  In ultrathin ferromagnets deposited on metallic substrates, excitation of
precessional motion of the spins produces a spin current in the substrate that
transports angular momentum out of the film. This phenomenon is referred to as
spin pumping, and is a source of damping of the spin motion. Spin pumping
enters importantly in the description of spin dynamics in other nanoscale and
subnanoscale systems as well. In this paper, we present an approach based on
the Kubo formalism that allows the explicit calculation of this spin current
and its spatial variation. We use the formalism to explore features of the spin
current generated by spin motions in a simple model system.
