Proximity effect in ferromagnet/superconductor hybrids: From diffusive to ballistic motion

Abstract

We present an analytical study of the proximity effect in ferromagnet/superconductor (F/S) heterostructures, allowing for an arbitrary magnetic exchange energy as well as arbitrary impurity and spin-flip scattering rates within a quasiclassical approach. While previous studies mainly have focused on the clean or dirty limits, our results grant access to the regime of intermediate impurity concentrations, thus allowing us to probe the crossover from the clean to the dirty limit. We find that in the crossover regime, all possible symmetry correlations of the proximity-induced anomalous Green's function are induced in the ferromagnet. We also point out that the local density of states oscillates spatially not only for an F/S bilayer but also for a normal/superconductor (N/S) bilayer in the diffusive limit, a fact which appears to have gone unnoticed in the literature. Within the weak proximity-effect regime, we present compact analytical expressions valid for arbitrary exchange fields and impurity-scattering rates for (i) the local density of states in an F/S bilayer, (ii) the Josephson current in an S/F/S junction, and (iii) the critical temperature in an F/S/F multilayer. For all cases, we study in particular the crossover regime between diffusive and ballistic motion. Our results may be useful for analyzing experimental data in cases when the dirty limit is not fully reached, thus invalidating the use of the Usadel equation.