Andreev reflection through a quantum dot coupled with two ferromagnets and a superconductor

We study the Andreev reflection (AR) in a three-terminal mesoscopic hybrid system, in which two ferromagnets ${(F}_1$ and $F_2)$ are coupled to a superconductor (S) through a quantum dot. By using nonequilibrium Green function, we derive a general current formula which allows arbitrary spin polarizations, magnetization orientations, and bias voltages in $F_1$ and $F_2.$ The formula is applied to study both zero bias conductance and finite bias current. The current conducted by crossed AR involving $F_1,$ $F_2,$ and S is particularly unusual, in which an electron with spin σ incident from one of the ferromagnets picks up another electron with spin $\overline{\sigma }$ from the other one, both enter S and form a Cooper pair. Several special cases are investigated to reveal the properties of AR in this system.