Spin-orbit-coupled transport and spin torque in a ferromagnetic heterostructure

Ferromagnetic heterostructures provide an ideal platform to explore the nature of spin-orbit torques arising from the interplay mediated by itinerant electrons between a Rashba-type spin-orbit coupling and a ferromagnetic exchange interaction. For such a prototypic system, we develop a set of coupled diffusion equations to describe the diffusive spin dynamics and spin-orbit torques. We characterize the spin torque and its two prominent—out-of-plane and in-plane—components for a wide range of relative strength between the Rashba coupling and ferromagnetic exchange. The symmetry and angular dependence of the spin torque emerging from our simple Rashba model is in an agreement with experiments. The spin diffusion equation can be generalized to incorporate dynamic effects such as spin pumping and magnetic damping.

Figure 1

Schematic view of the cross section of a typical ferromagnetic heterostructure that accommodates both a Rashba spin-orbit coupling and an exchange interaction. An ultrathin ferromagnetic metal film (e.g., Co) is sandwiched between an oxide (e.g., AlO${}_x$) and a heavy metal layer (e.g., Pt or Ta). A charge current is injected into the ferromagnetic layer along the $\widehat{\mathit{x}}$ direction. The dashed red arrow points to the direction of the effective Rashba field. We shall note that the system is not isolated but connected to an external source and drain.

Figure 2

Spatial profile of the nonequilibrium spin density $S_z$ (a) and (c) and $S_y$ (b) and (d) for various values of the Rashba constant. The width of the wire is $L=50$ nm. The magnetization direction is along the $\widehat{\mathit{x}}$ axis. Other parameters are momentum relaxation time $\tau ={10}^{-15}$ s, exchange splitting ${\tau }_{\Delta }={10}^{-14}$ s, spin relaxation time ${\tau }_{sf}={10}^{-12}$ s, and the Fermi vector $k_F=4.3{\mathrm{nm}}^{-1}$.

Figure 3

Magnitude of the out-of-plane torque $T_{\perp }$ (a) and in-plane torque $T_{\parallel }$ (b) as a function of Rashba constant for various exchange splitting. Other parameters are the same as in Fig. 2.