  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.
