T " \ ' , 11 4 1 T 1 UNCLASSIFIED ORNL TUT . . . . . Ww WV ...] * üm 297 ORNI pag17 CONF-690 -3 DTIE.S 6458 SEP2 : 1934 SEMIEMPIRICAL RULES OF SPIN ORDERING* M. K. Wilkinson Solid State Division, Oak Ridge National Laboratory Cak Ridge, Tennessee MAS INTRODUCTION The existence of magnetic coupling rules might lead to the assumption that such rules have had important appíications in the determinations of . magnetic structures by neutron diffraction. Actually, the reverse is true, and these rules have been developed on the basis of magnetic structures that have been experimentally determined. The main importance of these . A rules is not in the prediction of magnetic structures, but in the under- standing of the mechanisms which are important in indirect magnetic coupling. At the present time the semi empirical rules have been applied only to com- pounds which possess a sufficiently simple crystal structure that the = . . important magnetic exchange interactions are unambiguous. This does not . mean that the rules are not applicable to other structures, but merely that the analysis is much more complicated. Magnetic structures have been de- termined for about two hundred compounds, but relatively few structures have been correlated with specific coupling rules. HISTORICAL DEVELOPMENT . . . After the first neutron diffraction results, on MnO showed that the manganese ions appeared to be coupled antiferromagnetically to secona nearest neighbors through the intervening oxygen ions, many experiments , . have been performed in an attempt to find regularities in the magnetic an structures, and many theoretical ideas have been postulated to explain the = - *Research sponsored by the U. S. Atomic Energy Commission under contract with the Union Carbide Corporation. ". .2- important interactions. Perhaps the first rule was suggested in Anderson's first paper on superexchange, which showed that the general formalism of Kramers could be used to explain the primary interactions that were ob- served in Mno. It was suggested that superexchange would be antiferromag- r.etic between cations with d-shell.s that are at least half-filled, whereas the interaction would be ferroinagnetic between cations having d-shells that are less than half-filled. This conclusion was shown experimentally not to be correct, and in particular, many compounds were found in which Crºs ions were antiferromagnetically coupled. However, the present ideas of ligand field theory do predict rules of coupling which are directly related to Anderson's original prescription for the sign of superexchange. Most of the carly predictions for ordered magnetic lattices were based on the Weiss molecular field treatment, and this is still the most practical method for dealing with magnetic systems in which more than one magnetic interaction is considered. In problems involving antiferromagnetic inter- actions, it is necessary to divide the lattice of magnetic atoms into sub- lattices to take account of various possibilities of magnetic order. The method of subdivision depends on the symmetry of the specific lattice and on the number of magnetic interactions which are being considered. Calcu- lations by Van Vleck' for a body-centered cubic lattice, in which first and second nearest neighbor interactions were considered, showed that the two types of antiferromagnetic ordering in Fig. 1 are energetically possible. Similar calculations,' have been made for the face-centered cubic lattice and the four possible types of antiferromagnetic structures in this sym- metry are shown in Fig. 2. A significant basis for classification of the latter ordering schemes involves the numbers of parallel and antiparallel 4 -2 - S R AS F - moments in the nearest and riext-nearest neighbor configurations. In or- dering of the first kind, which is a luyer-type order, two-thirds of the nearest neighbors of any moment are coupled antiferromagnetically, while the other nearest neighbors and all next nearest neighbors are coupled ferromagnetically. Ordering of the second kind is characterized by having half the nearest neighbor moments parallel and half antiparallel, and all next-necrest neighbor moments are antiparallel. In ordering of the third kind, the nearest neighbor coupling is identical to that for ordering of - --. Pt - - - * * . . . . --- - - - - the first kind, but only two-thirds of the next-nearest neighbors are par- - -- - - allel. Because of this enhanced antiparallel correlation, ordering of the . a.