Current-induced torques and interfacial spin-orbit coupling

In bilayer systems consisting of an ultrathin ferromagnetic layer adjacent to a metal with strong spin-orbit coupling, an applied in-plane current induces torques on the magnetization. The torques that arise from spin-orbit coupling are of particular interest. Here we use first-principles methods to calculate the current-induced torque in a Pt-Co bilayer to help determine the underlying mechanism. We focus exclusively on the analog to the Rashba torque, and do not consider the spin Hall effect. The details of the torque depend strongly on the layer thicknesses and the interface structure, providing an explanation for the wide variation in results found by different groups. The torque depends on the magnetization direction in a way similar to that found for a simple Rashba model. Artificially turning off the exchange spin splitting and separately the spin-orbit coupling potential in the Pt shows that the primary source of the “fieldlike” torque is a proximate spin-orbit effect on the Co layer induced by the strong spin-orbit coupling in the Pt.

Figure 1

(a) System geometry for an ideal interface. (b) Example of an alloyed interface. Black (gray) dots represent Pt (Co), and the larger dots are in a layer above the smaller dots. (c) Spherical coordinate system used angle-dependent torque calculation.

Figure 2

Band structure of bulk Pt. Solid lines: with spin-orbit coupling, dotted lines: without spin-orbit coupling.

Figure 3

Magnetic moment versus layer for (a) 8 layers of Pt (P) and 2 layers of Co (C), and (b) 8 layers of Pt and 2 alloyed (A) layers and 1 layer Co [see Fig. 2 for the alloy coordinates]. The four atomic moments in each layer are plotted individually (since the disordered interface has inequivalent atoms in each layer), where filled (open) symbols represent Pt (Co). The moments are oriented in the $\widehat{z}$ direction.

Figure 4

Layer resolved effective field per current density for (a) 1, 2, and 3 ML of Co with an ideal interface, and (b) 2 alloyed interfaces.

Figure 5

The angular dependence of the effective field (solid blue) and odd “damping field” (dashed red) magnitude per current density. The left panel is for 2 ML of Co with an ideal interface, while the right panel is for 2 layers of alloy + 1 layer Co. The angles are conventional spherical coordinates for the coordinate system of Fig. 1.

Figure 6

(a) Plot of states at the Fermi level versus wave vector. (b) Zoom in of states near the zone center. The dark blue (light red) color indicates states with positive (negative) group velocity in the $x$ direction. The dot size of each state is proportional to the magnitude of the state's $z$ component of spin on the interface Co layer. (c) The $x$ component of spin on the interface Co layer. The magnitude of each state's spin is proportional to the dot size, and the colors (shading) indicate the sign of $S_x$. (d) The same plot for the $y$ component of each state's spin. The torque on the interface Co layer contributed by each state is proportional to the in-plane spin component for that state.

Figure 7

Layer resolved torque for 8 layer Pt and 2 layer Co system. “Default” refers to the system with full exchange and spin-orbit coupling. The other torques are calculated by removing the specified potential from the Pt atoms, as described in the text.

Figure 8

Similar plot as Fig. 7, but with an alloyed interface. The individual torques on the four atoms of each layer are shown (the torques from the bottom five Pt layers are omitted from the plot, as they vanish). Open (filled) circles represent Pt (Co).