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Spin-orbit torques in ferromagnet/nonmagnet/ferromagnet trilayers are studiedusing a combination of symmetry analysis, circuit theory, semiclassicalsimulations, and first-principles calculations using the non-equilibriumGreen's function method with supercell disorder averaging. We focus onunconventional processes involving the interplay between the two ferromagneticlayers, which are classified into direct and indirect mechanisms. The directmechanism involves spin current generation by one ferromagnetic layer and itssubsequent absorption by the other. In the indirect mechanism, the in-planespin-polarized current from one ferromagnetic layer ``leaks'' into the otherlayer, where it is converted into an out-of-plane spin current and reabsorbedby the original layer. The direct mechanism results in a predominantlydampinglike torque, which damps the magnetization towards a certain direction$\mathbf{s}_d$. The indirect mechanism results in a predominantly fieldliketorque with respect to a generally different direction $\mathbf{s}_f$. Similarto the current-in-plane giant magnetoresistance, the indirect mechanism is onlyactive if the thickness of the nonmagnetic spacer is smaller than or comparableto the mean-free path. Numerical calculations for a semiclassical model basedon the Boltzmann equation confirm the presence of both direct and indirectmechanisms of spin current generation. First-principles calculations revealsizeable unconventional spin-orbit torques in Co/Cu/Co, Py/Cu/Py, and Co/Pt/Cotrilayers and provide strong evidence of indirect spin current generation.

Direct and indirect spin current generation and spin-orbit torques in ferromagnet/nonmagnet/ferromagnet trilayers