--- - -- third kind has been referred to as "improved ordering of the first kind." - ... ----re: - . . - - . . . . . - r - - - ... VE.. Amor: - . . - ... .. . Finally, ordering of the fourth kind has the same nearest neighbor config- uration as ordering of the second kind, but only two-thirds of the next. nearest neighbors are antiparallel. Magnetic structures corresponding to all types have been found in the transition metal compounds hy neutron dif- fraction observations. However, as shown in Table I, which was taken from Smart," the types of order predicted from values of nearest neighbor and next-nearest neighbor exchange interactions, deduced from other experimental results, do not always agree with the observed structures. . - -. - - - - --- E V A F Y M - - - - - A very significant step in developing the presently-accepted ideas of magnetic coupling mechanisms in transition-metal compounds was made by Goodenough in his analysis of the magnetic structures that had been ob- served in a very comprehensive study of perovskite-type manganites by Wollan and Koehler. Goodenough and Loebt had applied a hybrid orbital approach to the study of distortions in spinel-type crystals and showed how covalent bonding influences the indirect magnetic-exchange interactions between two magnetic cations seperated by an anion. The magnetic inter- action was termed "semicovalent exchange," because it depended upon the concept of semicovalent bonds, which were defined as bonds due to the cou- pling of a single anion electron to the net spin of a cation. These ideas assumed that empty cation orbitals may have energies which are nearly de- generate with the atomic d-orbitals. Therefore, if these empty cation orbitals strongly overlapped the full orbi'uals of the neighboring anions, the anion p-electrons could spend part of their time in the cation orbitals. 0:1 this basis, Goodenough made the very significant observation that the unusual magnetic properties of the system (La Ca, ...)Mno, could be qualita- - tively explained by orbital overlaps involving hybrid orbitals of the planar dsp? type and of the octahedral a?sp3 type. In those cases where there was direct overlap between these orbitals and the p-orbitals, magnetic coupling was antiferromagnetic, but when the p-orbital was perpendicular to the planes of the dsp type so that the orbitals did not overlap, ferro- magnetic coupling resulted. Shortly after these ideas were formulated, Dunitz and Orgel's pointed out that the distortions observed in spinel structures could also be ex- plained on the basis of ligand field theory instead of tie special theory of semicovalent bonding, which did not appear to be generally applicable. These distortions were shown to be related in a simple fashion to the elec- tronic configuration of the metal ion and to arise as a consequence on a Jarn-Teller type of distortion. Wollan+4,+) then showed that the orbitals which resulted from ligand field splittings also provided a better explana- tion for the magnetic coupling in the perovskite-type manganites. These perovskites possess a particularly simple etructure for considering magnetic . 4 S coupling mechanisins through orbital overlaps. The magnetic ions are in octahedral sites, so that accurate calculations can be made of the crystal field splitting, and rurthermore, the crystal symmetry permits only one important magnetic interaction. Wollan was able to correlate the magnetic ordering in these compounds of the basis of three rules, which involved the population of the e, orbitals of the cations that overlapped the p-orbitals of the intervening oxygen ions. Similar ideas were also developed inde- pendently by Goodenough to and Kanamority at about the same time, and Kana nori presented a very detailed discussion of the relation between the symmetry of electron orbitals and the superexchange interaction for cations in both octahedral and tetrahedrel symmetry. DISCUSSION OF RULES It is now generally accepted that indirect magnetic interactions in compounds do take place hy overlap between the cation orbitals and inter- vening anion orbitals and that the cation orbitals to be considered are those stabilized by ligand fields. Consequently, it is possible to deter- mine which cation orbitals are populated with electrons, and rules can be formulated for the type of magnetic interaction on the basis of this popula- tion. Andersonº,+9 has showed that these rules can be explained by a revised approach to his original superexchange theury, and Marsha1120 has given a brief but comprehensive survey of the important types of exchange interactions which must be considered. The semi empirical rules that have been established by the various workers are identical, and the remaining part of this lecture will outline these rules, following the method taken by Marshall. No quantitative calculations will be made, and the discussion . K .