Spin-orbit torques induced by spin Hall and spin swapping currents of a separate ferromagnet in a magnetic trilayer

Abstract

Spin-orbit torques (SOTs) have been investigated most widely in normal metal/ferromagnet bilayers where the spin Hall effect of normal metal is a main source of spin currents. Recently, ferromagnets are found to also serve as spin-current sources through spin-orbit coupling. In this work, we theoretically investigate SOT acting on ferromagnet2 in ferromagnet1/normal metal/ferromagnet2 trilayers, which is caused by the spin Hall and spin swapping effects of ferromagnet1. Our result provides an analytical expression of SOT in the trilayers, which may be useful for quantifying the spin Hall and spin swapping effects of ferromagnets and also for designing and interpreting SOT experiments where a ferromagnet is used as a spin-current source instead of a normal metal.