Phase diagram and critical behavior of the Si-Ge unmixing transition: A Monte Carlo study of a model with elastic degrees of freedom

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Abstract
A statistical-mechanical model of binary semiconductor alloys, consisting of a distortable diamond lattice whose sites may be occupied by A atoms, B atoms, or vacancies, is studied by Monte Carlo computer simulations. By extending a grand-canonical lattice gas, the model allows for atomic displacements governed by the Keating valence force field. Unphysical boundary conditions are avoided by keeping the pressure constant. This model is similar to a compressible Ising model, but differs from it by the occurrence of a bilinear coupling between spin field and displacement field. The interplay between the chemical and translational degrees of freedom shows up in the form of the unmixing phase diagram of a system whose parameters were chosen in an attempt to mimic a Si-Ge alloy. Methods of thermodynamic integration to obtain the free energies of different phases are discussed. The critical behavior of the unmixing transition is studied by a multihistogram data analysis. The finite-size scaling of the data is in better agreement with mean-field-like critical behavior than with an Ising transition or Fisher-renormalized exponents. Vegard’s law is verified, and it is shown that the Keating potential leads to a negative coefficient of thermal expansion.