untitled © V.M. Pudalov, 2011 Fizika Nizkikh Temperatur, 2011, v. 37, No. 1, p. 12–24 David Shoenberg and the beauty of quantum oscillations V.M. Pudalov P.N. Lebedev Physical Institute, 53 Leninskii prospekt, Moscow 117924, Russia E-mail: pudalov@lebedev.ru Received August 24, 2010 The quantum oscillation effect was discovered in Leiden, in 1930, by W.J. de Haas and P.M. van Alphen in magnetization measurement, and by L.W. Shubnikov and de Haas — in magnetoresistance. Studying single crystals of bismuth, they observed oscillatory variations of magnetization and magnetoresistance with magnetic field. Shoenberg, whose first research in Cambridge had been on bismuth, found that much stronger oscillations are observed when a bismuth sample is cooled to liquid helium rather than to liquid hydrogen, which had been used by de Haas. In 1938 Shoenberg came from Cambridge to Moscow to study these oscillations at Kapitza In- stitute where liquid helium was available at that time. In 1947, J. Marcus observed similar oscillations in zinc, that persuaded Shoenderg to return to this research, and, since then, the dHvA effect had been one of his main research topic. In particular, he developed techniques for quantitative measurements of the effect in many met- als. Theoretical explanation of quantum oscillations was given by L. Onsager in 1952, and the analytical quantit- ative theory by I.M. Lifshitz and A.M. Kosevich in 1955. These theoretical advancements seemed to provide a comprehensive description of the effect. Since then, quantum oscillations were commonly considered as a tool for measuring Fermi surface extremal cross-sections and all-angle electron scattering times. However, in his pio- neering experiments in 1960s, Shoenberg revealed the richness and deep essence of the quantum oscillation ef- fect and showed how the beauty of the effect is disclosed under nonlinear conditions imposed by interactions in the system under study. It was quite unexpected, that under «magnetic interaction» conditions, the apparently weak effect of quantum oscillations may lead to such strong consequences as breaking the sample into magnetic (now called «Shoenberg») domains and the formation of an inhomogeneous magnetic state. Owing to his contri- bution to the field of quantum oscillations and superconductivity, Shoenberg is no doubt one of the 20th cen- tury's foremost experts. We describe the experiments on finding the quantitative parameters of electron–electron interaction, which are in line with the Shoenberg ideas that the quantum oscillations are modified by interactions and, hence, their analysis enables one to extract the quasiparticle interaction parameters. PACS: 71.30.+h Metal–insulator transitions and other electronic transitions; 72.15.Rn Localization effects (Anderson or weak localization); 73.40.Qv Metal–insulator-semiconductor structures (including semiconductor-to-insulator). Keywords: quantum oscillations, two-dimensional carrier system, electron–electron interaction. 1. Introduction: correspondence with David Shoenberg It is a privilege to be invited to contribute to this vo- lume in honour of Professor David Shoenberg with whom I had an opportunity to communicate and whose papers I studied thoroughly in the past. In the early 1970s, I was a graduate student at the famous Institute for Physical Problems in Moscow, which is now named after P.L. Ka- pitza. Though I have never met David Shoenberg personal- ly, of course, I knew much about his work with Kapitza in Cambridge and in Moscow and about his visits to Kapitza Institute in 1960s. A few years earlier, in 1966, my scien- tific supervisor, the outstanding experimentalist and Teacher, M.S. Khaikin, suggested me to develop a super- sensitive dilatometer, as a topic for gradate project. The dilatometer, i.e. a device for measuring small displace- ments, was intended for measuring changes in the sample size in magnetic field. The microwave technique was the favorite subject in the Khaikin laboratory, and, not surpri- singly, the dilatometer was a sort of a microwave cavity with a thin copper membrane and a needle-type coaxial conductor, which concentrated microwave energy near the region of the membrane deformation. David Shoenberg and the beauty of quantum oscillations Fizika Nizkikh Temperatur, 2011, v. 37, No. 1 13 When my dilatometer started working reliably, it ap- peared that I could easily observe the nice phenomenon of quantum oscillations of size of the metallic single crystals placed in varying magnetic field at low temperatures. The oscillatory magnetostriction had, of course, the same origin as the de Haas–van Alphen (dHvA) effect, and from its amplitude I anticipated to extract information on electron coupling to the lattice potential, that is the so-called de- formation potential. I was encouraged with this observa- tion and ignored a common wisdom that says «avoid quan- titative amplitude measurements when possible». Quite soon I realized that the measurements of the magnetostric- tion amplitude alone is insufficient to get quantitative in- formation. In order to extract components of the deforma- tion potential tenzor, one has also to measure the amplitude of the oscillatory magnetic moment, i.e., the dHvA effect. Experimentalists know perfectly that absolute ampli- tude measurements for a single effect is a hard task, and amplitude measurements for two effects are still much harder. In those times, David Shoenberg carried out nice absolute measurements [1,2] of the dHvA effect amplitude, and each of his papers represented a piece of experimental art and deserved careful reading. However, the technique he used was incompatible with my microwave cavity dila- tometer. Clearly, measurements of the two effects should be performed in situ during a single cooldown, because the oscillation amplitude is determined not only by controlla- ble variables (temperature, magnetic field), but also by the sample «quality» (more exactly, the Dingle temperature [3]), which may vary from one cooldown to another. The solution to the problem was found heuristically: it unexpectedly came