Quasichemical approximation in binary alloys

In this theoretical study, the quasichemical approximation for an $A_{1\mathrm{-}x}$ $B_x$ alloy is formulated for an arbitrary lattice and choice of cluster. The statistical problem of the average number of each class of clusters is collapsed into a polynomial equation. An n-atom cluster of type j is characterized by an excess energy ${\varepsilon }_j$, the number of B atoms $n_j$(B) and a degeneracy $g_j$. If ${\varepsilon }_j$ is a linear function of $n_j$(B) and $g_j$ is a binomial coefficient of n and $n_j$(B), then the cluster populations are random. Strains due to lattice-size mismatches, chemical (electron-ion interaction) differences, and differences between the electron-electron Coulomb interactions of the alloy constituents drive nonlinear variations of ${\varepsilon }_j$ on $n_j$(B). The $g_j$ is modified by coherent, externally applied stresses and temperature gradients present during crystal growth. We derive the conditions under which compounds are formed or spinodal decomposition occurs. We also discuss the possibility of materials consisting of arrays of two kinds of domains: one a random alloy and the other an ordered compound. The theory is specialized to semiconductor alloys $A_{1\mathrm{-}x}$ $B_x$C in a distorted zinc-blende structure; numerical results are presented for ${\mathrm{Ga}}_{1\mathrm{-}\mathrm{x}}$ ${\mathrm{In}}_{\mathrm{x}}$As and ${\mathrm{GaAs}}_{1\mathrm{-}\mathrm{x}}$ ${\mathrm{Sb}}_{\mathrm{x}}$ alloys. A major conclusion is that semiconductor alloys are almost never truly random.