Microscopic formulation of dynamical spin injection in ferromagnetic-nonmagnetic heterostructures

We develop a microscopic formulation of dynamical spin injection in heterostructure comprising nonmagnetic metals in contact with ferromagnets. The spin pumping current is expressed in terms of Green's functions of the nonmagnetic metal attached to the ferromagnet where a precessing magnetization is induced. The formulation allows for the inclusion of spin-orbit coupling and disorder. The Green's functions involved in the expression for the current are expressed in real-space lattice coordinates and can thus be efficiently computed using recursive methods.