not amena- ble to direct calculation without some approximations for the matrix elements and energy denominators. It has been found that in general |Prs"(k) is nearly con- stant across the Brillouin zone [9,10]. For the imagina- ry part of the linear dielectric constant eg, this approxi- mation leads to the conclusion that the dispersion of e2 is governed by the behavior of the joint density of states for the various possible transitions [9]. The dispersion can then be obtained from consideration of critical points in the density of states [11]. The corresponding approximation for Prº*(k)Psſº(k)Prº (k) can be justified only by a detailed examination of band structure calcu- lations [8], but with the aid of eq (4) it can easily be shown that to third order in k Trºi (k) = Im [P),(k)P.,(k)P}(k)] = F + F^k; (8) Then the assumption that |P, "(k) is approximately constant leads to the conclusion that at worst F = |F"|. Thus treating Trs (k) as a constant is no less (or more) justified than the assumption usually made in density-of-states calculations that |Prs”(k) is constant. If the Fermi-Dirac function frk is approximated by 1 for valence bands and zero for conduction bands, the contribution to x123°) of a valence band 1, k) and two other bands |2, k), 3, k) can be written |. Si [012(k) = nio][ois (k) – mol iF e \* Jº (*) = –– AX% 16h? (#) X. i = 1 dºk where F-Yº Pl, 11(0)P, 11(0)P|| ||(0) = iTsa(0) | 1, i = 1, 3 s—|| ||. (10) n; = – 2, 1, 1, -1. – 1, 2 m; = — 1. – 1, 2, — 2, 1, 1 Suitable approximations for the energy denominators in (9) are also needed. The upper valence and first two conduction bands of InAs are shown in figure 1 as cal- culated by the full-zone k"p method [12]. The main features of the band structure of other zincblende-type semiconductors have been found to be qualitatively similar [12-14]. The major contributions to ex(a)) in the range 0-4 eV have been found [12] to come from transi. tions at T(k=0) and along A(k= (k/V3) [11]]. These transitions are Tse – Tse (Eo); Tzo – Tac (Eo + A0); A40, A5 – Ago (E1); and Age - A6c (E -- A1). The band struc- ture model used was chosen to provide critical points of the proper type at the energies of these transitions, 758 }Tisc 21c ºrs, º/ X5v / KIV, K2V k=#(III) FIGURE 1. k=(OOO) k=# (IOO) k=# (##O) k=(OOO) Band structure of InAs as calculated by the full-zone k p method. and includes a second conduction band in order to per- mit calculation of three-band terms. The transitions at T are described by parabolic bands extended to infini- ty: k2 a 12(k) =-o-; J7l 12 •) | | owſk)=-o-º: (11) t 2m 13 The integrals in (9) then take the form ſ dºk = 8m 19 m 1877 ſolº (k) – nigol |a) is (k) – migo] 12 11t 13 CC o!/?do. | (2x4+ o) (2x} + O.) (12) where cy = k” xi = m12 (apo-H nigo) x = m13 (06-H I?l i (1) ). (13) A contour integration yields - ie” Fm 19 m . . ." S; A º - l,62 12 1 3 ! X123 V2. The mºo? 2. Vx- Vx; (14) The exact locations in the Brillouin zone of the A critical points are difficult to determine, and in fact reference to figure 1 indicates that the upper valence and lowest conduction bands are very nearly parallel in the [111] direction for perhaps 2/3 of the zone. We can therefore approximate these critical points by 2-dimen- sional minima while retaining a parabolic second con- duction band: k} 2m 1 012 (k) = — a 1– owſk)=-o-º-º: (15) 2m 18 where the subscripts on kl, kil, and ml refer to directions parallel and perpendicular to [111]. While this approximation is crude, it has been used, together with that of eq (11), to obtain a quite satisfactory description of the intrinsic piezobirefringence of Si, Ge. and GaAs [15]. For this case (9) becomes A (2) ie” F k 6 1 ſº, F – T. m. 1 m 13 K L X. XT93 Tº hºm” apº iT J – 1 f 1 J_*) . . . Si 3. -- *k}p} du ' — * / #k} 2 log —H- (16) z – x;' -- ###p. 3. i where x; ' - m 1(a) 1 + nio)) _*|| ki. and k = TV3/a is the magnitude of the wavevector at L. The integral in (16) cannot be evaluated in closed form, so a numerical integration is required. Before at- tempting to obtain numerical results from eqs (14) and (16) it is necessary to make an observation which will 759 lead to a rather extended digression. The results (14) and (16) have been derived from (2) and therefore describe the frequency dependence of the entire (com- plex) susceptibility—no attempt has been made to separate the real and imaginary parts. However, when an exactly analogous technique [15,16] is used to com- pute e1, the real part of the linear dielectric constant, it is found that a constant (frequency-independent) term must be added to the result in order to fit the experi- mental data. This term results from the neglect of the higher conduction bands which contribute to es at high frequencies and hence, through the dispersion (Kramers-Kronig) relations, to el at low frequencies. Since identical dispersion relations [16] apply to X*) one would anticipate that the results (14) and (16) un- derestimate the real part of X123°). If the neglected con- duction bands are assumed to lie high enough in energy that alig, Gola = 0 for a) in the region of interest then the contribution of these bands to Re X125°) can be regarded as a constant. The frequency dependence of X125%) is thus given by X\}(0) = AX}(0) + 6Xº, (17) where AX125%a)) is computed from (14) and (16), and ôX1939) is a real, frequency-independent correction given by ôXº, = Xº, (0) - AX}}(0) (18) (Here a) = 0 must always be interpreted as referring to frequencies well below those of the electronic transi- tions but above the lattice resonances). While 6x128% could be treated as an adjustable parameter or inferred from infrared (10.6p.) measurements of X125%), it is more satisfying to obtain this quantity from a direct calcula- tion of X125%0). 2.3. Low-Frequency Limit A number of calculations of X125%0) have been made for various zincblende-type materials [18-20], based on an approach originated by Robinson [7]. The results of these calculations differ significantly, so it is worthwhile to attempt yet another calculation along these lines but employing the same model as will be used to compute Axiºs!”(o). The basis for the calcula- tion is provided by the relations x}=2Neº ſºl. | |(lºal?)|*||(lxals) lº E Pa = Xº e8+ x; };egey-H e e is (19) and W=-ºxºeoes-àx},eaegey— . . . (20) for the polarization P and total energy W of the system in an applied field e (in the zero-frequency limit). If the Hamiltonian is taken to be H = Ho-Hex e (21) then for N unit cells per unit volume perturbation theory gives where Eo-E!")+E,0+Eº)+E})+ . . . (23) Holm ) = E!) m) (24) E})=ee,(0|xa 0) = ee,&a (25) •) e Q (0|xals) (x|xg|0) 6 E})=– e” eaeg X. # (26) E}=e^eaegey X'X' S t (0|xa|s) (s|xe–3.6 t () try|0) (27) (EQ)—E)) (EQ)—E®) A comparison of (20) with (26) and (27) would permit evaluation of X(1)(0) and X*)(0), given a method of per- forming the summations involved. Jha and Bloem- bergen [18] employ a single energy denominator and the closure property, while Flytzanis and Ducuing [19] use the method of Delgarno and Lewis in the form of a variational principle, and Phillips and Van Vechten [20] assume the existence of only two states. The method here will be to use the formalism of band theory and restrict the summations to states in the three bands described above. Since the lowest valence band lies well below any bands with which it can interact through the perturbation in (21), the perturbation energies (25) and (26) can be regarded as arising entirely from pertur- bation of states in the top valence band |1). Then (20), (26), and (27) give | (28) 21 31 760 | (2) = — 3 3 ------ Cy BY 3We ſ dºk |rº + (1|x, 3 ) ( 3|x|, |2 ) ( 2 |xy|l )) + ((2-2)-(1-1)+; alºlºl)(Glºſs)-(lºh)) These results can be expressed in terms of the well- known relation for off-diagonal matrix elements of P and x [21] (O |P|3)=imoog (o |x|3). (30) As before, |Prs"(k) and Tºſ(k) will be approximated as constant throughout the Brillouin zone. These quanti- ties will be evaluated at k = 0, employing a model proposed by Cardona [22] in which the zincblende- type material is regarded as derived from a fictitious diamond-type material by the action of an antisymmet- ric potential V. In the diamond-structure materials the top valence and first two conduction bands at k = 0 have the symmetries described by the representations T25'. To", and T15 of the full cubic group. If all other states are neglected, the effect of V is to mix the T25, and T15 states to produce two states belonging to the representation T15 of 43m. The T25" state (now T1) is unaffected, and the Hamiltonian for the Ti5 states (referred to T25', T15 basis) is E W H = IV+ 0 (31) where E(T25')=0, and E(TIs)=E. This Hamiltonian has eigenvalues E(Tisc) = }{E+[E”-- (2W)*]1/2} (32) E(T150) = }{E – [E”-- (2W)*]1/2} and eigenvectors l, 15c F all 15 - blºgs, (33) il 150 = bill 15 + alliz5. where a--- (*** y E V2 31 (34) ((I|x|2)(2xel3)(3|x|I) 1 , . Eğı (1|x. 2 ) ( 2 |x., 1) (29) and Eal = E(Tisc) – E(T150) = [E*-ī- (2W)*]1/2 (35) If E31° => E21° then in this model eqs (28) and (29) give _3eh o). OK X% (0) 777. Xaa, (E31 ) 3 (36) where i(T.s |P Tis) = Q. (37) While Xaa") can be determined experimentally, Q and V depend upon the fictitious diamond-structure materi- al referred to above. Since E31 is known from experi- ment, V can be found from (35) if E is known. Figure 2 shows E and Q as a function of the lattice parameter a for Si, Ge, and o-Sn. The values for E are those of the 4 | | 3 o C O * 2 > <[ 9 Cl– ., 3 H 2 Dr. 2 LL LL] 2 > LL! Lid —l Lil >< OC H. <ſ > 2 | | 5.O e 6.O a 6.5 LATT ICE CONSTANT (A) FIGURE 2: Dependence on lattice constant of the T35 - T15(E6) gap and the matrix elements in diamond structure materials. 761 Eo' peak in the electroreflectance spectra [23]. The matrix elements are determined from cyclotron resonance [24,25]. For a zincblende-type material of lattice constant a, figure 2 was used to determine the values of E and Q for a fictitious diamond-type material of the same lattice constant. Since Eo' varies in almost exactly linear fashion with the lattice constant from Si to o-Sn, E was obtained by linear interpolation. Since Q varies little (less than 8%) the method used to esti- mate it is not of great importance. Compounds of ele- ments from the same row of the periodic table were as- signed the value of Q belonging to the group IV material in that row. In general, compounds of elements from different rows were assigned the average of the values for the group IV materials in each row. The details of this procedure are given elsewhere [14]. The linear susceptibility X'...} (0) was obtained from Phillips and Van Vechten [20], and E31 from electroreflectance data [22]. Table 1 lists all the necessary parameters and the results of eq (36) for 9 zincblende-type semicon- ductors. TABLE 1. Parameters used in the calculation and re- sults for the low-frequency limit X.(0) (2) a(A) | E31(eV)|E(eV) X") () (a.u.) | 2/(eV)|(10-8 esu) GaAs.....] 5.62 4.44 3.15 ().79 || 0.535 3.12 97.7 GaSb..... 6.1 || || 3.27 2.63 | 1.07 ,518 1.94 199 In As...... 6.05 4.44 2.69 0.90 .518 3.53 122 InSb...... 6.49 3. 16 2.22 | 1.17 .50] 2.25 27] Gal’...... 5.45 || 4.78 || 3.34 || 0.65 .530 3.42 7(), () In P....... 5.87 4.72 2.89 .68 .535 3.73 83.7 AlSb..... 6.14 3.72 2.60 .73 .513 2.66 126 ZnTe..... 6.09 5.40 2.65 .5() ,518 4,7] 50.3 ColTe.....| 6.48 5.30 2.23 .49 .50] 4,8] 51.4 2.4. Dispersion The dispersion of X: (a)) can now be obtained from eqs (14), (16), and (17). Since experimental data are available [8.26] only for GaAs, InAs, InSb, and ZnTe, discussion will be restricted to these four materials. With the aid of eqs (33-35) we find that at k = 0 F= iP2OV/Eaſ (38) where i(T., |P I, ) = P. (39) Examinations of the matrix elements for the various possible transitions show that the spin-orbit splitting of the valence band at T may be taken into account by evaluating (14) with ha)0 = E0, ha)0' = E0 (for the light- hole and heavy-hole bands) and with ha)0= E0+ A0, hoo' = E0' + Ao (for the split-off band). The spin-orbit splitting of the Ti5 conduction states has been neglected. The valence band masses (mºth, m*nh, and mºsh for the light-hole, heavy-hole and split-off hole bands, respectively) were obtained from the average masses [22]. * = − 4 – C2 #–4–5 (1+（) (40) where A, B, and C are the inverse mass parameters of Dresselhaus, Kip, and Kittel [27], and A' is given by [22] —l EoF Y 4- (*, *26-21)+. (41) where p__{*(Eaſt E) E0E31 Y (E31 + E) (, = –0.8 ± =4- E31 (42) 20° M = — º . E31 The assumption that the shape of the second conduc- tion band near T is determined entirely by its interac- tion with the valence band leads (neglecting the spin- orbit interaction) to the result that the effective mass is approximately mºhn. In eq (16) the value m13– m”hh/2 has been used. The spin-orbit splitting along A can be treated in the same way as that at T, but the split-off valence band may be ignored. The interband mass m, is found by assuming that the shape of the valence band is deter- mined by the A3e-A1c interaction. Then m = m^1 (Lic)/2. - Table 2 lists the values used in the calculation. The energies E0, E1, Ao, and AI were obtained from elec- troreflectance measurements [22,28], with the excep- tions noted in table 2. The matrix element P was deter- 762 TABLE 2. Parameters used in the calculation of X;(0) GaAs InAs InSb ZnTe P(a, u.)..................... ().68 0.62 0.56 (). 62 EoſeV)..................... 1.43 8.4] b. 17 2.25 E1(eV)..................... 3.12. 2.5() 1.88 3.6] Ao(eV)..................... ().34 *(). 43 ().82 ().93 A1(eV)..................... .23 .28 .50 .57 m?.…. .067 .026 .()145 “. 180 m?....… ..] 16 | . 132 . 127 c. 199 A............................ — 7.39 — 17.1 – 26.5 – 4. () B............................ –4.93 – 14.9 | –24. I — 2.3 |Cl..…. 5. ()6 |().9 13.9 2.() A'..........…... “— 6.30 “— 8.7] “— 7.68 “— 3.23 “F. Matossi and F. Stern, Phys. Rev. l l 1, 472 (1958). "O. Madelung, Physics of III-P Compounds (J. Wiley and Sons, Inc., New York, 1964), p. 53. “Calculated as described in text. mined by the method described above for Q. The in- verse mass parameters for InSb and ZnTe are available from cyclotron resonance experiments [29,30]. Those for InAs and GaAs were obtained from the full-zone k'p band structure calculations [12,14], and the calcu- lated values for InSb [14] were found to give slightly better results than the experimental ones. The conduction band masses m”, (at T) were taken from magneto-optical absorption and Faraday rotation measurements [31-33] , except in the case of ZnTe where no experimental data are available. In the three- band approximation this mass is given by [22] E31 — E0 (43) J7? :: = 1 + |*(#1 | ) E31 3 Eo Eo-H Ao mo The results of (43) agree well with experiment (10% or better) in the three materials where m”, has been mea- sured, so the use of this approximation for ZnTe is probably justified. Finally, m*1 (L16) was obtained from the kºp calculations, again with the exception of ZnTe where (43) was used with Eo and Ao replaced by E, and A1, E31 replaced by E(L30) – E(L30), and E by the Lac – L3 gap in the corresponding homopolar material (average of Ge and cº-Sn)[22]. The frequencies appearing in (9) are complex; the real-frequency limit can be taken in the conventional way [3]. However, for the sake of comparison with ob- servations at room temperature, thermal broadening has been included by using the complex frequency a + im, with m = 0.05 eV (= 2 kT). Equation (17) was evaluated for energies from 0.05 to 2.0 eV at intervals of 0.05 eV by a computer pro- gram which required approximately 2 minutes per º & § >< | O O.5 |.. O |.5 2.O ENERGY (eV) FIGURE 3. Real and imaginary parts of X. (a) as calculated for y a HAS. material on an IBM 360 Model 50 computer. (The in- tegral in (16) was evaluated by a 6-point Gauss quadra- ture.) Figure 3 shows the real and imaginary parts of X125%a)) for GaAs. The modulus of the susceptibility 3- S § — O.5 H - O | | | | | | O O.5 |.. O |.5 E NERGY (eV) 2. O FIGURE 4. The modulus |X|}}(0)| as calculated for GaAs. 763 3.O - In As / S *—, / O \ 3. u) Q) to o 3 -, * *-*. & 8 -, (ſ) - Q) Q I - Q - 3 O ———1–1–1–1–1–1–1–1–1–1––––––. * O O.5 |.O |.5 2.O S & ENERGY (eV) & FIGURE 5. The modulus |x}}(0)| as calculated for InAs. |X125%) is given in figures 4-7, together with the available experimental data. The experiments of Chang, Ducuing, and Bloembergen [26] measured |X125%) relative to its value in KH2PO, (KDP). The result of Bjorkholm [34] for KDP, X125(*) = (1.6+ 0.4) X 10 °esu, was used to convert these measurements to absolute values. The results, indicated by circles in figures 4-7, have a total uncertainty of about 30%, most of which is contributed by the 25% error in the mea- surement for KDP. The relative measurements for the semiconductors have an error of about 15%. Also shown (by triangles) are the results of absolute mea- surements at 10.6p by Patel [35] and Wynne and Bloembergen [36]. These have uncertainties of the O | | | | l | | O O.5 |.O |.5 2.O ENERGY (eV) FIGURE 7. The modulus |x}}(0)| as calculated for InSb. 3. Discussion 3.1. Low-Frequency Limit Table 3 compares the results of the present calcula- tion with those of Flytzanis and Ducuing [19] and Phil- order of 30-50%. |2 | TABLE 3. Comparison of the results for the low- frequency limit X\}(0) with experiment and with the ° a calculations of Flytzanis and Ducuing (F–D) [19] and Phillips and Van Vechten (P-VV) [20]. Values are in units of 107* esu. - P–VV Present F—D work Expt. x(2) fºxº) GaAs...................... 190 256 122 97.7 a 90+ 30 GaSb...................... 160 || 433 236 199 a 302+ 100 InAs.......................] 410 | 380 157 122 a b 200+60 *- InSb....................... 650 | 611 282 27] 3 H O - Gap........................ 140 | 186 85 70.0 a 52 + 17 InP......................... 280 || 269 106 83.7 AlSb....................... 70 || 337 111 126 ZnTe....................... 257 53 50.3 b44 + 16 CaTe...................... 257 27 51.4 b80 + 30 9. as º º 2.O ENERGY (eV) * Ref. 36. FIGURE 6. The modulus |x}}(0)| as calculated for ZnTe. * Ref. 35. 764 lips and Van Vechten [20]. Two values of X125%0) are given for [20] since the authors argue that their results should be multiplied by a factor off.”, where f is the fraction of covalent character in the crystal bonding [37]. It is interesting to note that in five of the seven cases treated by Flytzanis and Ducuing their results agree well with the uncorrected values of X125%0) ob- tained by Phillips and Van Vechten. The present results, however, are in general agreement with the cor- rected values f.”X125°(0). Both of the calculations [19] and [20] employ pure spº hybrid molecular orbitals, and Phillips and Van Vechten suggest that the cor- rection f.” may arise from the neglect of other elec- tronic configurations. The use of a band-structure for- malism and experimentally determined momentum matrix elements seems to avoid this difficulty. 3.2. Dispersion The results for GaAs and InAs are qualitatively similar and will be discussed together. In each case the most notable feature in the dispersion is a peak in the vicinity of E1/2 and (E1 + A1)/2 which arises from the coincidence of the harmonic photon energy 2ha with the energy of the A30 – Aic transitions. The broken curve for InAs (fig. 5) was calculated by assuming the experimental value X125%)(0) = 2 × 10−6esu [35] rather than X125%0) = 1.22 X 10-6 as calculated above. The result indicates that the present calculation of the dispersion is consistent with the observed low-frequen- cy value and that the result of part 2.3 is probably too small. It has been suggested [26] that the sharp decrease in X125%a)) observed between 1.62 and 1.66 eV in these two materials is due to the variation of T(k) across the Brillouin zone, since no pronounced structure is ob- served in the linear dielectric constant e(a)) or e(200) at this energy. While it is likely that the wavevector de- pendence of T produces some structure in X125%0), the present calculation (which assumes T constant) reproduces the structure near 1.66 eV fairly well. As can be seen in figure 2 (for GaAs) the calculated decrease is caused primarily by a sharp drop in the real part of the susceptibility. The results are similar for InAs. It should be noted that it is possible for the non- linear susceptibility to show rapid variations as a func- tion of energy in regions where the linear susceptibility does not. While Imy” will in general exhibit rapid variation only where Imx") does also, the linear suscep- tibility is restricted by the requirement Im}( > 0 [38] while no such restriction applies to the nonlinear susceptibility. Thus, contributions to Im}(*) from dif- ferent points in the Brillouin zone may be of opposite sign. In an energy region where Imx") is (for example) increasing rapidly, Imx") may decrease or even change sign. The real part of the susceptibility, which is related to the imaginary part by the Kramers-Kronig relations, may then behave quite differently in the linear and nonlinear cases. The fact that most experiments measure only the modulus |X|28°Co) of the suscepti- bility further complicates the situation. While structure in X*)(a)) can be expected at energies where X")(a)) or X(1)(200) varies rapidly, X*) does exhibit additional structure unrelated to the matrix element product T(k). In ZnTe (fig. 6) the main peak near 1.8 eV is also due to the coincidence of the harmonic photon energy 2ho with that of the E1 and E1 + A1 transitions. The small peak near 1.15 eV is produced by transitions at T. The failure of this peak to appear experimentally casts some doubt on the accuracy of eq (14). Further experimental results in the 0.5-1.5 eV region for these materials would be needed to confirm (14). The calculation for InSb is less satisfactory than the others. The experimentally observed peak near 1.6 eV has been attributed to the coincidence of the funda- mental photon energy ha) with the El peak [8,26]. This structure appears in the calculation, but it occurs about 0.2 eV too high in energy. Band structure calculations [39] indicate that the critical point at Ei may occur ex- actly at L. In that case the eqs (15) would not apply, and (16) would have to be modified to take into account the correct symmetry of the critical point. 