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Spin-orbit torque switching of perpendicular magnetization in ferromagnetic trilayers

In ferromagnetic trilayers, a spin-orbit-induced spin current can have a spin polarization of which direction is deviated from that for the spin Hall effect. Recently, magnetization switching in ferromagnetic trilayers has been proposed and confirmed by the experiments. In this work, we theoretically and numerically investigate the switching current required for perpendicular magnetization switching in ferromagnetic trilayers. We confirm that the tilted spin polarization enables field-free deterministic switching at a lower current than conventional spin-orbit torque or spin-transfer torque switching, offering a possibility for high-density and low-power spin-orbit torque devices. Moreover, we provide analytical expressions of the switching current for an arbitrary spin polarization direction, which will be useful to design spin-orbit torque devices and to interpret spin-orbit torque switching experiments.