Structure and thermodynamics of ${\mathrm{Si}}_{\mathit{x}}$${\mathrm{Ge}}_{1\mathrm{-}\mathit{x}}$ alloys from ab initio Monte Carlo simulations

${\mathrm{Si}}_{\mathit{x}}$ ${\mathrm{Ge}}_{1\mathrm{-}\mathit{x}}$ alloys are studied with a new method based on density-functional theory and Monte Carlo sampling. Using perturbation theory with respect to the virtual crystal, we are able to map the alloy onto a lattice gas with long-range interactions, which are determined from first principles. Monte Carlo simulations show that ${\mathrm{Si}}_{\mathit{x}}$ ${\mathrm{Ge}}_{1\mathrm{-}\mathit{x}}$ is a model random alloy with a miscibility gap below ≊170 K. The bond-length distribution displays three well-defined peaks whose positions depend on composition, but not on temperature. The resulting lattice parameter follows Vegard’s law very closely.