4. Conclusion A highly simplified band-structure model has been found to predict the dispersion of the nonlinear optical susceptibility of four zincblende-structure semiconduc- tors with reasonable accuracy in the energy range 0-2 eV. Separate calculations were made of the low- frequency limit and of the contributions of critical points in the joint density of states for the top valence band and the first two conduction bands. The results for the low-frequency limit were found to agree well with the calculations of Phillips and Van Vechten [20], and the calculated dispersion is in substantial agree- ment with the experimental work of Chang, Ducuing and Bloembergen [26]. Variations in the susceptibility which had previously been regarded as arising from the dependence of interband momentum matrix elements are successfully predicted by the model in spite of the fact that the momenta are treated as constant throughout the Brillouin zone. WaVeVeCtOr 765 5. Acknowledgments The author would like to express his gratitude to Professor H. J. Gerritsen who suggested this problem and to Professor M. Cardona who gave valuable guidance and encouragement. Professor F. H. Pollak and Dr. C. W. Higginbotham were generous with advice and helpful criticism. Dr. J. C. Phillips, Professor J. A. Van Vechten, Dr. J. J. Wynne, and Professor N. Bloem- bergen made available their results prior to publication. 6. References [1] Armstrong, J. A., Bloembergen, N., Ducuing, J., and Pershan, P. S., Phys. Rev. 127, 1918 (1962). [2] Loudon, R., Proc. Phys. Soc. 80,952 (1962). [3] Butcher, P. N., and McLean, T. P., Proc. Phys. Soc. 81, 219 (1963): 83,579 (1964). [4] Kelley, P. L., J. Phys. Chem. Solids 24, 607 (1963). [5] Kelley, P. L., J. Phys. Chem. Solids 24, 1113 (1963). [6] Cheng, H., and Miller, P. B., Phys. Rev. 134, A683 (1964). [7] Robinson, F. N. H., Bell System Tech. J. 46,913 (1967). [8] Bloembergen, N., Chang, R. K., and Ducuing, J., in Physics of Quantum Electronics, P. L. Kelley, B. Lax, and P. E. Tannen- wald, Editors (McGraw-Hill Book Co., New York, 1966), p. 67. [9] Brust, D., Phys. Rev. 134, A1337 (1964). [10] Cardona, M., and Pollak, F. H., Phys. Rev. 142, 350 (1966). [11] Van Hove, L., Phys. Rev. 89, 1189 (1953). [12] Higginbotham, C. W., Pollak, F. H., and Cardona, M., Proc. IX Int. Conf. Phys. Semiconductors Moscow (Nauka, Leningrad, 1968). [13] Pollak, F. H., and Cardona, M., J. Phys. Chem. Solids 27, 423 (1966). [14] Pollak, F. H., Higginbotham, C. W., and Cardona, M., J. Phys. Soc. Japan Supplement 21, 20 (1966). [15] Higginbotham, C. W., Cardona, M., and Pollak, F. H., Bull. Am. Phys. Soc. 14, 416 (1969); Phys. Rev. 184, 821 (1969). Cardona, M., in Solid State Physics, Nuclear Physics, and Par- ticle Physics, I. Saaverdra, Editor (W. A. Benjamin, Inc., New York, 1968), p. 737. Kogan, Sh. M., Sov. Phys. JETP 16, 217 (1963). Jha, S. S., and Bloembergen, N., Phys. Rev. 171, 891 (1968). Flytzanis, C., and Ducuing, J., Phys. Rev. 178, 1218 (1969). Phillips, J. C., and Van Vechten, J. A., Phys. Rev. 183, 709 (1969). Schiff, L. I., Quantum Mechanics (McGraw-Hill Book Co., New York, 1968). Cardona, M., J. Phys. Chem. Solids 24, 1543 (1963). Cardona, M., Shaklee, K. L., and Pollak, F. H., Phys. Rev. 154, 696 (1967). Levinger, B. W., and Frankl, D. R., J. Phys. Chem. Solids 20, 281 (1961). Hensel, J. C., and Feher, G., Phys. Rev. 129, 1041 (1963). Chang, R. K., Ducuing, J., and Bloembergen, N., Phys. Rev. Letters 15, 415 (1965). Dresselhaus, G., Kip. A. F., and Kittel, C., Phys. Rev. 98, 368 (1955). Cardona, M., Shaklee, K. L., and Pollak, F. H., Phys. Rev. Let- ters 23, 37 (1966). Bagguley, D. M. S., Robinson, M. L. A., and Stradling, R. S., Phys. Letters 6, 143 (1963). Stradling, R. A., Solid State Comm. 6, 665 (1968). Cardona, M., Phys. Rev. 121, 752 (1961). Pidgeon, C. R., and Brown, R. N. Phys. Rev. 146, 575 (1966). Wrehen, Q. H. F., J. Phys. Chem. Solids 29, 129 (1968). Bjorkholm, J. E., IEEE J. Quant. Elec. QE-4, 970 (1968). Patel, C. K. N., Phys. Rev. Letters 16, 613 (1966). Wynne, J. J., and Bloembergen, N., Bull. Am. Phys. Soc. 14, 26 (1969); Phys. Rev., to be published. Phillips, J. C., Phys. Rev. Letters 22, 645 (1969). Landau, L. D., and Lifshitz, E. M., Electrodynamics of Continu- ous Media (Addison-Wesley Pub. Co., Reading, Mass., 1960). Higginbotham, C. W., Ph. D. thesis, Brown University, 1969 (unpublished). 766 Model Density of States for High Transition Temperatures Beta-Tungsten Superconductors” R. W. Cohen, G. D. Cody, and L. J. Vieland RCA Laboratories, Princeton, New Jersey O8540 We have applied a simple density of states model to the problem of superconductivity in high To beta-tungsten superconductors. If we assume that the interaction responsible for superconductivity is predominately between d-band carriers and acoustic phonons via a deformation potential matrix ele- ment, simple analytic expressions for the effective electron-electron coupling constant A and Tc(A) can be obtained. The quantities A and T. can then be estimated using parameters determined from an appli- cation of the density of states model to the cubic state elastic constants. We are able to establish the simple condition X > Aeri, as 0.7 for the existence of a cubic-tetragonal lattice transformation in these materials. Using our result for T., we find Te - 15 K for all materials which exhibit a lattice transforma- tion. Thus, we have established the relation between high Tc superconductivity and lattice transforma- tion in the 8-W compounds. Key words: Beta-tungsten; lattice transformation; model electronic density of states; superconduc- tivity. 1. Introduction In the field of superconductivity, it is of prime im- portance to determine the cause(s) of the observed high transition temperatures in the 8-tungsten compounds and to utilize this knowledge, if possible, to exceed the present maximum T. of about 21 K. In approaching the problem of the transition temperature, it can be as- sumed that there is little hope unless an understanding is achieved of the anomalous normal state properties [1] of these materials with particular attention to the cubic-tetragonal lattice transformation [2,3]. Recently attention has been focused on a unique structure in the electronic density of states for these materials as the source of their unusual normal state properties, and models have been developed which permit the correla- tion of this structure with high T. The Labbé-Friedel- Barišić [4,5] model used a one dimensional (3 indepen- dent degenerate sub-bands) tight binding calculation to obtain a density of states in energy N(E) for electrons with d-like character which has singularities at the band edges and a gradually decreasing N(E) as one proceeds away from the band edge. This model can ac- count for the tetragonal transformation in V3Si and *Work supported in part by the National Aeronautics and Space Administration under Contract No. NAS 8-21384. Nb5Sn. The authors further proposed [5] that the high T.’s in these materials result from the same source, i.e., the placement of the Fermi level in a high density of states region near a band edge. It was later shown [6,7] that singularities in N(E) are not necessary in order to explain the anomalous normal state properties, but that the idea of independent d-sub bands is essen- tial. It is possible to explain a wide variety of experimen- tal results in both the high temperature cubic and low temperature tetragonal lattice phases using simple rectangular energy bands [6,7] with just three impor- tant band structure parameters obtained from experi- ment. The results of calculations using this simple model are often in simple analytic form and are surpris- ingly successful in quantitatively understanding the ex- perimental data. Indeed the success of this model sug- gests its use in an attempt to calculate Tc from the basic parameters of the normal state. The usual calculations of superconducting transition temperatures require detailed information about the phonon spectrum, the Fermi surface, and the electron- phonon coupling [8]. Such information is clearly not available for the high T. (3-tungsten superconductors, so that accurate calculations of Te are not yet possible. However, as noted above, our analyses of the normal state properties of these materials has led to an experi- 767 mentally determined set of parameters that can be used in a first principles calculation of To. The primary as- sumption is that the predominant contribution to the electron-phonon interaction is between d-band parti- cles and acoustic phonons via a simple deformation potential matrix element. With this assumption, simple analytic expressions for the effective electron-electron coupling constant A and the transition temperature can be obtained. The quantities A and To can then be com- puted using parameters determined only from an analy- sis of the temperature dependent elastic constants [6]. Although such an approach is, admittedly, over sim- plified, we believe that it contains the essential in- gredients of the problem of superconductivity in these materials. It enables us to establish, in a simple way, the relation between high T. superconductivity and the cubic-tetragonal lattice transformation. In section 2, we sketch the important aspects of the electronic band structure model for 8-W superconduc- tors. In the next section we derive expressions for Te and N and relate the magnitude of A to the condition for the existence of the lattice transformation from the high temperature cubic to the low temperature tetragonal phase. The effect of the tetragonal transformation on the T.’s of VoSi and Nb5Sn is discussed and compared to the results of previous work. In a final section, we enumerate our conclusions. 2. The Model Density of States We present here the important aspects of the model electronic density of states. This model has been suc- cessful in explaining for Nb5Sn and VaSi (where data is available) [6,7,9,10] the behavior of the elastic con- stants and magnetic susceptibility in both the cubic and tetragonal lattice states, the magnitude of the tetragonal lattice distortion, the temperature depen- dence of the electrical resistivity, the acoustic attenua- tion, and the low temperature specific heat. As is usual in the case of transition metals, the rele- vant electronic band structure is presumed to consist of a narrow high density of states d-band overlapping a wide, low density of states s-band. The additional as- sumption is made that the d-band is almost entirely empty (or filled) so that the Fermi level at T=0 K would occupy a position E = EF(0)= kpſo - 10° K, very close to the d-band edge E=0. Over the energy range of in- terest (<108 K), the density of electronic states N(E) is regarded as constant both above and below the d-band edge. For the purpose of calculating the superconduct- ing transition temperature, we shall assume that only the d-band carriers contribute to superconductivity, so that we may ignore the s-band entirely. We consider, without loss of generality, the case of a nearly empty band. The density of electronic states for electrons of one spin in the cubic lattice state, including all interac- tion effects, is written in the form N(E) = No, E > 0 = 0, E -< 0. (2.1) The density of states in the cubic state is shown in figure la. In order to deal with the effect of homogene- ous uniaxial strains eii directed along the crystallo- graphic axes i (i = 1,2,3), we follow Labbé and Friedel [4] and assume that the d-band consists of three inde- pendent equal contributions (sub-bands) arising from the chains of the B-tungsten lattice. Because of the as- sumed independence of the sub-bands, under a uniaxial strain directed along a single transition metal chain, only the sub-band associated with that chain is per- turbed. The shift of the sub-band i under the strain eii is given by ôE;= Ueii (2.2) where U is a deformation potential. Thus, in the tetragonal lattice state in which the spontaneous strains N(E) a No LZ Z_2 z zºz O E F (d) N (E) No H N 5 No ſ I f -Ueo O EF + # Uso (b) FIGURE 1. The density of states configuration in (a) the cubic lattice state and (b) the tetragonal lattice phase with the sense of eo corresponding to that for Nb5Sn. At T=0 K, the Fermi energy EP = kBT, in the cubic state and EF = (– Ueo + 3ku To) in the tetragonal state. The sub-band displacement at T=0 K Ueo/kh = 2To [l-exp (-To/Tu)]-'. For Nb5Sn, To = 80 K, Tin - 43 K, and U = 4.1 eV, so that Ueolkh = 190 K and ea = 4.0 × 10−3. 768 are e11 = — eoſT) and e22 = e23 = 1/2 eoſT), the positions of the sub-band edges are 8E1 =– Ueo (T) (2.3) The density of states in the tetragonal state is shown in figure 1b for the case eoſT) > 0 (the case of Nb5Sn). C11 (T) – C12 (T) = —# NoU*[] – exp (— To/T)] +A11 – A 12 where A11 and A12 are temperature independent core contributions to C11 and C12, respectively. The quanti- ties WoOº, A11, A12 and To can be determined by fitting the above expressions to experimental data. As can be seen from eq (2.4a), the quantity [C11(T) – C12(T)) decreases with decreasing temperature. If the condi- tlOn # NoU* > (A11-A12) (2.5) is obeyed, the quantity (C11 – C12), which represents the restoring force against a tetragonal transformation, will vanish at a finite temperature. The lattice then transforms to a stable tetragonal phase, the sense of the transformation (the sign of eo) being determined by the higher order elastic constants. The transformation is predicted to be first order, occurring at a temperature Tin which is slightly higher than the temperature at which (C11 – C12) extrapolates to zero. We shall ignore this small temperature difference (about 2 K for Nb5Sn [To = 43 K]) and define Tm by setting the right side of eq (2.4a) equal to zero: l-exp (-To/Tm)=3(A11-A12)/2NoU*. (2.6) At temperatures below Tin, the spontaneous tetragonal distortion eoſT) grows rapidly, and the elastic constants Cij are soon restored to their lattice values Aij [7,10]. The values of the sub-band displacement at T=0 K for eo -> 0 (Nb5Sn) and eo - 0 (V3Si) are given by the expres- SIOIlS Ueo(0) =2kBTo [1 – exp (— To/Tm)]-1, (Nb5Sn) am – kBTo [1 — exp (— To/Tm)|-1. (V3Si) (2.7a) (2.7b) dk' X ()--ſº, TX ºr x. M.,(q,x)'aſa, x) i n" N In this equation, g;(k') is the temperature dependent propagator for quasi-particles of 4-momentum k' = (k', io') in sub-band j, and d(q, \) is the propagator for The experimental quantities which will concern us in superconductivity calculations are the temperature de- pendent elastic constants C11(T) and C12(T) and the sub-band displacement Ueo(T). These quantities have been calculated in previous work [6], and we merely state the results here. In the cubic lattice state, which exists above a lattice transformation temperature Tm, we have (2.4a) cºſt)=- NoD*[1 - exp (- To/T)] +A11, (2.4b) Even for fairly large reduced temperatures TT, is 0.7, the results (2.7) are good approximations for the spon- taneous strains. Thus, given the expressions (2.3) for the various sub-band displacements and the result (2.7), we can determine the Fermi level position in relation to the various sub-band edges. This is shown in figure 1b. The various energies at T = 0 K are given in the figure caption. It is noteworthy that at sufficiently low tem- peratures the final density of states is 1/3 No for Nb5Sn (2/3 No for V3Si). 3. Calculation of To for 8-W Superconductors 3.1. Formalism and Assumptions In order to treat superconductivity in these materi- als, we shall make certain reasonable simplifying as- sumptions that will allow us to give estimates of Tc from the elastic properties alone and to pinpoint the parame- ters which control To. First, we shall assume that the first order process for virtual phonon exchange is the essential interaction for superconductivity [11]. We shall also assume that the d-electrons are primarily responsible for superconductivity. The large penetra- tion depths [12] and small coherence lengths [13] in these materials are justifications for this assumption. We treat each sub-band separately; each sub-band has its own energy gap parameter and density of states function. In the Nambu [14] formalism, the self energy Xi of an electron in sub-band i is (3.1) phonons of 4 momentum q = (Q, ia) - io'). The summa- tion convention has been employed for the index j. The matrix element Mij(q, \) is the electron-phonon coupling 769 for transferring a quasi-particle from sub-band i to j using a phonon (q., A). The Ti are the Pauli spin matrices (with To the unit matrix), and all frequencies have the discrete values a = TT (2n + 1) with n integral (we use units where h = k = 1). We have, temporarily, ignored the Coulomb repulsion. Nambu’s ansatz for the self energy is X (k) =io(1–Z,(k)) to 4 x(k)T, Foſk)7. & l (3.2) Here X(k)|Z(k) is the contribution of (3.1) to the quasi- particle energy and diſk)|Zi(k) is the energy gap parameter. The quasi-particle Green's function at Te is In eq (3.4), e(k) is the quasi-particle energy without in- cluding renormalization effects, measured from the un- renormalized Fermi energy. For the phonon propagator, we use that of bare phonons of frequency on: 2004 d(q, \) = (3.5) (ia) – io')*-aſſ In order to evaluate the functions Z, X and d), we em- ploy the method of Koonce and Cohen [15], valid for the case where the Fermi energy is larger or of the order of the important phonon energies. We first per- form the sum on n' in eq (3.1) where the functions given by Z;(k’), Xi(k’) and dj(k’) are considered to be indepen- g;(k) = ia)Zi (k) To – Ei(k)Tait (p;(k)T, (3.3) dent of a)'. Performing the sum using the Poisson sum- Sl (ia)Z;(k))2 - €” (k) º sº mation formula and requiring the self-consistent condi- - z ºf t e º ºs tion that the T3 and T components of Xi(k) be indepen- where the energy Ei(k) is written in the form dent of a) and the To component be proportional to a) e; (k) = e(k) + Xi(k). (3.4) [16], we find dk' |Mij(q, \)| , d | 1 – Z; = ſ *** **** sgn & H = −III (3.6a) > (2+)" Z. 7 dé, on-Flé, dk' |Mij(q, \)|* Sgn ś i = — & Y . . . Y. * f f 3.6b x--> ſº Z, 04+|ć; (3.6b) dk' |Mij (q, \)|*A (f(- ) f(&) A = (Zi) T' > || * Y \ " |Mij f | #| Hº--— | (3.6c) - J (2+)" Z. é, loº-Fé, on-g In eqs (3.6), Z; = Z;(k) º Z} E Z;(k') , etc., Ai- bi/Zi, Be = To-1, f(&,') is the Fermi function, and £;= ej|Z;= E – EF (Tc) (3.7) is the quasi-particle energy; the observed density of states refers to the energies Ej. Thus, the functions X; do not appear explicity in eqs (3.6a) and (3.6c) for Zi and Ai, so that for the purposes of calculating Tc, it is not necessary to calculate the X. In deriving eqs (3.6), we have ignored the effect of real phonons, which are not expected to be important at low temperatures To sº on. We have also replaced Zi and Xi by their T–0 K values since these quantities do not vary significantly in the temperature range T's Tc. It is now necessary to make some statement about the matrix element Mij(q, \). We shall employ a defor- mation potential matrix element for longitudinal phonons [17]: |Mij (q, \)|*= q^U*|2poal for longitudinal phonons (3.8) = 0 for transverse phonons, where U is the deformation potential defined in section 2 and p is the atomic mass density. The actual matrix element is undoubtedly reduced by screening [17], but Umklapp processes and a nonspherical Fermi surface may bring in transverse phonons [17] and thereby in- crease the effective interaction. Therefore, the errors involved in using (3.8) tend to cancel and may permit (3.8) to give us a rough picture of the magnitude of the interaction. We shall also assume that this matrix ele- ment applies to both inter- and intra-sub-band transi- tions, a crucial assumption in considering the effect of 770 the lattice transformation on Te. Next, for oal we em- ploy a Debye spectrum: (3.9) COqL - SLQ, where SL is the longitudinal sound velocity which, at low temperatures, is given approximately by [7,10] SL = (A11/p)”. Finally, we replace the factor D = (a) -- |&;"|)-1 in eqs (3.10) Zi = 1 +% (U”/A11) ſ dé!N,(£) [6(§ – 6p.) +6(§ + 6p)] A = (AU’Anz) ſig-, dé, Nº.(6) (I/28) tanhº, Bºğ. Here Nº.(§) is the “bare” density of states of sub-band j. The renormalized density of states N(j), which was given in section 2, is related to Ni (; ;) through the formula N(£j) = Z;N,(& j). (3.13) 3.2. To in the Cubic State We first solve eqs (3.12) for the cubic state which will exist at the lowest temperature if the inequality (2.5) does not hold. Here the Ai and Zi = Z are the same for all sub-bands, and we easily obtain Z = 1 +} \ (3.14) 1 +3. A T. = 1.13 (6p To)” exp |- Hº! (3.15) where the effective electron-phonon coupling constant \ is given by X (1++ X) = No U”/A11. (3.16) In eq (3.15), we have included the term pºº, which represents the Coulomb repulsion [18]. In deriving eq (3.15) for Tc, we have assumed To is sufficiently low so that Epſ'To) = To = Tc. Equations (3.14) and (3.15) are our equations for Z and To for high To B-W superconductors which are analogous to those of McMillan [19] for ordinary super- conductors. The reduction of (Z – 1) by a factor of two from the expression of McMillan results from the fact that the band edge cuts off phonon interactions at ener- gies below the Fermi level. The factor Toº!” in the pre- exponential in eq (3.15) results for the same reason. Note the predicted isotope effect To Cº M-1/4 instead of the usual M-1/* dependence. Equations (3.14) and (3.15) would be of little use were it not for the fact that we (3.6) by the BCS-like model D= (0,1)- for lé, - 0, (3.11) = 0 for |&"| > 6p, where 6b is the Debye temperature. The choice of 0d as a cut-off energy is reasonable since 0D is approximately equal to the average longitudinal phonon energy. Using eqs (3.8)-(3.11) in eqs (3.6a) and (3.6c) we find that Zi and Ai are constant and are given by (3.12a) (3.12b) can use eq (3.16) to estimate A from quantities deter- mined from the cubic state elastic constants (sec. 2). For example, if we apply our equations to transforming Nb5Sn and V3Si, we find, respectively [20], A = 0.92, T. = 28 K and A = 0.72, To = 21 K. Our values of A, calcu- lated from eq (3.16) are quite close to those given by McMillan [19] from his equation for T. However, because of the difference between our eq (3.15) for Te and that of McMillan, there is only an approximate rela- tion between them. Our calculated value of To for V3Si is close to the observed value of 17 K, whereas the com- puted value is about 10 K too large for Nb5Sn. The sig- nificance of this discrepancy will be discussed below when we calculate Te for the tetragonal lattice configu- ration. 