The Challenge of d and f Electrons - American Chemical Society

Chapter 19

Response of Local Density Theory to the Challenge of f Electron Metals Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on April 15, 2018 | https://pubs.acs.org Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch019

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M . R. Norman and A. J . Freeman 1

Materials Science Division, Argonne National Laboratory, Argonne, IL 60439 Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208

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A brief review is given on the application of local density theory to the electronic structure of f-electron metals, including various ground state properties such as observed crystal structures, equilibrium lattice constants, Fermi surface topologies, and the electronic nature of known magnetic phases. A discussion is also given about the relation of calculated results to the unusual low energy excitations seen in many of these metals. Local density theory is an approximate method used to calculate ground state properties of a many-body system. The local density approximation (LDA) involves replacing the exact (unknown) exchange-correlation functional (representing the non-direct part of the Coulomb interaction) by that of a homogeneous electron gas (the condition being invoked at each point in space). Such an approximation would appear to be so gross as to be inapplicable in practical circumstances, but the LDA turns out to work remarkably well for most systems (1). One way to see why this is so is to note that the exact exchange energy density can be represented by a shape function times the density to the one-third power (2). The LDA simply replaces this shape function by a constant, which also occurs for the exact expression when the energy density is integrated to yield the total energy. On the other hand, the effective potential is obtained by functionally differentiating the energy, and thus samples the shape function. This in turn implies that one must be careful when interpreting one-particle eigenvalues obtained from the effective potential. In particular, it is now known that the LDA places valence d levels about 20 mRy too low at the beginning of the transition metal series (2), and valence f levels about 50 mRy too low for La, and 40 mRy too high for Tm (4). There is a deeper question that has been raised, however, in the case of f-electron metals. Given the large degree of localization of these electrons and the strong correlations among them, it has been suggested that the LDA is a poor starting point for understanding the ΟΌ97-6156/89/0394--0273$06.0Ό/0 ο 1989 American Chemical Society

Salahub and Zerner; The Challenge of d and f Electrons ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

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OF d A N D f ELECTRONS

electronic nature of these materials (5). This is a legitimate question, and it is our purpose here to address this problem.

Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on April 15, 2018 | https://pubs.acs.org Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch019

Bulk Properties Evaluation of the total energy as a function of atomic position can lead to such useful structural information as the equilibrium lattice constant, bulk modulus, and preferred crystal structure. Reviews of some of this work can be found in Refs. 6 and 7. For the 4f elemental metals from Eu to Lu, the observed equilibrium lattice constants are well represented by LDA calculations which treat the f electrons as core states (7) (The inapplicability of Fermi-Dirac statistics for treating highly localized electrons requires the imposition of a fixed occupation number). This need of forcing the f electrons into the core represents a well-known limitation, namely, that the LDA functional does not minimize the f count per site with integer occupation values in the localized limit. In atomic and molecular cases, this deficiency can be remedied by self-interaction corrections (8-9). but the latter terms are conceptually difficult to handle for extended systems. For the 4f elements Pr to Sm, there is about a 4-5% overestimation of the lattice constant, indicating the presence of some f bonding. For Ce metal, the lattice constant of γ-Ce is given fairly well by treating the f electron as a core state, whereas that for α-Ce is underestimated by about 5% when treating the f electron as a band state. Thus, the LDA calculations support the notion that the α to γ transition involves a localization of the f electron, whereas the above mentioned underestimate indicates that the LDA overestimates f bonding for itinerant f metals. As for the bulk modulus, the LDA well represents its trend across the 4f series, although the absolute errors are quite large (7). A careful analysis of the bulk modulus for the mixed valent metal TmSe indicates that the LDA error in that case can be corrected if the f levels are shifted down 40 mRy (4). This problem is connected to the fact that the LDA does progressively worse for higher angular momentum states (due to probable misrepresentation of the shape of the exchange-correlation hole). Self-interaction corrections may be able to explain this error (9-10). A great deal of work has been done for the 4f elemental metals concerning the relative stability of various crystal structures. For example, LDA predicts the correct fee structure for Ce in the 4f band case (11), whereas in the later lanthanides, the crystal structure is determined by the d states. The reader is referred to Refs. 11 and 7 for a further discussion of these results. The situation improves for the 5f elemental metals due to the greater derealization of the 5f as compared to the 4f electrons. In fact, the equilibrium lattice constants of the 5f elemental metals are well reproduced even with the f electrons treated as band states, as long as spin polarization is taken into account (12). The spin polarization "turns on" at Am, and correlates with an inferred localization of the f electrons. Even for such strongly correlated metals as the heavy fermion superconductors UPt3 and UBe-jo, the equilibrium lattice constant and bulk modulus are well-given oy LDA f band calculations oa=H)-