during the experiment and had a direct relation to David Shoenberg! The situation was as follows: in my measurements, for signal extraction from noise, I used not a conventional field modulation technique with a lock-in amplifier and a slow magnetic field sweep, but an alternative technique with a multichannel analyzer and fast multiple sweeps of magnetic field within several seconds [4]. The measuring system was based on the frequency modulation method, developed by M.S. Khaikin and re- presented a heterodyne microwave receiver. Variations of the crystal size were measured as changes in frequency of the microwave oscillator that had the cavity — displace- ment sensor — in the feedback loop. For convenience, in the Khaikin laboratory, the experimental data was not only recorded by electronics but was also available to experi- mentalists in visual and audio form. Correspondingly, the intermediate frequency signal transposed to the audio range was fed, besides a frequency detector, to an ordinary loudspeaker. Taking measurements of the oscillatory mag- netostriction late in the evening, in the silent laboratory, I heard that, upon cooling the sample down to 0.3 K, the tone of the audio frequency signal began to vary abruptly with magnetic field in a saw-tooth manner, rather than harmonically (see Fig. 1). The first thought that in magnetic field the sample was cracking was rejected at once, because on subsequent warming of the sample up to 1.4 K, the oscillations became harmonic again, as expected. The observed unharmonicity was much greater, than one could expect from the Lif- shitz–Kosevich (LK) theory [5]. Therefore, the more natu- ral assumption was to associate the saw-tooth oscillations with the so-called Shoenberg effect and magnetic domains [2]. Indeed, despite the smallness of the magnetic moment oscillations ,Mδ i.e., the amplitude of the dHvA effect, the oscillation period in good metals with a large Fermi surface (such as tin and indium under my investigation) is small and, therefore, the magnetic susceptibility | / |M B∂ ∂ becomes comparable with 1 / 4π . As a result, the magnetic induction in the sample B noticeably deviates from the external magnetic field H : = 4 (1 ) .B H D M+ π − (1) The deviation leads to the magnetic interaction or the Shoenberg effect [2] (here D is the demagnetization fac- tor), that is described by solution of the exact nonlinear equation [7] sin . 4 (1 ) r c r M A H D M ∞ ν ⎡ ⎤ω⎛ ⎞ ⎢ ⎥⎜ ⎟+ π −⎝ ⎠⎣ ⎦ ∑∑∼ v (2) As soon as the unharmonicity cause was identified, to make the next step was a matter of not too sophisticated though rather awkward calculations. Using sequential ap- proximation technique I have calculated the discrete spec- trum of oscillations. The obtained series converged rapidly and delivered a striking result: the desired amplitude of the magnetization oscillations could be found from the ratio of harmonics in the oscillatory magnetostriction spectrum. Therefore, the problem of simultaneous amplitude mea- surements for two effects reduced to measurements of the amplitude for only one of them supplemented with subse- quent analysis of its spectrum. Fig. 1. Typical shape of the quantum oscillations of magneto- striction 11u of tin single crystal versus magnetic field. Bracket next to the upper curve depicts the magnetostriction scale. Tem- perature 0.37T = K, 4 | / | 0.18dM dHπ ≈ : a) for || [001]H and b) for H tilted at 5° in the (010) plane. The lower curve reveals two groups of oscillations with frequency ratio 1: 2≈ , due to two extremal cross-sections of the FS. Note a saw-tooth shape of os- cillations. Reproduced from Ref. 6. а b 7160 7180 7200 7220 1 0 – 3 H, Oe V.M. Pudalov 14 Fizika Nizkikh Temperatur, 2011, v. 37, No. 1 This situation can be explained in the following way. Suppose we have a pendulum (oscillator) and wish to mea- sure the amplitude of its oscillations, but there is no cali- brated ruler to measure linear displacements. Then, for har- monic oscillator the problem has no solution. It appears, however, that for an unharmonic oscillator, when the phys- ics of its unharmonicity is known, the oscillation amplitude may be found by registering the oscillation spectrum and analyzing the spectrum with a linear but uncalibrated ruler. When the calculations of the oscillations spectrum were finished and successfully compared with experimental da- ta, I was proud and assured with the result. It seemed con- vincing to me, and I was about to send the paper for publi- cation. However, not all around me in the lab shared my assurance. The major expert in the field of quantum oscil- lations was certainly David Shoenberg and I decided to consult with him: I wrote a letter (e-mail didn't exist at that time) where I described my idea how the oscillation ampli- tude could be found from their spectrum under conditions of the magnetic interaction*. I was not sure that the famous scientist will answer to unknown graduate student. How- ever, I was pleased to get a short note from Shoenberg quite quickly: he wrote that currently was busy but in a couple of weeks will be able to answer. I waited for about two weeks and indeed received a very kind letter: Shoen- berg was positive about my idea, and also paid my atten- tion to a potential «underwater stones», such as deviations from the conventional LK theory of oscillatory effects [5], which might be related, e.g., with mosaic structure of crys- talline samples. These were the issues he studied at that time [8]. My paper was sent to the journal and published shortly [6]. I keep in memory the kind attention Shoenberg showed to an unknown graduate student, and try to educate my students in the same spirit of kindness and respect to others. 