3.3. Relation of A to the Tetragonal Transformation Using the foregoing results, we can easily relate the magnitude of the electron-phonon coupling A to the con- dition for the existence of the cubic-tetragonal lattice transformation. From eqs (2.5) and (3.16), and noting the empirical relation [20] A12 = 1/3 A11 which holds for Nb5Sn and V8Si, we have immediately, \ P 0.7, (3.17) for a tetragonal transformation. Equation (3.17) illus- trates the intimate relation between high Te’s and the lattice transformation; large A's favor high Te’s but lead to the lattice instabilities. 3.4. To in the Tetragonal State Let us now solve eqs (3.12) for the tetragonal lattice state of V3Si and Nb5Sn and compare the results with those for the cubic state. For Nb5Sn the relevant densi- ty of states configuration is shown in figure 1b. For our 771 purposes, the T = 0 K values of the various energies shown in the figure are sufficiently accurate. We find once again the result Z = 1 + \ for all sub-bands. However, there are now two values of the energy gap, i.e., the gap A1 for the single widened To = 1.13 (60/3To)!/? 6p |exp (To Tin) --- l] exp Hºl The factor 1/3 occurs in the denominator of the argu- ment of the exponential in eq (3.18) because the final density of states in the tetragonal state of Nb5Sn is reduced to 1/3 of the cubic state value. Note the large predicted isotope effect in the tetragonal state Teº M-84. Substituting our values of the parameters for Nb5Sn, we compute Te=8 K, and the energy gap parameters A and A2 differ by less than 1%. A similar calculation for V3Si yields To = 20K, close to the cubic state value. The small difference in the cubic and tegragonal Te’s for V8Si results primarily from its low [2] Trn = 22K; at T., the effective density of states at the Fermi level is still large, so that the interaction extends over roughly the same energy range in all sub-bands as it would if the material were cubic. Thus, we have the result that the approximations we have made work well for VaSi; we calculate Te’s close to the observed value and predict little difference between the cubic and tetragonal Te's. On the other hand, Nb5Sn is predicted to have a high Tc (28 K) in the cubic state and a relatively low one (8 K) in the tetra- gonal state. This is suprising because we would expect our approximations to be either uniformly good or uniformly bad for these materials. We believe that we have considerably underestimated the tetragonal Te for Nb5Sn. Evidence for this exists in the low tem- perature susceptibility data which indicates a fall-off of the final density of states to about 0.5 No rather than 1/3 No. The value 0.5 No is very close to the value predicted from the band model if we employ the effective mass approximation [10] (N(E) or E!'”) rather than rectangular energy bands. The final density of states is critical in determining Tc, since it occurs in the exponential, so that close agreement with the observed value of Te is obtained if we employ this modification to our model. On the other hand, we believe that the figure of 28 K for cubic Nb5Sn may actually represent the Tc that this compound would have if it did not first undergo a lattice transformation. There is some experimental [21] and other theoretical sub-band and the gap A2 for the two narrowed sub- bands. Equation (3.12b) is then actually two coupled homogeneous algebraic equations for A1 and A2. The transition temperature is obtained by setting the deter- minant of the coefficients equal to zero. If the energy difference (1/2 U.0=EF) = 3To (exp(To/Tm) – 1) Tº > Tc, we find 1 + # N (3.18) [22] evidence that Nb5Sn ought to have a much higher To than the observed 18 K. If we are correct, it is then a simple matter to show by a calculation similar to that of McMillan [19] that the maximum To occurs for N = 4 and is 45 K for Nb5Sn and 40K for V3Si. Thus, in contrast to McMillan, we find that Nb5Sn has an intrinsically higher To than that of V3Si. Of course, our result rests on the assumptions we have made in obtaining our equations for Tc and \; of particular importance is the assumption of equal intra- and inter- sub-band coupling in our model, Labbé, Barišić, and Friedel [5] did not calculate the difference in To in the two lattice states but felt that the difference between them ought to be small because Te in the cubic phase varies only slowly with No in their model. The ex- perimental situation with regard to the effect of the lattice transformation on To is not clear. A Nb5Sn crystal which showed lattice softening [23] but did not transform according to x-ray measurements had a Te only 0.2 Klarger than that for a transforming crystal [3]. However, analysis of the elastic constants [24] to obtain the quantities in eqs (3.15) and (3.16) yields the smaller computed values of N = 0.78 and T. = 20 K, so that the failure to realize a significantly higher Te in this particular nontransforming crystal is understood. The situation is further complicated by the fact that tetragonal crystals of Nb5Sn are heavily twinned [25,26], and the effect of the attendant inter- nal strains is not known. 4. Conclusion The results presented here and in previous papers show that a simple electronic density of states model can account for many properties of the normal tetragonal and cubic phases and the high T.’s of these materials. Our approach to the density of states differs only in detail from that of Labbé-Friedel-Barišić [4,5], although our model gives better numerical agreement * This crystal was found to contain 1.5% interstitial H. 772 with most of the experimental data [6]. The present treatment of superconductivity differs considerably from that of other authors [5,19,22] in that (a) we specifically treat the effect of the tetragonal transfor- mation, and (b) attempt to estimate the magnitude of the electron-phonon interaction from the elastic properties of the material. Despite the success of the model, certain important questions remain to be answered. First, and perhaps most important, is it true that transforming Nb5Sn would have a Tc of substantially higher than 18 K were it not for the lattice transformation? This question can only be answered experimentally. The question which is most relevant to this conference is the relation of the model density of states to the actual band structure in the vicinity of the Fermi surface. This question is very difficult to answer because ordinary band calculations overlook structure on the energy scale that we have considered. If our success in explaining the properties of these materials has a physical basis, then the essen- tial features of our model must result from a fundamen- tal treatment of the 8-W compounds. The early pioneer- ing calculation of Mattheiss [27] and the insights of Weger [28] and the Orsay group [4,5] provide the foundation for such a study. The technical and theoreti- cal importance of understanding the source of high T.’s should provide sufficient motivation. 5. Acknowledgments The authors acknowledge Drs. J. Gittleman, G. Webb, and A. Rothwarf for many helpful discussions. 6. References [1] A summary of extensive experimental work on Nb5Sn can be found in RCA Rev. 25, 333, F. D. Rosi, Editor (1964) (19 papers). [2] Batterman, B. W., and Barrett, C. S., Phys. Rev. Letters 13, 390 (1964); Phys. Rev. 145, 296 (1966). [3] Mailfert, R., Batterman, B. W., and Hanak, J. J., Phys. Letters applies to this material. On the other hand, since the magnetic susceptibility of V3 Ga is extremely temperature dependent (Clogston et al., Phys. Rev. Letters 9, 262 (1962), we have reason to believe that V3Ga can be treated by the present theory. [10] Cohen, R. W., Cody, G. D., Wieland, L. J., and Rehwald, W., to be published. [ll] If the effective Fermi energy To is actually of the order of or smaller than the important phonon energies, it is possible that the first order process may not be sufficiently accurate for strong coupling superconductors, Migdal, A. B., Sov. Phys. JETP 7, 505 (1958)). However, it is possible to treat (R. W. Cohen, thesis (unpublished)) the effect of higher order processes involving strong attractive interactions whose range in momentum space is greater than the Fermi momentum. It is found that the effect of these processes, in the t matrix ap- proximation, is only to renormalize the pre-exponential factor in the expression for T. We shall ignore this effect. [12] Cody, G. D., RCA Rev. 25, 414 (1964). [13] The coherence length is estimated to be approximately 130 Å. This estimate is arrived at using the penetration depth given in Reference [12] and the Ginzburg-Landau parameter K = 21.7 (Wieland, L. J., and Wicklund, A., Phys. Rev. 166, 424 (1968)). [14] Nambu, Y., Phys. Rev. 117,648 (1960). [15] Koonce, C. S., and Cohen, M. L., Phys. Rev. 177, 707 (1969). [16] Equations (3.6a) and (3.6c) for Zi and Ai are similar to McMillan's (ref. 19) eqs. (2a) and (2b) in limit (o-> 0 with the frequency o' put on the energy shell a)' = #'. Our result ignores real phonons (@g|To => 1) and explicitly treats the self energies in each sub-band. [17] See for example Ziman, J. M., Electrons and Phonons (Oxford University Press, London, England, 1962) Chapter V. [18] Morel, P., and Anderson, P. W., Phys. Rev. 125, 1263 (1962). [19] McMillan, W. L., Phys. Rev. 167,331 (1968). [20] For transforming Nb5Sn, we use the value 00 = 300 K (Wieland, L. J., and Wicklund, A., Phys. Rev. 166, 424 (1968)). The parameters obtained from elastic constant measurements [7,10] are T, = 80 K, and, in units of 10” erg-cm-3, NoU*= 7.86, A11 = 2.94 and A12= 0.84. For transforming VaSi, we use the values 0) = 500 K, To = 75 K, and, in units of 10° erg-cm−", NoU* = 6.33, A11 = 3.22 and A12 = 1.05. These parameters were obtained from an analysis of the elastic constant data of Testardi, L. R., and Bateman, T. B., Phys. Rev. 154, 402 (1967). For both materials, we use the value put = 0.13 for the Coulomb pseudopotential (ref. 19). 24A, 315 (1967). [21] Cody, G. D., Hanak, J. J., McConville, G. T., and Rosi, F. D., [4] Labbé, J., and Friedel, J., J. Phys. Radium 27, 153, 303 (1966) RCA Rev. 25, 338 (1964). (2 papers). [22] Hopfield, J. J., Phys. Rev. (in press). [5] Labbé, J., Barišić, S., and Friedel, J., Phys. Rev. Letters 19, [23] Keller, K. R., and Hanak, J. J., Phys. Rev. 154,628 (1967). 1039 (1967). [24] For this crystal, analysis of the elastic constants reported in [6] Cohen, R. W., Cody, G. D., and Halloran, J. J., Phys. Rev. Let- reference 23 yields To = 100 K, and, in units of 10° erg-cm−", ters 19, 840 (1967); Cody, G. D., Cohen, R. W., and Wieland, No U” = 6.46, A11 = 2.96, and A12 = 0.90. L. J., Proceedings of the 11th International Conference on [25] Batterman, B. W., presented at the Eighth General Assembly Low Temperature Physics (St. Andrews University, 1968) p. and International Congress of the International Union of 1009. Crystallography, University of New York (Stony Brook), Au- [7] Rehwald, W., Phys. Letters 27A, 287 (1968). gust 1969. [8] See for example Allen, P. B., Cohen, M. L., Falicov, L. M., and [26] McEvoy, J. P., presented at the International Conference on the Kasowski, R. V., Phys. Rev. Letters 21, 1794 (1968). Science of Superconductivity, Stanford University, August [9] Since Nb5 Al does not display the same kind of anomalous 1969; Conference proceedings, Physica (to be published). behavior as Nb5Sn and VaSi (Willens et al., Sol. State Comm. [27] Mattheiss, L. F., Phys. Rev. 138, All 2 (1965). 7, 837 (1969)), we do not believe the approach presented here [28] Weger, M., Rev. Mod. Phys. 36, 175 (1964). 417–156 O - 71 – 51 773 Summary of the Conference on Electronic Density of States” H. Ehrenreich Division of Engineering and Applied Physics, Harvard University, Cambridge, Massachusetts It is very difficult to summarize a conference such as this, involving as it did a many splendored array of top- ics, both experimental and theoretical, expounded in no less than ninety papers. The organization of a con- ference of such size is ordinarily impossible without resort to simultaneous sessions. When there are simul- taneous sessions, of course, it is easier for a single sum- marizer because he can only be at one place at one time and therefore can be excused for failing to do justice to half of the papers. It is also easier for the audience because if you permit people to resonate between ses- sions, you also allow them to become trapped in the halls, advertently or inadvertently as the case may be. However, through the devilish cleverness of the or- ganizers of the present conference, a goodly fraction of the papers were delivered by a rapporteur. Ac- cordingly, the entire audience, including the sum- marizer were exposed to everything during a period of three and a half hard-working and elaborately or- ganized days. Furthermore, I lose my excuse for having overlooked, as I undoubtedly did, some of the impor- tant new developments presented or presaged here. As it is, in the time of thirty-two minutes that have been allotted to me it is, of course, impossible to men- tion even a representative fraction of the contributions. Indeed, even the various areas discussed here can only be sketched in broad outlines. Fortunately my task is considerably eased by the various excellent review lec- tures and rapporteur summaries that punctuated the conference. In order to avoid the risk of offending a few, I have decided instead to offend everybody by not mentioning names in this talk, except when referring to work which was not explicitly reported at this con- ference which is appropriately referenced. *An invited paper presented at the 3d Materials Research Symposium, Electronic Density of States, November 3–6, 1969, Gaithersburg, Md. 'Supported in part by Grant No. GP-8019 of the National Science Foundation and the Advanced Research Projects Agency. At the outset, let me thank the organizers on behalf of everyone attending it for providing us with an out- standing scientific program that clearly focussed on the principal questions which those of us who are grappling with the electronic structure of condensed matter are facing. During the time when we were not riding buses or listening to papers, there was also a most pleasant social program, and mercifully, a few hours for sleep. I would regard the title of the present symposium, “Electronic Density of States” a leitmotif rather than an idée fixe because the subject matter presented at the conference in fact was far more general than might be implied by the title. Since the density of states is very influential in the determination of many basic physical properties, it provides an excellent focal point for presenting some of the more recent developments in the electronic properties of condensed matter. More importantly (and to my view this is one of the chief motivations of this conference) the density of states is a convenient central quantity for confronting theory and experiment even though, unfortunately, it never seems to be measured directly by any experiment. In the jargon of the modern theorist one might phrase this difficulty in the following way. The density of states is proportional to the imaginary part of the single particle Green’s function, whereas many ex- periments determine a response function, which in- volves Green’s functions of two or more particles. When suitable approximations are made, however, the state density enters in a fairly direct way into the theoretical interpretation of all of the various kinds of measurement described at this conference. The list of techniques available to the solid state physicist, which was extensively sampled here, is truly impressive and stands in contrast to the much more limited variety available to our colleagues in the ele- mentary particle field. We heard about optical absorp- 775 tion and reflectance, x-ray spectroscopy, photoemis- sion, Fermi surface experiments, tunneling, measure- ments of the electronic specific heat, magnetic suscep- tibility, superconducting critical fields, and transport properties, positron annihilation, Compton scattering, ion neutralization spectroscopy among others, and how some of these are influenced by pressure, strain, and temperature. The preceding list reflects the fact that photons con- tinue to be one of the favorite probes for studying the microscopic properties of matter. It is therefore tempt- ing to use the 1965 Conference in Paris on the Optical Properties and Electronic Structures of Metals and Al- loys [1] which dealt with the same class of materials as the present conference and similar ideas concerning the interpretations of experiments as a fiducial mark to give us some indication of what we have learned about metals and alloys during the interim. Pippard, in his summary of that conference, re- marked on the extraordinary number of times the au- dience was shown the Cu band structure. Since then the variety of band calculations, and, in particular the kinds of materials considered has proliferated greatly. Pol has received a great deal of recent attention largely as a result of excellent high field susceptibility and photoemission measurements. Other examples discussed here involved more exotic materials such as AuſAl2, EuO and GeO2. Evidently the machinery for doing such calculations on ordered alloys containing several atoms per unit cell, some of which are suffi- ciently heavy that relativistic effects become impor- tant, is now available at several laboratories in a readily usable form. However, as in all band calculations, even if one accepts the Hartree-Fock approximation, the result obtained is only as good as the potential that is used as input information. In metallic alloys one might expect some charge transfer among the atoms belong- ing to a single unit cell. In my view this possibility has not yet received adequate attention. For example, Au Al2 has the CaF2 structures and is one of the few metals in which a Raman frequency has been observed [2]. Remarkably, its magnitude is similar to that of CaF2. One might ask whether the very similar stiffness of the optical frequencies in these materials is purely an accident or whether there could be enough charge transfer among the atoms in the cell to result in appreciable ionic character. I realize that I am undoubtedly not saying anything that is not already familiar to band theorists. However, experimentalists should be warned that the construction of alloy poten- tials, even for ordered systems, is still a problem that requires attention. Indeed, the simpler problem of calculating band structures for monatomic metals on the single particle picture is still controversial. As we saw in connection with several of the contributions and much of the spirited discussion that followed them, we still don’t un- derstand clearly how, when, or why to localize the exchange interaction. To date no sufficient theoretical reason has been advanced for preferring either the Slater or the Gaspar-Kohn-Sham versions of this poten- tial. Proponents of either point of view, or those favor- ing intermediate values of the coefficients at present usually support their position by comparison with ex- periment rather than basic theoretical arguments. In this same connection we might note the debate fol- lowing the introductory lecture concerning the relative merits of first principles and pseudo- or model-potential band calculations. The essential point made in that lec- ture, in my view, is that “pseudism” is important because it provides elementary insight into the mean- ing of the results of the elaborate machine computa- tions. The two approaches, in fact, are complementary and the proponents of each have a genuine need for, and indeed ideally should merge with the other. There is no doubt whatsoever that band theory has done sufficiently well that it is worth using its results to calculate the density of states accurately. This is a dif- ficult numerical problem, particularly if one resignedly accepts spending ceilings that curtail the amount of available computer time, for one needs to know the energy at many millions of points in the Brillouin zone in order to construct adequate histograms that yield all the fine detail in the density of states that is often necessary to interpret experimental information re- liably. We saw several examples of advances in per- forming such calculations more economically at this conference. The QUAD scheme is one of these. Another, which I will call the IBM scheme, not after the machine but the workers, is similar to QUAD in its ability to generate very detailed E(k) curves, but it avoids the use of histograms. All these result in very finely grained structure in the