Theory of in-plane current induced spin torque in metal/ferromagnet bilayers

Abstract

Using a semiclassical approach that simultaneously incorporates the spin Hall effect (SHE), spin diffusion, quantum well states, and interface spin-orbit coupling (SOC), we address the interplay of these mechanisms as the origin of the spin-orbit torque (SOT) induced by in-plane currents, as observed in the normal metal/ferromagnetic metal bilayer thin films. Focusing on the bilayers with a ferromagnet much thinner than its spin diffusion length, such as Pt/Co with ∼10 nm thickness, our approach addresses simultaneously the two contributions to the SOT, namely the spin-transfer torque (SHE-STT) due to SHE-induced spin injection, and the inverse spin Galvanic effect spin-orbit torque (ISGE-SOT) due to SOC-induced spin accumulation. The SOC produces an effective magnetic field at the interface, hence it modifies the angular momentum conservation expected for the SHE-STT. The SHE-induced spin voltage and the interface spin current are mutually dependent and, hence, are solved in a self-consistent manner. The result suggests that the SHE-STT and ISGE-SOT are of the same order of magnitude, and the spin transport mediated by the quantum well states may be an important mechanism for the experimentally observed rapid variation of the SOT with respect to the thickness of the ferromagnet.

Spin-orbit torques from interfacial spin-orbit coupling for various interfaces

“…Ref. [43] reports the solution of the drift-diffusion equation in the non-magnetic layer to capture the spin Hall effect coupled to a quantum mechanical solution in the ferromagnetic layer to capture the effects of interfacial spin-orbit coupling.…”

Spin torque and persistent currents caused by percolation of topological surface states

“…2, suggesting that the percolation of the surface state is crucial to the magnitude of the spin torque. This predominantly field-like torque is similar to that occurs in the 2D version of this problem, 34 which has been attributed to the real wave functions of the percolated surface states that cannot accumulate a spindependent phase, unlike the spin-transfer torque in usual metallic heterostructures 51,52 and spin Hall systems 53,54 where the spin polarized plane waves accumulates a spindependent phase that eventually yields a damping-like torque. At a typical external electric current j c ∼ 10 11 A/m 2 , the spin polarization is basically the numerical values of χ b multiplied by GHz, which is close to that observed experimentally.…”

Persistent currents and spin torque caused by percolated quantum spin Hall state

“…If the magnetization lies in the xy-plane, then only the χ z F component is nonzero. This is very different from the STT in usual metallic heterostructures 32,33 or that induced by the spin Hall effect, 34,35 where the propagation of plane waves can accumulate a phase difference between spin up and down components that eventually contributes to a dampinglike torque. In contrast, the percolated edge state and the FMM quantum well state wave functions are completely real and hence do not support such a spin-dependent phase, rendering an entirely field-like torque (we neglect other complications such as spin-orbit torque 36,37 and spin relaxation).…”