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Fermi Surfaces One of the best ways to determine whether an itinerant or localized picture is applicable is to compare experimental and theoretical determinations of the Fermi surface topology and effective masses. It has been known for many years that the Fermi surfaces of most elemental 4f metals can be represented by an f core calculation (15). In the case of Pr, the calculated effective masses are a factor of ïour too small, a result which can be attributed to a polaronic drag of the conduction electrons caused by induced virtual excitations between f crystal field levels (16). This is a dynamic correlation effect, and is therefore not given when performing a ground state calculation, but can be represented by adding a self-energy correction to the (ground state) band results. It is now known that the heavy fermion metal CeBg is also describable by this picture, with the mass renormalization being of the order of fifty instead of four. This observation is of some importance, as it implies that large effective masses and f character in the Fermi surface are not equivalent statements. Also, calculations indicate that the Fermi surface topologies of both LaBg and CeBg can be improved by shifting the unoccupied f levels upwards Dy about 50 mRy. This error is analogous to the 20 mRy shift needed for d band metals such as V and Nb (3). These cases can be contrasted by most uranium intermetallics, which have Fermi surfaces in good agreement with LDA calculations which treat the f electrons as band states (17). In the one case where a mixed valent Fermi surface is known (OeSno), it is also in excellent agreement with an LDA f band calculation (18-19). with a mass renormalization of five due to a self-energy correction resulting from virtual spin fluctuation excitations (2Q). Notice the different dynamic correlations used to explain the mass renormalizations in the f core and f band cases. Of great relevance is the recent experimental determination of the Fermi surface of the heavy fermion superconductor UPt3 (21), which is found to be in excellent agreement witn LDA f band calculations (22-23). despite the fact that the Fermi surface is composed of five f bands. This, along with the above-mentioned structural information, indicates that the static charge correlations are handled quite adequately by the LDA. Again, there is a large mass renormalization (a factor of twenty in this case), attributable to a spin fluctuation self-energy correction (2Q). It is interesting to contrast the case of hep UPt3 to the case of U P d s , which has a closely related crystal structure (dhep). It turns out that the UPd3 Fermi surface is well-represented by an f core calculation (24). Thus one could say that UPt3 is on the verge of a localization (Mott) transition. In conclusion, it is clear that the LDA calculations have been very helpful in sorting out the nature of the ground states for those f electron metals which show anomalous behavior. Magnetic Properties The existence of strong antiferromagnetic correlations in heavy fermion metals has been confirmed by extensive neutron scattering studies (2f>), and these correlations are now thought to be responsible for both the large mass renormalizations and exotic superconductivity


