Strong-Coupling Theory of Impure Superconductors : Correspondence with Weak-Coupling Theory

TL;DR

This paper analyzes how impurities affect the superconducting transition temperature using Eliashberg theory, revealing the conditions under which strong and weak coupling theories correspond and how impurity scattering impacts electron interactions.

Contribution

It demonstrates that the correspondence principle between strong and weak coupling theories holds only with Anderson's pairing condition and explores impurity effects on $T_c$ in different phonon models.

Findings

Impurity scattering decreases $T_c$ proportionally to $1/(E_F au)$ in the dirty limit.

Phonon-mediated interaction is reduced by weak localization corrections similar to conductivity.

Strong localization causes the phonon-mediated interaction to become exponentially small.

Abstract

We reconsider the ordinary impurity effect on the transition temperature $T_{c}$ of superconductors using the Eliashberg formalism. It is shown that the correspondence principle, which relates strong-coupling and weak-coupling theories, works only when Anderson's pairing condition between the time-reversed scattered-states is used. For an Einstein phonon model, the change of the electron density of states caused by the impurity scattering leads to a $T_{c}$ decrease proportional to ${1/ E_{F}\tau}$ in the dirty limit. It is pointed out that the phonon-mediated interaction decreases by the same weak localization correction term as that of the conductivity. Accordingly, for strongly localized states the phonon-mediated interaction is exponentially small. We also discuss the case of Debye phonon model.