2. Interacting two-dimensional electron system The experiments by Shoenberg demonstrated how the hidden beauty and rich essence of quantum oscillations are revealed in nonlinear conditions introduced by magnetic interaction. With another example of the beauty and rich- ness of the quantum oscillations I faced much later, study- ing the electron–electron interaction effects in two-dimen- sional electronic system. This story is described below. For pedagogical purposes, the story is delivered in the sequence as follows: firstly, we consider how the inter- particle interaction modifies parameters of the electronic system, as compared with those for noninteracting gas. The interaction effects will be initially described, to a first ap- proximation, in terms of the Fermi-liquid interaction con- stants. Further, we will consider the experimental method and the results of measurements of the Fermi-liquid inter- action constants, where quantum oscillations are used as an experimental tool. We will presume first that the oscilla- tion amplitude is small and is not strongly affected by inte- raction. And finally, we will consider how the inter- electron interaction influences the quantum oscillation am- plitude and what consequences it leads. 2.1. Renormalization of the quasiparticle parameters As an Introduction, let us first recall that one of the ma- jor concepts for interacting Fermi systems is the Fermi liquid. It is the generalization of the Fermi gas, i.e. the sys- tem without interaction, to the case with interaction. The electrons in 3D metal or «metallic» 2D systems are charg- ed and, at first sight, seem to experience a great Coulomb repulsion forces 2 / eee r∼ , where eer is the interelectron distance; for 2D systems ~ 1/eer n , where n is the electron density. In fact, these classical forces are compensated by ion lattice, because the total system is neutral. At low energies which matter most to us, additional (or trial) charges introduced to the system do not interact via the Coulomb potential. This is so, since the charges polarize their environment. In 3D, the long-range interaction has effectively been reduced to a short range one with a po- tential sc( ) exp ( / ) /r r r rφ ∝ − where scr is the screening length. The Fourier transform of the screening effect reads 2 2 21 / 1 / ( )sq q q→ + , where sq is the Thomas–Fermi wave vector which is inverse proportional to the screening length sc .r In contrast to the 3D case, in 2D system the screening effect is much weaker. For large distance 1srq , the asymptotic form of the average potential seen by the elec- trons [9] does not show an exponential decay 2 3 ( ) s e r q r φ ∝ . (3) The essential idea on which Fermi-liquid theory is based was introduced by Landau. Even if the bare particles interact strongly, the low energy elementary excitations experience only a weak or moderate interaction. These elementary excitations are called quasiparticles. The quasi- particles can be labeled by the same quantum numbers as the excitations in the noninteracting system. In particular, in the vicinity of the Fermi surface they behave as if they were free fermions. In the absence of magnetic field the quasiparticles have the same charge and spin as free elec- trons, and for short, we shall call them «electrons». The in- teraction strength is characterized by the dimensionless ra- tio of the potential interaction energy eeE to the kinetic (Fer- mi) energy, 2 2= /2 = / 1/s ee Fr E E me n nκ π ∝ (here the factor of 2 takes the valley degeneracy in (100)-Si MOSFETs, = 2gv , into account). Within the framework of the Fermi-liquid theory, the interactions lead to renormalization of the effective quasi- particle parameters [10,11], such as the spin susceptibility * This was the lucky time, when physicists shared with each other unpublished results, thoughts, and samples. David Shoenberg and the beauty of quantum oscillations Fizika Nizkikh Temperatur, 2011, v. 37, No. 1 15 *χ , effective mass * ,m Landé g-factor *g , and compres- sibility *κ . Measurements of these renormalized parame- ters are the main source of experimental information on interactions. The renormalization is commonly described by harmonics of the Fermi-liquid interaction in the singlet (symmetric, (s)) and triplet (antisymmetric, (a)) channels [10,12,13]: * 0 1 = 1 ab g g F+ , * 1 1 = 1 2 s b m F m + , * * 0 1 = 1 sb b m m F κ κ + , * * 0 1 = 1 ab b m m F χ χ + . (4) Here bg , bm , bκ and bχ are the band (bare) values of the g-factor, mass, compressibility and spin susceptibility, respectively. In theory, the above Fermi-liquid interaction constants ,a siF are universal functions of the sr solely. Provided the Fermi-liquid constants are known, the charac- teristics of the interacting 2D electron system to the first order can be expressed as interaction quantum corrections to the characteristics for the noninteracting 2D electron gas. Though the results of numerical calculations of the re- normalized parameters [14–17] vary considerably, all of them agree qualitatively and suggest enhancement of *χ , *m and *g with sr . Earlier experiments [18–21] have shown growth of *m and * *g m at relatively small sr values, pointing to a ferromagnetic type of interactions in the ex- plored range 1 < 6.5.sr 2.2. Quantum oscillations in the 2D electron gas as a tool for extracting interaction constants For 3D noninteracting electron gas placed in quantizing magnetic field = zH H , besides the quantized spectrum = ( 1/2)N c Nε ω + in the ( , )x yk k plane there is a conti- nuous spectrum along the magnetic field direction ( )zkε . In contrast, for a 2D system of electrons placed in magnet- ic field H⊥ perpendicular the ( , )x y plane, the energy spectrum is fully quantized: 1 = , 2c Z N E⎛ ⎞ε ω + ±⎜ ⎟ ⎝ ⎠ (5) where the Zeeman energy 1 = 2Z B E g Hμ (6) and = 0,1, 2,...N is the Landau level number. Each of the Landau levels is 0= /H⊥ν Φ times degenerate, and the oscillation period in the inverse magnetic field has a fun- damental meaning being determined by the ratio of the electron density to the magnetic flux density, 0/ ( / ),n H⊥ Φ where 0 = /hc eΦ . For noninteracting 2D electron gas, the theoretical ex- pression for the Shubnikov–de Haas (SdH) effect, i.e. the oscillatory magnetoresistance, is as follows [5,22]: 0 = cos 1 ,xx s s s c n A s Z eH⊥ ⎡ ⎤⎛ ⎞δρ π π −⎢ ⎥⎜ ⎟ ρ ⎢ ⎥⎝ ⎠⎣ ⎦ ∑ (7) where 2 2 2 2 / = 4 exp 2 . sinh (2 / ) B cB D s c B c sk Tk T A s sk T ⎛ ⎞ π ω − π⎜ ⎟ ω π ω⎝ ⎠ (8) Here 0 = ( = 0)xx H⊥ρ ρ , = /( * )c