Electronic Transport in Alloys: Coherent-Potential Approximation

The coherent-potential approximation, which has been successfully used to describe the electronic structure of a nondilute binary alloy $A_x{B}_{1-x}$, is reformulated in a diagrammatic way suitable for the calculation of more complicated transport coefficients. This approach is applied to the calculation of three elementary transport coefficients: the conductivity $\sigma$, the thermoelectric power $Q$, and the low-field Hall coefficient $R_H$. The appropriate response functions are evaluated for a simple cubic tight-binding model. The rigid-band limit is considered in detail, with emphasis on the role of critical points. As the random alloy potential increases, deviations from rigid-band behavior — for example, Nordheim's rule — become more pronounced for unexpectedly small scattering strengths. However, the usual relations among the transport coefficients, e.g., Mott's equation between $Q$ and $\sigma$, are maintained. The conductivity is no longer symmetrical with respect to electron and hole concentrations. Furthermore, the change in sign of $Q$ and $R_H$ may not occur when the band is half-full. Therefore, the identification of the carrier sign becomes ambiguous. For the model treated, numerical calculations are quite tractable. Examples are given which illustrate the behavior for a wide range of alloy parameters.