Spin-torque current induced by topological Berry phase in a two-dimensional system with generic k -linear spin-orbit interaction

The Berry phase on the Fermi surface and its influence on the conserved spin current in a two-dimensional system with generic k-linear spin-orbit interaction are investigated. We calculate the response of the effective conserved spin current to the applied electric field, which is composed of conventional and spin-torque currents by using the Kubo formula. We find that the conventional spin current is not determined by the Berry-phase effect. Remarkably, the spin-torque Hall current is found to be proportional to the Berry phase, and the longitudinal spin-torque current vanishes because of the Berry-phase effect. When the k-linear spin-orbit interaction dominates the system, the Berry phase on the Fermi surface maintains two invariant properties. One is that the magnitude of the spin-torque current protected by the Berry phase is unchanged by a small fluctuation in energy dispersion. The other one is that the change in the direction of the applied electric field does not change the magnitude of the spin-torque current even if the energy dispersion is not spherically symmetric, i.e., the Berry-phase effect has no dependence on the two-dimensional material orientation. The spin-torque current is a universal value for all k-linear systems, such as Rashba, Dresselhaus, and Rashba-Dresselhaus systems. The topological number attributed to the Berry phase on the Fermi surface represents the phase of the orbital chirality of spin in the k-linear system. The change in the topological number results in a phase transition in which the orbital chirality of spins s sub(z) and -s sub(z) is exchanged. We found that the spin-torque current can be experimentally measured.