Coherent alloy phase separation: Differences in canonical and grand canonical ensembles

Coherency stresses affect the stability of lattice-mismatched alloys that phase separate. In order to understand the effect of coherency on semiconductor alloys such as Si1-xGex and Ga1-xInxP, we have studied a microscopic model of an A(1-x)B(x) alloy on a square lattice. It is assumed that the energy of an alloy bond has two parts. The first depends only on the identity of the atoms the bond joins; this leads to nearest-neighbor couplings in the alloy Hamiltonian. The second is the bond's elastic energy. Most alloy bonds are stretched to accommodate the different natural lengths of A-A and B-B bonds. The energy of this distortion is modeled by a Keating-type potential. With the Monte Carlo technique, we numerically investigate the alloy's statistical mechanics in the canonical and constant-pressure grand canonical ensembles. Because of the coherency stresses, these two ensembles are physically inequivalent. The canonical ensemble, corresponding physically to an isolated coherent A(1-x)B(x) alloy, yields Ising universality. The constant-pressure grand canonical ensemble, corresponding physically to a coherent A(1-x)B(x) alloy in contact with a fluid of A and B materials, yields mean-field universality and a higher critical temperature.