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{\ensuremath{-}}\mathit{x}}$${\mathit{B}}_{\mathit{x}}$ alloy in any given substitutional arrangement of A and B on a given lattice is expanded in a series of interaction energies {${\mathit{J}}_{\mathit{f}}$(V)} of ``figures'' f. (ii) The ${\mathit{N}}_{\mathit{s}}$ functions {${\mathit{J}}_{\mathit{f}}$(V)} for ``figures'' f are determined as a function of volume V by equating the total energies of a set of ${\mathit{N}}_{\mathit{s}}$ periodic structures, calculated in the local-density formalism, to a series expansion in ${\mathit{J}}_{\mathit{f}}$ with known coefficients. The calculation includes in a natural way atomic relaxation and self-consistent charge transfer, hence providing a link between the electronic structure and the interaction energies which decide phase stability. (iii) The number ${\mathit{N}}_{\mathit{s}}$ and range of the interaction energies needed in such an Ising description is determined by the ability of such cluster expansions to reproduce the independently calculated total energy of other structures.We find that this requires extending the expansion for zinc-blende-based alloys up to the fourth fcc neighbors. (iv) Using such a ``complete'' set of ${\mathit{N}}_{\mathit{s}}$ interaction energies, {${\mathit{J}}_{\mathit{f}}$}, we find approximate solutions to the corresponding Ising Hamiltonian within the cluster-variation method, retaining up to four-body and fourth-neighbor terms. A renormalization procedure, whereby distant-neighbor correlations are folded into a compact set of effective near-neighbor correlations, is used and tested against Monte Carlo solutions. This yields the phase diagram and thermodynamic properties. The set {${\mathit{J}}_{\mathit{f}}$} is also used to perform a ground-state search of all stable structures. This identifies stable and metastable phases. This approach has been applied to five III-V pseudobinary alloys (${\mathrm{Al}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Ga}}_{\mathit{x}}$As, ${\mathrm{GaAs}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{P}}_{\mathit{x}}$, ${\mathrm{In}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Ga}}_{\mathit{x}}$P, ${\mathrm{In}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Ga}}_{\mathit{x}}$As, and ${\mathrm{GaSb}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{As}}_{\mathit{x}}$) and three II-VI pseudobinary alloys (${\mathrm{Cd}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Hg}}_{\mathit{x}}$Te, ${\mathrm{Hg}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Zn}}_{\mathit{x}}$Te, and ${\mathrm{Cd}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Zn}}_{\mathit{x}}$Te). We have calculated (i) excess enthalpies, (ii) phase diagrams, (iii) clustering probabilities, and (iv) equilibrium bond lengths. We discuss in detail the chemical trends in these properties and offer a simple (``\ensuremath{\varepsilon}-G'') model which reveals the underlying physical factors controlling such trends.