Spin-orbit torque in a thin film of the topological insulator Bi

The spin-orbit torque provides an efficient method for switching the direction of a magnetization by using an electric field. Owing to the spin-orbit coupling, when an electric field is applied, a nonequilibrium spin density is generated, which exerts a torque on the local magnetization. Here, we investigate the spin-orbit torque in a thin film of topological insulator $\text{Bi}_{2}\text{Se}_{3}$ based upon a Boltzmann equation, with proper boundary conditions, which is applicable from the ballistic regime to the diffusive regime. It is shown that due to the spin-momentum interlocking of the electron surface states, the magnitude of the field-like torque is simply in linear proportion to the longitudinal electrical current. For a fixed electric field, the spin-orbit torque is proportional to the sample length in the ballistic limit, and saturates to a constant in the diffusive limit. The dependence of the torque on the magnetization direction and exchange coupling strength is also studied. Our theory may offer useful guidance for experimental investigations of the spin-orbit torque in finite-size systems