Tunneling spin current and spin diode behavior in bilayer system

The coherent tunneling spin current in the bilayer system with spin-orbit coupling is investigated. Based on the continuity-like equations, we discuss the definition of the tunneling current and show that the overlaps between wavefunctions for different layers contribute to the tunneling current. We study the linear response of the tunneling spin current to an in-plane electric field in the presence of nonmagnetic impurities. The tunneling spin conductivity we obtained presents a feature asymmetrical with respect to the gate voltage when the strengthes of impurity potentials are different in each layer.