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seen in these metals (Ref. 20 and references therein). It is interesting to note that an antiferromagnetic tendency was predicted from band structure calculations before the neutron scattering experiments were done (12). In reality, however, many heavy fermion metals, although showing strong magnetic fluctuations, exhibit either a small or zero ordered moment. This fact has been used to question the validity of the local spin density (LSD) approximation (5). The importance of the above-mentioned criticism has led us to undertake polarized calculations on the following metals: the heavy fermion magnets N p S n (22), TmSe (27), and UCU5 (Norman, M. R.; Min, Β. I.; Oguchi, T.; Freeman, A. J. Phvs. Rev. B. in press.), and the heavy fermion superconductors UPts URU2S10 (Norman, M. R.; Oguchi, T.; Freeman, A. J . Phvs. Rev. B. in press.). At the outset, it must be stated that there does not exist a proper LSD treatment in the presence of spin-orbit coupling, due to the fact that spin is no longer a good quantum number. The approximation we employ is to include the spin-orbit terms as a perturbation within each self-consistent iteration. The orbital contribution to the magnetic moment can then be determined by the method suggested by Brooks and Kelly (2g). We note, however, that the orbital magnetic moment does not make any contribution to the energy functional in this formalism, which would not be true in an exact theory. Despite this, the formalism works quite well in determining the total magnetic moment for various uranium compounds (2£), and shows the proper Hund's rule effect that the spin and orbital moments are antiparallel in the first half of a series, and parallel for the second half. In the cases of NpSn3 and UCuc, magnetic moments in good agreement with experiment were found, wnereas the discrepancy with experiment in TmSe can be attributed to multiplet effects (which are not included in the LDA). Moreover, in all these cases, one finds strong reductions of the electronic specific heat coefficients at the magnetic phase transition which are also explained quite well by the calculations. Therefore, a standard LSD treatment of magnetism seems to work quite well. This can be contrasted with the case of the above-mentioned super conductors. We calculate a magnetic moment of 0.8 μ ο for UPto and 1.2 μ β for URU2S12 as compared to the 0.02 μ β (2Q) and 0703 μ β (21) values actually observed. This is the origin of the justifiable criticism mentioned in Ref. 5. The essential difference in the band calculations between the magnets and the superconductors is that in the former, the Stoner criteria for magnetism is greatly exceeded, whereas in the latter cases it is only weakly exceeded, indicating that in these cases, some other effect occurs which acts to greatly suppress the LSD predicted moment. We believe that this is the Kondo effect and note that the band structure moment is recovered by either doping the metal or subjecting it to a strong magnetic field. The inclusion of this "Kondo suppression" effect in an LSD calculation is a major challenge. This suppression is of extreme importance, as without it, these heavy electron metals would be strong magnets instead of superconductors. 3

a

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Conclusions Despite several problems, the local density approximation has turned out to be quite useful in elucidating information about the ground state properties of f electron metals. For further information, the reader is


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referred to reviews given in Refs. 3, 6, 7,12,15,17, and 19. For a further discussion of some problems with local density theory as regards to f electron metals, see Ref. 32. Finally, an excellent discussion of the relation of band calculations to observed low energy excitations seen in heavy electron metals has been given by Pickett

(32).


