Relativistic band structures

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. 

Conceptually, the formation of the energy bands is most easily described within the ASA, the atomic-spheres approximation. [4,30,73] A pure band of character (, is the angular momentum), pure in the sense that hybridization is omitted, extends over the energy range Eb () to Ea ) where the logarithmic derivative 

There are cases where the inclusion of the relativistic shifts of the bands is essential, but where inclusion of spin-orbit coupling is not needed. In 

Other methods of formulation a scalar relativistic scheme based on a differential equation where SO coupling is eliminated have been developed. [74,75] 

In the free atom a spin-orbit parameter, is associated with each one-electron state. In the solid -states broaden into bands of non-zero widths, and the spin-orbit parameter varies with energy over the band. We define e E) through 

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The non-relativistic band structures of silver and gold are very similar so that, were it not for relativistic effects, gold would look silver . 

Whereas Johansson, following Herring s argument assumes that Uh should always be less than Ujt, Herbst and Watson from relativistic band structure calculation found that while this is true for light actinides, the situation is reversed above Pu, thus favoring the transition between Pu and Am. 

Early band structure calculations for the actinide metals were made both with and without relativistic effects. As explained above, at least the mass velocity and Darwin shifts should be included to produce a relativistic band structure. For this reason we shall discuss only the relativistic calculations. There were some difficulties with the f-band structure in these studies caused by the f-asymptote problem , which have since been elegantly solved by linear methods . Nevertheless the non-self-consistent RAPW calculations for Th through Bk indicated some interesting trends that have also been found in more recent self-consistent calculations ... 

Tung, Y. W., and M. L. Cohen (1969). Relativistic band structure and electronic properties of SnTe, GeTe, and PbTe. Phys. Rev. 180, 823-26. 

In addition to the OP formalism, several alternative schemes have been suggested in the past to account, within a relativistic band-structure calculation, for correlation effects not incorporated within the local approximation to SDFT (LSDA). For example, the LDA+U scheme has been applied to the compound CeSb (Antropov et al. 1995), a system that has a maximum Kerr-rotation angle of 90° (Pittini et al. 

Ebert, H. (2000) Fully relativistic band structure calculations for magnetic solids—formalism and application. In Electronic Structure and Physical Properties of Solids (ed. H. Dreyssd), p. 191. Lecture Notes in Physics, vol. 535. Springer. 

For solids with heavy atoms, relativistic shifts may affect the bonding properties, and also optical properties may be influenced. The relativistic shifts of the 5d bands relative to the s-p bands in gold change the main inter band edge more than 1 eV. Already Pyykko and Desclaux mentioned [1] that the fact that gold is yellow is a result of relativistic effects. These are indirect [2] (see also the introduction. Sect. 1), and the picture was confirmed by relativistic band structure calculations [3,4]. Also the optical properties of semiconductors are influenced by relativistic shifts which affect the gap between occupied and empty states, see for example Ref. [5]. Two additional examples may be mentioned where relativistic shifts in the energy band structure drastically influence the physical properties. First,... 

Figure 4. Relativistic band structure of Ge obtained by a pseudopotential calculation, Ref. [13]. The valence band maximum (at k=0) is the Fg state.

For applications, see Refs. [84-89]. Reference [89] describes the results of relativistic band structure calculations for CeRu2Si2, and also in this case it was found that the topology of the Fermi surface is well described by the LDA, although the T-linear specific heat coefficient is very large, 7 350 mJ/molK. This, and the similar observation made for UPts were explained [85,86] by showing that the Fermi surface topologies derived from renormalized bands and an LDA calculation for this kind of systems... 

Magnetism is a central field in condensed matter research, basic as well as applied. Several physical effects such as, for example, the magnetooptic Kerr effect, are caused by the simultaneous occurrence of spin polarization and spin-orbit coupling. It is therefore necessary to include spin polarization in the (fully) relativistic band structure formalisms. Feder et al. 

Fig. 1. Total energy of fee thorium in dependenee of the lattiee eonstant, ealeulated with the relativistie FPLO method (RFPLO) using the Perdew-Wang 92 version of LDA [25]. The position of the minimum is indieated by the dashed line. Further, the experimental lattice constant is given by a box, where the width shows the scatter of the experimental data. Calculated equilibrium lattice constants with other relativistic band structure codes are denoted by arrows. Figure taken from Ref. [26].

Despite the great variety of calculational schemes employed, relativistic band structure codes have by now achieved a high level of accuracy. While for example the calculated lattice constant of fcc-Th in early publications covered a broad range of values (Fig. 1), a number of state-of-the-art relativistic full potential methods give reliable values very close to each other, about 2.5 percent below the experimental lattice constant (which is the systematic error of the LDA functional used in the calculations). Moreover, the most accurate schemes coincide in their total energies within a few mHartree per atom, a level of accuracy almost comparable to non-relativistic band structure schemes. 

Self-consistent Dirac-Slater calculations of molecules and embedded clusters have been recently reviewed by Ellis and Goodman. Relativistic band structure calculations have also been carried out. Dirac scattered-wave calculations have been carried out on a number of inorganic complexes such as W(CO)fi and WjQg - The electronic structure and geometries of X2H2 (X = O,S, Sc and Te) have also been investigated recently. ... 

Fig. 3.20. The relativistic band structure and Fermi surface of hep Yb (Jepsen and Andersen, 1971).

The four-component DHF LCAO equations for ID-, 2D- and 3D-periodic systems were at first presented by Ladik [547]. The resulting somewhat compUcated generaUzed matrix eigenvalue equation for solids is described (for details we refer the reader to [547]). It was also shown for ID and 2D systems how MP2 methods could be applied iu their relativistic form. With the help of these, on the one hand, the total energy per unit cell (including correlation effects) can be computed. On the other hand, the relativistic band structure can also be corrected for correlation. Note that the symmetry of crystaUine orbitals changes, compared with the nonrelativistic case, as the symmetry of the DHF Hamiltonian is described by double space groups. Finally,... 

Relativity affects the kinetic term and the exchange-correlation potential in the Kohn-Sham equation. As investigated in detail for the uranium atom and the cerium atom, the relativistic effect on the exchange correlation potential is rather small and therefore we use /Ac[ ( )] in n relativistic band structure calculation. The relativistic effect on the kinetic term is appreciably large and can be taken into account by adopting the Kohn-Sham-Dirac one-electron equation instead of eq. (3) as follows ... 

Self-consistent calculation. It is now possible to carry out a relativistic band structure calculation self-consistently. Following Mattheiss method (Mattheiss 1964), a starting electron density for a crystal is constructed by superposing the self-consistent atomic electron densities, which are calculated for the neutral atoms using the method of Liberman et al. (1965). In calculations both for the atom and the crystal, the exchange... 

The relativistic band structure of HfN (78) is characterized by a N-25 band at the bottom of the valence band region. This band lies well below the very narrow Hf-4/bands. Above the Hf-4/bands, and nearly touching the band, the N-2p bands are found. They are separated... 

TaN has the stoichiometric composition and no range of homogeneity, and 8-TaN exists in the composition range from TaNo g to TaNo.9. The most significant feature of the relativistic band structure of B1 TaN is the overlap of the band with the Ta-4/ band in the [111] and... 

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