Non-linear spin current in the surface states of topological insulators

Spin Hall effect is one of the primary sources of pure spin current in spintronic devices, observed in spin-orbit coupled materials. In this work, we investigate linear and non-linear pure spin currents in the hexagonally warped surface states of topological insulators like $\rm Bi_2Te_3$, using the formalism of conserved spin transport. The hexagonal warping effect enables the application of this formalism by providing a well-defined Berry curvature without breaking the time-reversal symmetry. The absence of in-plane spin currents in the linear response motivates the extension of the formalism to the non-linear regime, by combining the perturbation of Bloch states with the Boltzmann transport equation. The second-order response exhibits both intrinsic and extrinsic in-plane spin currents induced by the spin-weighted Berry curvature, Berry connection polarizability (BCP) dipole and Berry curvature dipole (BCD) respectively. The band anisotropy due to the $C_{3v}$ symmetry of the material ($\rm Bi_2Te_3$) propagates through the spin textures and the band geometric quantities, resulting in the selection of certain components of the spin conductivity tensors.