2. SALIENT PHYSICAL PROPERTIES
2.3 Fermi surfaces
(4) If the atomic model were rigorously correct, a lanthanide element would have the same magnetic moment in every material and crystalline environment. Of course, this is not the case, and more advanced methods have to be employed to determine the effect of the crystalline and chemical environment on mag- netic moments with numerical precision. Examples of materials where the theory described above fails to give an accurate value for the lanthanide ion magnetic moment are ubiquitous. They include NdCo5(Alameda et al., 1982), fullerene encapsulated lanthanide ions (Mirone, 2005), and lanthanide pyro- chlores (Hassan et al., 2003).
In conclusion, it is clear that the standard model makes an excellent first approximation to the magnetic properties of lanthanide materials, but to under- stand lanthanide materials on a detailed individual basis, a more sophisticated approach is required.
4fs. BothSkriver (1983b) and Eriksson et al. (1990a)have performed a successful calculation of the cohesive properties and the work of Skriver reproduces well the crystal structures which are sensitively dependent on the details of the valence band. For example, the partial 5d occupation numbers decrease across the series with a corresponding increase in the 6s occupation. The increase in 5d occupation is what causes the structural sequence hcp!Sm structure!dhcp!fcc as the atomic number decreases or the pressure increases (Duthie and Pettifor, 1977). A detailed analysis of Jackson’s work (Jackson, 1969) on Tb yields a Fermi surface that shows features with wave vector separations corresponding to the character- istic wave vector describing the spiral phase of Tb.
One of the successes of this approach is the comparison of calculated electronic structures with de Haas van Alphen measurements. The investigations of the Fermi surface of Gd using this technique among others were pioneered by Young and co-workers (Young et al., 1973; Mattocks and Young, 1977) and extended by Schirber and co-workers (Schirber et al., 1976). These studies found four frequencies along thec-direction and three in the basal plane and later work found a number of small frequencies (Sondhelm and Young, 1977). All these frequencies could be accounted for on the basis of the band structure calculations of the time. Sondhelm and Young also measured cyclotron masses and mass enhancements in Gd and found values in the region of 1.2–2.1 which were in agreement with the band structure calculations, but were rather smaller than those derived from low-temperature heat capacity measurements. More recently, angle-resolved photoemission have been used for Fermi surface studies of Tb (Do¨brich et al., 2007) and also Dy and Ho metals were studied (Schu¨ßler- Langeheine et al., 2000). Fermi surfaces of Gd-Y alloys were studied with positron annihilation (Fretwell et al., 1999; Crowe et al., 2004). In particular, the changes to the Fermi surface topology upon the transition from ferromagnetism to helical anti-ferromagnetism could be followed. Concerning the light lanthanides, the Fermi surface areas and cyclotron masses in Pr were measured (Wulff et al., 1988) with the de Haas van Alphen technique. Large mass enhancements, espe- cially for an elemental metal, were measured and the understanding of the Fermi surface needed an approach well beyond the standard model (Temmerman et al., 1993).
This standard model does have several major drawbacks:
(1) It can never get the magnetic properties correct.
(2) It cannot describe systems with large mass enhancements.
(3) There is one overarching shortcoming of the standard model, which is slightly more philosophical. It treats the f-electrons as localized and the sd-electrons as itinerant, that is, it treats electrons within the same material on a completely different basis. This is aesthetically unsatisfactory and furthermore makes it impossible to define a reference energy. In principle, one has to know,a priori, which electrons to treat as band electrons and which to treat as core electrons.
It would be infinitely better if the theory itself contained the possibility of both localized and itinerant behaviour and it chose for itself how to describe the electrons.
For two elements, it is possible to perform LDA calculations including the f-electrons in the self-consistency cycle. The first is cerium where the f-electrons are not necessarily localized. Cerium and its compounds are right on the border between localized and itinerant behaviour of the 4f electrons. There has been much controversy over the nature of the f-state in materials containing cerium.
The key issue is the g–a phase transition where the localized magnetic moment associated with a 4f1configuration disappears with an associated volume collapse of around 14–17%, but no change in crystal symmetry. The f electron count is thought to be around one in both phases of Ce. What appears to be happening here is a transition from a localized to a delocalized f-state (Pickett et al., 1981;
Fujimori, 1983). BothPickett et al. (1981) and Fujimori (1983)highlight the limita- tions of the band theory picture for cerium. In particular, Fujimori discusses the calculation of photoemission spectra and the reasons why band theory does not describe such spectroscopies well. The a-phase can be described satisfactorily using the standard band theory, while there is more difficulty in describing the g-phase and the energy difference between the two phases (which is very small on the electronic scale) is not calculated correctly. The second lanthanide element where band theory may hope to shed some light on its properties is gadolinium.
Here the f-levels are spilt into seven filled majority spin and seven unfilled minority spin bands, neither of which are close to the Fermi energy. It is feasible to perform a DFT calculation and converge to a reasonable result. This has been done by a number of authors (Jackson, 1969; Ahuja et al., 1994; Temmerman and Sterne, 1990) with mixed results. Temmerman and Sterne were among the first to indicate that the semi-band nature of the 5p levels could significantly influence predicted properties.Sandratsakii and Ku¨bler (1993)took this work a bit further by investigating the stability of the conduction band moment with respect to disorder in the localized 4f-moment.
The experimentally determined Fermi surface of Gd (Schirber et al., 1976;
Mattocks and Young, 1977) has been used by several band structure calculations to determine whether the f-states have to be treated as core-like or band-like. Gd metal is the most thoroughly studied of all the lanthanides and one of the few lanthanides that have been studied by the de Haas van Alphen technique (Schirber et al., 1976; Mattocks and Young, 1977). Being trivalent, it has, through the exchange splitting, a half-filled f-shell. The Gd f-states are well separated from the Fermi level, and therefore f-states are not contributing to the Fermi surface. This was seen in the Fermiology measurements (Schirber et al., 1976; Mattocks and Young, 1977). Both f-core (Richter and Eschrig, 1989; Ahuja et al., 1994) and f-band (Sticht and Ku¨bler, 1985; Krutzen and Springelkamp, 1989; Temmerman and Sterne, 1990; Singh, 1991) calculations claim to be able to describe this Fermi surface.
While these methods provide some useful insight into Gd and Ce, they yield unrealistic results for any other lanthanide material as the f-bands bunch at the Fermi level leading to unphysically large densities of states at the Fermi energy and disagreement with the de Haas van Alphen measurements. It is clear that a satisfactory theory of lanthanide electronic structures requires a method that treats all electrons on an equal footing and from which both localized and itinerant behaviour of electrons may be derived. SIC to the LSDA provide one such theory.
This method is based on DFT and the free electron liquid, but corrects it for electrons where self-interactions are significant and takes the theory towards a more localized description.