Effects of Off-Diagonal Randomness in the Tight-Binding Model of Random Binary Alloys: Weak-Coupling Limit

The single-band tight-binding model of random binary alloys is studied in the limit of second-order self-consistent perturbation theory, allowing the off-diagonal, as well as the diagonal, matrix elements of the model Hamiltonian to be dependent upon alloy composition. Fluctuations of both types of matrix elements about their configurational averages are assumed to be small but comparable, so that the randomness introduced into both parts of the Hamiltonian must be treated on an equal footing. Only nearest-neighbor hopping between constituent atoms (type $A$ or $B$) is considered, with the hopping integrals parametrized by the three numbers $\alpha$, $\beta$, and $\gamma$ for $A-A$, $B-B$, and $A-B$ hopping, respectively. The single-particle alloy spectrum and the alloy density of states are obtained from an equation for the effective single-particle self-energy using standard techniques and with the dependence on model parameters explicitly displayed. With the assumption that $\gamma =\frac{1}{2}(\alpha +\beta )$, the theory is found to be rather simply characterized by the wave-vector-dependent displace ment $E_A\left(\mathrm{k}\right)-E_B\left(\mathrm{k}\right)$ of the pure constituent spectra $E_A\left(\mathrm{k}\right)$ and $E_B\left(\mathrm{k}\right)$.