eeH m m c⊥ω is the cyc- lotron frequency, *m is the dimensionless effective mass, em is the free electron mass. This is the famous Lifshitz– Kosevich formula [5], modified for the 2D case [22]. An additional exponential factor describes Landau level broa- dening due to temperature-independent scattering by short- range impurity potential, / 2D DT ≡ πτ is the so-called Dingle temperature [3], and Dτ is the «all angle» scatter- ing time (in contrast to the transport time τ determined by large angle scattering). In the limit of weak oscillations 0/ 1xxδρ ρ when temperature is not too low, Eq. (8) can be simplified: 2 2 * ( ) 4 exp 2 4B D Bs cc k T T k T A s s ⎛ ⎞+ ≈ − π π⎜ ⎟ ⎜ ⎟ ωω⎝ ⎠ . (9) In Eqs. (7), (8) the valley splitting is assumed = 2gv (that corresponds to the 2D layer of electrons at the (100)-Si surface). The Zeeman factor in Eq. (7) 2= cos [ ( ) / ( )]sZ s c n n eH⊥↑ ↓π − (10) for = 0H reduces to a field-independent constant. Here ( )n n↑ ↓− is the difference in population of the two spin subbands. In case the spin magnetization is a linear func- tion of the total field totH , the nonzero difference in sub- bands populations, i.e. the spin polarization P , can be related to the renormalized spin susceptibility *χ as fol- lows: * tot tot**= = , b B n n H eH P g m n g n nhc ↑ ↓− χ≡ μ (11) where 2bg is the bare g-factor for Si, and totH = 2 2H H⊥= + . Bychkov and Gorkov [23] have shown that the period of oscillations is not affected by interactions, whereas oscillation amplitude (Eqs. (9), (8)) is determined by * *= /( )c eeH m m cω with the renormalized mass * ,m ra- ther than the band mass bm , and the Zeeman splitting (8) — by the renormalized g-factor. 2.3. The idea of measurements *χ , *m , and *g Experimental studies of magnetooscillations go back to the end of 60s when the high mobility 2D structures be- came available [20,24]. Information on the renormaliz- ed effective mass *m is provided by the amplitude of the Shubnikov–de Haas (SdH) effect [23]. The effective mass is usually found from the so-called Dingle-plot, V.M. Pudalov 16 Fizika Nizkikh Temperatur, 2011, v. 37, No. 1 0ln (| | / )xxδρ ρ as a function of temperature. According to the LK theory, the damping factor can be expressed as * ** 1ln ( ) . LK c c DA T T mT ⎛ ⎞ω − ω ≈ +⎜ ⎟ ⎜ ⎟ ⎝ ⎠ (12) As follows from Eq. (12), in the limit of weak oscilla- tions, 0| | / 1xxδρ ρ , the slope, 0ln(| | / ) /xxd dTδρ ρ , is nearly proportional to the effective mass *m , whereas extra- polation of the 1ln [ ( / )]cA Tω to = 0T enables to deter- mine DT . Here, both, the effective mass and the g-factor are thermodynamic quantities and includes all many-body interaction effects. The Zeeman factor Eq. (10) carries information on the renormalized spin susceptibility * * *m gχ ∝ . Conventional technique to measure the effective *χ [20,21] is based on the SdH measurements in magnetic fields tilted with re- spect to the 2D plane. In these measurements, the cyclotron energy related to H⊥ is compared with the Zeeman split- ting, which depends on the total field, totH . To have a good control of both fields, the angle is to be measured with a very high accuracy, a difficult task at mK tempera- tures. To probe separately orbital and spin degrees of free- dom, it is convenient to apply two independently varied magnetic field components: (i) H⊥ , normal to the 2D electron plane, which causes quantization of the orbital motion, and (ii) the in-plane field H , which couples only to spins. Application of H should facilitate the analysis of SdH oscillations, especially near the 2D metal–insulator transition, when the number of observable oscillations is small. The idea of measurements with two field components is explained by Fig. 2. The parallel field H shifts the spin- up and spin-down subbands relative to each other in accord with Eqs. (5), (6), and produces an unequal population of the two subbands (see Eq. (11)). The role of the perpendi- cular field component is to provide measurements of the difference in subband population, and thus to extract the spin susceptibility according to Eq. (11). The perpendicular field causes quantization of the energy levels in both sub- bands and enables to count the difference in their individu- al population, because all spin-split Landau levels are ex- actly /( / )H hc e⊥ times degenerate. 2.4. Crossed field technique In order to facilitate measurements, we developed a «crossed-field technique» [25] by adding the second sole- noid and taking data in crossed magnetic fields, which can be varied independently of each other (see Fig. 3). The conventional technique of measuring * *g m in tilted mag- netic fields [20,21] is not applicable when the Zeeman energy is greater than half the cyclotron energy [26]. The crossed field technique removes this restriction and allows us to extend measurements over the wider range of elec- tron densities. Development of this novel experimental technique enabled us to explore the effect of H on the electron spectrum, as well as to measure directly *m and *g in strongly correlated selectron systems [27]. Typical traces of the longitudinal resistivity xxρ as a function of H⊥ are shown in Fig. 4. Due to the high elec- Fig. 2. Schematic diagram of the Landau levels in the presence of the Zeeman splitting * totBg Hμ . The left and right ladders of Landau levels are for spin-up and spin-down subbands. 