Magnetic impurities in a strongly coupled superconductor

We revisit certain aspects of a problem concerning the influence of carrier scattering induced by magnetic impurities in metals on their superconducting properties. Superconductivity is assumed to be driven by strong electron-phonon interaction. We use the self-consistent solution of the Nagaoka equations for the scattering matrix together with the Migdal-Eliashberg theory of superconductivity to compute the energy of the in-gap bound states, superconducting critical temperature and tunneling density of states for a wide range of values of the Kondo temperature and impurity concentrations. It is found that similar to the case of the weak coupling Bardeen-Cooper-Schrieffer superconductors there is only one pair of the bound states inside the gap as well as re-entrant superconductivity for the case of antiferromagnetic exchange coupling between the conduction electrons and magnetic impurities. In addition, we find that strong electron-phonon coupling significantly reduces the decay length of the impurity-induced bound state, leading to enhanced localization compared to the weak-coupling case, which in turn narrows the spatial overlap between impurities. In agreement with the earlier studies, we find that the gapless superconductivity can be realized, which in the case of antiferromagnetic exchange requires much smaller impurity concentration. Surprisingly, in contrast with the weakly coupled superconductors we find that superconducting transition exhibits two critical temperatures for the ferromagnetic exchange coupling.