state density. In this same connection, I think the importance of learning how to sum functions of k over constant energy surfaces in the Brillouin zone efficiently and reliably needs emphasis. This is important not only for calculat- ing the state density, but also for computing Green’s functions that are central to the solution of alloy band problems, frequency dependent dielectric functions that can be compared with optical data, susceptibili- ties, and many other quantities. While many theorists may think such problems as insufficiently dignified, I would, nevertheless, stress that their solution is impor- 776 tant if one ever expects to confront theory and experi- ment realistically on a more complete basis for more complicated systems. At the same time let me temper this call to computer and numerical analysis handbooks by reminding you of the obvious fact stressed by many speakers that band calculations are single particle descriptions involving electrons or holes as ideal quasi-particles which in- teract with a self-consistent field that is in practice determined more or less self-consistently. However, due to electron interactions including those involving phonons, real quasi-particles acquire finite lifetimes, except right at the Fermi surface. Some of the papers presented here reflected the fact that methods of taking quasi-particle effects into account more systematically in band calculations are now being developed. I would look towards greater exploitation of such techniques in the band calculations of the near future. It is important to remember that when we speak of dressing effects, say, due to electron-phonon interactions, we ought to be dressing the right bare object, namely the correctly cal- culated quasi-particle appropriate to the stationary lattice. One should again be reminded of the fact that very few if any of the experiments discussed at this meeting correspond to creation of just one quasi-particle near the Fermi surface. This fact was also stressed already at the Paris Conference. Optical experiments, for ex- ample, correspond to the creation of two quasi-particles and ion neutralization measurements to three. These may interact with each other as well as with the other particles in the system. This was illustrated in the discussion of the various types of phenomena that can occur in x-ray emission, which lead to the conclusion that the observed spectrum of the valence band may bear less resemblance than one would hope to what is calculated from band theory. In addition to the long familiar Landsberg or Auger tails that smear out the lower valence band edge, there are recently predicted elementary excitations such as the plasmaron and other broad structures that also result from interactions with plasmons. Another effect that was reported on here results from exciton type interactions between elec- trons near the Fermi surface and the core holes with which they combine in an x-ray emission process. These may strongly affect the transition rate and lead to substantial enhancement or dimunition of the ob- served intensity near the Fermi surface. As has been pointed out to us, these effects must be quantitatively understood before information concerning band struc- ture can be reliably extracted from such experiments. As a result of particle interaction effects, the informa- tion supplied, even what would in a simple minded view correspond to the same theoretical quantity, often dif- fers from experiment to experiment. The Fermi surface effective mass determined from electronic specific heat and Pauli susceptibility measurements is an example of this. Even within a single particle framework, the state density function is only characteristic of a particular type of experiment. For example, in x-ray emission ex- periments, optical selection rules pick out only those components of the valence band state density having appropriate symmetries with respect to the core hole. We saw that this fact has particular utility in providing insight into the character of the wave function overlap and hybridization among different components in both ordered and disordered alloys. Since the core hole is lo- calized in a given atomic site, the study of say the Al-L2, 3 emission spectrum in systems such as AuAl2 and others discussed here, provide an indication of the amount of d wave function in these systems located on the Al sites. As another example, we might mention the k-conver- sation rule entering interband optical processes which implies that the state density appearing in theoretical expressions for the optical constants is the so-called “joint density of states.” This seemingly innocuous fact has led to a spirited controversy in connection with the interpretation of photoemission experiments which was already in full bloom at the Paris Conference. As you all know by now, there are two schools of thought whose proponents we might call the k-conservationists and the k-nonconservationists. The latter group has main- tained, on the basis of a considerable body of experi- mental evidence, that particularly in materials having narrow bands such as the noble and transition metals, the energy distribution of the photoemitted electrons should directly reflect the structure in the density of states pertaining to these bands. The k-conservationists on the other hand have asked, “Why should this con- servation law be violated?” Indeed, one of the papers, which represents the first attempt at the formulation of a systematic theory of the photoelectric effect in solids, points to ways in which this might come about. Several of the other contributions point to progress towards a reconciliation of these viewpoints. For exam- ple, we have heard in connection with Cu that a direct transitions analysis using constant matrix elements ac- counts quite well for the observed energy distribution. Similar conclusions have been reached on the basis of very detailed calculations for Pa. The essential point, which was emphasized by both camps, is that the stron- gest peak in the joint density of states coincide with 777 peaks in the calculated state density, particularly in the case of narrow valence bands. While in many cases the photoemission technique is a very useful tool, this may not be the case universally. It was suggested, for example, that it is less successful in providing information concerning f states in the Eu chalcogenides and rare earth metals since these states are seen to give rise to abnormally low quantum yield relative to, for example, p states. It is also clear that the variation of optical matrix elements with energy and selection rules, which can also lead to structure in the observed spectra, needs further attention because in many calculations this matrix element is still regarded to be a constant. Before leaving the subject of optical properties of crystal, two other points are worth making. Despite the fact that one learns only about the joint density of states in such experiments, it is, in fact, possible to derive the conventional state density from optical data by a more circuitous The of differential reflectance techniques is now well established and was illustrated in several of the contributions presented here which even extended to the x-ray case. Informa- tion from such measurements can be used as input for pseudopotential band calculations or those based on the k p approximation. The problem of constructing potentials which plague first principle band calcula- tions is thereby avoided. Since the secular equations for such problems are generally smaller, they can be solved at sufficiently large numbers of points in the Brillouin zone to obtain the state density. The other point concerns another recent develop- ment. A fact that has been distressing to many theorists is that, while in semiconductor calculations of optical coefficients there was always good agreement between theory and experiment in regard to both the position and magnitude of the observed structure, this has not been the case in metals. For the case of Al we saw quite convincing evidence that such discrepancies are on the point of disappearing, largely as a result of better calcu- lations which deal more adequately with the k-depend- ence of the momentum matrix element. Indeed, other recent investigations have shown that electron-electron scattering effects, which lead to vertex corrections that might be expected to be stronger in metals than semiconductors, are, in fact, very weak in this material [3]. Even though the so-called Mayer-El Naby resonance is probably no longer with us [4], our un- derstanding of the alkali metals is unfortunately still not in as good a shape. It is regrettable that relatively few papers presented at the conference attempted to provide a detailed com- rOute. usefulness parison between the results obtained by different types of experiments. There was only one noteworthy excep- tion, which was concerned with efforts to confront Knight shift data with those of soft x-ray emission ex- periments. There is a real need for more such detailed comparisons, even on the basis of band theory alone. According to a paper count, superconductors and semiconductors received less attention than the simple, transition, and rare earth metals. However, there are good reasons for mentioning them even in this broad summary. As appropriate, superconductivity was not discussed as a phenomenon, but rather as a tool to ex- tract information relevant to the state density. On the positive side, we heard how strong coupling theory can be used together with other measurements to obtain the electron-phonon coupling constants, and how measure- ments of the critical field at very low temperatures can be made to yield the electronic specific heat as a func- tion of pressure with high accuracy. On the negative side, it was pointed out in connection with a general review of the information provided by tunneling experi- ments, that such measurements for superconductors do not really tell us all that much about the normal state properties of metals and semiconductors. It is clear from the exquisitely detailed interpretable information being currently obtained from cyclotron resonance, magneto-optical, and even nonlinear optical data in simple semiconductors and semimetals, that our understanding of these materials is still in a somewhat more mature state than that of most metals. Of course this applies only to the classical and long stud- ied materials like Ge and InSb and not to amorphous semiconductors which were discussed in only a single paper. However, the very beautiful interplay and agree- ment between theory and experiment must still be re- garded as serving as a standard of excellence which solid state physics in general must continue to emulate. Surfaces also were not discussed extensively here, largely, I think, because the theoretical ideas and techniques for dealing with such problems in realistic systems are only beginning to be developed. However, some very promising experimental techniques, notably ion neutralization spectroscopy and resonance tunnel- ing which shed light on the nature of surfaces and im- portant phenomena like chemisorption were described. This area will surely see a great deal of activity in the near future. Since the discussion of a large variety of alloy systems occupied so much of the conference, let me conclude this summary with some remarks concerning this subject. We were exposed to a wide variety of data concerning many alloy systems, most of them involving 778 transition metals. Certainly there are many more than were considered at the Paris Conference. However, this comparison is somewhat unfair since even now, with one or two notable exceptions which were presented here, there is still a dearth of optical informa- tion concerning disordered alloys. This is to be con- trasted with the situation involving specific heat and transport measurements about which we heard a great deal. Indeed, most of the papers dealing with the elec- tronic specific heat were concerned with alloy systems. A most interesting effect that was described dealt with the recently discovered magnetic clusters in NiCu al- loys which can make their own appreciable contribu- tion to the specific heat. This conjecture is quite new and deserves detailed theoretical treatment. There was also an intriguing discussion concerning rare earth metals which raised the question as to whether the f electrons could possibly be at least partially itinerant in some of these systems. One fairly obvious thing that needs emphasis in con- nection with these papers and that was stressed in a number of them is the need for data of single crystal specimens having known phases, and how crucial it is to avoid samples involving mixture of phases. Without these precautions, the overanxious theorists will, as they did in the case of the Mayer-El Naby anomalies, find themselves in the awkward position of explaining what Pippard already warned in 1965 might be non- facts. As we saw, many of the experimental techniques ap- plicable to pure metals are relevant for disordered al- loys as well. We have already mentioned optical and x- ray data in this connection. The fruitful and relatively easily interpretable Fermi surface experiments, alas, seem to be much more difficult for many alloy systems. However, measurements such as those involving positron annihilation which also probe the Fermi sur- face geometry are not restricted by such criteria. They have already been very successfully used to investigate detailed Fermi surface changes in Cu-Al [5] and as we heard here, to brass. A great deal of progress in this area since the Paris Conference has come along the theoretical front. Until a few years ago the only theoretical models available for describing alloy behavior involved perturbation theory, the virtual crystal, or the rigid band models. However, recently a number of rather effective techniques based on scattering theory have been adapted to this problem and implemented by calculations for both model and realistic systems. These all transcend the earlier, more limited approaches. The first incisive contributions to electronic theory, made by Edwards and Beeby [6], in- volved the so-called average t-matrix approach which Soven [7] applied to brass. An extension of such calcu- lations was discussed here. Subsequently, Soven [8] formulated a more general self-consistent effective field approach that he termed the coherent potential approximation which is more general than the other. While this was mentioned in several of the papers, it is perhaps worthy of some additional commentary because of its possible applicability to realistic alloy systems. In this approximation the alloy is replaced by an ef- fective medium described by a single particle non-Her- mitian and complex Hamiltonian which, however, is still periodic in the case of substitutional alloys. The self-consistency condition determining this Hamiltoni- an is simply that an effective electron wave travelling through the crystal which impinges on an atomic site suffers no further scattering due to the random character of the crystal potential. Put another way, the effective wave, just like a Bloch wave in a crystal, is not scattered by the atoms. However, unlike the Bloch wave, the effective wave may be damped as it propagates through the crystal. The present limitation of this description is that it is only applicable to certain classes of Hamiltonians in which the random character is cell localized. The theory has the virtue of correctly reducing to the known results for small impurity con- centrations and arbitrary scattering strengths on the one hand, and for arbitrarv concentrations but small scattering strengths on the other. It interpolates in a physically reasonable way between these limits, yield- ing results that are valid for arbitrary alloy concentra- tions and reasonably strong scattering strengths. It fails in predicting band tailing effects, experimental evidence for which we heard described here in connec- tion with semiconductor tunneling experiments. Also, it does not yield strictly localized states except in the limit of very small impurity concentrations. We should note parenthetically that while the ex- istence of such states near band edges is generally be- lieved, some questions were raised here as to whether or not such states can exist in the middle of a tight bind- ing band and whether or not the frequently made hypothesis that there exists a sharp demarcation between localized and nonlocalized states is correct. As we were reminded, localized states such as those due to f-electrons, can exist even in periodic systems when the Coulomb interaction is sufficiently strong. As was shown, conventional band descriptions break down under these circumstances. To obtain the effects omitted by the coherent poten- tial description of disordered alloys it is necessary to 779 allow for the possibility of statistical clustering effects. This is a much more difficult problem. But as we heard in two of the papers presented here, some very promis- ing progress is beginning to be made in these directions. Indeed, the early work of I. M. Lifshitz [9] has already given us an indication of the sorts of results to be expected. Because of the previously stated limitations, the coherent potential theory in its present form is strictly speaking applicable only to isoelectronic alloys like GeSi, where the random part of the potential is substan- tially confined to the core region at each site, or to 3d transition-noble metal alloys in which the d-states that are most affected by the disorder are substantially localized. —º N —10% CU —20% CU |- % CU ſ | | | - 4 – .3 –.2 – O ENERGY (Ry) 40% CU CPA (ALCULATION FOR NICU To give these remarks a sharper focus, I should like to show you by means of one example the results of an application of this theory to Cuni alloys. Figure 1 ex- hibits S. Kirkpatrick’s calculations [10] for the density of states of these alloys and also the results of photoemission experiments by Seib and Spicer [11], which, for reasons already mentioned, should only be compared qualitatively with the theoretical results. It should be emphasized that these calculations do in- volve approximations, most of them probably not too serious. Since it would be inappropriate to discuss these in the present context, I would like to confine my remarks to a few brief comments. The first is that the coherent potential approximation is evidently applica- ble to quite complicated density of states functions \ # PURE N # ||| 9, CU - 19% CU — : as 39% CU | | | ºw — –3 – 2 — O ENERGY (R) PHOTO EMISSION RESULIS FOR NICU (Seib Gnd Spiceſ) FIGURE 1. The hybridized d state densities, calculated in the coherent potential approximation, are compared with the optical state density obtained from photoemission experiments [10, 11 |. 780 which include degenerate d-bands and hybridization with conduction bands. Second, the only input informa- tion needed for nickel-rich alloys is the hybridized nickel state density, the positions of the resonant Ni and Cu d-levels which give rise to the d-bands, and the concentration. Third, the distortion in the state density curve with increased alloying shows the rigid band model, which has been particularly popular for this alloy system, does not really apply. This is seen even more clearly in the results of calculations of the mag- netic properties [12]. Finally, the prominent calculated structures and their behavior is qualitatively in accord with the experimental observations. The principal peaks remain stationary, but they change in intensity and shape in both the calculations as well as the experi- ment. The contribution of copper to the state density turns out to be broad and relatively structureless and comes principally from the lower regions of the d-band. While this kind of theory predicts a wealth of detail, its quantitative validity remains an open question that must be explored further. More important, some of its present limitations must be overcome to render it ap- plicable to a wider class of alloys. However, given the improving theoretical and experimental situation that is clearly evidenced from this conference, it seems clear that at the next meeting of this type we will surely hear about further, more extensive developments which will place the theory of alloys on a firmer footing. After these somewhat discursive concluding re- marks, let me close the conference by once again thanking everyone; organizers, speakers, and rappor- teurs, questioners, commenters, and listeners for hav- ing made it as stimulating as it turned out to be. References [1] The Proceedings of that Conference were edited by F. Abeles and published under that title by North-Holland Publishing Company (Amsterdam) in 1966. [2] Feldman, D. W., Parker, J. H., and Ashkin, M., Phys. Rev. Let. ters 21, 607 (1968). [3] Beeferman, L., and Ehrenreich, H., (Phys. Rev., to be published). [4] Mayer, H., and El Naby, M. H., Z. Physik 174, 289 (1963): Smith, N. V., Phys. Rev. Letters 21, 96 (1968). [5] Fujiwara, K., Sueoka, O., and Imura, T., J. Phys. Soc. Japan 24, 467 (1968). [6] Beeby, J. L., Proc. Roy. Soc. (London) A279, 82 (1964) and cited papers. [7] Soven, P., Phys. Rev. 151,539 (1966). [8] Soven, P., Phys. Rev. 156, 809 (1967). [9] Lifshitz, I. M., Soviet Physics—Uspekhi 7,549 (1965). [10] Kirkpatrick, S., Velicky, B., and Ehrenreich, H. (to be published). [11] Seib, D. H., and Spicer, W. E., Phys. Rev. Letters 22, 711 (1969). [12] Kirkpatrick, S., Velicky, B., Lang, N. D., and Ehrenreich, H., J. Appl. Phys. 40, 1283 (1969). 