2 2 tot =H H H⊥ + . � �F � �c H� g* H�B tot Si-MOSFET Mixing chamber of dilution fridge Solenoid H|| Split coils, H� Fig. 3. The crossed magnetic field set-up. The main supercon- ducting solenoid produces the in-plane magnetic field H up to 8 T. The superconducting split coils, positioned inside the main solenoid, produce the normal field H⊥ , which can be as large as 1.5 T at 4H ≤ T, and decreases gradually down to 0.6 T at = 7.5H T. The sample (Si-MOSFET) is attached to the cold finger of the mixing chamber, with its plane perpendicular to the axis of the coils. Represented from Ref. 25. David Shoenberg and the beauty of quantum oscillations Fizika Nizkikh Temperatur, 2011, v. 37, No. 1 17 tron mobility, oscillations were detectable down to 0.2 T and at temperatures up to 1.6 K; a large number of oscilla- tions and wide range of temperatures enabled us to extract *m and DT with a high accuracy. As Fig. 4 shows, application of H induces beating of SdH oscillations. This is because the uppermost levels in the two spin ubbands move with field at different rates and cross the Fermi energy either in phase or out of phase. The beatings are observed as a function of H⊥ and the beat frequency is proportional to the spin polarization of the interacting 2D electron system .P In experiment, we ob- served a well pronounced beating pattern at a nonzero H (see Figs. 4 and 5), in agreement with Eq. (7). The phase of SdH oscillations remains the same between the adjacent beating nodes, and changes by π through the node. The interference pattern (including positions of the nodes) is controlled by sZ in Eq. (7) and is defined solely by * *.g m Systematic study of this pattern enabled us to determine * *g m with high accuracy ( 2%∼ ). The * *g m values are independent of T (at < 1 K)T within our accuracy [28]. We have observed a weak dependence of * *g m on H in strong H . To determine * *g m in the linear regime, we systematically measured, for each n , the beating pattern at decreasing values of H until * *g m becomes independent of H . Evolution of the beating pattern with H is illu- strated in Figs. 5,a and b. 2.5. Data analysis Comparison between the measured and calculated de- pendences 0/xxδρ ρ versus H⊥ , both normalized by the amplitude of the first harmonic 1A is shown in Fig. 5 for three carrier densities; for the sake of clarity, the oscilla- tions are plotted as a function of the filling factor 1 / H⊥ν ∝ . The normalization assigns equal weights to all oscillations. We analyzed SdH oscillations over the low- field range 1 TH⊥ ≤ ; this limitation arises from the as- sumption in Eq. (7) that c Fω ε and 0| | / 1xxδρ ρ . The latter condition also allows us to neglect the inter-level interaction which is known to enhance *g in stronger fields [29]. The amplitude of SdH oscillations at small H⊥ can be significantly enhanced by applying H (see Fig. 6) [27], which is another advantage of the cross-field technique. Indeed, for low H⊥ and n , the electron energy spectrum is complicated by crossing of levels corresponding to dif- ferent spins/valleys. By applying H , one can control the energy separation between the levels, and enhance the am- plitude of low-field oscillations. We have verified that ap- plication of H (up to the spin polarization 20%∼ ) does not affect the extracted *m values (within 10% accuracy), (the insets to Fig. 6 show that the values of *m measured at = 0H and 3.36 T do coincide). Fitting of the data provides us with two combinations of parameters: * *g m and *( )DT T m+ . The first combination, * * / 2 bg m m normalized by the band values, represents the sought-for renormalized spin susceptibility. The measured values of * *g m , as well as *m which are discussed below, were similar for different samples. Figure 7,a shows that for small sr (high densities), our * *g m values agree with Fig. 4. Shubnikov–de Haas oscillations for 11= 10.6 ·10n cm–2 (Si-MOSFET sample) at = 0.35 KT [27]. H = 0 T|| H = 4.5 T|| 0.4 0.8 1.2 H , T� R , k � � R , k � � Fig. 5. Examples of fitting with Eq. (7): (a) 11= 10.6 ·10n cm–2, = 0.35T K, = 4.5H T, = 6P % (the data corresponds to Fig. 1,b); (b) 11= 9.75 ·10n cm 2− , = 1.5H T, = 2% ;P (c) 11= 2.02·10n cm–2, = 0.2T K, = 0.34H T, = 4P %. The data are shown as the solid lines, the fits (with parameters shown) as dashed lines. All are normalized by 1( )A H⊥ . 1 1 0 0 –1 –1 a b c m* = 0.218 m* = 0.298 m* = 0.22 g* = 2.63 g* = 3.164 g* = 2.625 H|| = 4.5 T H|| = 1.5 T H|| = 0.34 T P = 6% P = 2% P = 4% n = 10.63 n = 9.75 n = 2.018 0.5 0 –0.5 � / A 1 � / A 1 � / A 1 1 2 3 4 H , T� –1 –1 V.M. Pudalov 18 Fizika Nizkikh Temperatur, 2011, v. 37, No. 1 the earlier data by Fang and Stiles [20] and Okamoto et al. [21]. For 6sr ≥ , * *g m increases with sr faster than it might be expected from extrapolation of the earlier results [21]. The second combination, *( )DT T m+ , controls the am- plitude of oscillations. In order to disentangle DT and * ,m we analyzed the temperature dependence of oscillations over the range = 0.3–1.6 KT (for some samples 0.4–0.8 K) . The conventional procedure of calculating the effective mass for low sr values ( 5 ), based on the assumption that DT is T-independent, is illustrated by the insets in Fig. 6. In this small-rs range, our results are in a good agree- ment with the earlier data by Smith and Stiles [19], and by Pan et al. [18]. The assumption of temperature independent DT , however, becomes dubious at low densities (high sr ), where the resistance varies significantly over the studied temperature range; in this case, the two parameters DT and *m become progressively more correlated. The open dots in Fig. 7,b were obtained by assuming that DT is T-in- dependent over the whole explored range of n : *m in- creases with sr , and the ratio * / bm m becomes 2.5∼ at = 8sr ( = 0.19bm is the band mass). As another limiting case, one can attribute the change in ( )Tρ solely to the temperature dependence of the short-range scattering and request ( 1 / )D DT ∝ τ to be proportional to ( )( 1 / )Tρ ∝ τ . In the latter case, the extracted dependence * ( )sm r is weaker (the solid dots in Fig. 7,b). Our data shows that the combination *( )DT T m+ is al- most the same for electrons in both spin-up and spin-down subbands (e.g., for 11= 3.76 ·10n cm –2 and = 2.15H T ( = 20%P ), the DT values for spin-up and spin-down le- vels differ by 3%≤ ). This is demonstrated by the observed almost 100% modulation of SdH oscillations (see, e.g., Figs. 5,a and 5,b). Thus, the carriers in the spin-up and spin-down subbands have nearly the same scattering time. 2.6. Comparison with other data 2.6.1. High density/weak interaction regime. As seen from Fig. 8, the data on n-channel Si-MOS samples are in a reasonable agreement with the data obtained by Zhu et al. [30] for n-type GaAs/AlGaAs samples from mea- surements