Pair Effects in Substitutional Alloys. I. Systematic Analysis of the Coherent-Potential Approximation

A single-band model is used to study the electronic structure of disordered binary alloys. Functional-derivative techniques are used to generate an expansion for the electron self-energy that is free of all "multiple-occupancy" corrections. This analysis reveals that the relevant small parameter for the coherent-potential approximation (CPA) is $Z^{-1\mathrm{}}$, where $Z$ is the number of nearest neighbors. In addition to being exact to first order in the concentration $x$ and third order in the impurity potential $\delta$, the CPA retains just those contributions of higher order in $x$ and $\delta$ that are independent of $Z^{-1\mathrm{}}$. Various methods have been suggested to calculate corrections to the CPA due to two-atom clusters. While all of these are exact to order $x^2$ and ${\delta }^5$, we argue that a proper generalization of the CPA must also be correct to higher orders in $Z^{-1\mathrm{}}$. The appropriate equations are derived and shown to imply the existence of satellite levels on either side of the impurity subband. A formalism is developed to examine the departure from the usual assumption of complete compositional disorder. To order $x^2$, the single-band Hamiltonian is found to imply the existence of short-range order in the alloy. The influence of this short-range order on the density of states is discussed and is shown to modify the clustering effects previously evaluated.