CeCu ,(tore)
5. Conclusions
74 M.R. NORMAN and D.D. KOELLING TABLE 3
Calculated symmetry of the gap function for UPt 3 from the semi-phenomenological spin fluctuation theory assuming pairing by either high-frequency spin fluctuations (F ~ 5 meV) or low-frequency spin fluctuations [ F ( Q ) ~ 0.3 meV]. Tabulated is the group representation, the functional form that the order parameter transforms under group operations (the actual order parameter is determined numerically), and the nodal structure. Note that two possible solutions are found for high-o» pairing depending on the ratio of near-neighbor to next-near-neighbor interactions. Also listed are the two most likely order parameters from phenomenological fits to
experimental data.
Rep. Form Nodes
Theory (high m) AI« kzz line
Elu (k x +_ iky)z point
Theory (low co) AI~ k~x + kyy point
Experiment (?) Elg kz(k x +_ iky) line, point
E2u k~(k x + iky)2z line, point
a highly simplistic fashion. Any of these effects could lead to a qualitative change in the solutions found. For instance, the fact that odd-parity solutions are found for AF correlations between planes may be a total artifact of the assumptions made on the matrix elements mentioned above. The problem, of course, is that a simple theory (U on-site, J between sites) may no longer be sutticient once one decides to include matrix element effects, and one is thus forced to work with a truly microscopic model, as opposed to the semi-phenomenological assumptions of the current models.
Thus, the problem of heavy-fermion superconductors is rar from being solved, although we feel at this stage that a spin fluctuation approach has the correct physics for constructing the "ultimate" theory. A likely scenario is one that uses the semi- phenomenological developments as a guide to construct a more realistic K o n d o lattice theory. We might note that the dynamic susceptibility calculated from a slave-boson approach has many things in common with the experimentally observed neutron scattering data (Auerbach et al. 1988), although as emphasized repeatly by Varma, an intrinsic multi-site approach is probably necessary to get an adequate description of the m o m e n t - m o m e n t response function (Varma 1991). Given an adequate micro- scopic theory for the dynamic susceptibility, a pairing self-energy can be constructed and a proper strong-coupling gap equation solved. The resulting solution will then give us some guidance about the validity of a spin fluctuation treatment, and whether it should be pursued or abandoned in favor of a more promising theory, possibly based on quadrupolar or inter-site phonon exchange.
Even the strongly enhanced metals CeSn 3 and U P t 3 are weil described - with regard to Fermi surface topology. But there is room for improvement in numerous cases.
There are three basic concerns: large mass enhancements, magnetic moments and localization.
Enhancements: In the strongly enhanced materials, approximating the quasi- particle masses by the density functional band masses fails dramatically. One of the interesting aspects of the L12 structured materials is that they offer sufficient examples to be able to see the mass enhancements grow (i.e. this approximation breaks down) as the f states become more localized and then diminish with the advent of antiferro- magnetism. Magnetic fluctuation derived enhancements have been successfully calculated for a number of strongly enhanced materials. In seeking to develop a band structure reftecting the quasiparticle spectrum, the renormalized band structures start from D F - L D A results and phenomenologically incorporate the enhancements.
The successes of the renormalized band structures a r e a clear indication of the accuracy of the D F - L D A charge densities even for these strongly enhanced materials. Thus, the observation of large enhancement factors alone should not be interpreted as a failure of D F - L D A but merely the indication that self-energy (excited state) effects are important.
Moments: F o r magnetically polarized metals, although the LSDA appears to work well for some materials (Gd, U C u » NpSn3), it predicts sizable moments for USn3 and UBe13 where none exist. It also predicts large moments for U P t 3 and URu2Si 2 which actually have very small moments. There are several concerns that make the calculation of moments less secure. (1) The basic functionals to be used for the spin dependence are less well known (the overtendency to mägnetism may be due to the fact that Kondo-like correlations are not included in the LSD functional). (2) The local approximation is even less valid for the spin dependence than for the charge dependence. (3) The only orbital contributions incorporated within the LSDA are those induced by spin-orbit coupling (note the discussion of Gd). Consequently, the moments of lanthanide materials which have sizable orbital moment contributions (7-Ce, TmSe) are poorly determined. This failure is largely a result of improper treatment of Hund's rule effects by LSDA. It can be remedied either by going to an open-shell Hartree Fock formalism, or by using the H a r t r e e - F o c k formalism as a guide to construct a correction to the LSDA. There have now been enough successful applications of this approach to give it considerable credibility.
