Tunnelling spin current and spin diode behaviour in a bilayer system.

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