Dependence of the upper critical field on the spectral density for arbitrary impurity concentrations

Abstract

We have developed equations for, and calculated the functional derivatives with respect to, the electron-phonon spectral density of the upper critical field for arbitrary impurity concentrations and any temperature. Comparison of our results in the dirty limit agree well with previous work. The functional derivatives are found to change shape significantly at all frequencies as a function of impurity content. Also, there is a qualitative change in behavior at low frequencies over the dirty limit. In the second part of the paper a weak-coupling version of the theory is presented. In the clean and dirty limits universal behavior of the functional derivatives is found with an overall scaling of 1+λ, as was the case for previous derivatives.