Spin-Orbit Torques beyond the Spin-Diffusion Model in Ferromagnet/Normal-Metal/Ferromagnet Trilayers*

Spin-orbit torques in Co/Cu/Co, Co/Pt/Co, and Py/Cu/Py trilayers are studied using the first-principles non-equilibrium Green’s function method with supercell disorder averaging. Trilayers with a Cu spacer exhibit strong current-in-plane giant magnetoresistance, and the torques exhibit features that can not be captured by the spin-diffusion model. In the parallel configuration there is strong disorder-dependent dampinglike torque which can greatly exceed the torque in bilayers with Pt. This torque is strongly reduced in the antiparallel configuration. We also consider the case where the magnetizations in the two layers are orthogonal to each other. In addition to dampinglike and fieldlike torque components, we find a new torque with the angular dependence (s·m₂)m₁ where s=E×z while m₁ and m₂ are the magnetizations in the spin-orbit source and detector layer, respectively. For further insight, we develop a semiclassical model based on the Boltzmann equation. Numerical calculations show that spin torques on one ferromagnetic layer are modulated by the other through interlayer scattering. Thus, in contrast to ferromagnet/heavy-metal bilayers, ferromagnetic trilayers can exhibit unconventional torques that can not be captured by the spin-diffusion model.