Relativistic APW calculations

Table 12. Charges inside the atomic spheres Qy and the ratios Qy VOy of the 1 partial charges for LiTl and NaTl. For both compounds the self consistent scalar relativistic APW calculations have been performed for both structures B2 and B32. The lattice constants are chosen aB32 = 2 agi where aB32(NaTl) = 7.47 A and aB2(Lin) = 3.42 A are the lattice constants found experimentally. The volumes of the muffin-tin sphere are Wu(LiTl) = w-nCLiTI) = 12.47 A and ti)nj(NaTl) = o)Ti(NaTl) = 16.84 A ...

The band calculation of Dimmock and Freeman was done for paramagnetic Gd, i.e. by putting equal numbers of 4f electrons in spin-up and spin-down states in constructing the potential. Two different starting potentials were used in the non-relativistic APW calculation, but the results were remarkably similar. The energy bands obtained from one of these calculations are plotted along high symmetry axes of the Brillouin zone as shown in fig. 3.1. The two bottom bands... 

It was revealed by a relativistic APW calculation (Yamagami and Hasegawa 1992) that the main Fermi surface of LaRu2Si2 is a large closed hole sheet which is centered at the Z point in band 14. It is essentially a sphere but highly distorted. Although its volume is... 

As we have seen, on the whole the agreement with theory for the localized form factor associated with the 4f electrons in lanthanide metals and compounds is satisfactory provided one is careful to use relativistic calculations. The situation for the conduction electron polarization distribution is less clear. Conduction electron form factors were obtained for Gd by Moon et al. (1972) and for Er by Stassis et al. (1976). In both cases, these were obtained by separating from the measured form factor the localized 4f contribution, and in both cases appear to be different from either a 5d or 6s atomic form factor. A spin-polarized augmented-plane-wave (APW) calculation of the conduction electron polarization in ferromagnetic Gd was performed by Harmon and Freeman (1974). Their results are, however, only in qualitative agreement with the results of Moon et al. The theoretical form factor of Harmon and Freeman is in somewhat better agreement with the experimental results of Stassis et al. on Er. 

For studies of the Fermi surface in the lanthanide compounds, it is necessary to develop a reliable theoretical method in which hybridization of the 4f electrons with other electrons as well as the relativistic effect can be taken into account quantitatively. For that purpose, the relativistic APW method proposed by Loucks (1967) provides a good starting basis. Loucks derived his original method from the Dirac one-electron equation, which is a natural extension of Slater s non-relativistic APW method (Slater 1937). It proved to be a powerful method comparable to a relativistic KKR method (Onodera and Okazaki 1966, Takada 1966). Loucks method does not accocunt for the symmetrization of the wave functions by group theory, nor it is a self-consistent method. These shortcomings are serious limitations for calculations of the energy band structure in the lanthanide compounds. 

Yamagami and Hasegawa carried out a self-consistent calculation of the energy band structure by solving the Kohn-Sham-Dirac one-electron equation by the density-functional theory in a local-density approximation (LDA). This self-consistent, symmetrized relativistic APW approach was applied to many lanthanide compounds and proved to give quite accurate results for the Fermi surface. 

Electron density function. In order to carry out a self-consistent eneigy band calculation, it is necessary to calculate the electron density function associated with a symmetrized relativistic APW function. For the Bloch state k. A), the spherically averaged radial density function within the j-type APW sphere is given by... 

Fig. 4. A flow chart for a self-consistent energy band structure calculation by a symmetrized relativistic APW method in LDA. In each cycle of the iteration, the total electron potential V-r(r) is given by the sum of the nuclear potential v(r), the direct Coulomb potential FhW the exchange-correlation potential / xcW To construct a new input potential, the parMneter/(< 1) is introduced to optimize the convergence.

