Theory of Electronic Transport in Disordered Binary Alloys: Coherent-Potential Approximation

The coherent-potential approximation (CPA) for single-particle properties of electrons in a disordered alloy $A_x{B}_{1-x}$ (Soven and others) is extended to complex admittances. The one-electron Kubo formula is used. The CPA is viewed as a single-site decoupling of the averaged multiple-scattering expansion. It properly gives the exact formulas in the limits of weak scattering (Edwards) and of dilute alloys (Langer). For any $x$ and any random-potential strength, CPA satisfies a number of physical conditions, including energy and particle-number conservation. The CPA equations are exactly soluble for a single-band model with short-ranged random scatterers. The vertex corrections are related to the response of local densities to a given disturbance. For the electrical conductivity $\sigma$, they vanish. Variation of $\sigma$ with the randompotential strength is studied numerically. A low-mobility region appears well before the band splits. In the split-band limit, CPA yields a reasonable finite $\sigma$ in the host band, but it fails in the impurity band.