Self-consistent screening in a simple model

The self-consistent screening of a charged impurity by conduction electrons in a simple cubic metal is calculated in the tight-binding approximation. In order to satisfy screening it is not sufficient to add to the impurity-site potential an impurity-induced hopping perturbation, rather, perturbations on neighboring lattice sites must also be included. If the impurity potential is truncated a few lattice sites away from the impurity, the self-consistent Hartree screening problem can be reduced to a small set of simultaneous equations which can be solved numerically. Using techniques developed by Callaway and by Mann, Seeger, and co-workers, calculations are performed of the residual resistivity and specific heat of a dilute alloy for a set of self-consistent screening parameters. In addition, the forward-scattering amplitudes of electrons on the Fermi surface are calculated. The results show significant numerical differences from the non-self-consistent highly localized potential model used in current coherent-potential-approximation calculations of the properties of disordered binary alloys. The conclusion is that quantitatively accurate calculations of disordered alloys must include screening contributions to the potential. Predictions concerning the range of validity of the forward-scattering approximation introduced by Stern are also discussed. In addition, it is discovered that a new approximation, representing a hybrid between first-order perturbation theory and the highly localized potential model, can adequately describe self-consistent screening of the charged impurity investigated here.