781 Thermcil Electron Effective Mass of Rubidium and Cesium D. L. McIrtin Division of Physics, National Research Council of Canada, Ottawa, Canada Key words: Cesium; effective mass; electronic density of states; rubidium; specific heat. Specific heat measurements in the range 0.4 to 3.0 K on 99.99% pure rubidium give a thermal electron effec- tive mass of 1.37 ± 0.01 and on 99.97% pure cesium give an effective mass of 1.80 +0.04. These results are much closer to the previous specific heat results of Lien and Phillips [1] than to those of Martin, Zych and Heer [2]. The result for rubidium is in good agreement with the theoretical results of Ashcroft [3] (1.38 + 0.09) and Wasserman and deVitt [4] (1.36). The cesium result is in fair agreement with Wasserman and deVitt [4] (1.69) and Mahanti and Das [5] (1.63). This work has been published in Canadian Journal of Physics 48, 1327 (1970). References [1] Lien, W. H., and Phillips, N. E., Phys. Rev. 133, A1370 (1964). [2] Martin, B. D., Zych, D. A., and Heer, C. V., Phys. Rev. 135, A671 (1964). [3] Ashcroft, N.W., Phys. Rev. 140, A935 (1965). [4] Wasserman, A., and deVitt, H. E., J. Phys. Chem. Solids 29, 2113 (1968). [5] Mahanti, S. D., and Das, T. P., Phys. Rev. 183, 674 (1969). 783 Density of States and Numbers of Carriers from the dHvA Effect * S. Hornfeldt, J. B. Ketterson, and L. R. Windmiller Argonne National Laboratory, Argonne, Illinois 60439 With the dBiv A effect, one can determine the angular dependence of the extremal area and effec- tive mass over all sheets of the Fermi surface. Using recently developed techniques this data can be in- verted and the angular dependence of the Fermi radius and Fermi velocity determined. Techniques have been developed to allow the inversion of both open and closed surfaces. For closed surfaces we use an expansion in symmetry adapted spherical harmonics (to order l = 60) while for open surfaces a three- dimensional Fourier series representation is used. With this information one may determine, for a given sheet of the surface, the number of carriers n(EF) and density of states N(EF) by performing the ap- propriate integrations. Key words: Electronic density of states; de Haas van Alphen effect; Fermi surface; Fermi velocity; Fourier series; spherical harmonics; symmetrized techniques. 1. Introduction In this paper we will discuss methods for deducing the Fermi radius and Fermi velocity from measure- ments of the de Haas van Alphen (dhvA) effect. The period of the dBiv A. oscillations determines the ex- tremal cross-sectional areas of the Fermi Surface (FS) A(6,40), where 6 and go are the polar angles of the mag- netic field. In addition, dHvA measurements allow a determination of the associated cyclotron effective masses m” = 1/2T 6A(6,40)/6E. We will discuss the mathematical techniques that allow one to determine the Fermi radius and velocity from a knowledge of A(6,40) and m”(0,0). If we have the radius k(6,40) then by integration we can determine the number of carries, n(E), contained within a given sheet of surface, i.e.: 2 n (EP) -siºnſ k” (6, p) sin 6d.6d.p. (1) Furthermore, a knowledge of k(0,0) and v(0,0) allows us to determine the density of states, N(E), for each sheet of the surface, i.e., 6, p) sin 6d.6dp vº (6, go) º (2) 2 ſ k” ( N(E)-ºnſ *Work performed under the auspices of the U.S. Atomic Energy Commission. It is well known that the inclusion of many-body ef- fects decreases the magnitude of the Fermi velocity from the single particle value that would be deduced from a band structure calculation. As can be seen from eq (2), this results in an increase in N(EP). Such many- body effects include electron-electron and electron- phonon interactions. The paramagnon contribution, which is really a form of electron-electron interaction, has received considerable attention recently. Accord- ing to the Landau's Fermi liquid theory, the density of states determined according to eq (2) (using techniques to be described here) should be identical to that deter- mined in a heat capacity experiment, i.e., the quasi-par- ticle density of states accounts for all of the density of states. Actually the Fermi velocity information should be regarded as potentially much more valuable than the density of states since there is probably a wealth of in- formation in the anisotropy of the velocity reduction and its variation from sheet to sheet of the surface. In order to accomplish this “inversion” it is useful to have a mathematical representation for the surface (or surfaces in cases where the FS consists of several sheets). Two cases arise quite naturally; either the sur- face is open or it is closed. For a closed surface our representation must be invariant to the operations of the point group of the surface. For an open surface the representation must be invariant under the operations of the space group of the crystal. We will discuss these two cases separately. 785 2. The Point Group Representation Lifshitz and Pogorelov || 1] have shown that closed surfaces containing a center of inversion symmetry are invertable providing the radious vector measured from this center is single valued. The theorem, in its original form, was awkward to apply and Mueller [2] has refor- mulated it in a manner which considerably simplifies its application. A comparison of the Lifshitz-Pogorelov and the Mueller formulation has been given by Foldy [3]. For the representation of the surface we use sym- metry adapted spherical harmonics. Since the radius vector is a real quantity we deal with the real spherical harmonics. | Cº-º; [Yl, m (6, go) + Yi, —m (0. o)] (3a) –– in T; V; [Yi, m (6, p) - Yi, -m (6, ©)] (3.b) Clo = Yio (6, p) (3.c) We expand the area A(6,40) and the square of the radius vector k”(6,40) as follows: k” (6, go) – X. [y. ,C#. , (0, go) –H y o,C# }}l (6, go) | l, m (4) A (0, c) =X [B,C#,C6, e) + 8,0}, (0, 0)]. l, ºn (5) What Mueller showed was that Bl, m = TP (0) yi, n. (6) where P1(0) is the Legendre function of order l. A unique connection between the radii and areas requires even l since P(0) = 0 for odd l. Even l is the same as requiring inversion symmetry since the parity of spheri- cal harmonics is (–1)'. Of the 32 crystal point groups 10 contain the inversion element, these being Si, C2h, D2h, Can, Dan, C3i, D3a, Cºn, D6h, and Oh. The values of m to be summed over in eqs (4) and (5) depend on which of these 10 point groups the surface in question belongs [4]. The z-axis is chosen as the axis of highest sym- metry. The inclusion of even (g) and uneven (u) coeffi- cients are required when the z-axis does not lie in a mir- ror plane. If the z-axis does lie in a mirror plane, then the z-axis is also chosen to lie within the mirror plane and only even coefficients are required. All values of m (0 – l) are required for the group S2 along with both even and odd coefficients. Only even coefficients are required for the groups D2h, Dah, Dad, and D6h and m = 0 (mod 2), m = 0 (mod 4), m = 0 (mod 3), and m = 0 (mod 6) respectively. Both even and odd coefficients are required, and the x-axis may be chosen arbitrarily, for the groups C2n, Can, Cai, and C6), and m = 0 (mod 2), m = 0 (mod 4), m = 0 (mod 3), and m = 0 (mod 6) respec- tively. The group On requires special consideration [5]. The symmetrized harmonics K1,i (6,40) which transform according to the cubic group are called Kubic har- monics. These harmonics are linear combinations of real even spherical harmonics. K= X. a', ,C#. , (0, ©) º }}l (7) The coefficients aff, have been tabulated to order l- 30 by Mueller and Priestley [5] and to order l = 60 by Aurbach, Ketterson, Mueller, and Windmiller [6]. The actual inversion proceeds as follows. The coeffi- cients Bin are found by least squares fitting the availa- ble experimental areas to eq (5). The Fermi momentum then follows by combining eqs (4) and (6). It is necessa- ry that the available experimental areas be measured over a wide range of angles, otherwise the number of coefficients which can be determined is limited [4,5]. Often one encounters surfaces which are nearly ellip- soidal in shape. This is particularly true in semi-metals. Equations (4) and (5) do not terminate for ellipsoids and converge slowly if the ellipsoid is quite elongated. For such surfaces it is convenient to perform a “spherical mapping” on the coordinates in order to map the “ellip- soid” into a “sphere” [4]. We do this by the following transformation x'= ox y' = y (8) z' = yz. Let T represent this transformation, i.e., T(x,y,z) = (x',y',z") or in polar coordinates T(Y,0,0) = (y',0', 6'). We also need the inverse transformation TT'(x',y',z')=(x,y,z) or equivalently T-1 (y',0', p")= (y,6,9). It is easy to show the following will r” (6', p") = r^ (6, p.) [sin” 0(sin” (p + O.” cos” (p) + y” cos” 6] (9a) ' = O. COs ºp 9b COS (p (sin” p + oº cos” (p) /* (9b) cos 0 = y cos 6 (9c) [sinº. 6 (sinº e-Foy, cosº cy-Fºy, cosº. 6]72 786 The question arises as to what the relation is between the area in the transformed system and the untransformed system. It can be shown that [4] A' (T-19, T-1.p) = oy[sin” 9(sin” p + o-º cos” (p) + y-3 cos” 0]1/*A (0, p). (10) What one does in practice is map the areas measured mapped back to the untransformed system using eq in the untransformed to the transformed system using (9a). eq (10). The coefficients in eq (5) are then determined We now turn to the inversion of areas and effective by the least squares procedure. The radii in the trans- masses into the Fermi velocity vr(0,0)[7]. The velocity formed system (which follow from eq (6)) are then is given by (h = 1): ÖF < 1 / (E A. | ÖF A. v (6, •)–(#). k+; (#). 0+, sin 6 (#). Ø (11) We expand 6A(6,42)/6 E (= 27tm”(0,p)) in a series of the form given in eq (5) ôA (0, p) 9E - X. {{3|{!,C#, m (6, ©) + £8;"n | m (6. ©)] (12) l, m ôk” (6, p) / (; /T ( ! {{ /T ºf w =#--> |y|{{!,C#, m (6, ‘p) + yº",C#, m (6, ©)]. (13) where 8.0 - TF1(0) y},. vº(0,0) we have immediately the k component of the Since 6 k2(0,0)/6 E = 2k(0,p) 6 k(0,p)/0E = 2k(0,0)/ velocity (since k is known from eq (4)), i.e., 1/2 2| X. [y!. mCl, m (6. go) +y|| mCl, m (6, go)] | wk = l, m f f e (14) X. ſyſ",C#, m (6, ‘p) + yº",C#. m (6, go)] l, m The other two components may easily be calculated using the laws for the differentiation of implicit functions, i.e., (#).--(#)./(#)...and(*).--(;)./(#) 66/k, a 30), 2/\ºe), , “"“Vºc), , 60/E, 6/ \óE/2, 9. %) ; X ſy,C#,0, e) + y,C#,0, c)] l, m vg = — 1/2 |X. ſy!",C#, (0, 0) + y!",C#,C6, •))) |X [Y/,C#,C0, c) + y,C#,C6, •))) l, m l, m (15) Thus we find: - 6 l, m vp = – > 1/2" (16) |> bº.º.o.º.º.º.o.)|x||y|.º.º.o.º.º.º.o.) l, m l, m 787 It is also possible to obtain similar formulas when using the spherical mapping procedure [6]. 3. The Space Group Representation A representation which is invariant under the opera- tions of the space group is a three-dimensional Fourier series of the form: F(k)=S. Ce"" (17) R where the CR are the Fourier coefficients. This representation was used by Roaf [8] to construct the FS of Cu, Ag and Au from the dFivA data of Shoenberg [9]. The vectors R are the vectors of the real space lattice: Rimn= la + m b + n c (18) where l, m, and n are integers and a, b, and c are the primitive translation vectors of the lattice. The vectors Rumn may be factored into sets, which we will call stars, according to their length [Runn|[10]. Thus eq (17) may be written in the form F(k)=S.C.S,(k) (19) where the j number the stars of increasing length, and S;(k) is the sum of eikº over all R in the jth star. In eqs (17) and (19) F(k) is not necessarily identical to the band structure E(k). If the coefficients are deter- mined by fitting to a band structure calculation over some range of energy then F(k) will approximate the band structure in that interval. In this case the Fermi velocity is given by v= i S. RCne"" (20) R If on the other hand the Ci are determined only from in- formation involving the shape of the FS (e.g. dBiv A areas) then F(k) = EF, is a number which may be chosen arbitrarily and the velocity is not given by eq (20). If, however, we have both area and effective mass data then we can construct both the Fermi radii and Fermi velocities. We discuss first the inversion of areas to radii. The calculated area is given by 4–4 || 400 (21) where k l measures the component of the momentum in the plane of the orbit, and l, measures the angle of klin the plane of the orbit. Since eq (19) is an implicit func- tion of k we must solve using the Newtron-Raphson technique. Assume that we have a set of N areas Ai and an initial set of coefficents CŞ. The problem is to vary the C; until the areas calculated using eqs (19) and (21) minimizes the rms error A between the calculated and measured areas where A is given by: * | JA A – Ai Nº AC)= S(***). i- 1 1. The amount 6C; by which we must correct C; follows from minimizing eq (22), i.e.: ôA” ôA* ac," > 6C;6C (22) 6C = 0. (23) It can be shown that the vector and tensor coefficients in eq (23) are given by [11]: ôA2 2 \, (A - Ai) 0A; |T = - N' –––– 24 and ôA° 2 º' [0A, 8A £ – A d’A; | l (25) WCWC Nº. |. act ( ;-A) ôC;0C, A, where 64° ſº" ki ôC; | k . WF S;(k) dil, (26) and d'A' ſ" | S(k)sſk), k? S;(k) k, V S (k) + Si (k) k . W. S.; (k) ôC;0C, Jo + (k, VF)2 “* (k, VF)* S;(k)SI (k) k, WWF . * - – k? “ T duli. + (k, WF)3 tly (27) 788 Using the above equations we can determine a set of C; which fit the data and thus construct the shape of the surface. This technique is applicable to both open and closed surfaces. We will now show how this technique can be ex- tended to determining Fermi velocities from area and effective mass data. As mentioned previously V F will not yield the Fermi velocity when the C; are determined using eqs (23) through (27). While it is possible to deter- mine a set of coefficients C; such that VE = vp it is sim- pler to introduce a new set of coefficients C. We define these coefficients by writing the velocity in the following form V F(k) v= — R-------. 28 X, CS,(k) (28) j ÖA” 2 \, (mº – mº) () mº icº TW º º id: (31) j i- 1 i j and ôA* 2 \, [/0m;eV/ðmic ÖCſöCT N' > | ôC. )( ÖC' J * = 1 J J These derivatives follow immediately from eq (29) (re- calling k1 does not change on varying Cſ) Omº I fºr ki 62m *C *— = —— ==- ~ s → = 0. (33 2T k; The important point is that | R. VF S;(k) dil, was already evaluated when doing the area to radii in- version. Thus it is only necessary to save these coeffi- cients in order to perform the effective mass inversion. To evaluate the number of carriers and density of states using the Fourier series representation it is con- venient to evaluate the integrals differently than in eqs (1) and (2). We rewrite (1) and (2) in the following form: n(E)-º: ſ A (kil) dk| (34) and N(E)-ºr ſ 2Tm”(k) dk. (35) where k is measured normal to the plane of the orbit. A problem arises when the surface is open in that the orbits can change from “hole like” to “electron like.” This problem is solved by subtracting the area of the )+ (mº – mº) The effective mass follows from ##ſ ºf d ” T ºr ET2: 0 k, VE l, __ ] º ſº" kiCfS;(k) =-3. X. 0 k, VF dili. (29) J We assume that the C; which yield F(k) are already known and that we vary the C until a best fit is achieved to the effective masses. We define a new function A' similar to eq (22) by , , , , 1 \, (mº - m^* A (C)=y. X. ("..") (30) i– 1 m; which on minimizing leads to an equation similar to (23). Thus we need Ó2m * | | WCWG, (32) 32" J77. l hole orbits from the corresponding area of the Brillouin zone, and selecting the direction of ki such that no open orbits occur. 4. Applications The techniques involving the symmetrized spherical harmonic representations have been applied to the T centered electron surfaces in Pt [7] and Pa [12]. For Pt the number of carriers in the 6th band electron sur- face is found to be n (Er) = 0.419 electrons/atom while the corresponding density of states if N(EP) = 6.35 electrons/atom Ry. The data are not yet sufficient to allow an inversion of the heavy “d like” open hole sur- face of Pt but should be in the near future. Fermi radii in Cu, Ag and Au were deduced by Roaf using the Fourier series approach [8]. This was extended by Halse to calculate the Fermi velocity in Cu and Ag [13]. The improved Fourier series techniques developed here will shortly be applied to recent dFiv A data in Au [14]. An inversion of the As and Sb electron Fermi surface was reported recently using the sym- metrized spherical harmonic approach [4]. 5. References [1] Lifshitz, I. M., and Pogorelov, A. V., Dokl. Akad. Nauk. SSSR 96, 1143 (1954). 417–156 O - 71 - 52 789 [2] [3] [4] [5] [6] [7] [8] [9] Mueller, F. M., Phys. Rev. 148,636 (1966). Foldy, L. L., Phys. Rev. 170, 670 (1968). Ketterson, J. B., and Windmiller, L. R., to be published in Phys. Rev. Mueller, F. M., and Priestley, M. G., Phys. Rev. 148, 638 (1966). Aurbach, R., Ketterson, J. B., Mueller, F. M., and Windmiller, L. R., Argonne National Laboratory report to be published. Ketterson, J. B., Windmiller, L. R., Hornfeldt, S., and Mueller, F. M., Solid State Communications 6,851 (1968). Roaf, D. J., Phil. Trans. Roy. Soc. (London) A255, 135 (1962). Shoenberg, D., Phil. Trans. Roy. Soc. (London) A255, 85 (1962). [10] In the fec lattice this test is insufficient. There are Rim, whose lengths are the same which do not, however, transform into each other under the operations of the point group. Those are distinguished by comparing R, lº-F|R, lº-F||R2|*. [ll] Ketterson, J. B., Mueller, F. M., Windmiller, L. R., to be published in Phys. Rev. [12] Windmiller, L. R., Ketterson, J. B., and Hornfeldt, S., Journal of Applied Phys. 40, 1291 (1969). [13] Halse, M. R., to be published. [14] Bosacchi, B., Ketterson, J. B., Shaw, J., and Windmiller, L. R., to be published. 790 A Note on the Position of the “Gold 5d Bands” in Au/Al2 and AuCa2 * P. D. Chan and D. A. Shirley Department of Chemistry and Lawrence Radiation Laboratory, University of California, Berkeley, California 94720 Switendick and Narath [1] recently proposed a solution for the “AuGaº dilemma” pointed out by Jaccarino, et al. [2]. This solution was based on the results of band structure calculations [1]. The den- sity of states for AuAl2 derived from these band-structure calculations were presented at this con- ference [3]. A surprising result of this calculation was the position of the “gold d-band” states. These states were located at about –7 eV in Auals and at similar energies in AuCa2 and Auſng [1]. The in- teresting optical properties of gold intermetallic compounds (e.g., AuſAlg is violet) are often attributed to the proximity of the gold d-bands to the Fermi energy, Ep. If these states really lay at EF-7 eV, and were as flat (i.e., the p(E) peak was as narrow) as the calculation indicated, then they could scarcely affect the compounds’ optical properties [4]. To help resolve this “d-band dilemma,” we undertook measurements of the valence-band spectra of AuſAl2 and AuCag by x-ray photoelectron spectroscopy (XPS). This method has been described elsewhere [5]: accordingly we describe below only those experimental fea- tures of this work that were peculiar to the AuſAlg-AuCaº problem. The two compounds are first treated in separate sections. The results are then discussed in the final section. Key words: Electronic density of states; gold aluminum two (AuſAl2); gold gallium two (AuCa2); x- ray photoelectron spectroseopy. 1. AUAl2 This compound prepared by heating stoichiometric amounts of metallic Au and Al to ~ 1000 °C in an induction furnace which was flushed a few times with argon and then pumped down to 2 × 10 °. One of the compounds (sample B.1) was further re- melted at ~ 1000 °C in an arc furnace also flushed with argon and then pumped down to 10 °. The resulting al- loys were spark-cut and polished with size 600 sand paper and, in two cases, with one micron diamond paste and kerosene on a canvas wheel; but they were not etched. They were rinsed with absolute ethanol be- fore mounting. A total of three samples were used to collect the data that were accepted for final analysis. The samples were studied in a H2 atmosphere (p = 0.01 torr) at temperatures indicated in table 1. The signal-to- noise ratio for energies near Er was unusually low, and it was not possible to make a detailed study of the den- sity-of-states function pauaº (E). The Spectra showed W 3 S *Work performed under the auspices of the U.S. Atomic Energy Commission. one dominant peak at Er – 6 eV, however. A least- squares analysis was made on each spectrum. The trial function consisted of a linear, sloping background plus a Gaussian with a constant tail on the low energy side. This combination usually fits XPS spectra quite well, and it did in this case. For some of the spectra the “d- band” peak appeared on visual inspection to be slightly asymmetric, and two-Gaussian fits were made. These fits give a very weak higher-energy peak of variable position and shifted the main peak only a few tenths eV. We therefore did not regard the experimental evidence for a second peak as being conclusive, and the question was of no particular interest, so only the one- Gaussian fits are presented in table 1. One of the AuſAl2 spectra is shown in figure 1. Our final values for the position and width of the “d-band” peak are E = Er – (6.0+0.3) eV, AE = (4.1 +0.5) eV FWHM. The Al 2p peak was found to be at E = Er – 75.05 + 0.20 eV. 791 TABLE 1 Au/Al2 sample no. T(°C) Reference — E a AE — E AE peak (FWHM) (5d-band) (FWHM) B.I. 25 ......... . ......... . ...... 5.99(14) 4.6(3) B.2 25 Al 2p 75.0(2) ...... 5.71(19) 4.6(4) B.3 600 Al 2p 75.2(2) ...... 6.42(14) 3.3(4) Adopted values" | ...... ......... 75.1(2) ...... 6.0(3) 4.1(5) AuGag sample no.: - A.3 400 Ga 3d(1) “ 20.1 d 1.8 5.62(21) 5.9(5) Ga 3d(2) 18.2 1.8 A.4 400 Ga 3d(1) 20.8 1.95 5.81(14) 4.6(3) Ga 3d(2) 18.9 1.95 A.5 375 Ga 3d(1) 20.5 2.0 5.90(8) 4.2(4) Ga 3d(2) 18.7 1.5 Adopted values" | ...... . ......... 