Diamagnetic property and optical absorption of conventional superconductors with magnetic impurities in linear response

By solving the renormalization of the $s\text{\ensuremath{-}}d$ interaction from magnetic impurities embedded in conventional superconductors at low concentration, we derive the macroscopic superconducting phase fluctuation and electromagnetic linear-response properties within the path-integral approach. It is found that there exist two superconducting phase modes, both exhibiting similar behaviors of the Nambu-Goldstone mode. The existence of two phase modes suggest that in addition to the conventional free Cooper pairs as in the BCS case, there emerges a small part of the localized Cooper pairs around magnetic impurities due to the quantum correlation by the $s\text{\ensuremath{-}}d$ interaction, acting as Josephson islands. The emerging impurity Shiba bands inside the superconducting gap then correspond to the excitation of the ground state of the localized Cooper pairs, associated with the breaking of these Cooper pairs. In the diamagnetic response, the state of the free Cooper pairs gives rise to the conventional real contribution in the generated supercurrent, whereas the one of the localized Cooper pairs results in an imaginary contribution, leading to the superconducting Friedel oscillation, i.e., oscillation in the decay of the vector potential in the Meissner effect. As for the optical absorption of a conventional superconductor lying in the anomalous-skin-effect region, it is found that besides the conventional interband transition of Bogoliubov quasiparticles as revealed by Mattis-Bardeen theory, there also exist the interband transition between the impurity Shiba bands as well as all interband transitions between Bogoliubov quasiparticle and impurity Shiba bands. These transitions exhibit clear and separate resonance characters, providing a feasible scheme for the experimental detection.