-.- .. will attempt to point out the types of exchange from a purely qualitative viewpoint. The important interactions, which are designated by Marshall as Superexchange I, Superexchange II, and Virtual Double Exchange, will be assumed to be of the same omer of magnitude. At the present time this approach appears to be logical, because calculations of these effects are extremely difficult. Those calculations, which have been made for specific cases, are at best only semi quantitative estimates, and they are frequently revised as more details are considered in the calculations . Superexchange I: The Superexchange I mechanism is the process which was originally mentioned by Kramers* and discussed in detail by Anderson. This process involves the formation of a state in which only one electron in the anion p-orbital is excited into the d-orbital of one of the cations. Consequently, it can be considered phenomenologically as a "single transfer" process, in which one p-electron is transferred to the d-orbital of one cation. As a specific illustration of this process, consider ions with one magnetic electron in each of the cation d-orbitals which overlap an inter- vening p-orbital. The p-orbital is, of course, doubly occupied, and the orbital occupation, when the electron spins in the cation orbitals are parallel, can be represented schematically as: ∞ ∞ ∞ In this diagram and all subsequent diagrams of this type, the small arrows represent individual electron spins, and the largte arrows represent the "S" - , it total spin on the atom. Since each cation nas other orbitals which are occupi.ed, 1t 18 assumed that these other orbitals point in directions which give very small overlap effects and can be ignored. However, the other electrons in these orbitals are important in determ'.ning the total electrori population of the atom and, consequently, in determining the manner in which the electron in the orbital of interest is coupled to the other elec- trons by Hund's rule. In the diagram shown above, it is assumed that the spin of the electron 1s parallel to the total spin of the atom. For this same example, when the electroz spins in the cation orbitals are antiparallel, the representation is: ∞ ∞ In looking for possibilities for a single electron to be transferred - from the p-orbital to the d-orbitals in these two cases, two methods exist - - - - in both the parallel and antiparallel arrangement. In the parallel case, these are and and where the schematic diagrams of the orbitels involved have been omitted. For the antiparallel case, the two methods are: C: 7 = … , and and Anderson's arguments show that the energies are slightly different when the remaining electron in the p-orbital 18 parallel or antiparallel to the election left unpaired in the d-orbital. Specifically, he has shown that the energy is lower when these two electrons are antiparallel, so that in this particular one-electron transfer process, there is antiferromagnetic coupling between the magnetic ions. As a second example of a single-electron transfer process, examine the coupling when one magnetic ion has an orbital containing an electron and the other has an empty orbital. Transferring the negative electron to the orbital containing an electron, such as gives negligible coupling, because the remaining p-electron cannot couple to the empty d-orbital. Therefore, it is only possible for coupling to occur when the p-electrons are transferred to the empty orbital, and two methods exist. These are and . Sun As mentioned previously, the energy is lower when the remaining electron in the p-orbital has its spin antiparallel to the unpaired electrons in the d- orbitals. The second process, therefore, has the lower energy, and the coupling between magnetic ions is ferromagnetic. These are the only two situations in which single electron "transfer" can cause magnetic coupling between the d-electrons. Obviously, a single electron cannot cause an interaction when both