Angular dependence of spin-orbit spin-transfer torques

In ferromagnet/heavy-metal bilayers, an in-plane current gives rise to spin-orbit spin-transfer torque, which is usually decomposed into fieldlike and dampinglike torques. For two-dimensional free-electron and tight-binding models with Rashba spin-orbit coupling, the fieldlike torque acquires nontrivial dependence on the magnetization direction when the Rashba spin-orbit coupling becomes comparable to the exchange interaction. This nontrivial angular dependence of the fieldlike torque is related to the Fermi surface distortion, determined by the ratio of the Rashba spin-orbit coupling to the exchange interaction. On the other hand, the dampinglike torque acquires nontrivial angular dependence when the Rashba spin-orbit coupling is comparable to or stronger than the exchange interaction. It is related to the combined effects of the Fermi surface distortion and the Fermi sea contribution. The angular dependence is consistent with experimental observations and can be important to understand magnetization dynamics induced by spin-orbit spin-transfer torques.

Figure 1

Fermi surface and spin direction for a free-electron model. (a) $r=0$ (only exchange splitting), (b) $r=\infty$ (only Rashba spin-orbit coupling and nonmagnetic), (c) comparison of Fermi surfaces (majority band) for $r=0$ and $1.0$, (d) $0.8$, (e) $1.0$, and (f) $1.2$. We assume $\widehat{\mathbf{M}}=(0,1,0),E_{\mathrm{F}}=10\text{eV},m=m_0$, and $J=1\text{eV}$. Here, $m_0$ is the free electron mass. The outer red (inner blue) Fermi surface corresponds to the majority (minority) band. Arrows are the eigendirections of spins on the Fermi surface. The coordinate system is shown on the right.

Figure 2

Polar angle $\left(\theta \right)$ dependence of fieldlike spin-orbit torque coefficient ${\tau }_{\mathrm{f}}$. Free-electron model [(a)and (b)]: (a) azimuthal angle of magnetization $\varphi =0$ and (b) $\varphi =\pi /2$. Tight-binding model [(c) and (d)]: (c) $\varphi =0$ and (d) $\varphi =\pi /2$. For both models, we use $E_F=4\text{eV},m=m_0$, and $J=1\text{eV}$. For a tight-binding model, we use $a=0.25$ nm.

Figure 3

Angular dependence of dampinglike spin-orbit torque coefficient ${\tau }_{\mathrm{d}}$. Free-electron model [(a) and (b)]: (a) polar angle dependence at $\varphi =0$ and (b) azimuthal angle dependence at $\theta =\pi /2$. Lines for $r=-1.2,-1.6$, and $-2.0$ are not clearly visible since ${\tau }_{\mathrm{d}}=0$ for these cases. Tight-binding model [(c) and (d)]: (c) polar angle dependence at $\varphi =0$ and (d) azimuthal angle dependence at $\theta =\pi /2$. Same parameters are used as in Fig. 2.