Random transfer integrals and the electronic structure of disordered alloys: Equilibrium and transport properties

A tight-binding model Hamiltonian is used to study the electronic properties of the disordered alloy $A_x{B}_{1-x}$. Both the local energy levels and the interatomic transfer integrals are allowed to vary randomly. The problem simplifies considerably if it is assumed that the values of the transfer integrals are additive, i.e., that $h^{\mathrm{AB}}=\frac{(h^{\mathrm{AA}}+h^{\mathrm{BB}})}{2}$. In this case the conventional methods of perturbation theory are applicable and a mean-field approach, analogous to the coherent-potential approximation, is easily formulated. Our equations are compared to exact results concerning the dilute and split-band limits and also to the work of previous authors. In the latter connection they are shown to be equivalent to the appropriate limit of the equations proposed by Blackman, Esterling, and Berk. Finally, we consider the effects of off-diagonal disorder on the problem of electronic transport. Again working within the additive approximation, the vertex corrections to the two-particle Green's function are evaluated exactly and a closed expression relating the static conductivity to the electron self-energy is derived.