d-orbitals are empty, and it also cannot cause an interaction when either orbital already contains two electrons . Superexchange II: The Superexchange II mechanism is a double transfer process in which excited states are formed by "transferring" both electrons in the p-orbital to the d-orbitals of the cations. This process was suggested relatively recently by Nesbit, 21 and both Yanashita and Kondo22 and Keffer and Oguchi 23 have considered these double transfer integrals. This process, like the single electron transfer, cannot produce any coupling when one of the d-orbitals contains two electrons, but unlike the single electron transfer, it can cause an interaction when both orbitals are empty. In this situation, which can be represented schematically as: the excited state becomes -* 1* * 10. and the coupling between magnetic ions is antiferromagnetic. For the two cases in which the single electron transfer process caused a magnetic interaction, there is also a possible interaction due to double transfer. Simple considerations of the type outlined previously show that in each case the Pauli principle restricts the double transfer process to only one excited state. The resulting courling is antiferromagnetic with one electron in each d-orbital and ferromagnetic between an empty d-orbital and one containing a single electron. Therefore, the Superexchange II mechanism causes the same type of interaction as the Superexchange I mech- anism for both cases. Virtual Double Exchange: The mechanism that Marshall has designated virtual double exchange is actually an extension of the process of double exchange, which was first proposed by Zener24 and which has received more complete treatment by Anderson and Hasegawa 25 and by de Gennes. 26 Zener's theory was first ap- plied to certain compounds of manganese with the perovskite structure to explain the correlation between ferromagnetism and relatively large elec- trical conductivity on the basis of an electron exchange between trivalent ard tetravalent manganese. In these compounds, it is possible to have a ground state described by the following orbital occupation Mn+3 0-2 1 2 AY . 2.. " - ! MIR Ji the electron on the Mn*lon 18 transferred to the 0-2 ion simultaneously with the transfer of an electron from the oxygen ion to the Mnion, the following state results 2 0 Mn *3 which has the same energy as the initial state. Since the same type of electron (the spin-up electron in the above diagram) must reside on both manganese ions as the only electron in the d-orbital that 18 Involved, it minü couple identically to the other electrons on the ions. Such coupling requires that the magnetic moments of the ions must be ferromagnetically aligned. This particular case involves exchange between a cation with an empty d-orbital and one with a d-orbital containing one electron. An identical argurent, also giving rise to ferromagnetic exchange, can be pre- sented for coupling a cation with a filled d-orbital to one with a d-orbital containing only one electron, such as Mn to Mns. These two types of situations were the only ones that were possible according to the original ideas of double exchange. Marshall“ pointed out that there probably is never a true electron transfer as suggested by double exchange, because some type of crystal dis- tortion would undoubtedly exist in the ground state and energy would be required to transfer the electron. Consequently, the coupling must take place by an admixture of an excited state, and the process can better be termed "virtual double exchange." The latter process can then be important in coupling magnetic ions (such as Mn's and Cr>) where it is apparent that -12- no real electron transitions can be made. The primary difference between this process and superexchange 18 that in virtual dourle exchange, one of the magnetic ions must lose an electron. The ferromagnetic coupling rules for the conditions of orbital population discussed for true a'ouble exchange still exist, but in addition, it is possible for virtual double exchange to occur between a cation with a filled d-orbital and one with an empty do orbital. In this case, the configuration permits an excited state and the coupling is antiferromagnetic due to the different manner in which the transferred electron 18 coupled by Hund's rule to the two magnetic ions. SUMMARY Table II is a summary of the semiempirical rules for coupling thiet have been discussed. The occupation