First-principles calculation of temperature-composition phase diagrams of semiconductor alloys

The three-dimensional spin-1/2 Ising model with multiple-site interactions provides a natural framework for describing the temperature-composition phase diagram of substitutional binary alloys. We have carried out a ‘‘first-principles’’ approach to this problem in the following way: (i) The total energy of an ${\mathit{A}}_{1\mathrm{-}\mathit{x}}$ ${\mathit{B}}_{\mathit{x}}$ alloy in any given substitutional arrangement of A and B on a given lattice is expanded in a series of interaction energies {${\mathit{J}}_{\mathit{f}}$(V)} of ‘‘figures’’ f. (ii) The ${\mathit{N}}_{\mathit{s}}$ functions {${\mathit{J}}_{\mathit{f}}$(V)} for ‘‘figures’’ f are determined as a function of volume V by equating the total energies of a set of ${\mathit{N}}_{\mathit{s}}$ periodic structures, calculated in the local-density formalism, to a series expansion in ${\mathit{J}}_{\mathit{f}}$ with known coefficients. The calculation includes in a natural way atomic relaxation and self-consistent charge transfer, hence providing a link between the electronic structure and the interaction energies which decide phase stability. (iii) The number ${\mathit{N}}_{\mathit{s}}$ and range of the interaction energies needed in such an Ising description is determined by the ability of such cluster expansions to reproduce the independently calculated total energy of other structures.