Relativistic LMTO calculations

Figure 9. Spin splittings of the heavy hole hh) and heavy electron (he) bands in GeSi for k at the A symmetry line as calculated with the relativistic LMTO method (Sect. 4.1).

Earlier it was mentioned that the relativistic theory of electronic states in solids in many respects is identical to that of atoms. Since this is well described elsewhere, this section will only deal with some features of specific implementations of the theory in actual calculation methods used for solids, and the importance of relativistic effects — apart from those already discussed — will be illustrated by examples. Although Section 3 did refer to results of LMTO calculations, we did not describe how these included relativity. This section will deal with these items in the form of an overview, and the basic band structure calculations described relate to the density-functional theory [62,63]. Since magnetism is one of the most important solid state physics fields we shall discuss the simultaneous inclusion of spin-polarization and relativistic effects, in particular the spin-orbit coupling. In that context it appears that several of the materials where such effects are particularly large and interesting are those where electron... 

The relativistic LMTO and LAPW methods were used to calculate [77-80] the Fermi surface of UPta. This is a heavy fermion compound, and its physical properties axe strongly influenced the presence of the narrow U-/ bands at the Fermi level. The shape of the Fermi surface is then sensitive to relativistic effects, in particular the SO-coupling. The results of the calculations [78] were surprising since they showed that the topology of the Fermi surface was well described by these band structures although they were obtained within the LDA. A similar precision was not found for the effective cyclotron masses which were off by up to a factor of 30 when compared to experiments. The crystal potential enters in the LMTO via the potential parameters [30,73] for each (or each j in the relativistic version [4]), including the mass parameters fi (eq.(49)). A convenient way... 

Figure 13. UPts Fermi surface calculated within the LDA using the Dirac-relativistic LMTO method. The width of the hatched stripes is proportional to the U-/5/2 component. The U-/7/2 content is very low all-over the Fermi surface. Dotted stripes show the m, =l/2 contribution, right-hatched the [mj 1=3/2, and left-hatched the the mj =5/2 projections. Note that band 1 and 2 have regions with very low /-character. On these parts of the Fermi surface there is a strong hybridization with other states, mainly Pt-p and -d. (Ref. [80]).

PdTe2 [1.39], AuCu [1.40], AgMg and AuMg [1.41], A Cu [1.42] and NiSi [1.43], Jarlborg and Arbman applied the LMTO method to V Ga and other A15 compounds [1.44], and Christensen developed the first completely relativistic LMTO code which was subsequently used in calculations on Pd, Ag, Pt, and Au [1.45]. From this period date also the Chevrel-phase calculations by Andersen et al. [1.46], in which the physically simplifying assumptions of the LMT0-ASA and KKR-ASA methods are exploited to the full. 

The volume contribution by the metallic 5f-5f and the covalent cation 5f-N2p bonds to a virial-theorem formulation of the equations of state of a series of light actinide nitrides was calculated in the self-consistent linear muffin tin orbital (LMTO), relativistic LMTO, and spin-polarized LMTO approximations [46]. The results for ThN give the same lattice spacing in all three approximations higher by ca. 3% than the experimental value, which discrepancy is attributed to the assumed frozen core ions [47]. 

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... 

In the PP method applied to transition metals one normally treats only the valence s, p, and d electrons, which total five per atom in V and Ta, and six per atom in Mo. Here special pseudopotentials in the Troulher-Martins form [48] have been constructed from scalar-relativistic atomic calculations to be accurate in the pressure range below 400 GPa. An important advantage of the PP method is that it provides accurate forces so that fuUy relaxed atomic configurations can be considered. We have used this capabihty here to obtain accurate relaxed 110 and 211 y surfaces for Ta, Mo, and V. It is also possible to use relaxed PP configurations to perform vahdating FP-LMTO calculations on relaxed defects and y surfaces, as was done previously at ambient pressure [13,45]. 

Second, using the fully relativistic version of the TB-LMTO-CPA method within the atomic sphere approximation (ASA) we have calculated the total energies for random alloys AiBi i at five concentrations, x — 0,0.25,0.5,0.75 and 1, and using the CW method modified for disordered alloys we have determined five interaction parameters Eq, D,V,T, and Q as before (superscript RA). Finally, the electronic structure of random alloys calculated by the TB-LMTO-CPA method served as an input of the GPM from which the pair interactions v(c) (superscript GPM) were determined. In order to eliminate the charge transfer effects in these calculations, the atomic radii were adjusted in such a way that atoms were charge neutral while preserving the total volume of the alloy. The quantity (c) used for comparisons is a sum of properly... 

We conclude that more work is need<. In particular it would be useful to repeat the TB-LMTO-CPA calculations using also other methods for description of charge transfer effects, e.g., the so-called correlated CPA, or the screened-impurity modeP. One may also cisk if a full treatment of relativistic effects is necessary. The answer is positive , at least for some alloys (Ni-Pt) that contain heavy elements. 

The results of the LMTO and RLMTO calculations are very similar to those of the RKKR calculations just described. The principle difference appears to come from the different exchange approximations used to create the potentials, rather than the band calculations themselves. The recent calculations by Podloucky and Weinberger for US were made with a similar exchange and correlation approximation to the LMTO and RLMTO calculations. In Fig. 9 a and b we show the TX bands of the actinide nitrides NpN-PuN from self-consistent LMTO and RLMTO calculations, which may be compared with the RKKR calculations of Podloucky and Weinberger. The agreement between the two sets of relativistic bands is very good. 

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. 

Brooks, Johansson and Skriver (1984) investigated the band structure of UC and ThC by nonrelativistic and relativistic (based on the Dirac formalism) LMTO methods. They analysed the electron density changes in the compounds as compared with free atoms, as well as the influence of pressure on the band structure. Crystal pressures as a function of lattice constants (equations of state) were calculated as well as theoretical values of the lattice constants. The calculated trends in the variations of lattice constants and bulk moduli agree well with the available experimental data. Some of the most important results of these calculations are shown in Figs. 2.20 and 2.21. 

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