20.5(4) 1.9 5.8(2) 4.6(3) 18.6(4) 1.8 * All energies in eV, relative to the Fermi Energy. Errors in last place are indicated parenthetically. "Adopted values and assigned errors reflect some systematic errors. * The Ga 3dſ2) peak was 0.4–0.6 times as intense as the Ga 3d(1) peak. * For samples A.3 and A.4, peak widths were constrained to be equal. T- | —I I I I I T- 5|OOH- I § Au 4 f 69 É 8.O H. gº - 50 OOH * 7.6 °. - o • * * º A | 2p 49 OOH : º * : •' - § § ºvº’, B 5. O *** - G asoo- A u 5d 9 O.O 85.0 800 75.O 70.O 8: Binding energy (Et-E) co 47OO - CNJ ^. º 46OO º - # • Au 5d from O 45 OO M M9 Kaz a. - 44OOH-–––– 2^ - 43O OH- | | | | | | | 5. O | O.O 5. O O.O – 5.O – | O.O Binding energy (Et-E) FIGURE 1. Typical x-ray photoemission spectrum for AuſAl2 near the Fermi energy. Filled circles represent data points. Top solid curve was fitted to the data. It was composed of background, indicated by sloping solid line, plus a response curve indicated by dashed curve. The main peak of this dashed curve arises through photoemission from the “gold 5d.” bands by Mgko. 12 radiation. Core-level peaks in inset indicate chemical purity of sample. 2. AUGd2 Dr. Narath kindly provided one of our AuCa2 samples: the other two were prepared in our laboratory as described above for AuAl2. The experimental and data reduction procedures were also similar to those used for Au Al2. Again a single Gaussian was fitted to the “d-band” peak; in this case the peak was quite sym- metrical and no attempt was made to fit two Gaussians to it. The Ga 3d peak, which lay about 20 eV below Er, was asymmetric, however. It appeared to consist of two peaks, each of width 2 eV FWHM, and spaced about 2 eV apart, with the higher-energy peak about 0.4 as in- tense as the lower-energy peak. We attributed the high- energy peak to free Ga, and auxiliary experiments on nonstoichiometric samples seemed to support our as- signment, although we were not able to eliminate this peak. Further support is given by the Au 4f;/2, 4f.9 doublet in AuCa2, which showed no evidence of a second phase. It is also possible that the asymmetric 3d |.7 I | # took | yº - - # 9.O. H. : ". - § § 80| ‘. . *E |.6 - % 7.O- ... . - G 2 *. * . 8 |.5 – # 5 of * - C.) 4. l I I O 95.0 900 850 800 75.O o | .4 Gd 3d Binding energy (Et-E) ^. ====<--- T. s______." Sºº-ºº- 25.O 2O.O |5.O |O.O 5.O O.O Binding energy (Et-E) FIGURE 2: Typical x-ray photoemission spectrum from AuCay near the Fermi energy. 792 peak shape reflects real broadening arising from its proximity to Ep. A spectrum of AuGaº is shown in figure 2. Our final values for the “d-band” peak are E = Er – (5.8–H 0.2) eV, AE = (4.6 = 0.4) eV. 3. Discussion The peaks at ~ Er – 6 eV in the photoemission spec- tra from Au Al2 and AuCa2 can be attributed to the “gold 5d bands” with considerable confidence. We in- terpret these spectra as giving strong support for Switendick's band structure calculations, in a qualita- tive way: the positions of these peaks are in good agree. ment with his predictions. On closer inspection, how- ever, there are points of difference. The experimental peaks are not quite so deep (EE – 6 eV) as the theoreti- cal value (EF – 7 eV), and the experimental line width (~ 4 eV) is about twice theoretical. In fact the combina- tion of less depth and greater width for the experimen- tal d bands somewhat weakens the conclusion that these bands are not important in optical phenomena. Comparison with the Al(L2,3) soft x-ray emission spectrum from AuAl2, reported by Williams, et al. [6], is interesting. These workers studied x-ray spectra from transitions in which holes in the 2p shell of Al were filled by electrons from s-like states at Al sites. States derived from Al 3s atomic states are of course s- like at Al sites, but so also are some states that are d- like on Au sites. Thus the strong peak that Williams et al., found at EF – 8 eV could be attributed in part to the “gold 5d bands,” and they took this as support of Switendick's calculation. Our results are basically in very good agreement with Switendick’s band structure results and the x-ray spectra results of Williams et al., but the small differences are also of interest. The soft x-ray work shows a peak at EF – 8.3 eV, while the XPS peak in the AuſAl2 is at EF – 6.0 eV. We attribute this difference to the fact that XPS is most sensitive to d electrons and thus determines the position of the “gold 5d bands” directly, while the soft x-ray emission work is sensitive to bands with s-like symmetry at the Al sites. Switendick [3] has calculated, for Au/Al2, peak positions of Er – 8.0 eV and Er – 7.1 eV, respectively, for the peak in the Al 3s-like states and the peak in the total density of states. This would account for some, but not all, of the difference between the positions of the experimental peaks obtained by the two methods. Another more direct point of disagreement is the posi- tion of the Al 2p state. Switendick calculated E=Er – (73.8 + 0.1) eV for this state, and he quoted an experi- mental value of Er – (73.5 + 0.5) eV. Our XPS spectra showed this peak at Er – (75.1 + 0.2) eV. While these points of disagreement in detail are worth further stu- dy, it now seems clear that the three investigations mentioned here — Switendick's theoretical work, the x- ray emission work of Williams et al., and our XPS results – all concur on one major point: the gold 5d bands in the AuſAl2-type alloys lie 6-8 eV below the Fermi energy. 4. Acknowledgments Several people contributed significantly to the suc- cess of this work, Dr. Albert Narath suggested the problem. Dr. Jack Hollander and Mr. Charles Fadley gave valuable advice during the experiments. Miss Carol Martinson assisted in preparing the alloys. Dr. A. C. Switendick elucidated the relationships among his work, that of Williams, et al., and ours. Dr. R. E. Wat- son and Dr. L. H. Bennett encouraged us to prepare this note for the Proceedings during a discussion of the AuAl2 problem. Mr. Charlie Butler and Mr. Lee John- son provided technical assistance in the preparation of the experiments. 5. References [1] Switendick, A. C., and Narath, A., Phys. Rev. Letters 22, 1423 (1969). [2] Jaccarino, V., Weber, M., Wernick, J. H., and Menth, A., Phys. Rev. Letters 21, 1811 (1968). [3] Switendick, A. C., these Proceedings, p. 63. [4] A wavelength of 3000 A corresponds to an energy of 4.14 eV. [5] The application of XPS to valence-band studies in metals is discussed by C. S. Fadley and D. A. Shirley, NBS J. Res. 74A, 543 (1970) and these Proceedings, p. 163. [6] Williams, M. L., Dobbyn, R. C., Cuthill, J. R., and McAlister, A. J., these Proceedings, p. 303. 793 The Reliability of Estimating Density of States Curves from Energy Band Calculations* E. B. Kennard,” D. Koskimaki, J. T. Waber,” and F. M. Mueller” Materials Science Department, Northwestern University, Evanston, Illinois 60201 The density of states curve for aluminum was calculated for different initial energy bands using the quadratic interpolation method (QUAD) developed by Mueller et al. In one case, a true parabolic energy band was used as input and in the second, the E(k) values were those obtained for aluminum by Snow using the APW method. Deviations from the parabolic density of states curve were found to be inversely proportional to the number of E(k) values per histogram box and hence inversely proportional to the square root of the number of random k points in the Brillouin zone. It was necessary to use 100,000 points to obtain a relative deviation of 0.3%. In the second case, the self consistent band calculations for 2048 points in the full Brillouin zone and for a subset of 256 of these were used as input data. The effect of increasing the number of input values was assessed for 25,000 random points in the Brillouin zone. The relative errors were 26 and 9 respectively for 256 and 2048 points. The effects of “smoothing” as an alternative method of reducing statistical error in computing den- sity of states curves are also discussed. Key words: Aluminum; electronic density of states; free electron parabola; QUAD scheme; relia- bility of smoothing procedures. It has been known by many workers in the field of energy bands that substantial statistical errors could occur in drawing the histograms for a density-of-states curve when a small set of E(k) values was used. We can imagine a discrete spectrum of calculated E(k) values falling randomly within the set of n boxes given by Emin + nAE. Therefore, the relative errors would be depen- dent upon not only on the width of the energy intervals AE but also on their relative location. Waber and Snow [1] discussed a smoothing procedure in their paper on copper, in which several sets of N(E) curves were com- puted. The value of E at the location of the first histo- gram was given by Emin + (AEp/q) where q is an integer and p = 1.2 . . . (q-1). The resulting set of q density of states curves were than averaged at energy points which were separated by AEſq. Despite this procedure, Snow [2] observed that the N(E) curve for aluminum was very ragged when only 256 input k values were used. He observed that a smoother N(E) curve resulted *Research supported by the Advanced Research Projects Agency of the Department of Defense through the Materials Research Center of Northwestern University. **Work done in partial fulfillment of the requirements of the degree of Doctor of Philosophy, Materials Science Dept., Northwestern University, Evanston, Illinois 60201. ***Research Assistant and respectively, Materials Science Dept., Northwestern University, Evanston, Illinois 60201, ****Solid State Physics Division, Argonne National Laboratory, Argonne, Illinois 60439. Professor, when he employed 2048 points in his study of the self. consistent APW bands. An N(E) curve drawn from the data of Snow [2] using only 256 points is presented as figure l; it may be compared with his published curve for 2048 points. 0.40 | | | | I | | | | j DENSITY OF STATES FOR ALUM | NUM O. 35 – BASED ON 256 POINTS IN THE -º-º-º: BR LLO U |N ZONE O.30 – - O.25 F- *: >E O.2O H. - O H- <ſ > O. | 5 || | - tud \ * 0.10 – - <ſ H. of) O.O.5 H. --- O | | | | | | | | º | | — |.5 –|.4 -].3 —1.2 — 1. I —i.O ~ O.9 —O.8 —O.7 —O.6 —O.5 — 0.4 ENERGY |N R Y DBE RGS FIGURE 1. Density of States curve drawn from the E(k) data which Snow [2] obtained in his self-consistent band study of aluminum. The subset with f- 1 was employed. The smoothing procedure of Snow and Waber [1] was employed with q = 5. 795 An idealized free-electron band in a body-centered cubic Brillouin zone was studied by Wood [3] to assess the efficacy of this technique. A fixed number of k points and three different widths of the histograms were employed. The strong dependence of the N(E) on the number of values per box was demonstrated. A N(E) curve which almost reproduced the parabola was found for the largest AE width. The difficulty is that although a smooth N(E) could be obtained by increas- ing AE, one simultaneously becomes less able to esti- mate the value of Er accurately. It would appear that the smoothing procedure would also tend to level peaks and valleys and thus eliminate some of the fine-struc- ture in N(E), which is legitimately there. Thus, one is lead to the necessity of using small energy intervals and many E(k) values. Of course, it is impractical to carry out full scale band calculations for 100,000 points in the Brillouin zone in order to obtain more precise values for both the Fermi level Er and the density-of-states at the Fermi level W(E)). An alternative method of estimating N(E) curves has been developed by Mueller et al. [4]. In this quadratic interpolation scheme (QUAD), the E(k) surface is fitted locally by a quadratic expression to a set of input ener- gies which have been calculated by any independent method for fixed values of k. A Monte-Carlo method is used to generate a large number of random k points, the appropriate E(k) values are found from the local fitting expression, and these are used to construct an im- proved density-of-states curve. Because an energy surface is locally fitted to a given set of E(k) values, the reliability of the curved surface directly depends on how accurately the actual set of E(k) can represent the surface when the k points have high local symmetry and when therefore several sheets of the E(k) surface intersect at k or nearby. This question is quite apart from the precision of the QUAD method. The interpolated values cannot be superior to the input information. One way to improve the accura- cy of the density-of-states curve and hence, of informa- tion derived from it is to increase the number of mem- bers in the set of calculated E(k) which are used as input. To evaluate the effectiveness of the QUAD scheme in reducing statistical error, the input values of E(k) were taken from an ideal parabolic band. As a further test, a set and a subset of E(k) values obtained an actual band structure calculation were also used. 1. Methods In the original QUAD method, the values of E(k) were calculated in terms of a model Hamiltonian matrix of nine basis functions representing the interaction of S and d bands of a foc transition metal. The method was adapted in the current study to permit one to use an input E(k) values obtained in any separate inde- pendent manner. For example, in the idealized parabolic band (using atomic units where h = m = e = 1) the energy E(k)=k*/2 Rydbergs. We used as the coordinates of k the set of evenly spaced points in the Brillouin zone which were given by (m/2, n/2, pſ2) Tſaaf. Herein ao is the lattice parameter of the direct lattice, fis an integer and the in- tegers m, n, p individually range from 0 to 8. A unit value of f corresponds to 256 points in the Brillouin zone which is associated with a face centered cubic metal and f = 2 yields 2048 points. For the Brillouin zone associated with the body centered cubic metals, a unit value of f yields 128 and f- 2 yields 1024 k points. The QUAD method was applied to free-electron E(k) values with f set equal to unity and only a single E(k) curve was used. Only those grid points were used for which k was less than or equal to the distance from T to Å. In the study of the smoothing procedure, single ener- gy band was used and f was set equal to 2. For the input from an APW calculation, we employed 2048 points arranged in the coordinate grid given above as well as a subset of 256 of these. In order to shorten the machine time required, the random k values were obtained by a Monte-Carlo method in only 1/48 of the Brillouin zone. The remainder of the zone was “filled” by symmetry operations and the resulting E(k) values were sorted into the energy “boxes.” In the case of aluminum, Snow [2] did not find the E(k) roots for all of the 2048 k values where the energies were well above the Fermi level. Those needed to complete the four s plus p bands were obtained in two ways. In the first case, all the unknown E(k) values were set equal to zero. This choice had a strong effect on the N(E) curve at values near the Fermi level and produced an abnormally high spike at E = 0. In the second case, the unknown values were estimated by the formula E(k) = Eo-H a (k - ko)” where ko and Eo correspond to values already found in a given band. This method was used for the construc- tion of figure relating to the effect of the number of points used in the Brillouin zone. 2. Results Three “smoothed” density-of-states curves obtained using the parabolic band are shown in figures 2a, 2b, 796 ... 2 I SMOOTHED DENSITY OF STATES cuRVE PLANE WAVE FOR BCC ZONE (16 POINTS PER BOX) .1O- *- F e-J O8- *d > | O H. <[ N C X. * 0.4| – 2 [[] 2. FREE ELECTRON PARABOLA O2– - Q; i # —l _l I i # 8 3 4 ENERGY (RYD) t —I T i I i T SMOOTHED DENSITY OF STATES CURVE O8- PLANE WAVE FOR BCC ZONE - (32 POINTS PER BOX) º O l l— 1— 1– 1– 3 4 5 ENERGY (RYD) ë O 2 # 8 O7 º: Ol l -1– —l SMOOTHED DENSTY OF STATES CURVE PLANE WAVE FOR BCC ZONE (64 POINTS PER BOX) 3 4. ENERGY (RYD) 5 ă 7 FIGURE 2, Smoothed Density of States curve for a free electron parabola. The effect of increasing the energy interval for a fixed set of E(k) values in a single band. (a) For AE = 0.125, (b) for AE = 0.250, (c) for AE = 0.500. and 2C. These pertain to using 1024 k values in the full Brillouin zone of a bec metal and three energy inter- vals, AE = 0.125, 0.25 and 0.5 Ryd. The smoothing in- teger q was set at 5. The mean number of points per in- terval was obtained by dividing the number of k values by the number of histogram intervals before smoothing. The results in terms of points per histogram box are presented in table 1. The precision, as indicated by the maximum [N(E) — k”]/k”, is clearly dependent on the number of values per “box.” In a rather similar way, the ability to reproduce the ideal free-electron parabola was investigated with the QUAD method. The results are presented in figures 3a. 3b, and 3c. The three graphs pertain to using v equal to 5,000, 25,000, and 100,000 random k values in a cell which is 1/48 of the foc Brillouin zone. The root mean square relative deviation, from the parabola, namely (26N(E)/k”) were computed using all of the histograms. These results are given on these graphs. The ragged ap- pearance of these curves is apparent. It is clear that 797 EFFECT OF THE NUMBER OF POINTS IN THE BRLLQUIN ZONE ON THE DENSITY OF STATES CURVES EFFECT OF THE NUMBER OF POINTS IN THE BRILLOUN ZONE ON THE DENSITY OF STATES CURVES 5,000 pts. I—I-T—I-I-I I H–H–H |, | H OT = 9.84% 25,000 pts. 22 O = 5.59 % |.OH- 5 O.9}- O º º 2. | > e-se -: 32 O.7- / - à - É O,6 º 0- > * o 99F 32 Lil Or 5 O,4}- H (ſ) O.3 {ſ} ** 3 - H / ; " - O.2– O, i * Ok—H-H-H-H-H-H 2 6 |O || 4 |8 22 26 3O 34 38 ENERGY ENERGY EFFECT OF THE NUMBER OF POINTS |N THE BRILLOUIN ZONE ON THE DENSITY OF STATES CURVE º IOO,000 pts. O = 3.56% |. ||— |.O *4 3 os O 5 os i . O.7 - Dr. O.6 sº LL] 0– O.5 Oſ) # 0. - 5 O.3 - O.2 - O. * O I-1—1—1–1 l—1–1—1–1 l 1–1–– 1–1 H-TăT: ENERGY | 1 3O FIGURE 3. Illustration of the effect of increasing the number of E(k) values found by the QUAD procedure for a fixed number of input values. (a) v = 5,000 pts, (b) v = 25,000 pts, (c) v= 100,000 pts. one must employ a large number of random k values to obtain a smooth parabolic curve. The precision de- pends upon how many E(k) values are found within each energy box of width AE. Table 2 presents the values found for the ideal parabolic band. In figure 3, only the first band of a free electron metal was used. However, in a realistic solid, the E(k) curves are disjoint and piecewise continuous—the separation being caused by the lattice potential. Thus, one would anticipate that when the wave vector (k+ g) takes on critical values which correspond to touching the sur- face of the Brillouin zone, there would be large peaks in the N(E) curve. The reason is that dE/dk becomes ap- proximately equal to zero on “each side” of the critical k value. Thus, when one includes the four s plus p bands of a metal like aluminum several peaks in the 798 TABLE 1. Deviation of the smoothed N(E) curve from the free electron parabolic curve (1024 k points) Energy Mean Maximum interval points per deviation (Ryd) histogram percent 0.125 16 50.5 .250 32 15.0 .500 64 8.8 | I | | | | | | | | FREE ELECTRON O.45 H DENSITY OF STATES FOR FCC LATT| CE - 256 POINTS QUAD |NPUT O.4O H - O.35 H. — > O º O.30 T - à \ 0.25 H _ _ -- ~ T * _ -- ~ T < _ -- ~ 5 o.20 H __ ]--T - : oſs I- --~1 -- T - # _2~ o O.I.O H _* - O.05 T — O | | | | | | | | | | O O. : 0.2 0.3 O.4 O.5 O.6 O.7 O.8 O.9 I.O |, | ENERGY RYDBERGS I | I | | | | | | I 0.30 L FREE ELECTRON DENSITY OF STATES FOR FCC LATTIce 2048 POINT S QUAD INPUT O.2O – >- O. I O - ºv - H. | wn 2. ; C) O.O5 – W. XI W2. - Li X.4? W, O | | | | | | | | | | O O. O.2 O.3 0.4 O.5 O.6 O7 O.8 O.9 |.O |...} ENERGY RY DBERGS FIGURE 4. Effect of increasing the number of input k values from 256 to 2048 on the standard deviations with a total of 25,000 points found by the QUAD scheme. For 4 bands if an FCC metal. (a) For subset with f- 1, (b)fuller set with f-2. N(E) curve would be observed even though the N(E) curve would confirm to a parabola for small k values. TABLE 2. Deviation of calculated density of states curve from a parabolic band (120 energy intervals) Total k Values per | Standard points histogram deviation | percent 5,000 42 9.84 25,000 210 5.59 100,000 840 3.56 The effect of increasing the set of E(k) values before using QUAD was investigated. In figure 4, the calcu- lated N(E) curves are presented for 256 and 2048 input values using the free-electron model for a foc metal. The standard deviations are 26 and 9 percent respec- tively. This was done to make the fourth test more valid. The location of various states is also illustrated. The fourth test made was of the ability of the QUAD scheme to reproduce not only an idealized parabola but also a real N(E) with its peaks and valleys. Aluminum was chosen, since Snow [2] and Harrison [5] have shown it to approach very closely to being a free-elec- tron metal. The close agreement with a parabola is seen at low energies. In a similar test on aluminum, the N(E) curves are presented for 2048 points and the subset of 256 of these. Both parts of figures 4 and 5 relate to extending the calculations to 25,000 points by the QUAD method. For comparison with figure 4, the location of specific states and their relation to the peaks in the N(E) curve are indicated. 