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We investigate the extrinsic spin-orbit torque (SOT) on the surface oftopological insulators (TIs), which are characterized by two-dimensional warpedDirac surface states, in the presence of an in-plane magnetization. Theinterplay between extrinsic spin-orbit scattering and the in-planemagnetization results in a net spin density leading to a SOT. Previous theorysuggested that the SOT could only be generated by an out-of-plane magneticfield component, and any in-plane magnetic contribution could be gauged away.However, we demonstrate theoretically that with an in-plane magnetization, theSOT can be finite in TIs due to extrinsic spin-orbit scattering. In the case ofa 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 tomodifications to other scattering terms. We further show that the SOT dependson the azimuthal angles of the magnetization and an external electric field. Byadjusting the extrinsic spin-orbit strength, Fermi energy, magnetizationstrength and warping strength, the resulting SOTs can be maximized. Thesefindings shed light on the interplay between spin-orbit coupling andmagnetization in TIs, offering insights into the control and manipulation ofspin currents in these systems.

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