Dirac spin-orbit torques and charge pumping at the surface of topological insulators

We address the nature of spin-orbit torques at the magnetic surfaces of topological insulators using the linear-response theory. We find that the so-called Dirac torques in such systems possess a different symmetry compared to their Rashba counterpart, as well as a high anisotropy as a function of the magnetization direction. In particular, the damping torque vanishes when the magnetization lies in the plane of the topological-insulator surface. We also show that the Onsager reciprocal of the spin-orbit torque, the charge pumping, induces an enhanced anisotropic damping. Via a macrospin model, we numerically demonstrate that these features have important consequences in terms of magnetization switching.

Figure 1

(a) Bilayer consisting of a TI substrate with a ferromagnetic overlayer. The black arrows represent the local magnetic moment with overall magnetization direction $\mathbf{m}$. The electric field $\mathbf{E}$ is applied along $\mathbf{x}$ and generates two spin-density components, ${\mathbf{S}}_{\perp }\sim \mathbf{z}\times \mathbf{E}$ (red arrow) and ${\mathbf{S}}_{\parallel }\sim m_z\mathbf{E}$ (green arrow). (b) Schematics of the two-dimensional band structure at the magnetic surface of the TI when $\mathbf{m}=\mathbf{z}$. The red (blue) arrows represent the spin direction in the conduction (valence) band, and the perpendicular magnetization opens a gap $2\mathrm{\Delta }$.

Figure 2

Calculated switching phase diagram with an applied in-plane field along $\mathbf{x}$ for a current-induced (a) Dirac torque and (b) spin Hall torque (retrieve results from Ref. [8])