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Antidamping spin-orbit torque driven by spin-flip reflection mechanism on the surface of a topological insulator: a time-dependent nonequilibrium Green function approach
Motivated by recent experiments observing spin-orbit torque (SOT) acting on the magnetization m of a ferromagnetic (F) overlayer on the surface of a three-dimensional topological insulator (TI), we investigate the origin of the SOT and the magnetization dynamics in such systems. We predict that lateral F/TI bilayers of finite length, sandwiched between two normal metal leads, will generate a large anti-damping-like SOT per very low charge current injected parallel to the interface. The large values of anti-damping-like SOT are spatially localized around the transverse edges of the F overlayer. Our analysis is based on adiabatic expansion (to first order in dm/dt) of time-dependent nonequilibrium Green functions (NEGFs), describing electrons pushed out of equilibrium both by the applied bias voltage and by the slow variation of a classical degree of freedom [such as m(t)]. From it we extract formulas for spin torque and charge pumping, which show that they are reciprocal effects to each other, as well as Gilbert damping in the presence of SO coupling. The NEGF-based formula for SOT naturally splits into four components, determined by their behavior (even or odd) under the time and bias voltage reversal. Their complex angular dependence is delineated and employed within Landau-Lifshitz-Gilbert simulations of magnetization dynamics in order to demonstrate capability of the predicted SOT to efficiently switch m of a perpendicularly magnetized F overlayer.