We use a perturbative approach to study the effects of interfacial spin-orbit coupling in magnetic multilayers by treating the two-dimensional Rashba model in a fully three-dimensional description of electron transport near an interface. This formalism provides a compact analytic expression for current-induced spin-orbit torques in terms of unperturbed scattering coefficients, allowing computation of spin-orbit torques for various contexts, by simply substituting scattering coefficients into the formulas. It applies to calculations of spin-orbit torques for magnetic bilayers with bulk magnetism, those with interface magnetism, a normal metal/ferromagnetic insulator junction, and a topological insulator/ferromagnet junction. It predicts a dampinglike component of spin-orbit torque that is distinct from any intrinsic contribution or those that arise from particular spin relaxation mechanisms. We discuss the effects of proximity-induced magnetism and insertion of an additional layer and provide formulas for in-plane current, which is induced by a perpendicular bias, anisotropic magnetoresistance, and spin memory loss in the same formalism.

Center for Nanoscale Science and Technology National Institute of Standards and Technology Gaithersburg Maryland 20899 USA

Department of Materials Science and Engineering Korea University Seoul 02841 Korea

Institut für Physik Johannes Gutenberg Universität Mainz Mainz 55128 Germany

Institute of Physics Academy of Sciences of the Czech Republic Cukrovarnická 10 162 53 Praha 6 Czech Republic

KU KIST Graduate School of Converging Science and Technology Korea University Seoul 02841 Korea

Maryland NanoCenter University of Maryland College Park Maryland 20742 USA

PCTP and Department of Physics Pohang University of Science and Technology Pohang 37673 Korea

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Large Damping-Like Spin-Orbit Torque in a 2D Conductive 1T-TaS2 Monolayer