Effects of Nonmagnetic Impurities upon Anisotropy of the Superconducting Energy Gap

Abstract

The influence of the presence of nonmagnetic impurities upon the anisotropy of the superconducting energy-gap parameter is considered. Using a factorable BCS-like model for the effective electron-electron matrix element, ${V}_{p{p}^{\ensuremath{'}}}=(1+{a}_{p})V(1+a_{p}^{}{}_{}{}^{\ensuremath{'}})$, within the context of an earlier theory by Markowitz and Kadanoff, it is shown that when impurities are present the wave-vector-dependent gap parameter ${\ensuremath{\Delta}}_{p}$ is replaced by a complex, wave-vector- and energy-dependent gap parameter $\ensuremath{\Delta}(\mathbf{p},\ensuremath{\omega})={\ensuremath{\Delta}}_{i}(\ensuremath{\omega})+{a}_{p}{\ensuremath{\Delta}}_{a}(\ensuremath{\omega})$. The behavior of ${\ensuremath{\Delta}}_{i}(\ensuremath{\omega})$ and ${\ensuremath{\Delta}}_{a}(\ensuremath{\omega})$ is extensively examined as a function of impurity concentration; it is found, for example, that the magnitude of the anisotropic part ${\ensuremath{\Delta}}_{a}(\ensuremath{\omega})$ of the gap parameter tends to zero in the limit of large impurity concentration. A model calculation, assuming a rectangular shape for the anisotropy distribution function $P(a)$, illustrates the behavior for small and moderate impurity concentrations. The behavior for large impurity concentrations is found to depend, to lowest order, only upon the mean-squared anisotropy $〈{a}^{2}〉$. The behavior of the effective density of states is also examined; it is shown to become isotropic as the impurity concentration increases. The precise shape of the effective density of states for energies near the gap is obtained for the large-impurity-concentration limit. Experimental manifestations of the reduction of the anisotropy by impurity scattering are briefly discussed.