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Electronic structure and self energies of randomly substituted solids using density functional theory and model calculations

We describe procedures to obtain the electronic structure of disordered systems using either tight binding like models or quite directly from ab inito density functional band structure calculations. The band structure is calculated using super cells much larger than those containing a single minority component atom. We average over a large number of different super cell calculations containing different randomly positioned minority component atoms in the super cell as well as a varying total number of minority component atoms, weighted by the statistical probability. We develop an efficient and simple algorithm for unfolding of these bands, based on Bloch's theorem. The unfolded band-structure obtained in this way exhibits momentum and energy broadened structures replacing the gaps observed in often used single super cell calculations. Using the super cell averaged band-structure one can introduce a self-energy, resulting from the scattering of randomly positioned alloy components. The self-energy is causal, and shows strong energy and some momentum dependence. The self-energy shows rather non-trivial behavior and is in general non-zero at the Fermi-energy, resulting in an ill or undefined Fermi surface. The real-part of the self-energy at the Fermi-energy relates to an apparent violation of Luttinger's theorem. There is no simple relation between the apparent Fermi-surface volume and the electron count. Examples introducing these effects both for model and real binary alloy systems are presented.