Magnetic impurity in a superconductor: Local phase transitions and finite size effects

The influence of a single magnetic impurity on an s-wave superconductor is investigated by solving the Bogoliubov-De Genne equations with the impurity modeled by a spin-dependent square potential well of radius a and strength J. Unlike the one with a $\delta$ -function potential, our model shows multiple phase transitions with increase of J or a, which manifest themselves through a change of the nature of impurity bound states. By considering a finite size superconducting host (impurity on a superconducting grain) we suggest a way of observing the transitions in experiment.