Interplay of anisotropy and energy dependence in electron density of states in strong-coupling superconductors.

Abstract

Using a separable model for anisotropy, we have derived the generalized Eliashberg equations applicable to anisotropic superconductors when the electron density of states (EDOS) has structures on the scale of the Debye energy. We have solved them numerically to study the interplay of anisotropy and energy dependence in EDOS in determining the behavior of the critical temperature ${T}_{c}$ as a function of normal impurity concentration. For the EDOS, we have used two models described by parameters whose values are chosen to make the models resemble very closely the structures obtained in recent band-structure calculations. For as high and narrow a peak in the EDOS placed close to the Fermi level as is obtained in the band-structure calculations, the effects due to the energy dependence in EDOS through broadening of its peak, dominate the overall reduction of ${T}_{c}$ with increasing impurity concentration and mask any effects arising from the disappearance of anisotropy. It makes any estimation of the degree of anisotropy difficult. If, however, the peak is made wider and placed well away from the Fermi level, a clear separation between the effects of energy dependence in EDOS and those of washing out of anisotropy occurs, making an estimation of the anisotropy feasible. These conclusions are similar to those drawn by Whitmore and Carbotte in a previous study based on a BCS-type theory.