Spin-orbit torques due to warped topological insulator surface states with an in-plane magnetization

We investigate the extrinsic spin-orbit torque (SOT) on the surface of topological insulators (TIs), which are characterized by two-dimensional warped Dirac surface states, in the presence of an in-plane magnetization. The interplay between extrinsic spin-orbit scattering and the in-plane magnetization results in a net spin density leading to a SOT. Previous theory suggested that the SOT could only be generated by an out-of-plane magnetic field component, and any in-plane magnetic contribution could be gauged away. However, we demonstrate theoretically that with an in-plane magnetization, the SOT can be finite in TIs due to extrinsic spin-orbit scattering. In the case of a TI model with a linear dispersion relation, the skew scattering term is zero, and the extrinsic spin-orbit scattering influences the side-jump scattering, leading to a finite SOT in TIs. However, when considering the warping term, finite intrinsic and skew scattering terms will arise, in addition to modifications to other scattering terms. We further show that the SOT depends on the azimuthal angles of the magnetization and an external electric field. By adjusting the extrinsic spin-orbit strength, Fermi energy, magnetization strength and warping strength, the resulting SOTs can be maximized. These findings shed light on the interplay between spin-orbit coupling and magnetization in TIs, offering insights into the control and manipulation of spin currents in these systems.