Spin-orbit torques from intrinsic spin-orbit couplings in a periodically buckled honeycomb lattice

Abstract

In this paper, we study the spin-orbit torques (SOTs) originated from the spin-orbit coupling (SOC) of intrinsic type in a periodically buckled honeycomb nanoribbon such as silicene. Using the Green function formalism with density matrix expressions, we analyze the SOT contributions from the Fermi-level and -sea electrons. We find that the intrinsic-SOC-induced inverse spin galvanic effect generates spin accumulations perpendicular to the honeycomb plane. The anti-damping torque results purely from the Fermi-level contribution, while the field-like torque from both the Fermi-level and -sea contributions. At zero bias voltage, the SOT is symmetric with respect to the Fermi energy (i.e., an even function of the Fermi energy), whereas the presence of bias further introduces the anti-symmetric torque components. When the ferromagnet magnetization lies in the honeycomb plane, all Fermi-sea contributions disappear. The maximum SOT among different Fermi energies is also inspected. For large enough in-plane magnetization compared to the staggered potential, the field-like torque has a maximum at zero Fermi energy. When the magnetization is smaller than some value characterized by the intrinsic SOC, the maximum of the anti-damping torque occurs at the cross point of the two Dirac linear bands in the leads. More importantly, we find that the anti-damping torques responsible for magnetic switching as well as the torkance require staggered potential from an applied out-of-plane electric field, i.e., lattice buckling is essential to the switching. When the electric field is reversed, the anti-damping torque and the torkance are also reversed. Accordingly, full electric control of the SOTs can be realized in this buckled system.