Spin currents in metallic nanostructures: Explicit calculations

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.