The use of self-consistent band calculations to understand the electronic structure of solids is reviewed. The formal basis for the approach is first examined to emphasise that not all band calculations are applications of the same theory. Similarly the critical points, both fundamental and mundane, of the self-consistency process are examined. Again, the central theme is the idea that perhaps some aspects of the self-consistent calculation are not completely established. On the other hand, band structure technique, while still developing, has been rather well explored. Thus one can examine it from the general point of view of seeing what the various techniques imply about the nature of the band structure problem. Finally, some applications of self-consistent band structure calculations are examined.

http://iopscience.iop.org/article/10.1088/0034-4885/44/2/002/pdf

Self-consistent energy band calculations

On the interpolation of eigenvalues and a resultant integration scheme

The self-consistent full-potential total-energy Korringa-Kohn-Rostoker electronic-structure method is generalized to include all relativistic effects. The Dirac equation for a general anisotropic 4\ifmmode\times\else\texttimes\fi{}4 potential is solved inside Voronoi polyhedra surrounding each basis atom. As an illustration, the method is used to calculate the self-consistent electronic band structure, the Fermi surface, the equilibrium lattice constant and the bulk modulus of the fcc transition metals Pd, Ir, Pt, and Au. If the cutoff of the multipole expansions of the wave functions is at least ${\mathit{l}}_{\mathrm{max}}$=4, the calculated equilibrium lattice constants of the transition metals deviate from experiment by less than 1%, and the calculated bulk moduli deviate between 6% and 20%, which is comparable to results of other local-density calculations. In addition, the method is used to calculate the self-consistent electronic band structure of the semiconductors GaAs, InSb, and InN, and the equilibrium lattice constant and the bulk modulus of InSb. We find that the inclusion of both spin-orbit coupling and full-potential effects influences the size of the valence-band-width and the band gap in comparison with scalar relativistic local-density calculations. Interestingly, if after self-consistency has been achieved in scalar relativistic calculations, spin-orbit coupling is taken into account by the so-called second variation, the energy bands are found to agree very well with the results obtained here with the full relativistic treatment. \textcopyright{} 1996 The American Physical Society.

Self-consistent relativistic full-potential Korringa-Kohn-Rostoker total-energy method and applications

An efficient step-forward way to solve the Schrödinger eigenvalue equation in self-consistent calculations

Because a solid can be viewed as a collection of interacting atoms, it will exhibit all the relativistic effects known to occur in atoms. In fact, the relativistic contributions in solids arise exclusively from the atomic-like regions of space near the nucleii with the interstitial region being quite well described in a non-relativistic approximation. Thus, for those band structure techniques such as the APW, KKR and related methods which partition space into atomic like spheres and an interstitial region, it is actually possible to “switch kinematics” at the boundary as well. Such a procedural step not only tells one how to include the relativistic contributions but also how to predict their effects: they will be the atomic results only slightly modified by the differing boundary conditions. We will discuss this using simplified views of the APW and LCAO methods as examples.

Relativistic Effects in Solids

A method to calculate relativistic energy bands and wave functions is presented, in which rapidly converging basis sets are created from linearly combined RAPW functions. By constructing other basis sets, convergence of the band structure of complex materials (e.g. compounds) can be studied with a matrix of moderate size. Results of an application to fcc praesodymium show the usefulness of the method and indicate a considerable potential for future developments.

http://iopscience.iop.org/article/10.1088/0031-8949/5/6/008/pdf

D D Koelling

Papers

Combined Basis Function Method for Energy Band Calculations