Spin transport at interfaces with spin-orbit coupling: Formalism

Spin transport at interfaces between nonmagnets and ferromagnets plays an important role in spintronic devices. Lately, there is an increasing suspicion that spin-orbit coupling, which couples the spin and momentum of carriers, might contribute significantly to this process. Unfortunately, the existing description of spin transport at such interfaces, magnetoelectronic circuit theory, is not valid when spin-orbit coupling is present at the interface. This paper presents a generalization of magnetoelectronic circuit theory to interfaces with spin-orbit coupling. Like the original theory, this generalization describes spin transport in terms of drops in spin and charge accumulations across the interface, but also includes responses to in-plane electric fields and offsets in spin accumulations. The most important result is a description of the way in-plane electric fields generate spin accumulations, spin currents, and torques at the interface. The effects described by this generalized circuit theory impact the interpretation of experiments involving spin-orbit torques, spin pumping, spin memory loss, the Rashba-Edelstein effect, and spin Hall magnetoresistance.