Local and nonlocal Andreev reflections in double barrier junctions

We study spin dependent charge transport through a junction, consisting of a superconductor connected to two normal or spin-polarized electrodes. Special attention is paid to a process known as a crossed or nonlocal Andreev reflection. The nonlocal reflection occurs when an incident electron (hole) from one electrode, injected with a subgap energy onto o superconducting layer, transforms into an Andreev hole (electron) in the second lead. When two electrons (holes), forming a Cooper pair, come from the same electrode, the process of the Andreev reflection is local. The coherent spin polarized transport in ferromagnet/superconductor/ferromagnet (F/S/F) double barrier junctions is analyzed using the Bogolubov-de Gennes (BdG) equation with appropriate boundary conditions. We calculate and discuss probabilities of the normal, local and crossed Andreev reflections, as well as probability of the elastic co-tunneling. These processes contribute to tunneling current when the distance between the magnetic electrodes is comparable to the superconducting coherence length. The dependence of the tunneling charge transport on the strength of the exchange field in the ferromagnetic electrodes, and on the height of the tunnel barriers are presented. We also discuss the local and nonlocal spin quantum entanglements of quasiparticle pairs induced by the Andreev reflections in the junction.