numbers of the important d-orbitals involved in the exchange are given by n, and ng, and the type of coupling is shown for each of the ihree processes. Although the rules for indirect magnetic coupling given in Table II are fairly well accepted, it is still very difficult to use these rules for predicting magnetic order in most compounds. First of all, it is necessary to obtain details of the impor- tant orbitals which result from ligand field splitting. The specific directions of these orbitals and the amount of orbital overlap must also be carefully considered to determine which orbitals play a prominent part in coupling processes. It must be remembered that some processes take place . USA mmm...aren y YANAIR ! .. Ni MAI .1 4 -13- through more than one path and that the orbitel overlap changes with angle. Finally, it is necessary to know the magnitude of the various interactions, and only crude estimates can be made. Early calculations by Anderson suggested that the superexchange interaction between two magnetic ions would be strongest when the anion was directly between them, but recent calcula- tions by Marshall and Stuart, and by Casselman and Kefferão have raised doubts concerning this 1,80° rule. Consequently, except in the very simplest magnetic structures, a prediction of the magnetic ordering 18 pure specula- tion. It is better to determine the structure and to analyze the experi- mental observations in terms of the semiempirical rules. .14. Table I* Values of J, and J, for f.c.c. Magnetic Lattices Compound Type of Order -J7(®3) Observed Mno 7.2 -Jalºk) 3.5 8.2 20 21.6 sz/, Predicted 0.48 2,3 1. 052 20 Feo Coo 7.8 1.0 6.9 150 no no Bm N10 no 50 no o & -Mns B-Mns - 1.0 0.69 n MnS, 10.5 5.6 7.4 7.2 5.9 1.2 1.1 no Mnre, 0.2 w *from Ref. 9 Table II. Rules for Indirect Magnetic Coupling Orbital Population _n2 naŠE I Types of Exchange SE II VE o baina beste o b o 11' 111 1 " EL - 16- REFERENCES Shull, Strauser, and Wollan, Phys. Rev. 83, 333 (1951). C. G. Smil and J. 8. Smart, Phys. Rev. 76, 1256 (1949). P. W. Anderson, Phys. Rev. 29, 350 (1950). H. A. Kramers, Physica 1, 182 (1934). J. H. Van Vleck, J. Phys. et Rad. 12, 262 (1951). F. W. Anderson, Phys. Rev. 79, 705 (1950). 7. J. S. Smart, Phys. Rev. 86, 968 (1952). 8. See, for example, tabulation by L. M. Corliss and J. M. Hastings, American Institute of Physics Handbook McGraw-H11.. Book Co., PP 5-200, (1963). 9. J. 8. Smart in Magnetism (H. Suhl and G. Rado, eds.) Academic Press, New York, p 63 (1963). 10. J. B. Goodenough, Phys. Rev. 100, 564 (1955). 11. E. O. Wollar and w. C. Koehler, Phys. Rev. 100, 545 (1955). 12. J. B. Goodenough and A. L. Loeb, Phys. Rev. 98, 391 (1955). 13. J. D. Dunitz and L. E. Orgel, J. Phys. Chem. Solids 3, 20 (1957). 14. Wollen, Child, Koehler, and Wilkinson, Phys. Rev. 112, 1132 (1958). 15. E. O. Wollan, Phys. Rev. 117, 387 (1960). 16. J. B. Goodenough, J. Phys. Chem. Solids 6, 287 (1958). J. Kanamori, J. Phys. Chem. Solids 10, 87 (1959). 18. P. W. Anderson, Phys. Rev. 115, 2 (1959). 19. P.W. Anderson in Solid State Physics, (F. Seitz and N. Turnbull, ed.), Vol. 14, P 99 (1962). 20. W. Marshall in Perspectives in Materials Research, (1. Himnel, J.J. Harwood and W. S. Harris, eds.), U. S. Govt. Print. Off., p 76 (1963). 21. R. K. Nesbit, Annl. Phys. 4, 87 (1958). 22. J. Yamashita and J. Kondo, Phys. Rev. 109, 730 (1958). L . ' 5. 1. W " IN 11.11. vu AC 2 .13 -17- 23. F. Keffer and T. Oguchi, Phys. Rev. 115, 1428 (1959). 24. C. Zener, Phys. Rev. 82, 403 (1951). 25. P. W. Anderson and H. Hasegawa, Phys. Rev. 100, 675 (1955). 26. P. G. de Gennes, Phys. Rev. 118, 141 (1960). 27. W. Marshall and R. Stuart, Phys. Rev. 123, 2048 (1961). 28. T. N. Casselman and F. Keffer, Phys. Rev. Ltrs. 4, 116 (1960). W. Mo R. St - - . - -_ - .. - . -.. Pi erre Mato il o monte , UMCLASSIFIED ORNL-DWG 64-6550 ORDERING OF THE FIRST KIND ORDERING OF THE SECOND KIND ORDERING OF THE THIRD KIND ORDERING OF THE FOURTH KIND Fig. 9. Possible Kinds of Ordering for a Face- Centered Cubic Lottice. UNCLASSIFIED ORNL-DWG 64-6551 _ - -. ORDERING OF THE FIRST KIND . - ORDERING OF THE SECOND KIND Fig. 2. Possible Kinds of Magnetic Ordering for a • Body-Centered Cubic Lattice. DATE FILMED 1123 /64 C - 1 . : - - LEGAL NOTICE This roport was propared as an account of Government sponsored work. Nellhor the United States, por the Commission, nor any person acting on behalf of the Commission: A. 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