of SdH effect in tilted magnetic field. Because of a smaller (by a factor of 3) electron effective mass in Fig. 6. Shubnikov–de Haas oscillations versus H⊥ for n = 112.2·10= cm–2 (i.e. = 5.6sr ) and = 0.4T ; 0.5 ; 0.6 ; 0.7 ; 0.8 K: = 0H (a) and 3.36 T (b). The insets show the temperature de- pendences of fitting parameters *( )DT T m+ . a b H = 0|| H = 3.36 T|| H , T� R , k � � R , k � � 0 0.5 1.0 T, K T, K 0.4 0.4 0.6 0.6 0.8 0.8 0.35 0.25 1.8 1.6 1.4 2.2 2.0 1.8 0.4 K 0.4 K 0.8 K 0.8 K 0.3 0.2 m* = 0.27 m* = 0.27 T = 0.42 KD T = 0.42 KD (T + T )m * D (T + T )m * D Fig. 7. Parameters * *g m , *m , and *g for different samples as a function of sr (dots). The solid line in Fig. 7,a shows the data from Ref. 21. The solid and open dots in Figs. 7,b and 7,c corres- pond to two different methods of finding *m (see the text). The solid and dashed lines in Fig. 7,b are polynomial fits for the two dependences * ( )sm r . The values of *g shown in Fig. 7,c were obtained by dividing the * *g m data by the smooth approxima- tions of the experimental dependences * ( )sm r shown in Fig. 7,b. 5 4 3 2 1 2.5 2.0 1.5 1.0 4 3 2 g * m * /2 m b m * /m b g * a b c rs David Shoenberg and the beauty of quantum oscillations Fizika Nizkikh Temperatur, 2011, v. 37, No. 1 19 GaAs, similar sr values have been realized for the electron density 10 times lower than in Si-MOS samples. The width of the confining potential well in such GaAs/AlGaAs heterojunctions is greater by a factor of 6 than in (100) Si-MOS, due to a smaller mass zm , lower electron densi- ty, and higher dielectric constant. This significant differ- ence in the thickness of 2D layers may be one of the rea- sons for the 20% difference between the χ*-data in n-GaAs and n-Si-MOS samples seen in Fig. 8; at the same time, the minor difference indicates that the effect of the width of the potential well on renormalization of *χ is not dramatic. The SdH experiments provide direct measurement of *χ in weak perpendicular and in-plane magnetic fields c FEω , * totB Fg H Eμ [27]. Under such condi- tions, the quantum oscillations of the Fermi energy may be neglected, and, in the clean system, the magnetization should remain a linear function of H , ** tot 0( )Hχ ≈ χ . Also, under such experimental conditions, the filling factor is large, = / ( ) 1nh eB⊥ν and the amplitude of oscilla- tions is small | | / 1xx xxδρ ρ . Figure 9 shows, on the H⊥ρ− plane, the domain of the weak magnetic fields, > 6ν , where the SdH oscillations have been measured in Refs. 27, 31. As the perpendicular magnetic field increases further (and ν decreases), the SdH oscillations at high density cn n transform into the quantum Hall effect; for low densities, cn n≈ , the SdH oscillations transform into the so-called «reentrant QHE–insulator» (QHE–I) transi- tions [32]. The uppermost curve (open circles) presents the ( )Hρ variations in the regime of the QHE–I transitions [32], measured for a density slightly larger (by 4%) than the critical density value cn for the metal–insulator transi- tion. This diagram is only qualitative, because the cn value is sample (disorder)-dependent. 2.6.2. Low density/strong interaction regime. In the vi- cinity of the critical density cn n≈ , the number of ob- served oscillations decreases, their period increases, and the interpretation of the interference pattern becomes more difficult, thus limiting the range of direct measurements of * ( )srχ . The horizontal bars in Fig. 8 are obtained from considera- tion of the sign and period of SdH oscillations [31] as ex- plained below. They show the upper limit for *χ , calculated from the data reported in Refs. 27, 31, 32. Figure 9,b demon- strates that in the density range –2 11 –20.7 cm < < 1·10 cmn , the oscillatory xxρ (beyond the magnetic field enhanced = 1ν valley gap) has minima at filling factors = (4 2), = 1, 2, 3...,i iν − (13) rather than at = 4iν (in (100) Si-MOSFETs, the valley degeneracy = 2gν ). The latter situation is typical for high densities and points to the inequality ** < / 2B cg Bμ ω . In other words, the sign of oscillations at low densities is reversed. This fact is fully consistent with other observa- tions (see, e.g., Fig. 2 of Ref. 31, Fig. 1 of Ref. 33, and Fig. 8. Renormalized spin susceptibility measured by SdH effect in tilted or crossed fields on n-Si-MOS by Okamoto et al. [21], Pudalov et al. [27], and on n-GaAs/AlGaAs by Zhu et al. [30]. Horizontal bars depict the upper and lower limits on the *χ values, determined from the sign of SdH oscillations, measured at = 0.027 mKT in Ref. 31. Dashed and dotted lines show two examples of extrapolation of the data [27]. 2 2 4 4 6 6 8 8 10 [27] [30] [21] rs * / b Fig. 9. (a) Overall view of the SdH oscillations in low fields at different densities. Empty circles show the xxρ oscillations for high-mobility Si-MOSFET sample in high fields, corresponding to the reentrant QHE–insulator transitions [32]. (b) Expanded view of one of the ( )xx Hρ curves ( 11= 1.04 ·10n cm–2 (right axis) and its oscillatory component normalized by 1( )A H (left axis)) [27]. Dashed line confines the region of the SdH measure- ments in Refs. 27, 31. n = 0.85 0.98 0.99 1.04 1.10 1.20 1.30 1.98 0 0.2 0.4 0.6 0.8 1.0 100 10 1 0.1 a b In su la to r In su la to r Q H E Q H E 0.8 0.7 0.6 0.5 1 0 –1 0 0.2 0.4 0.6 0.8 1.0 H, T � � � /( A ) 1 � /( h /e ) 2 � /( h /e ) 2 1/ = eH/(nh)� � = 14 10 6 V.M. Pudalov 20 Fizika Nizkikh Temperatur, 2011, v. 37, No. 1 Figs. 1–3 of Ref. 34). As Fig. 8 shows, the ratio * / bχ χ exceeds 1 / 2 = 2.6bm at 6sr ≈ ; the first harmonic of os- cillations disappears at this density (so-called «spin-zero»), and the oscillations change sign for lower densities. The sign of the SdH oscillations is determined by the ratio of the Zeeman to cyclotron splitting [5,23] * * * cos cos ,B b bc g H m ⎛ ⎞ ⎛ ⎞μ χ π ≡ π⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ χω ⎝ ⎠⎝ ⎠ (14) therefore, we concluded in Ref. 31 that, in order to have negative sign in the range 10 > > 6sr , the spin susceptibil- ity *χ must obey the following inequality: *1 3 2.6 = < < = 7.9. 