Acknowledgments Work at Argonne National Laboratory was supported by the U. S. Dept. of Energy, Office of Basic Energy Sciences, under Contract No. W-31-109-ENG -38, and at Northwestern University by the National Science Foundation (DMR Grant No. 85-18607). Literature Cited 1. Theory of the Inhomogeneous Electron Gas: Lundqvist, S.; March, N., Eds.; Plenum: New York, 1983. 2. Harris, J . Phys. Rev. A 1984, 29, 1648. 3. Koelling, D. D. In The Electronic Structure of Complex Systems: Phairseau, P.; Temmerman, W. M., Eds.; Plenum: New York, 1984; p. 183. 4. Jansen, H. J . F.; Freeman, A. J.; Monnier, R. Phvs. Rey. Β 1986, 33, 6785. 5. Lee, P. Α.; Rice, T. M.; Serene, J . W.; Sham, L. J.; Wilkins, J . W. Comments on Cond. Mat. Phys. 1986, 12, 99. 6. Norman, M. R.; Koelling, D. D. J . Less-Common Metals 1987 127, 357. 7. Freeman, A. J.; Min, B.I.; Norman, M. R. In Handbook on the Physics and Chemistry of Rare Earths: Gschneidner, Κ. Α., Jr.; Eyring, L.; Hufner, S., Eds.; North Holland: New York, 1987; Vol. 10, p. 165. 8. Perdew, J . P. In Density Functional Methods in Physics: Dreizler, R. M.; daProvidencia, J., Eds.; Plenum: New York, 1985; p. 265. 9. Norman, M. R.; Koelling, D. D. Phys. Rev. Β 1984, 30, 5530. 10. Harrison. J . G. J . Phys. Chem. 1983, 79, 2265. 11. Skriver, H. L. Phys. Rev. Β 1985, 31, 1909. 12. Johansson, B.; Eriksson, O.; Brooks, M. S. S.; Skriver, H. L. Physica Scripta Τ 1986, 13, 65. 13. Sticht, J.; Kubler, J . Solid State Comm. 1985, 58, 389. 14. Pickett, W. E.; Krakauer, H.; Wang, C. S. Physica Β 1985, 135, 31. 15. Liu, S. H. In Handbook on the Physics and Chemistry of the Rare Earths: Gschneidner, Κ. Α., Jr.; Eyring, L., Eds.; North Holland: New York, 1978; p. 233. 16. Fulde, P.; Jensen, J. Phys. Rev. Β 1983, 27, 4085. 17. Arko, A. J . ; Koelling, D. D.; Schirber, J . E. In Handbook on the Phvsics and Chemistry of the Actinides: Freeman, A. J . ; Lander, G. H., Eds.; North Holland: New York, 1985; Vol. 2, p. 175. 18. Koelling, D. D. Solid State Comm. 1982, 43, 247. 19. Norman, M. R. In Theoretical and Experimental Aspects of Valence Fluctuations and Heavy Fermions: Gupta, L. C.; Malik, S.; K., Eds.; Plenum: New York, 1987; p. 125. 20. Norman. M. R. Phys. Rev. Lett. 1987, 59, 232. 21. Taillefer, L.; Lonzarich. G. G. Phys. Rev. Lett. 1988, 60, 1570.


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22. Oguchi, T.; Freeman, A. J . ; Crabtree, G . W. J. Magn. Magn. Matls. 1987, 63-64, 645. 23. Wang, C. S.; Norman, M. R.; Albers, R. C.; Boring, A. M.; Pickett, W. E.; Krakauer, H.; Christensen, Ν. E. Phys. Rev. Β 1987, 35, 7260. 24. Norman, M. R.; Oguchi, T.; Freeman, A. J . J . Magn. Magn. Matls. 1987, 69, 27 25. Goldman, A. I. Jap. Jour. Appl. Phys., Suppl. 1987, 26-3, 1887. 26. Norman, M. R.; Koelling. D. D. Phys. Rev. Β 1986, 33, 3803. 27. Norman, M. R.; Jansen, H. J . F. Phys. Rev. Β 1988, 37, 10050. 28. Brooks, M. S. S.; Kelly, P. S. Phys. Rev. Lett. 1983, 51, 1708. 29. Brooks. M. S. S. Physica Β 1985, 130, 6. 30. Aeppli, G . ; Bucher, E.; Broholm, C.; Kjems, J . K.; Baumann, J . ; Hufnagl, J . Phys. Rev. Lett. 1988, 60, 615. 31. Broholm, C.; Kjems, J . K.; Buyers, W. J . L.; Matthews, P.; Palstra, T. T. M.; Menovsky, Α. Α.; Mydosh, J . A. Phys. Rev. Lett. 1987, 58, 1467. 32. Norman, M. R. In Condensed Matter Theories; Vashishta, P.; Kalia, R. K.; Bishop, R., Eds.; Plenum: New York, 1987; Vol. 2, p. 113. 33. Pickett, W. E. In Novel Superconductivity; Wolf, S. Α.; Kresin, V. Z., Eds.; Plenum: New York, 1987; p. 233. RECEIVED October 24,1988


The Challenge of d and f Electrons - American Chemical Society

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