Spintronic properties of topological surface Dirac electrons with hexagonal warping

Abstract

We study spin-torque and electrical transport properties of Dirac electrons on the surface of magnetic topological insulators. Usually, surface Dirac electrons are described by a linear-dispersion model, which exhibits various peculiar features originating from the spin-velocity equivalence or spin-momentum locking. In this paper, a focus is placed on the effects of hexagonal warping (parametrized by $\ensuremath{\lambda}$) of the otherwise linear dispersion, which removes such peculiar features. Focusing on perpendicular magnetization, we calculate Gilbert damping, spin-orbit torque (SOT), and electrical conductivity, which are ``degenerate'' in the absence of hexagonal warping, $\ensuremath{\lambda}=0$, but this degeneracy is lifted in its presence, $\ensuremath{\lambda}\ensuremath{\ne}0$. The electrical conductivity was found to be greatly enhanced by $\ensuremath{\lambda}$ at large doping, whereas the other two remain moderate. Their reactive counterparts (spin renormalization, reactive SOT, and Hall conductivity), which are nonzero even in the undoped (insulating) state and also mutually degenerate at $\ensuremath{\lambda}=0$, become differentiated by $\ensuremath{\lambda}$, with the Hall conductivity remaining unchanged at the universal value. We next study current-induced torques with magnetization gradient. We first classify all possible forms allowed by symmetry, and find there are three ways of representation (SOT, spin-transfer torque, and gradient torque), which we call ``viewpoints,'' each of which forms a complete set of torques. Explicit calculation is then performed in the SOT viewpoint. While such torques are exhausted by two SOT-type ones at $\ensuremath{\lambda}=0$, examination of magnon Doppler shift and effects on domain-wall motion indicates that the spin-transfer torques (in a narrow sense) do exist even at $\ensuremath{\lambda}=0$. Finally, the current-induced Dzyaloshinskii-Moriya interaction is shown to arise starting linearly with $\ensuremath{\lambda}$, in contrast to the other effects mentioned above that start quadratically $(\ensuremath{\sim}{\ensuremath{\lambda}}^{2})$.