2 2b b bm m χ χ (15) Thus, Eqs. (13) and (15) enable us to set the upper and lower limits for *,χ which are shown by horizontal bars in Fig. 8 at = 7.9–9.5.sr As density decreases (and sr in- creases), due to finite perpendicular fields, in which the SdH oscillations were measured, the condition of Eq. (15) becomes a bit more restrictive, which leads to narrowing the interval between the upper and lower bars [31]. 3. Magnetooscillations in strongly interacting 2D electron system In Sec. 2.2 above we used the semiclassical LK formu- la for noninteracting 3D case [5] and have made only transparent changes for the 2D electron spectrum [22]. We assumed that for the interacting system, the LK-formula remains applicable, when bare (band) quasiparticle para- meters are replaced with their values renormalized by inte- raction. This assumption is examined in this section. We shall consider only the case of weak oscillations when they are exponentially damped by either disorder broadening of the Landau levels or temperature smearing of the Fermi energy. For simplicity, we call this the «low-magnetic field regime». In higher quantizing fields, deviations from the LK formula were found earlier [9,35,36] and attributed to the magnetic field dependent oscillatory renormalization of the effective g-factor and mass, due to the inter-Landau level interaction [37]. Magnetooscillations in the interacting 2D system were studied theoretically over the last 50 years. The main issue under investigation is whether the oscillations frequency, phase, and damping factor for the strong interaction case remain the same as in the noninteracting system Eq. (7) and whether the FL parameters in the LK formula can be taken at zero magnetic field. Bychkov and Gorkov [23] have found that the amplitude of oscillations Eq. (9), rather than their frequency, is renormalized by interaction. Fow- ler and Prange [38] and Engelsberg [39] showed that the electron–phonon scattering rate does not appear in the os- cillations amplitude; Martin et al. [40] have shown that the inelastic electron–electron scattering does not contribute to the damping of magnetooscillations. Recently, an important advancement has been made in theory [40,41] of magnetooscillations. It was shown that the LK formula, in general, is still applicable when the os- cillations are exponentially small. However, due to the in- terference between electron–electron and electron–impur- ity interactions, damping factor in oscillations acquires an additional term in both the diffusive and ballistic regimes as follows [41]: [ ]1 2 *ln ( , ) = ( ) ( ),2 De B eH A T H T T T m m ck ⊥ ⊥− + − α π (16) where ** * * * * ( ) = ,DD D m m T T T m m ⎛ ⎞δτδ δ α − − −⎜ ⎟ ⎜ ⎟τ⎝ ⎠ * * ( ) = ln F Em T Tm δ ⎛ ⎞ − ⎜ ⎟ ⎝ ⎠ A , * * = 2 ln ,D F D E T T δτ ⎡ ⎤⎛ ⎞ π τ−⎢ ⎥⎜ ⎟ ⎝ ⎠τ ⎣ ⎦ A 0 2 0 15 1 = 1 , 1 4 a a D F F ⎛ ⎞ +⎜ ⎟ ⎜ ⎟+ π σ⎝ ⎠ A (17) and the factor 15 in the last line is the number of triplet terms for a system with two degenerate valleys. Our numerical simulations show that within the relevant interval = 0.03–0.8T K and 6sr ≤ , the ln T terms in Eq. (17) can be replaced with a T-independent constant. By combining the LK result Eq. (7) with the interaction-induc- ed corrections and replacing all ln T terms by a constant within our limited T range, we obtain the following linea- rized equation [42] in the ballistic regime for the short- range scattering (i.e. trDτ ≈ τ ): [ ]1 2 *ln ( , ) = (1 2 )2 De B eH A T H T T T m m ck ⊥ ⊥− + + π τ = π A 1 ( ) = 1 2D D T T T ⎛ ⎞δσ + −⎜ ⎟ σ⎝ ⎠ . (18) This remarkable result means that the T-dependent cor- rection to the Dingle temperature, ( ) /D DT T Tδ is just one- half of the interaction correction to the conductivity [43] ( ) / DTδσ σ . The factor 1/2 originates from the difference between the interaction corrections to the momentum re- laxation time ( tr ( )Tδτ ) and quantum scattering time [44] ( ( )D Tδτ ). We note that the empirical procedure used for finding *m in our earlier paper (Ref. 27) was based on the assumption that [ ]* = 1 ( ) / ,D D DT T T−δσ σ which differs from Eq. (18) by a factor of 1/2 . For the interacting case, Eq. (18), the linear T-de- pendence of 1ln ( , )A T H⊥ holds to the first approximation. This is in agreement with our experiments [27] which David Shoenberg and the beauty of quantum oscillations Fizika Nizkikh Temperatur, 2011, v. 37, No. 1 21 show that 1ln ( , )A T H⊥ for high sr varies linearly with temperature within the experimental range = 0.2–1 KT (see the inserts to Fig. 6). 3.1. Refinement of the extracted 0 aF values At relatively high densities (which correspond to < 4)sr , the corrections to the LK result are insignificant within the studied T range. As sr increases, the tempera- ture dependences of the oscillation magnitude predicted by the LK theory Eq. (7) and the interaction theory Eq. (17) start deviating from each other. The values of 0| | aF ex- tracted from SdH data using Eq. (18) are smaller than those obtained with the LK theory but larger than 0| | aF obtained with the empirical procedure used in Ref. 27. E.g., at = 6.2sr , the 0 aF values obtained according to Eq. (18) and the empirical procedure of Ref. 27 are –0.40 and –0.45, respectively. In Ref. 42 we have reanalyzed the data of Ref. 27 using Eq. (18) and compared them with available results from other transport measurements. The 0 aF values obtained in Ref. 42 from the analysis of SdH oscillations using the theories [5,41] (see Sec. 3) are plotted in Fig. 10. For comparison, we have also plotted the 0 aF values calculated in Ref. 42 from fitting the mono- tonic temperature and magnetic field dependences of the conductivity ( )TΔσ and ( )HΔσ [42,45,46] with the theory [43]. There is a good agreement between all data. The spin susceptibility * * *g mχ ∝ obtained from SdH measurements appears to be almost T-independent [47], in apparent disagreement with the interaction correction the- ory [48] and renormalization group (RG) theory [49]. This contradiction could be resolved, provided the T-depen- dence of *g is exactly compensated by the opposite T-de- pendence of *m , so that * * *g mχ ∝ remains almost con- stant. The compensation, however, seems rather unlikely. We believe that the absence of temperature dependence in * *g m values from SdH is simply a consequence of the cut- off that is imposed by finite magnetic fields *tot > / BH kT g μ which are applied in SdH measurements. 