The procedure that we have described above differs from the early[3,4] TB CPA calculations in two respects. First, our SK Hamiltonians which are fit to self-consistent relativistic augmented plane wave (APW) calculations of the stoichiometric hydrides are much more accurate than those used by other workers. Our rms error is typically less than 5 mRy for the first 7 bands. Second, previous[3,4] TB-CPA calculations freeze the s- and p-bands in transition metals and apply a so-called one-level TB model which ignores the coupling of t and e orbitals. In our calculations we use five coupled CPA conditions corresponding to the self-energies of s, p, t g, Og, and s symmetry. 

Fermi surface for ThC and UC using the relativistic APW method. The calculated results are in good agreement with the de Haas-van Alphen measurements. The UC Fermi surface appears to consist of three hole pockets in the region of the valence C2p states and six electron pockets in the region of the U5/ states. It was shown that UC is a semimetal and contains almost the same number of holes and electrons. 

The first accurate band structure calculations with inclusion of relativistic effects were published in the mid-sixties. Loucks published [64-67] his relativistic generalization of Slaters Augmented Plane Wave (APW) method. [68] Neither the first APW, nor its relativistic version (RAPW), were linearized, and calculations used ad hoc potentials based on Slaters s Xa scheme, [69] and were thus not strictly consistent with the density-functional theory. Nevertheless (or, maybe therefore ) good descriptions of the bands, Fermi surfaces etc. of heavy-element solids like W and Au were obtained.[3,65,70,71] With this background it was a rather simple matter to include [4,31,32,72] relativistic effects in the linear methods [30] when they (LMTO, LAPW) appeared in 1975. 

The enormous progress in the calculation of solid state properties in the past decades has been pushed by the development of a nrunber of distinct band structure schemes like KKR-, ASW-, LMTO-, LCAO-, PP- and (L)APW-methods which differ essentially in their representation of basis functions. For all of the mentioned methods there exist by now full potential codes which also incorporate relativistic effects in one way or another. 

The APW method is a variational method for calculating the eigenfunctions and eigenvalues of the periodic MT potential. There are a number of comprehensive reviews of this method and its relativistic generalization (RAPW) (Mattheiss et al., 1%8 Loucks, 1%7 Dimmock, 1971 Callaway, 1974). We will briefly mention the philosophy of the method and some practical considerations that arise in connection with rare-earth metals. 

The energy bands of bcc Eu were calculated by Freeman and Dimmock (1%6) using APW method and by Andersen and Loucks (1%8) using RAPW method. The non-relativistic bands are shown in fig. 3.14 and the relativistic bands in hg. 3.15. In both papers the authors stressed the similarity between the structure of... 

An LDA band structure calculation is expected to yield a good description of the ground state properties of rather extended 4f-band Ce metal, provided it is carried out to self-consistency. Kmetko and Hill (1976) performed the first self-consistent APW band structure calculation for y- and a-Ce and pointed out the increase in hybridization of the 4f-states with the conduction band with reduction of the atomic volume. Glotzel (1978) reported the cohesive and magnetic properties of fee Ce obtained with the self-consistent relativistic LMTO method (Andersen 1975) and... 

Among the band calculations, there were two attempts, by Ivashchenko (1984) and Schalder and Monnier (1989), to study the influence of structural defects on the electronic structure of solid solutions. Ivashchenko used the APW-LCAO-CPA method for analysis of the DOS distribution in the ZrC jN D (x + y + z = 1) alloy. It was found that metallization of the alloy takes place as z increases. Some Hf carbonitrides with a variable number of p atoms and cation and anion vacancies were studied by Schalder and Monnier (1989) making use of the relativistic KKR-GF method. Reasonable agreement was obtained between the calculated and experimental photoelectron spectra. It was shown that the presence of defects in the solid solutions leads to the formation of additional vacancy states (Fig. 5.12), which are typical for binary phases (Chapter 4), and to partial charge polarisation towards metallic centres. The latter effect is more pronounced for C vacancies (see Table 5.1). One of the most interesting conclusions of this paper was the establishment of the nonmonotonic variation of the N(Ep) in the series HfC HfCj,Nj HfN (Table 5.1). 

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