Effects of Nonmagnetic Impurities upon Anisotropy of the Superconducting Energy Gap

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, $V_{p{p}^{\prime }}=(1+a_p)V(1+a_p^{}{}_{}{}^{\prime })$, 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 ${\Delta }_p$ is replaced by a complex, wave-vector- and energy-dependent gap parameter $\Delta (\mathbf{p},\omega )={\Delta }_i\left(\omega \right)+a_p{\Delta }_a\left(\omega \right)$. The behavior of ${\Delta }_i\left(\omega \right)$ and ${\Delta }_a\left(\omega \right)$ is extensively examined as a function of impurity concentration; it is found, for example, that the magnitude of the anisotropic part ${\Delta }_a\left(\omega \right)$ 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 $P\left(a\right)\mathrm{}$, 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.