Current induced torques and interfacial spin-orbit coupling: Semiclassical modeling

In bilayer nanowires consisting of a ferromagnetic layer and a nonmagnetic layer with strong spin-orbit coupling, currents create torques on the magnetization beyond those found in simple ferromagnetic nanowires. The resulting magnetic dynamics appear to require torques that can be separated into two terms, dampinglike and fieldlike. The dampinglike torque is typically derived from models describing the bulk spin Hall effect and the spin transfer torque, and the fieldlike torque is typically derived from a Rashba model describing interfacial spin-orbit coupling. We derive a model based on the Boltzmann equation that unifies these approaches. We also consider an approximation to the Boltzmann equation, the drift-diffusion model, that qualitatively reproduces the behavior, but quantitatively differs in some regimes. We show that the Boltzmann equation with physically reasonable parameters can match the torques for any particular sample, but in some cases, it fails to describe the experimentally observed thickness dependencies.

Figure 1

Currents, spin currents, and spin accumulations. The left panels [(a), (d), and (g)] show the current density (heavy lines) $j_x$, which is flowing in the plane of the sample and the spin current $Q_{xz}$ (lighter lines), flowing in the $x$ direction with spins aligned with the magnetization in the $z$ direction. The dotted lines indicate the bulk values. All currents and spin currents are dimensionless; currents are scaled by the bulk current in the nonmagnet and spin currents are scaled by the bulk current in the nonmagnet and an additional factor of $\hslash /2e$. The spin densities are scaled by the same two factors and $v_{\mathrm{F}}$. The middle panels [(b), (e), and (h)] show the spin currents $Q_{zy}$ (heavy lines) and $Q_{zx}$ (lighter lines), flowing perpendicular to the layers ($z$ direction) with spins pointing perpendicular to the magnetization, i.e., the $x$ and $y$ directions. The right panels [(c), (f), and (i)] show the accumulation of spin perpendicular to the magnetization $s_y$ (heavy lines) and $s_x$ (lighter lines). The top panels [(a)–(c)] are for the case in which there is no interfacial spin-orbit coupling, the bottom panels [(g)–(i)] for the case with interfacial spin-orbit coupling $u_{\mathrm{R}}=0.04$ and no spin Hall effect in the nonmagnet, and the middle panels [(d)–(f)] for the case when both are present. In (h) and (i) the spin accumulations have been scaled by the indicated factors.

Figure 2

Average current density in the nonmagnetic layer. For three thicknesses of the ferromagnetic layer, the average current density in the nonmagnetic layer is shown scaled by the bulk value as a function of the thickness of the nonmagnetic layer.

Figure 3

Torques as a function of nonmagnetic layer thickness. The solid curves are the full Boltzmann equation calculation and the dashed curves give the analytic approximation based on the drift-diffusion model and the circuit theory. The more negative curves show the fieldlike torque ${\tau }_{\mathrm{f}}$ and those closer to zero show the dampinglike torque ${\tau }_{\mathrm{d}}$. For both torques, the Boltzmann results have been calculated for three different mean free paths (labeled on the dampinglike torques) in the ferromagnet.

Figure 4

Scaled torques as a function of the interfacial Rashba interaction. The solid curves are calculated with a bulk spin Hall effect and the dotted curves without. The nonmagnetic layer is 6 nm thick and the ferromagnetic layer is 4 nm. The dampinglike and fieldlike torques are labeled with ${\tau }_{\mathrm{d}}$ and ${\tau }_{\mathrm{f}}$, respectively. The jitter is due to numerical instabilities.

Figure 5

Dimensionless torque components as a function of the thickness of the nonmagnetic layer. (a) The torques in the absence of the Rashba contribution from the interfacial spin-orbit coupling. (c) The torques in the absence of the spin Hall effect in the nonmagnet. (b) The torques with both present. In each part, the solid lines show the dampinglike torque and the dotted lines the fieldlike torque.

Figure 6

Dimensionless torque components as a function of the thickness of the ferromagnetic layer. (a) The torques in the absence of the Rashba contribution from the interfacial spin-orbit coupling. (c) The torques in the absence of the spin Hall effect in the nonmagnet. (b) The torques with both present. In each part, the solid lines show the dampinglike torque and the dotted lines the fieldlike torque.