  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 function 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.