3. Discussion and Conclusions As Mueller et al. [3] pointed out, when the mesh points of the Brillouin zone are a simple fraction of the distance from T to X, a number of peaks in the N(E) curves will occur. This is illustrated for figures 4 and 5 drawn for face centered cubic metals. A large prime value off would have been desirable in such an analy- sis, but suitable input values were not available. The use of the free electron curve for N(E) forms a reliable test of the interpolation method since the correct answer for a single band is known a priori; procedures for choosing the number of E(k) values are identical to those used in most band structure calculations. The present study emphasizes two points about reliability; there is a joint dependence on both the fineness of the 799 O.50 | | | | | | | | | | O45 H. ALUM | NUM - DENSITY OF STATES O.40 H 256 polnTS QUAD INPUT --- > - # 0.35 H. º > N 0.30 P- Uſ) * 0.25– < H- of) O.2O H. £ 0.15 k— ty) z ul C O.10 – O,05 H- O | | | | | | | | | | | -1, I –1 O – 0.9 –0.8 -0.7 —O.6 –0.5 – O.4 -0.3 –0.2 -0.] O O.] ENERGY RYDBERGS I | | I | I I | I | | I O.30— ALUM [NUM – DENSITY OF STATES FROM QUAD ź 2O48 POINTS QUAD NPUT H. * O.25 H. > Lil N Uſ) Lu O.2O H. Hi- <ſ H. ty) O.] 5 H- O.IO I- i —O.9 – O.8 —O.7 —O6 – O.5 - O.4 —O.3 —O2 — O. O O.] ENERGY RYDBERGS FIGURE 5. A similar calculation for aluminum using the Snow’s values of E(k) at 2048 vectors of the Brillouin Zones and completing the values for the 4 bands where necessary. (a) For subset of 256 points (f= 1), (b) for 2048 points (f= 2). mesh of points in the Brillouin zone and on number of values available per histogram interval. The first point is illustrated by the effect of the number of input values on either the smoothed curve from Snow’s data or the subsequent tests in figures 4 and 5. The effect of the number of values per interval is shown in table 2. The two parameters are not wholly independent of each other but both are important if one wishes to know simultaneously the value of N(EP) and the location of Er to better than 1 percent precision. Wood [6] recently called our attention to the local gradient method of Gilot and Raubenheimer [7]. He re- ports that his study of this method by a scheme similar to that outlined herein indicates an equivalent (or su- perior) reliability can be achieved at a similar cost. It has been shown that the reliability of estimating N(E) curves from calculated data can be improved so that a precise measure of the error for each E can be given by increasing the number of points for which E(k) values are obtained and by increasing the number of values per histogram box. A procedure such as the QUAD interpolation scheme can contribute signifi- cantly the reliability by gaining more E(k) values for AE interval without incurring an excessive cost in machine time that would be necessary to solve the det H-E(k) = 0 at many points. 4. Acknowledgments The authors are indebted to Dr. John Wood who made his study of the smoothing method available to us prior to publishing. In addition, he offered valuable ad- vice. We appreciate the cooperation of Dr. Edward Snow, also of Los Alamos Scientific Lab, who made detailed sets of E(k) values available to us. 5. References [1] Snow, E. C. and Waber, J. T., “Self Consistent Energy Bands of Metallic Copper by the Augmented Plane Wave Method,” Phys. Rev. 157, 570 (1967). [2] Snow, E. C., “Self Consistent Energy Bonds of Metallic Copper by the Augmented Plane Wave Method,” Phys. Rev. 158, 683 (1967). [3] Wood, J. W., unpublished work, Los Alamos Scientific Lab, Communicated June 1968. [4] Mueller, F. M., Garland, J. W., Cohen, M. H., and Bennemann, K. H., submitted to Phys. Rev. [5] Harrison, W. W., “Pseudopotentials in the Theory of Metals,” Frontiers in Physics, (Benjamin Press, New York, 1966). [6] Wood, J. W., private communication, December 1969. [7] Gilat, H., and Raubenheimer, B., Phys. Rev. 144, 390 (1966). 800 Appendix I ELECTRONIC DENSITY OF STATES SYMPOSIUM COMMITTEES Elio Passaglia, General Chairman, Metallurgy Division Harry C. Burnett, Committee Coordinator, Metallurgy Division PROGRAM COMMITTEE Lawrence H. Bennett, Chairman, Metallurgy Division Russell C. Casella John R. Cuthill Albert Feldman H. P. R. Frederikse John W. Gadzuk Arnold H. Kahn Calvin S. Koonce Archie J. McAlister Elio Passaglia Cedric Powell James F. Schooley John W. Cooper Center for Radiation Research Metallurgy Division Inorganic Materials Division Inorganic Materials Division Atomic and Molecular Physics Division Inorganic Materials Division Heat Division Metallurgy Division Metallurgy Division Atomic and Molecular Physics Division Heat Division Atomic and Molecular Physics Division REGISTRATION COMMITTEE Robert R. Stromberg, Chairman, Polymers Division ARRANGEMENTS COMMITTEE Robert T. Cook, Chairman, Office of Technical Information and Publications SOCIAL COMMITTEE Howard T. Yolken, Chairman, Institute for Materials Research LADIES” HOSPITALITY COMMITTEE Mrs. Evelyn Brady, Chairman Mrs. Edna Passaglia, Vice Chairman Mrs. Joan Yolken Mrs. Alice Ambler Mrs. Leatrice Gevantman FINANCE COMMITTEE Ronald B. Johnson, Chairman, Institute for Materials Research 801 Appendix II Session Chairmen and Rapporteurs BAND STRUCTURE I Chairmen: A. J. Freeman, Northwestern University E. Passaglia, NBS BAND STRUCTURE II Chairmen: F. Herman, IBM Research Center, San Jose R. C. Casella, NBS Rapporteur: R. E. Watson, Brookhaven National Laboratory OPTICAL PROPERTIES, BAND STRUCTURE III Chairmen: E. A. Stern, University of Washington J. R. Cuthill, NBS Rapporteur: G. W. Pratt, Jr., Massachusetts Institute of Technology PHOTOEMISSION Chairmen: E. T. Arakawa, Oak Ridge National Laboratory C. J. Powell, NBS Rapporteur: S. B. M. Hagström, Chalmers University, Göteborg, Sweden MANY-BODY EFFECTS Chairmen: L. N. Cooper, Brown University J. W. Cooper, NBS EXCITONS: SOFT X-RAY I Chairmen: R. A. Farrell, University of Maryland C. S. Koonce, NBS Rapporteur: L. Hedin, Chalmers University, Göteborg, Sweden SOFT X-RAY II; DISTRIBUTIONS IN MOMENTUM SPACE Chairmen: F. M. Mueller, Argonne National Laboratory A. J. McAlister, NBS Rapporteur: D. J. Fabian, University of Strathclyde, Glasgow, Scotland ION-NEUTRALIZATION: SURFACES: CRITICAL POINTS; ETC. Chairmen: E. Callen, American University - R. R. Stromberg, NBS Rapporteurs: D. E. Aspnes, Bell Telephone Laboratories B. Lax, National Magnet Laboratory, Massachusetts Institute of Technology DISORDERED SYSTEMS I Chairmen: L. F. Mattheiss, Bell Telephone Laboratories A. Kahn, NBS 802 DISORDERED SYSTEMS II Chairmen: L. M. Roth, General Electric Research and Development Center H. P. R. Frederikse, NBS Rapporteur: M. H. Cohen, University of Chicago ALLOYS; ELECTRONIC SPECIFIC HEAT I Chairmen: J. R. Anderson, University of Maryland J. H. Schooley, NBS ELECTRONIC SPECIFIC HEAT II; KNIGHT SHIFT: SUSCEPTIBILITY Chairmen: A. Narath, Sandia Laboratories H. C. Burnett, NBS Rapporteurs: J. Rayne, Carnegie-Mellon University I. D. Weisman, NBS TUNNELING; SUPERCONDUCTORS; TRANSPORT PROPERTIES Chairmen: J. R. Leibowitz, Catholic University of America J. W. Gadzuk, NBS TRANSPORT PROPERTIES: APPLICATIONS Chairmen: A. I. Schindler, Naval Research Laboratory A. Feldman, NBS Rapporteur: R. J. Higgins, University of Oregon 803 Appendix III Registrants–Electronic Density of States Symposium Ajami, F. University of Pa. Philadelphia, Pa. 19104 Albers, W. A. General Motors Research Labs. 12 Mile & Mound Rds. Warren, Mich. 48090 Albert, F. Laboratoire de Phys. Solides Giorsay, France Alexander, M. N. Materials Research Laboratory Army Materials & Mechanical Research Center Watertown, Mass. 02172 Amar, H. Temple University Philadelphia, Pa. 19102 Anderson, J. R. Dept. Phys. & Astronomy University of Maryland College Park, Md. 20740 Anderson, O. K. University of Pennsylvania Philadelphia, Pa. 19104 Arakawa, E. T. Oak Ridge National Lab. Bldg. 4500–S, H–160 Oak Ridge, Tenn. 37830 Arlinghaus, F. J. General Motors Research Labs. Warren, Mich. 48090 Artman, G. Northwestern University Evanston, Ill. 60202 Ashcroft, N. W. Laboratory of Atomic and Solid State Physics Cornell University Ithaca, N.Y. 14850 Aspnes, D. E. Bell Telephone Labs. Murray Hill, N.J. 07974 Ausman, G. A. Harry Diamond Laboratories Washington, D.C. 20438 National Bureau of Standards November 2–6, 1969 Austin, R. AFOSR Wilson Blvd. Arlington, Va. 22207 Azaroff, L. V. University of Connecticut Storrs, Conn. 06268 Baarle, C. V. Kamerligh Onnes Labs. Leiden, Netherlands Bambakidis. G. NASA, Lewis Research Center Cleveland, Ohio 44135 Baratoff, A. Brown University Providence, R.I. 02912 Barbe, D. F. Westinghouse Electric Corp. Baltimore, Md. 21203 Barker, R. C. Yale University New Haven, Conn. 06520 Baumgardner, C. A. University of Idaho Moscow, Idaho 83843 Beck, Paul University of Illinois Metallurgy Department Urbana, Ill. 61801 Beeby, John L. Atomic Energy Research Establishment Theoretical Physics Div. B.8.9 Harwell, Didcot, Berks England Bell, M. I. Brown University Physics Department Providence, R.I. 02912 Bennett, H. National Bureau of Standards Washington, D.C. 20234 Bennett, L. H. National Bureau of Standards Washington, D.C. 20234 Blatt, F. J. Michigan State University Physics Department East Lansing, Mich. 48823 Blewer, R. S. Sandia Labs. Albuquerque, N.M. 87106 Bohm, Horst Owens-Illinois Tech. Cent. Toledo, Ohio 43601 Boyle, John J. Night Visions Lab. Ft. Belvoir, Va. 22060 Bradley, D. C. Queen Mary College London, England Brailsford, Alan D. Ford Motor Co. Science Lab. Dearborn, Mich. 4812] Breedlove, H. Ch. Tng. and Dev. Div. CPO Ft. Belvoir, Va. 22060 Brodersen, Robert W. M.I.T. Cambridge, Mass. 02139 Brouers, F. Institut de Physique Universite de Liege Sart-Tilman, Liege l Belgium Brown, J. S. University of Vermont Burlington, Vt. 05401 Burke, J. Richard Naval Ordnance Lab. Silver Spring, Md. 20910 Burnett, H. C. National Bureau of Standards Noordwijk, Holland Washington, D.C. 20234 Butler, W. H. Auburn University Auburn, Ala. 36830 Butrymowicz, D. B. National Bureau of Standards Washington, D.C. 20234 Callaway, J. Louisiana State University Baton Rouge, La. 70803 804 Callen, E. American University Washington, D.C. 20016 Campbell, W. J. U.S. Bureau of Mines College Park, Md. 20740 Campi, Morris Harry Diamond Labs. Washington, D.C. 20438 Cardona, M. Deutsches Elektronen-Synchrotran (DESY) F 41 Notkestieg 1 Hamburg 52 German Federal Republic Carlson, F. Night Visions Lab. Ft. Belvoir, Va. 22060 Caron, L. G. Universite de Sherbrooke Quebec, Canada Carr, W. J. Westinghouse Res. and Dev. Pittsburgh, Pa. 15235 Carter, Forrest L. Naval Research Labs. Washington, D.C. 20390 Carter, G. C. National Bureau of Standards Washington, D.C. 20234 Casella, R. C. National Bureau of Standards Washington, D.C. 20234 Caskey, George R. National Bureau of Standards Washington, D.C. 20234 Cate, Robert National Bureau of Standards Washington, D.C. 20234 Chang, K. Night Visions Lab. Ft. Belvoir, Va. 22060 Chen, An-Bun College of William & Mary Williamsburg, Va. 23185 Cho, Sang-Jean National Research Council Ottawa, Ontario, Canada Choyke, W. J. Westinghouse Res. Labs. Pittsburgh, Pa. 15235 Christman, J. Richard Tufts University Medford, Mass. 02155 Church, E. L. Brookhaven National Labs. Physics Department Upton, N.Y. 11713 417–156 O - 71 – 53 Clark, Howard National Bureau of Standards Washington, D.C. 20234 Clune, Lavern C. Ohio University Physics Department Athens, Ohio 45.701 Cohen, Morrel H. University of Chicago Chicago, Ill. 60637 Cohen, Roger W. RCA Labs. Princeton, N.J. 08540 Çollings, E. W. Battelle Memorial Inst. Columbus Labs. Columbus, Ohio 4320] Collins, Thomas C. Aerospace Res. Labs. Area B, Bldg 450 Wright Patterson Air Force Base, Ohio 45433 Colwell, J. H. National Bureau of Standards Washington, D.C. 20234 Conklin, James B. University of Florida Physics Dept., Williamson Hall Gainesville, Fla. 32601 Connolly, John W. D. Advanced Materials Research & Development Laboratory Pratt and Whitney Aircraft Middletown, Conn. 06458 Cooper, J. W. National Bureau of Standards Washington, D.C. 20234 Cuthill, John R. National Bureau of Standards Washington, D.C. 20234 Cutler, Paul H. Pennsylvania State University 101 Osmond Lab. University Park, Pa. 16802 Damon, Dwight H. Westinghouse Research Labs. Beulah Road Pittsburgh, Pa. 15235 Davis, H. L. Oak Ridge National Metals & Ceramics Div. Oak Ridge, Tenn. 31830 Deegan, R. A. University of Illinois Urbana, Ill. 61801 DePorter, Gerald L. Los Alamos Scientific Lab. Los Alamos, N.E. 87544 Dobbyn, Ronald C. National Bureau of Standards Washington, D.C. 20234 Dow, John D. Palmer Physical Lab. Princeton University Princeton, N.J. 08540 Dresselhaus, Mildred M.I.T. Cambridge, Mass. 02139 Driscoll, T. E. U.S. Bureau of Mines College Park, Md. 20740 Dryer, Joseph E. Ohio State University Res. Found. Columbus, Ohio 43220 Duff, K. J. Ford Motor Company Dearborn, Mich. 4814] Eastman, Dean E. IBM Yorktown Heights, N.Y. 10598 Ehrenreich, Henry Division of Pure & Applied Physics Harvard University Cambridge, Mass. 0.2138 Eisenberger, Peter Bell Telephone Labs. Murray Hill, N.J. 07974 Erlbach, E. City College of CUNY 251 Ft. Washington Avenue New York, N.Y. 10032 Esaki, L. - IBM, Thomas J. Watson Research Center Yorktown Heights, N.Y. 10598 Eschenfelder, A. H. IBM Res. Lab. San Jose, Calif. 95114 Etzel, H. National Science Foundation Washington, D.C. 20550 Evenson, William E. University of Pennsylvania Philadelphia, Pa. 19104 Fabian, Derek J. Laboratorium fur Festkorperphysik Gloriastrasse 35 8006 Zurich, Switzerland Falge, Raymond L. National Bureau of Standards Washington, D.C. 20234 Faulkner, J. S. Oak Ridge National Lab. Oak Ridge, Tenn. 37830 Fehrs, Delmer NASA-ERC Cambridge, Mass. 02139 Feldman, A. National Bureau of Standards Washington, D.C. 20234 805 Ferris-Prabhu, A. IBM, Dept. 392 Essex Junction, Vt. O5452 Fert, Albert Laboratoire de Phys. Solides Orsay, France Finegold, L. X. University of Colorado Boulder, Colo. 80302 Fitton, B. European Space Res. Organization Noordwijk, Holland Foner, Simon M.I.T. Francis Bitter Natl. Magnet Lab. Cambridge, Mass. 02139 Forest, H. Zenith Radio Corp. Chicago, Ill. 60639 Forman, Richard A. National Bureau of Standards Washington, D.C. 20234 Frederikse, H. P. R. National Bureau of Standards Washington, D.C. 20234 Freeman, A. J. Northwestern University Evanston, Ill. 60201 Frolen, Lois J. National Bureau of Standards Washington, D.C. 20234 Fromhold, Al National Bureau of Standards Washington, D.C. 20234 Gadzuk, J. W. National Bureau of Standards Washington, D.C. 20234 Gerstner, J. Operation Res. Inc. State College, Pa. 16801 Goff, James F. Naval Ordnance Lab. Silver Spring, Md. 20910 Gordon, William L. Case Western Reserve Univ. Cleveland, Ohio 44106 Grabner, Ludwig H. National Bureau of Standards Washington, D.C. 20234 Grant, Wesley Yale University New Haven, Conn. 06520 Gray, A. Watervliet Arsenal Watervliet, N.Y. 12189 Gray, D. M. Watervliet Arsenal Watervliet, N.Y. 12189 Greenfield, Arthur J. BAR-Ilan Univ. Ramat-Gan, Israel Greer, William National Bureau of Standards Washington, D.C. 20234 Grobman, Warren D. IBM, Watson Research Center Yorktown Heights, N.Y. 10598 Gross, Chris NASA-Langley Res. Center Hampton, Va. 23366 Hacker, K. University of Pittsburgh Pittsburgh, Pa. 15213 Haeringen, W. Van Philips Res. Labs. Eindhoven, Netherlands Hagstrum, Homer D. Bell Telephone Labs. Mountain Avenue Murray Hill, N.J. 07974 Hagstrom, S. Chalmers University Göteborg, Sweden Halder, N. C. State University of New York Albany, N.Y. 12203 Hammond, Robert H. University of California Berkeley, Calif. 94720 Handler, Paul University of Illinois Urbana, Ill. 61801 Hanzely, Stephen Youngstown State University Department of Physics 410 Wick Avenue Youngstown, Ohio 44503 Harrison, Michael J. Michigan State University East Lansing, Mich. 48823 Harvey, W. W. Kennecott Copper Corp. Lexington, Mass. 02173 Hayes, Timothy M. Xerox Corp. Res. Labs., Rochester, N.Y. 14603 Hedin, L. Chalmers University of Tech. Göteborg, Sweden Hensel, J. C. Bell Telephone Labs. Murray Hill, N.J. 07974 Herman, Frank IBM Research Lab. San Jose, Calif. 95114 Herzog, Donald C. College of William & Mary Williamsburg, Va. 23185 Hicks, Jay Charles 684 N.W. 18th Corwallis, Oreg. 97330 Higgins, Richard J. University of Oregon Eugene, Oreg. 97403 Hindley, Norman K. Corning Glass Works Corning, N.Y. 14830 Hirst, Robert G. General Electric Co. Pittsfield, Mass. 01201 Ho, James C. Battelle Memorial Institute Columbus, Ohio 4320] Hofer, Alvin D. TAM-National Lead Corp. Niagara Falls, N.Y. 10013 Hoffman, Herbert J. NASA-E RC Cambridge, Mass. 02139 Holliday, Jerome E. E. C. Bain Lab. Fundamental Research U.S. Steel Corp. Res. Cen. Monroeville, Pa. 15146 Horowitz, Emanuel National Bureau of Standards Washington, D.C. 20234 Houston, Bland U.S. Naval Ordnance Lab. Silver Spring, Md. 20910 Hsia, Yukun Litton Systems Inc. Guidance and Control Systems Div. 5500 Canoga Avenue Woodland Hills, Calif. 91364 Hyde, G. R. U.S. Bureau of Mines College Park, Md. 20740 Islar, William E. Harry Diamond Labs. Washington, D.C. 20438 Janak, James IBM Yorktown Heights, N.Y. 10598 Jan, Jean-Pierre National Research Council Ottawa, Ontario, Canada Jarrett, Howard S. DuPont and Co. Wilmington, Del. 19898 Johnson, Keith H. M.I.T. Cambridge, Mass. 02139 Johnson, Ronald B. National Bureau of Standards Washington, D.C. 20234 806 Joshi, S. K. University of Roorkee Roorkee, India Kahan, Daniel National Bureau of Standards Washington, D.C. 20234 Kahn, A. H. National Bureau of Standards Washington, D.C. 20234 Kane, Evan O. Bell Telephone Labs. Murray Hill, N.J. 07974 Karo, Arnold M. Lawrence Radiation Lab. Livermore, Calif. 94.550 Kasowski, Robert V. DuPont and Co. Wilmington, Del. 19898 Kaufman, Larry A. M.I.T. National Magnet Labs. Cambridge, Mass. 02139 Ketterson, John Argonne National Labs. Argonne, Ill. 60439 Keyes, Robert W. IBM Thom. J. Watson Res. Cent. Yorktown Heights, N.Y. 10598 Kjeldaas, T. Polytechnic Inst. of Brooklyn Brooklyn, N.Y. 1120.1 Klein, Michael W. Wesleyan University Middletown, Conn. 06457 Klein, W. Night Visions Lab. Ft. Belvoir, Va. 22060 Kmetko, Edward A. Los Alamos Sci. Lab. University of California Los Alamos, N.M. 87544 Kohn, Walter University of California La Jolla, Calif. 92037 Koonce, Calvin S. National Bureau of Standards Washington, D.C. 20234 Korringa, J. Ohio State University Columbus, Ohio 83210 Krieger, J. B. Polytechnic Inst. of Brooklyn Brooklyn, N.Y. 11201 Kunz, Christof University of Maryland College Park, Md. 20740 Kunzler, J. E. Bell Telephone Labs. Murray Hill, N.J. 07974 Kurrelmeyer, B. Brooklyn College of CUNY Brooklyn, N.Y. 11210 Lalevic, Bogoljub Franklin Inst. Res. Lab. Philadelphia, Pa. 19103 Langer, Dietrich W. J. Aerospace Res. Labs. Wright Patterson Air Force Base Ohio 45433 Lapeyre, Gerald J. Montana State University Bozeman, Mont. 59715 Laufer, Jeffrey University of Michigan Department of Chemistry Ann Arbor, Mich. 48104 Lax, Benjamin M.I.T. National Magnet Lab. Cambridge, Mass. 02139 Libelo, Louis U.S. Naval Ordnance Lab. White Oak, Md. 20910 Long, Jerome R. V.P.I. Department of Physics Blacksburg, Va. 24061 Lopez-Escobar, Albert H. M. Goddard College Plainfield, Vt. 05667 Lundqvist. Bengt I. Chalmers Univ. of Tech. Göteborg 5, Sweden Lundqvist, Stig Chalmers University Institute of Physics Göteborg, Sweden Lye, Robert G. Head, Metal Physics Group Martin Marietta Corp. 1450 South Rolling Rd. Baltimore, Md. 21227 Mackintosh, A. R. Lab. for Electrophysics Technol. Univ. Lungby, Denmark Maeland, Arnulf Worcester Polytechnic Inst. Worcester, Mass. 01609 Mahan, G. D. Clare-Hall Herschnell Road Cambridge, CB39A1 England Mapother, D. E. University of Illinois Physics Department Urbana, Ill. 61801 Marcus, Paul M. IBM Research Center Yorktown Heights, N.Y. 10598 Martin, D. L. National Research Council Ottawa, Ontario, Canada Martin, Richard M. Bell Telephone Labs. Murray Hill, N.J. 07974 Massey, Walter E. University of Illinois Department of Physics Urbana, Ill. 61801 Masuda, Yoshika Nagoya University Department of Physics Chikusa-Ku, Nagoya, Japan Mattheiss, Leonard F. Bell Telephone Labs. Murray Hill, N.J. 07974 Mayroyannis, C. National Research Council Ottawa, Ontario, Canada McAlister, Archie J. National Bureau of Standards Washington, D.C. 20234 McNeil, M. B. Department of Materials Engineering Mississippi State University P.O. Drawer CM State College, Miss. 39762 Mebs, R. National Bureau of Standards Washington, D.C. 20234 Menth, A. Bell Telephone Labs. Murray Hill, N.J. 07974 Meyer, Axel N. Illinois University Physics Department DeKalb, Ill. 601.15 Misetich, Antonio National Magnet Lab., M.I.T. Cambridge, Mass. 02139 Mitchell, Dean L. Naval Research Labs., Washington, D.C. 20390 More, Richard M. Department of Physics Faculty of Arts and Sciences University of Pittsburgh Pittsburgh, Pa. 15213 Mozer, Bernard National Bureau of Standards Washington, D.C. 20234 Mueller, Fred M. Argonne National Labs. Argonne, Ill. 60439 Muldawer, Leonard Temple University Physics Department Philadelphia, Pa. 19122 807 Mura, Toshio National Bureau of Standards Washington, D.C. 20234 Narath, A. Sandia Labs. Albuquerque, N.M. 87115 Narsing, G. R. U.S. Bureau of Mines College Park, Md. 20740 Natapoft, Marshall Newark College of Eng. Newark, N.J. 07102 Neddermeyer, H. 8035 Gauting, Waldpromenade 61 8 Munchen 22, Germany Nedoluha, Alfred NWC Corona Labs. Corona, Calif. 91720 Neustadter, H. NASA Cleveland, Ohio 44135 Nilsson, Olof Chalmers Univ. of Tech. Physics Department 40220 Göteborg 5, Sweden Novak, Robert L. Westinghouse Astronuclear Pittsburgh, Pa. 15228 Ortenburger, Irene B. IBM Research Labs. San Jose, Calif. 95114 Overhauser, A. W. Ford Motor Co. Dearborn, Mich. 48121 Parsons, Brian J. Michelson Labs., N.W.C. Chiff, Lake, Calif. 93555 Passaglia, Elio National Bureau of Standards Washington, D.C. 20234 Paul, William Harvard University Pierce Hall Cambridge, Mass. 0.2138 Penchina, Claude University of Massachusetts Physics Department Amherst, Mass. 01002 Petroff, Irene University of California Department of Electrical Sciences Los Angeles, Calif. 