Structure and Thermodynamics of SiGe Alloys from Computational Alchemy

We present a new method for studying theoretically the structural properties of semiconductor alloys. The alloy is considered as a perturbation with respect to a periodic virtual crystal, and the relevant energies calculated by density-functional perturbation theory. We show that —up to second order in the perturbation— the energy of the alloy is equivalent to that of a lattice gas with only two-body interactions. The interaction constants of the lattice gas are particular linear response functions of the virtual crystal, which can be determined from first principles. Once the interaction constants have been calculated, the finite-temperature properties of the alloy can be studied rather inexpensively by MonteCarlo simulations on the lattice gas. As an application, we consider the case of SixGe1-x. A comparison with traditional self-consistent calculations for some simple ordered structures demonstrates that the accuracy of the perturbative approach is in this case of the same order as that of state-of-the-art density-functional calculations. Ignoring lattice relaxation, the range of the interactions is very short. Atomic relaxation renormalizes the interactions and makes them rather long range, propagating mainly along the bond chains. Monte Carlo simulations show that SixGe1-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 very closely Vegard’s law.