3.2. Other quasiparticle parameters extracted from SdH data 3.2.1. Valley splitting. The analysis of SdH oscillations using Eq. (7) also allowed us to estimate the value of the energy splitting [9] VΔ between two valleys in the (100)- Si-MOSFET samples. A nonzero valley splitting causes beating of SdH oscillations. Figures 11,a and b show the SdH oscillations for two different samples. The electron densities are 116.1·10 and 121·10 cm–2, respectively. The amplitude of weak SdH oscillations normalized by the first harmonic 1A is expected to be field independent if = 0.VΔ A noticeable reduction in the SdH amplitude observed for both samples at small fields can be attributed to a finite valley splitting. Although the node of SdH oscillations expected at 0.15 TH⊥ ≈ cannot be resolved for samples with mobilities 2∼ m2/(V·s), VΔ can still be estimated from fitting of the H⊥ -dependence of the SdH amplitude with Eq. (7) modified for the case of a finite VΔ : = 0.4VΔ K for sample Si6-14 and 0.7 K for Si1-46. This estimate provides the upper limit for VΔ at = 0H⊥ : in nonzero H⊥ fields, VΔ may be enhanced by the interlevel interaction effects [9,35,37]. 3.2.2. Drude scattering time. The momentum relaxa- tion time τ needed for calculating the interaction correc- tions was determined from the Drude conductivity =Dσ 2= e / ;b en m mτ the latter was found by extrapolating the quasi-linear ( )Tσ dependence observed in the ballistic regime to = 0T [43,50]. Note that in order to extract τ from the Drude conductivity, one should use the bare bm rather than the renormalized effective mass: according to the Kohn theorem, the response of a translationally-in- variant system to the electromagnetic field is described by bm in the presence of electron–electron interactions; this result also holds for weak disorder ( 1FE τ ). It is worth mentioning that several prior publications [45,46,50] incor- rectly used *m instead of bm to estimate τ from Dσ ; as a b –0.1 –0.2 –0.3 –0.4 –0.5 –0.1 –0.2 –0.3 –0.4 –0.5 –0.6 F 0s F 0s 1 2 3 4 5 6 7 8 rs Fig. 10. (a) The dashed curve corresponds to 0 ( ) a sF r extracted from the SdH data [27] using the LK theory, the dash-dotted curve — to the empirical approach used in Ref. 27. The symbols depict 0 aF values obtained from fitting the transport data with the theory [43]. The shaded regions in panels (a) and (b) show the 0 ( ) a sF r dependence (with the experimental uncertainty) ob- tained from fitting our SdH data [27] with the theory [41]. (b) Comparison of 0 aF values recalculated from available ( , = 0)T Hδσ data: dots — Ref. 42, triangles and diamonds — Refs. 45 and 46, respectively. V.M. Pudalov 22 Fizika Nizkikh Temperatur, 2011, v. 37, No. 1 shown in Ref. 42, this affects the value of the fitting para- meters extracted from comparison with the theory [43]. The textbook value [9] for the light electron mass in bulk Si is (3 ) 0.19Dbm ≈ . For inversion layers on (001) Si- surface [51], (2 ) = (0.19–0.22) 0.02Dbm ± was found from tunneling measurements. The * ( )m n data obtained from the analysis of SdH oscillations over a wide range of densi- ties = 1.4–8.5sr [27], can be fitted with a polynomial 4* ( ) = 0.205(1 0.035 0.00025 )s s sm r r r+ + . These *m data ag- ree well with earlier values of *m extracted from SdH oscil- lations [19,20,24,52] in narrower ranges of densities. By extrapolating the polynomial * ( )sm r to = 0sr we obtain (2 ) = 0.205 0.005Dbm ± ; following Ref. 42 we adopted this value throughout the paper. 4. Conclusion David Shoenberg was a great Master in experimental low-temperature physics. Besides the de Haas–van Alphen effect, he has made substantial contribution to understand- ing of the magnetic properties of superconductors. His book «Superconductivity» written in Moscow in 1938 is one of the reference books on my bookshelf and is used as a textbook by Russian students. Shoenberg's experiments on studying quantum oscillations in metals and the fine experimental techniques developed by him for this re- search represent a piece of experimental art. David Shoen- berg has shown how the properties of interacting systems can be revealed by measuring quantum oscillations under nonlinear conditions imposed by interactions. In line with this approach, we performed studies of the Shubnikov–de Haas effect for strongly interacting two-di- mensional system of electrons. From the amplitude of quantum oscillations, we determined the interaction-induc- ed renormalization of the quasiparticle parameters, such as the effective mass, spin susceptibility, and g-factor. The Fermi-liquid interaction parameter 0 ( ) aF n obtained from the analysis of SdH oscillations agrees well with the 0 ( ) aF n values obtained by fitting the monotonic transport data with the interaction correction theory [43]. However, it remains so far unclear how to reconcile the 0 aF values obtained at low electron densities from fitting the ( )Tσ and SdH data (by using the interaction correction theory) with the corresponding values [53–55] obtained by fitting the ( , )T Hσ data with the RG theory. Possibly, for a quantitative description of the interaction effects in low temperature transport, the RG theory should be extended to a more realistic case of a finite intervalley scattering rate and to higher orders. 5. Acknowledgments The author benefited from fruitful collaboration with M. Gershenson, H. Kojima, E.M. Dizhur, G. Bauer, G. Brun- thaler, O.E. Omel'yanovskii, N.N. Klimov, D.A. Knyazev, A.Yu. Kuntsevich in performing measurements. The work was partially supported by grants from RFBR, Russian Acad- emy of Sciences, and Russian Ministry for Education and Science (under contracts Nos. 02.552.11.7093, 14.740.11.0061, P2306, P798, P1234). 1. D. Shoenberg and J. Vanderkoy, J. Low Temp. 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