90024 Piller, Herbert Louisiana State University Physics Department Baton Rouge, La. 70803 Platzman, P. M. Bell Telephone Labs. Murray Hill, N.J. 07974 Plummer, Ward National Bureau of Standards Washington, D.C. 20234 Pollard, John Night Visions Lab. Ft. Belvoir, Va. 22060 Powell, Cedric National Bureau of Standards Washington, D.C. 20234 Praddaude, Hernan M.I.T. National Magnet Lab. Cambridge, Mass. 02139 Price, Peter J. IBM Watson Lab. New York, N.Y. 10025 Rayne, J. A. University of Lancaster England Redfield, David RCA Labs. Princeton, N.J. 08540 Reilly, M. H. U.S. Naval Res. Lab. Washington, D.C. 20390 Richardson, P. E. U.S. Bureau of Mines College Park, Md. 20740 Rooke, G. A. University of Strathclyde Glasgow, Cl, Scotland Ross, Gaylon S. National Bureau of Standards Washington, D.C. 20234 Ross, Marvin University of California Lawrence Radiation Lab. Livermore, Calif. 94.550 Roth, Laura M. General Electric Corp. P.O. Box. 8 Schenectady, N.Y. 12301 Rothberg, Gerald M. Stevens Inst. of Tech. Hoboken, N.J. 07030 Rousseau, Cecil C. Baylor University Department of Physics Waco, Tex. 76703 Rowe, J. E. Bell Telephone Labs. Murray Hill, N.J. 07974 Rowell, J. M. Bell Telephone Labs. Mountain Avenue Murray Hill, N.J. 07974 Rowland, T. J. University of Illinois Metallurgy Department Urbana, Ill. 61801 Rubin, Robert J. National Bureau of Standards Washington, D.C. 20234 Rudge, William E. IBM Research San Jose, Calif. 95114 Sacks, Barry H. University of California Berkeley, Calif. 94720 Saffren, M. M. California Institute of Technology Jet Prop. Lab. Pasadena, Calif. 91103 Sagalyn, Paul L. Army Materials & Mechanical Res. Center Watertown, Mass. 02172 Sanford, Edward R. Ohio University Clippinger Research Labs. Athens, Ohio 45701 Satya, A. V. S. IBM New York, N.Y. 12533 Savage, Howard T. Naval Ordnance Labs. Silver Spring, Md. 20910 Schaich, William Cornell University Clark Hall Ithaca, N.Y. 14850 Schindler, A. I. Naval Research Laboratory Washington, D.C. 20390 Schirber, J. E. Sandia Labs. Albuquerque, N.M. 87115 Schone, Harlan E. College of William & Mary Physics Department Williamsburg, Va. 23185 Schooley, James F. National Bureau of Standards Washington, D.C. 20234 Schrieffer, J. R. University of Pennsylvania Physics Department Philadelphia, Pa. 19104 Schwartz, Lawrence Harvard University Cambridge, Mass. 62138 Shaw, R. F. Energy Conver. Dev. Inc. Troy, Mich. 48084 Shaw, Robert W. Bell Telephone Labs. Murray Hill, N.J. 07974 Sher, M. A. College of William & Mary Williamsburg, Va. 23185 808 Shimizu, Masao Nagoya University Department of Applied Physics Nagoya, Japan Shirley, David A. University of California Chemistry Department Berkeley, Calif. 94720 Sievert, Paul R. Battelle Memorial Inst. Columbus, Ohio 4320.1 Simmons, John A. National Bureau of Standards Washington, D.C. 20234 Smith, John R. NASA Cleveland, Ohio 44135 Smith, Neville V. Bell Telephone Labs. Murray Hill, N.J. 07974 Sonntag, Bernd Physikalisches Staatsinstitut Hamberg, Bahrenfeld, Germany Soonpaa, Henn H. Department of Physics The University of North Dakota Grand Forks, N.D. 5820] Spain, Ian Inst. for Mol. Phys. University of Maryland College Park, Md. 20740 Speiser, Rudolph Ohio State University Columbus, Ohio 4322] Spicer, William E. Stanford University Stanford, Calif. 93405 Stanley, D. A. U.S. Bureau of Mines College Park, Md. 20740 Stearns, Mary Beth Ford Sci. Lab. Dearborn, Mich. 48121 Stern, Edward A. University of Washington Physics Department Seattle, Wash. 98105 Stern, Frank IBM, Watson Res. Center Yorktown Heights, N.Y. 10598 Steslicka, Maria University of Waterloo Department of Applied Math Waterloo, Ontario, Canada Stewart, A. T. Queen’s University Physics Department Kingston, Ontario, Canada Stilwell, E. P. Clemson University Physics Department Clemson, S.C. 29631 Stromberg, R. R. National Bureau of Standards Washington, D.C. 20234 Swartzendruber, Lydon J. National Bureau of Standards Washington, D.C. 20234 Swerdlow, Max A. F. Office of Sci. Res. Arlington, Va. 22209 Switendick, A. C. Sandia Labs. Albuquerque, N.M. 97115 Suzuki, K. Bell Telephone Labs. Murray Hill, N.J. 07974 Tahir-Kheli, R. A. Temple University Department of Physics Philadelphia, Pa. 19122 Tauc, J. Bell Telephone Labs. Murray Hill, N.J. 0797] Teitler, Sidney Naval Research Labs. Washington, D.C. 20390 Thompson, James C. Energy Conv. Dev. Inc. Troy, Mich. 48084 Thornber, Karvel K. Bell Telephone Labs. Murray Hill, N.J. 07974 Torgesen, John L. National Bureau of Standards Washington, D.C. 20234 Toxen, A. M. IBM Yorktown Heights, N.Y. 10598 Triftshauser, Werner Queen’s University Department of Physics Kingston, Ontario, Canada Tsang, Paul Sa-Min IBM Hopewell Junction, N.Y. 12533 Uriano, George A. National Bureau of Standards Washington, D.C. 20234 Vechten, James A. Van Bell Telephone Labs. Murray Hill, N.J. 07974 Violet, Charles E. Lawrence Radiation Lab. Livermore, Calif. 94.550 Vosko, S. H. Westinghouse Res. Labs. Pittsburgh, Pa. 15235 Waber, James T. Northwestern University Materials Science Department Evanston, Ill. 6020.1 Wachtman, J. B. National Bureau of Standards Washington, D.C. 20234 Walford, Lionel K. McDonnel Douglas Corp. St. Louis, Mo. 63166 Warren, William W. Bell Telephone Labs. Murray Hill, N.J. 07974 Watson, R. E. Brookhaven National Lab. Upton, N.Y. 11973 Weisman, Irwin D. National Bureau of Standards Washington, D.C. 20234 Weisz, Gideon College of William & Mary Williamsburg, Va. 23185 Welch, Gerhard University of Utah Physics Department Salt Lake City, Utah 84112 Wei, C. T. Michigan State University East Lansing, Mich. 48823 Wiech, Gerhard 8000 Munchen Fl Engadinerstr 34, Germany Willens, R. H. Bell Telephone Labs. Murray Hill, N.J. 07974 Williams, A. R. IBM Yorktown Heights, N.Y. 12535 Williams, Morgan L. National Bureau of Standards Washington, D.C. 20234 Williams, Ronald W. Oak Ridge National Labs. Oak Ridge, Tenn. 37830 Windmiller, Lee Argonne National Labs. Argonne, Ill. 60439 Winogradoff, Nicholas National Bureau of Standards Washington, D.C. 20234 Wolcott, Norman M. National Bureau of Standards Washington, D.C. 20234 Wood, John K. Utah State University Physics Department Logan, Utah 84321 Woods, Joseph F. IBM, T. J. Watson Res. Cent. Yorktown Heights, N.Y. 10598 809 Wooten, Frederick Lawrence Radiation Lab. Livermore, Calif. 94.550 Yelon, Arthur Yale University 10 Hillhouse Avenue New Haven, Conn. 06520 Yolken, H. Thomas National Bureau of Standards Washington, D.C. 20234 Young, Russell National Bureau of Standards Washington, D.C. 20234 Zeisse, Carl R. Naval Elec. Lab. Cent. San Diego, Calif. 92.152 Ziman, John M. University of Bristol Physics Laboratory Bristol, England 810 Subject Index Following each subject is the page number of the first page of the article in which the subject appears. core hole—233,269 core polarization—601, 649, 659 correlation—47,233,665 crystal potential–63, 755 Curie temperature — 105, 557, 57.1 Curie-Weiss law—601 cyclotron resonance–423,431, 757 D Davydov splitting—261 Debye temperatures— 139, 557, 565, 579,597,601, 685, 741, 767 de Haas-van Alphen effect–57,339, 649, 673,685, 717, 737, 785 — and the Knight shift—601 delta— —function formulation of density of states—437 —function potential—509, 515 – potential impurity—473 depletion layer barrier tunneling—693 diamond–335 dielectric constant—33, 77, 111, 115, 125, 223,757 diffusion equation—473 Dirac equation, one-dimensional–477 disordered— — solid solution alloys — 19 — systems—1,453,465,505, 509, 521 Drude model—33 ductility—483 E actinide metals— 17 absorption— — potential—359 — spectra—223, 253, 275, 319 alkali– — ammonia solutions—601 – metal alloys – 191 — metals—67, 103,223, 253,269,275,287,359,601, 651, 783 alloy phase — —local density of states–307, 313 — stability—19 amorphous semiconductors– 1, 493, 505, 521 Anderson transition—493 anthracene—261 anti-ferromagnetism—27,493, 527, 557, 571, 741 anti-phase domain—571 aromatic crystals—261 Auger– — effect— 179 —transitions—281, 287 augmented plane wave method (APW)–1, 19, 53, 63, 67, 105, 111, 115, 191,297,493,565 B band— – absorption—693 — population effects—407 — tail—493, 505, 579, 693 – transitions, direct-non-direct— 159, 181, 191,205,217 benzene (C6H6)–26] bond, chemical–385 brass, BCu-Zn—339, 509 carbon contamination—199 cellular disorder—521 chalcogenide glasses—493 chalcogenides—601 charge density—359 charging effect—313, 587 chemisorption—375 cluster theory—505 coherent potential approximation—505, 551 compressibility—67, 731,741 Compton scattering—233, 345 conduction electron spin resonance (CESR)—601 electroreflectance—407,411,417, 757 electron— — donation model—727 —electron scattering—129, 139, 767,785 — enhancement—571 —interaction—63,233,493 — phonon — coupling constant—681, 731, 741, 767, 785 — enhancement—571 —interaction–63, 227, 233, 493, 565, 587,685 — paramagnetic resonance—601 — synchrotron light—275 electronic specific heat—57, 63,297,565, 579, 651,685, 731 electrons in a box– 129 electrical resistivity—483, 665,685, 737 —temperature dependent–737 energy distribution curves (EDC)— 139 enhancement— — electron-electron—571 811 indirect transition—111 interband transition—33, 39, 125, 181 intermetallic compounds, CsCl structure — 19, 303 interpolation method—27 itinerancy—47 ion-neutralization spectroscopy (INS)— 163, 349 Ising model—381 J joint density of states—19,93, 129, 192,209 — magnetic—431 Josephson tunnelling—l K k-conservation—19, 139, 181, 191, 199,205, 437 Knight shift—303,601, 649, 651,659, 665 Korringa-Kohn-Rostoker method (KKR)—19, 57,63, 181,509 Kondo effect— 1,601 k p method — 115,335, 757 Kronig-Penney model – 129,477 L enhancement— Continued — electron-phonon–557, 571,681 — exchange—601, 657, 665 Esaki diode–673, 693 exchange—43,659, 665 exchange— — Kohn-Sham–93 — Liberman–93 – Slater – 19, 57,93, 111, 359 exchange potential—105,233, 741 excitons—253,261, 717 excluded terms—515 F f bands—105 Fermi-Dirac — 253,287,685, 741, 757 Fermi– — edge singularity—269 — energy–53 — surface—39, 385,735, 785 ferro-magnetic spiral structure—571 ferromagnetism—27,47 field-emission resonance tunneling—375 Franck-Condon principle—493 Franz-Keldysh theory–93, 417 Friedel– —oscillations—253,359, 601 – screening theory—551 G G. P. zone—587 galvanomagnetic– — experiments — 737 — properties—717 Ginzburg-Landau coherence length — 587 glass– — chalcogenide–493 — tetrahedral— I graphite–717 Green’s functions – 1, 233, 465, 509, 515, 527, 551, 767 Gruniesen parameter, electronic—731, 741 Gunn effect—437 H Hall effect—483, 717 Hamiltonian– — Anderson—375 — Ashcroft–33 — Hubbard–527 Hartree-Fock method—67, 93,233,271, 527,601 Hartree-Fock-Slater potential–63, 181,601 Hellman-Feynman theorem –27 Hume-Rothery electron compound—19 hydrocarbons — 13, 261 hydrogen— — absorption–727 —in palladium–737 hyperfine fields—601,649 Landau levels—39,431,437,673 Landau-Peierls diamagnetism—601 lanthanides—67,217,571 lasers— — semiconductor—437 LCAO – 19, 27, 115,329 LEED spectra— 1, 349 liquids—l local– — field cancellation—557 – density of states—465 localized states—493 - low-temperature specific heat—57, 483,571, 579, 587, 597, 681, 685, 767 luminescence experiments—693 M magnetic– — clusters—557 — critical points—431 — specific heats—571, 741 — susceptibility — 483,587,601, 645, 649, 651,685, 737, 767 magneto — —optical density of states—437, 757 — reflection—39 —resistance–717 Matsubara approximation—527 mechanical behavior—483 melting—483 membrane, two dimensional classical–473 metal— —insulator-metal (MIM)—673 — insulator-semiconductor (MIS)—407,673 – vacuum interface—359, 473 812 model— – density of states–767 – potential of Heine and Abarenkov–483 modulation technique–281 molecular crystals— 1,261 Mott insulator—493,527,601 muffin-tin potential—1, 19, 57, 181,297, 349,453,473, 509 multiple-scattering series–515 N near-free electron model–359 nearly free electron approximation—19 naphthalene —261 Néel temperature—493,527,601 noble metal alloys– — Ag–1, 303, 307, 483,545, 551, 601 – Au– 19,303, 483,493, 545, 601,673, 727,791 — Cu- 139, 339, 385, 483, 509, 557, 601 noble metals— 17, 19, 67, 105, 139, 163, 181, 191, 205, 253, 281, 303, 319,349,359,545, 551, 597,601,651, 795 nonlinear optical susceptibility–757 nuclear magnetic resonance (NMR)—601, 659 O —superconducting transition temperature–681, 731, 735 pressure effects—63, 717 pseudopotential–1, 33, 57, 93, 115, 125, 269, 483, 601, 649, 651,659, 755 omega phase—587 one-dimensional Dirac equation—477 Onsager model—381 optical– — absorption—77, 111, 125,233,253,279 — constants–545 — density-of-states— 139, 159, 191, 205, 209, 437,493 – properties— 19, 33, 111, 115, 125, 191, 209, 233, 303, 758, 791 — reflection—47 order-disorder–483 orthogonalized plane wave (OPW)—27,47, 53,93, 335,601, 649, 651, 659 oscillatory dielectric function—417 P Pauli spin susceptibility—601, 645, 649, 737, 757 Pauling radii–385 Peierls barriers—483 phonon–673, 767 photoabsorption—275 photoconductivity—493 photo-electric effect— 129 photoemission spectroscopy (PES)–27,47, 53, 105,159,163, 181, 191, 199,205, 209, 217,493,551,597 piezo soft x-ray effect—281 plasmeron—233 plasmon—125, 181,287 polaron–293 positron annihilation—269,339, 345 potential— — coherent approximation—505, 551 — deformation—767 — exchange — 105,233, 741 — model of Heine and Abarenkov—483 – muffin-tin— 1, 19, 57, 181,297,453,473, 509 pressure dependence of the – —band structure–67, 77 — Knight shift–601, 651 QUAD scheme — 17, 181, 741,795 quasi- —localized state–521 — particle—125, 227, 381 R random phase approximation (RPA)–115,233,659 rare-earth metals—67,217,571 reflectivity–33, 39, 115, 209, 493, 545 resistivity ratios —651 rigid-band approximation – 19, 53, 163, 313, 329, 515, 551, 587, 597, 601, 685, 727, 737 S satellite— — emissions—319 — structure — 125 scattering phase shifts—515 Schottky barrier–693 Schrödinger equation— 1,253,521,651 secondary emission – 181 self absorption—319 semiconductor laser—437 semiconductors— — amorphous–1, 493, 505, 521 — compounds— 135,483 — diamond or zinc blende—93, 335,411,693, 757 short-range-order parameters—509 small particles—473 soft x-ray spectroscopy (SXS) — 139, 163, 217, 233,253,269,273, 275, 281, 287, 297, 303, 307, 313, 319, 327, 335, 555, 597, 601, 665, 791 specific heat— — electronic—57, 63,297,565, 579, 651,685, 731, 737 — low temperature—57, 483,571,579,587,597,681, 767,783 — magnetic–571, 741 spin– — dipolar interactions—601 — orbit splittings—39, 67, 163,221, 303,411, 545, 717, 756, 757 — wave theory—571 strain dependence of superconducting transition temperature–731, 735 superconducting transition temperatures—557, 565, 587, 731, 735 superconductivity–565, 731, 735, 767 surface— — Auger spectroscopy (SAS)—349 — energy—359 — potential–407 synchrotron light—279 T temperature dependent resistivity–737 thermal— — conductivity—685 — effective mass—233 — expansion, electronic—731 thermoelectric power—651,685, 737 813 tight-binding approximation–27, 47, 115, 359, 453, 521, 527, 579, 601 tetrahedral— — glass–1 — semiconductor–93, 335, 411, 693, 757 t-matrices— 129, 253, 509, 527, 551 transition-metal behavior—57, 527 transition metals—17, 19, 163, 191, 349, 385, 483,565, 756, 767 transport properties—651, 685, 693 tunnel junctions—737 tunneling— 1, 375, 693 U uv photoemission spectroscopy (UPS)— 163, 199, 217, 233, 349 V Van Vleck temperature independent paramagnetism—601 virtual crystal approximation—551 volume dependence of density-of-states–731 W Weiss model–381 Wigner-Seitz cell–53, 67, 601, 74] X x-ray– — absorption—233 — emission—47 — photoemission spectroscopy (XPS)–163, 217, 233, 319, 349, 791 Z vacuum ultraviolet reflectance spectra— 111 Van der Waals type interaction—261 zinc blende structure—163, 693, 757 814 Materials Index Following each material is the page number of the first page of the article in which the material appears. All materials in the index are arranged alphabetically by their chemical symbols. Compounds (e.g., oxides, nitrides, etc.) are also listed alphabetically. Example: NaCl is listed under ClNa. A hyphen is used to indicate various compositions which may include compounds or solid solutions. Example: Al-Cu. A Actinide metals— 17 Ag–63, 163, 181, 205, 551, 601 AgAl–303, 307 Ag-Al–303,601 Ag-Au–1, 545, 551, 601 Ag-In-551 Ag-Pol—551, 601 Ag-Sb-601 Ag-Sn—601 Agº Te—483 alkali— — ammonia solutions—601 – metal alloys — 191 Al–33, 159, 233,253,269, 275,287, 303, 313, 601, 673, 731, 795 Al-Al2O3-semiconductor–673 Al2-Au-297, 303, 665, 791 Al2Au-AuCa2–791 AlCl3–665 AlCo– 19, 601 Al-Cu-307 Al-Cu-Ni–557 AlFe—601 Al-Fe-557 Al-Fe-V –557 Al-insulator-Ni–673 Al-insulator-NiPol—673 Al-Mg-287, 313, 329, 665 AlNi–19, 27, 483, 601, 665 Al-insulator-Pa–191, 737 Al2O3–665 Alp —93 AlSb–757 AlTi5–483 Al-Ti-329, 483 Al2U–601 Al-Zn—307 amorphous semiconductors— 1, 493, 505, 521 anthracene—261 aromatic crystals—261 As –39 AsGa–77, 93, 139, 407, 411, 693, 757 Asin–77, 673, 757 AsNi–385 As-Si (As doped)–693 Au – 17, 63, 139, 163, 205, 303, 545, 551, 601 AuCu3–483 Au-Cu-483, 545 AuFe—545 AuCa2–303, 601, 791 AuGe–493 Au-Ge-673 Auln2–303, 791 Au-Pol— 727 Au-Pol-H – 727 Auzn – 19 BTi–329 Ba—67, 217, 375 Be—233,275, 601, 649, 659 benzene (C6H6)—261 Bi-601, 673 Biln— 601 BiºMg3–483 BizTes sºmºn 483 brass, 3Cu-Zn-339, 509 C–335, 597, 717 CTi–329, 598 CSi —601 CW — 597 Ca–57, 67, 375 Col-359, 659 CaCl2–163 Co-In (Cd dilute)—601, 735 ColS – 139 ColSb – 483 ColSe— 139 CaTe– 139, 757 Ce–67, 601 chalcogenides— — glass—493 Co- 105, 163,319 CoTi–329, 483 Cr–27, 209, 221, 571, 587, 598, 685 Cr-Cu-601 Cr-insulator-Pb —673 815 Cr2Ti–329 Cr-W – 601 Cs—67, 233, 601, 651, 783 Cu–33, 63, 74, 105, 139, 163, 181, 191, 205, 253,281, 319, 339, 349, 555, 598, 601, 651, 795 Cu (cesiated)–139, 191 Cu-Mn—385, 557 Cu-Ni– 139, 557 Cu-Pol—601 CuPt – 483 Cu-Pt–483 CuZn—339, 509 D diamond–335 diamond semiconductors—335, 411 E Er–571 hydrocarbons—13, 261 He—345, 349 Hg–359, 587 HgC) – 181 Ho–571 In — 159, 601 InP — 757 In-Pb —601 In-Pol—26 InSb – 407, 693, 757 In-Sn—601 Ir— 163, 685 K–103, 233, 287, 651 Kr – 279 L Er-Tm—571 Eu – 105, 217 Eu-chalcogenides — 105, 221 EuO – 105 EuO-EuSe — 105 EuS – 105 EuSe — 105 EuS-EuSe — 105 F Fe–1, 17, 27, 47, 67, 105, 163, 319, 587, 641 Fe2O3–319 FeTi – 483 Fe-W –557, 601 Ga–303, 601, 731 Gap – 757 GaSb —411, 757 Gd– 105, 221, 571 Gd-Pr–571 GdS – 105 Ge–1, 33, 77, 93, 139, 253, 335, 349, 375, 411, 417, 431 GeO2 – 111 graphite–717 glass — chalcogenide–493 —tetrahedral—l H2 in Au-Pol—727 H2 La – 601 H-Nh —601 H-PG – 727 H-SC – 601 H-W – 601 H2Y – 601 Hf-Ta-565 lanthanides—67, 217, 57.1 Li–67, 139, 253, 269, 275, 359, 493, 651 M Mg-233,253, 275, 313, 601, 659, 673 MgO –493 MgsSb2–483 Mn–587 Mn-Ni–557 Mo–53, 209, 221, 598, 673, 685 Mo-Ti–221, 587 molecular crystals – 1, 261 N Na–103, 125, 233,269, 275, 359, 601, 651 naphthalene —261 Nb – 221, 598, 685 Nb5Sn–601, 767 Ne—358 NTi — 329 Ni– 17, 105, 139, 163, 281, 319, 349, 555, 601, 673 NiO – 221,493 NiTi – 329, 483 noble metal alloys— —Ag–1, 303, 307, 483, 545, 601 — Au – 19, 303, 483, 493, 545, 601, 673, 727, 791 — Cu-139, 339, 385, 483, 509, 557, 601 O OsRe–27 OTi — 329 Os– 163 P P-Si (P doped)—685 Pb –67, 233, 557, 601, 673 816 Pb chalcogenides—77 PhTe–77, 115, 139, 601 Pa–17, 91, 139, 163, 181,571, 601, 645, 673, 685, 735, 737 Pa-Rh–579, 601 Pt—17, 163,359, 601, 658, 685 Pu–67 R rare-earth metals—67, 217, 57.1 Rb —601, 651, 783 Re—587, 685 Re-Ta-565 Re-W – 565 Rh–163, 579, 685 Rh-Ru–579 Ru – 163 Sb2n–483 Sb-Si (Sb doped)–693 Sc– 17, 57, 205, 587, 598 semiconductors– — amorphous–1, 493, 505, 521 — compounds— 135, 483 – diamond or zinc blende—93, 335, 411, 693, 757 SeZn–93 Si–1, 77, 93,253, 335, 349, 693 Si (As, P, Sb, doped)–693 SiVa —601, 767 Sn–673 Sn-Ti-483 SnáU–601 Sr–67 Ta–53, 565, 587, 685 TaW – 565 TeTl2–483 TeZn–757 Th–731 Ti–205, 221, 329, 587, 597 Tizr–483 Tl–601, 681 Trm — 571 transition metals—17, 19, 163, 191, 349, 385, 483, 556,565, 767 U U–67 V V–221,587,601 W W–53, 67, 209,375, 387, 685, 691 W (8-W compounds)–767 X Xe — 279 Y Y — 598 Yb —221 Z Zn–163, 209, 287, 319, 359, 659 zincblende structure—163, 693, 757 Zn2Zr—601 ZnTe — 757 Zr—205, 375, 587, 598 817 FoEM NBS-1 14A (1-71) U.S. DEPT. OF COM M. 1. PUBLICATION OR REPORT NO. 3. Recipient’s Accession No. B|B LiOGRAPHIC DATA SHEET NBS-SP-323 4. TITLE AND SUBTITLE 5. Publication Date December 1971 Electronic Density of States 7. AUTHOR(S) Lawrence H. Bennett, Editor 9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. Project/Task/Work Unit No. NATIONAL BUREAU OF STAND ARDS DEPARTMENT OF COMMERCE ll. Contract/Grant No. WASHINGTON, D.C. 20234 12. Sponsoring Organization Name and Address 13. Type of Report & Period Covered Final National Bureau of Standards 15. SUPPLEMENTARY NOTES 16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant bibliography or literature survey, mention it here.) This volume is based on materials presented at the Third Materials Research Symposium of the National Bureau of Standards, held November 3–6, 1969. It provides a review of various experimental and theoretical techniques applied to the study of the electronic density of states in solids and liquids. The topics covered in a series of invited and contributed papers include theory of and experiments to obtain the one-electron density-of-states; many-body effects; optical properties; spectroscopic methods such as photoemission (x-ray and UV), ion neutralization, and soft x-ray; obtaining the density-of-states at the Fermi level by specific heat, magnetic susceptibility, and the Knight shift; the disordered systems of alloys, liquids, dirty semiconductors, and amorphous systems; and superconducting tunneling and the application of density of states to properties such as phase stability. An edited discussion follows many of the papers. 17. KEY WORDS (Alphabetical order, separated by semicolons) Band structure; disordered systems; electronic density of states; ion neutralization; Knight shift; magnetic susteptibilit many-body effects; optical properties; photoemission; soft x-ray; specific heat; super- 18. AVAILABILITY STATEMENT conductivity; transport proper- 3. SECURITY CLASS 21. NO. OF PAGES ties; tunneling. (THIS REPORT) [x] UNLIMITED. X 834 UNCL ASSIFIED [ ] FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE 20. SECURITY CLASS 22. Price TO NTIS. 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