Effects of linear and non-linear electron–phonon interactions on impurity bound states in BCS superconductor

We have considered a quantum impurity with local electron–electron, linear, and quadratic electron–phonon interactions embedded in a Bardeen-Cooper-Schrieffer (BCS) superconducting medium, modeled by a single spin-degenerate impurity Anderson-Holstein model Hamiltonian. We have employed the canonical transformations combined with the Kikuchi–Morita Cluster variation (CV) method to analyze the conditions necessary in the superconductor for the formation of the magnetic moment, subgap Yu-Shiba-Rusinov bound state associated with the magnetic impurity, and transition from single to bipolaron at the impurity site as a function of electron–electron, linear and nonlinear electron–phonon interactions, and the superconducting gap. We have observed that the effective coupling between the electrons and phonons is reduced (enhanced), and the effective phonon frequency is hardened (softened) by a slight positive (negative) quadratic electron–phonon interaction. These results show that nonlinear electron–phonon interaction can significantly impact the physics of many electron–phonon coupled systems. They are critical for magnetic quantum impurities deposited on a metal with typical nonlinear coupling strengths.