# Superconductors with magnetic impurities: Instantons and subgap states

# Abstract

When subject to a weak magnetic impurity potential, the order parameter and quasiparticle energy gap of a bulk singlet superconductor are suppressed. According to the conventional mean-field theory of Abrikosov and Gor’kov, the integrity of the energy gap is maintained up to a critical concentration of magnetic impurities. In this paper, a field theoretic approach is developed to critically analyze the validity of the mean-field theory. Using the supersymmetry technique we find a spatially homogeneous saddle point that reproduces the Abrikosov-Gor’kov theory, and identify instanton contributions to the density of states that render the quasiparticle energy gap soft at any nonzero magnetic impurity concentration. The subgap states are associated with supersymmetry broken field configurations of the action. An analysis of fluctuations around these configurations shows how the underlying supersymmetry of the action is restored by zero modes. An estimate of the density of states is given for all dimensionalities. To illustrate the universality of the present scheme we apply the same method to study “gap fluctuations” in a normal quantum dot coupled to a superconducting terminal. Using the same instanton approach, we recover the universal result recently proposed by Vavilov et al. Finally, we emphasize the universality of the present scheme for the description of gap fluctuations in d-dimensional superconducting/normal structures.