Superconducting spin filter

Consider two normal leads coupled to a superconductor; the first lead is biased while the second one and the superconductor are grounded. In general, a finite current $I_2(V_1,0)$ is induced in the grounded lead 2; its magnitude depends on the competition between processes of Andreev and normal quasiparticle transmission from the lead 1 to the lead 2. It is known that in the tunneling limit, when normal leads are weakly coupled to the superconductor, $I_2(V_1,0)=0$, if $|V_1|<\Delta$ and the system is in the clean limit. In other words, Andreev and normal tunneling processes compensate each-other. We consider the general case: the voltages are below the gap, the system is either dirty or clean. It is shown that $I_2(V_1,0)=0$ for general configuration of the normal leads; if the first lead injects spin polarized current then $I_2=0$, but spin current in the lead-2 is finite. XISIN structure, where $X$ is a source of the spin polarized current could be applied as a filter separating spin current from charge current. We do an analytical progress calculating $I_1(V_1,V_2), I_2(V_1,V_2)$.