Impurity-induced bound states in BCS superconductor: The role of Rashba spin-orbit and non-linear electron-phonon interactions

We have theoretically studied the influence of linear, non-linear electron-phonon interactions and Rashba spin orbit interaction on formation of the localized magnetic moment and subgap Yu-Shiba-Rusinov bound state for a single-level quantum impurity embedded in an s-wave Bardeen-Cooper-Schrieffer superconductor, modeled by single-impurity Anderson-Holstein Hamiltonian. We have used timescale separation approach combined with Bogoliubov transformation and modified Lang-Firsov transformation to decouple quadratic and linear electron-phonon interactions present on impurity. Then we used the Kikuchi-Morita cluster variation method for the calculation of ground state energy of the system. We theoretically studied the influence of Rashba spin-orbit interaction on the correlation effects of magnetic impurity embedded in an s-wave superconductor. Further, we analyze transition from single to bipolaron phase transition at the impurity site as a function external tunable Rashba spin orbit interaction strength.