Boundary conditions for micromagnetism with spin currents

Abstract

In this paper we show how the spin current in insulating ferromagnets can be treated in the micromagnetic framework by generalizing the boundary conditions in order to allow for the transport of magnetic moment and energy at the interface with other spin carrying media. As specific examples we demonstrate the spin Hall torque and spin pumping effects for a ferromagnetic layer in contact with a spin Hall metallic layer. In particular we show how by this method we can reproduce the known literature result in the case of thin films and make predictive examples for thick ferromagnets.