Modeling the Interface of a Heavy Metal/Ferromagnetic Bilayer Using the Boltzmann and Drift-Diffusion Equations

Accurate models for spin torques due to electrical currents in heavy metal/ferromagnetic bilayers allow for the meaningful extraction of parameters from measurements and optimization of devices.~Descriptions of coherent spin transport across magnetic interfaces used to date typically take the form of resistor-like components that neglect the spin-orbit interaction. Using the Boltzmann equation, we develop new boundary conditions for the drift-diffusion equations to include both interfacial spin-orbit coupling and the exchange interaction due to the magnetization.~These boundary conditions capture interfacial spin-flip scattering, coupling between longitudinal and transverse spin components, and significant modifications to the interfacial torques on the magnetization. They address the spin-orbit interaction at the interfaces of heavy metal/ferromagnetic bilayers, and allow for the calculation of interfacial spin-orbit torques within the drift-diffusion approach.