Spin Transport at Interfaces with Spin-orbit Coupling: Formalism
Vivek P. Amin, Mark D. Stiles
Spin transport remains poorly understood in multilayer systems with interfacial spin-orbit coupling. Currently, drift-diffusion models cannot accurately treat this phenomenon, since the important consequences of interfacial spin-orbit scattering remain uncharacterized in a systematic way. Here we present boundary conditions suitable for drift-diffusion models that capture the phenomenology of spin-orbit coupling at interfaces. To assess their viability, we compare solutions of the drift-diffusion equations using these boundary conditions to solutions of closely related Boltzmann equations for a heavy metal/ferromagnet bilayer. The latter approach treats a momentum-dependent distribution function equipped to describe momentum-dependent spin-orbit scattering. A key result is that in-plane electric fields create spin accumulations and spin currents at the interface that are polarized in all directions, generalizing the Rashba-Edelstein and spin Hall effects. In heavy metal/ferromagnet bilayers, this phenomenon provides mechanisms for the creation of damping-like and field-like torques; it also leads to possible reinterpretations of experiments in which interfacial torques are thought to be suppressed. We reproduce magnetoelectronic circuit theory in the appropriate limit, and discuss the interpretation of experiments involving spin orbit torque, spin pumping and memory loss, the Rashba-Edelstein effect, and the spin Hall magnetoresistance.