Large antidamping-like spin-orbit torque driven by spin-flip reflection mechanism on the surface of a topological insulator

Motivated by recent experimental observations of spin-orbit torque (SOT) on the magnetization $\vec{m}$ of a ferromagnetic (F) overlayer attached to the surface of a three-dimensional topological insulator, we analyze conditions which can maximize its magnitude. We predict that tuning the Fermi energy close to the Dirac point can generate a large antidamping-like SOT driven by low injected charge current. The large values of SOT are spatially localized around the boundaries separating the region of TI metallic surface with and without proximity-induced exchange field due to the F overlayer. They originate from spin-momentum-locked Dirac electrons reflecting off these boundaries while their spins are flipped. Our analysis is based on adiabatic expansion (to first order in $\partial \vec{m}/\partial t$) of time-dependent nonequilibrium Green functions 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 $\vec{m}(t)$]. When applied to SOT, this framework determines its components based on their behavior (even or odd) under the time and bias voltage reversal.