Localization: The pervasive question for f electron materials is: "Under what circumstances do the f electrons behave as itinerant or as localized in the ground stare?" Localization occurs when the energy gain from Coulomb U and Hund's rule effects exceeds the kinetic energy gain achieved through hybridization. Unfortunately, it is not clear how well current ab initio methods treat these localizing effects for realistic solids. Thus, we are forced to resort to ad hoc schemes to properly describe localized systems. As examples, for Pr or for U P d » one artifieially localizes the f electrons in the core (i.e. manually suppresses the hybridization) to obtain a good representation of the Fermi surface. But such an approach is not always appropriate.
Gd should fit the requirements of the f core model, yet continuing to incorporate the f states in the band structure is necessary to understand the minority carrier
76 M.R. NORMAN and D.D. KOELLING
Fermi surface. And for Ceß6, while the topology of the Fermi surface is generally consistent with an f core model, (1) the observed extremal areas are larger than allowed by an f core model, and (2) one finds a mass enhancement of 50 implying f orbital involvement in the conduction bands. Similarly, for CeA12 f core modelling is approximately correct but not correCt in detail. In a related way, neither an f core nor an f band treatment appears to work for C e C u 6. This metal may satisfy the conditions discussed by Zwicknagl where all the f stares but the ground state one are effectively projected out if the crystal field splitting of the ground state multiplet is larger than the effective K o n d o temperature. A related projector method was used by Kasuya's group to get a good representation of the Fermi surface of CeSb suggesting that projection onto the suspected crystal field ground state may also work for CeB6, CeA12 and CeCu 6. (It should be remembered that the occupied and unoccupied f levels in CeSb were shifted to lie where they would be in an Anderson model - i.e. several eV away from the Fermi energy.) Incorporation of Hund's rule effects might provide the basis for this effective projection.
Several improvements to density functional formalism would help clarify the situation for localized materials although very possibly not resolve it. (1) Improve- ments discussed for the moments apply as well for localization. After all, the question of localization and moment formation is highly intertwinded in these metals. (2) Self-interaction corrections (a way of including U effects in LDA) act to further localize the more locall orbitals. SIC would also reduce the tendancy of the LDA to overbond the higher ~ orbitals. SIC also acts to provide an energy separation between occupied and unoccupied states that are orten incorporated in phenomenological descriptions. With all these desirable features, it is unfortunate that there are serious unsolved conceptual issues when applying it to solids [there has been a recent application of this method, though, to Pr metal: Szotek et al. (1991)]. (3) Further correlations due to the strong intra-atomic Coulomb interactions could be significant.
Anderson model results indicate that the closer restriction to f orbital integral occupation would significantly reduce the f level broadening due to hybridization.
Each of the improvements which could probably be incorporated into a density functional formalism would act to permit greater localization with reduced hybridi- zation. But would it be enough? Localized states are particularly difficult to describe in an ab initio formalism, not being possible even at the level of Hartree Fock, for example. The issue of how far one can get with the evolutionary improvements discussed above and when one must seek a new approach (such as a slave-boson treatment of the Anderson model using realistic parameters determined by D F - L D A ) is a very real one. Despite some progress, there is yet no definitive way from any ab initio calculation to determine whether the f states will behave as an f band metal or as an f core orte. Until this can be done, we will not have truly understood the f electron problem.
Finally, we would also like to say that after more than a decade's worth of work, the origin of superconductivity of heavy-fermion metals is still a mystery. It will probably remain a mystery until we get a truly rigorous understanding of the normal ground stare of these metals.
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