Theory of electrical conductivity of random binary alloys in the averaget-matrix approximation

Abstract

A formulation of the average $t$-matrix approximation (ATA) for the electrical conductivity of a disordered binary alloy is presented on the basis of the method of kinetic equations, with the help of a projection operator to denote the configuration average. It is shown that this ATA preserves all qualitative properties of the exact average density operator $〈\rho \mathrm{}\left(z\right)\mathrm{}〉$ and that it yields the correct behavior of the conductivity for small concentrations of either type of atom and for small scattering potential (the difference between the two atomic potentials). It is proved further that when the reference medium is taken to be the "virtual crystal," the "coefficients" of the kinetic equation for $〈\rho \mathrm{}\left(z\right)\mathrm{}〉$ can be given very simply in terms of the $t$ matrix of a scaled scattering potential. For a general one-band model of a random alloy it is demonstrated that the distribution function $f_k$, that suffices to determine the conductivity, satisfies an integral transport equation of the Boltzmann-Bloch type that is familiar from the well-known theory of slightly impure metals and semiconductors. Furthermore, this equation is not any more complicated than that for the case of low-concentration impurity scattering. This makes this formulation of the ATA for transport a practical scheme.