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.

自旋轨道扭矩 (SOT) 在普通金属/铁磁体双层中得到了最广泛的研究，其中普通金属的自旋霍尔效应是自旋电流的主要来源。最近，发现铁磁体还通过自旋轨道耦合用作自旋电流源。在这项工作中，我们从理论上探讨作用于铁磁SOT ₂在铁磁₁ /正常金属/铁磁_2层三层膜，它是由铁磁体的自旋霍尔和自旋交换作用造成₁. 我们的结果提供了三层中 SOT 的分析表达式，这可能有助于量化铁磁体的自旋霍尔和自旋交换效应，也可用于设计和解释 SOT 实验，其中铁磁体用作自旋电流源而不是正常金属。