Spin-orbit torques from topological insulator surface states: Effect of extrinsic spin-orbit scattering on an out-of-plane magnetization

The origins of the spin-orbit torque (SOT) at ferromagnet/topological insulator (FM/TI) interfaces are incompletely understood. Theory has overwhelmingly focussed on the Edelstein effect due to the surface states in the presence of a scalar scattering potential. We investigate here the contribution to the SOT due to extrinsic spin-orbit scattering of the surface states, focusing on the case of an out-of-plane magnetization. We show that spin-orbit scattering brings about a sizable renormalization of the field-like SOT, which exceeds 20$\mathrm{\%}$ at larger strengths of the extrinsic spin-orbit parameter. The resulting SOT exhibits a maximum as a function of the Fermi energy, magnetization, and extrinsic spin-orbit strength. The field-like SOT decreases with increasing disorder strength while the damping-like SOT is independent of the impurity density. With experimental observation in mind we also determine the role of extrinsic spin-orbit scattering on the anomalous Hall effect. Our results suggest extrinsic spin-orbit scattering is a significant contributor to the surface SOT stemming from the Edelstein effect when the magnetization is out of the plane.