Spin-torque generation in topological insulator based heterostructures

Heterostructures utilizing topological insulators exhibit a remarkable spin-torque efficiency. However, the exact origin of the strong torque, in particular whether it stems from the spin-momentum locking of the topological surface states or rather from spin-Hall physics of the topological-insulator bulk, remains unclear. Here, we explore a mechanism of spin-torque generation purely based on the topological surface states. We consider topological-insulator-based bilayers involving ferromagnetic metal (TI/FM) and magnetically doped topological insulators (TI/mdTI), respectively. By ascribing the key theoretical differences between the two setups to location and number of active surface states, we describe both setups within the same framework of spin diffusion of the nonequilibrium spin density of the topological surface states. For the TI/FM bilayer, we find large spin-torque efficiencies of roughly equal magnitude for both in-plane and out-of-plane spin torques. For the TI/mdTI bilayer, we elucidate the dominance of the spin-transfer-like torque. However, we cannot explain the orders of magnitude enhancement reported. Nevertheless, our model gives an intuitive picture of spin-torque generation in topological-insulator-based bilayers and provides theoretical constraints on spin-torque generation due to topological surface states.

Figure 1

The heterostructures we consider in this work: (a) TI / FM bilayer [1, 2] with a topological surface state at the interface and (b) TI/magnetically doped TI bilayer [3] with surface states at the two opposite surfaces (indicated in red). The current in both cases runs in $x$ direction and the in-plane magnetization $\vec{M}=M\vec{m}$ is along the in-plane diagonal.

Figure 2

Spin accumulation in the ferromagnet ($d=8\mathrm{nm}$) as a function of distance $z$ from the TI/FM boundary, where the solid (dashed) line denotes $S_{\perp }$ ($S_z$). For these plots, we used a spin decoherence length of ${\lambda }_{\varphi }=1\mathrm{nm}$ and the spin diffusion length of Permalloy ${\lambda }_{\mathrm{sf}}=5\mathrm{nm}$ [19]. Green (red) curves correspond to a spin-precession length ${\lambda }_{\mathrm{J}}=1\mathrm{nm}$ (${\lambda }_{\mathrm{J}}=0.5\mathrm{nm}$).

Figure 3

Integrated torque as a function of the FM thickness $d$. We again set ${\lambda }_{\mathrm{sf}}=5\mathrm{nm}$, and the solid (dashed) lines denote the in-plane (out-of-plane) torque.

Figure 4

The two torque components as a function of the TI thickness $d_1$ for ${\vec{S}}_1=-{\vec{S}}_2$ for ${\lambda }_{\mathrm{J}}={\lambda }_{\varphi }=1\mathrm{nm}$ (in the mdTI) and ${\lambda }_{\mathrm{sf}}=5\mathrm{nm}$ (on both sides) and $d_2=6\mathrm{nm}$. The solid (dashed) line denotes the in-plane (out-of-plane) torque. Panel (b) shows the two components for fixed $d_1=3\mathrm{nm}$ [gray bar in (a)] as a function of the ratio $|{\vec{S}}_1|/|{\vec{S}}_2|$ for $|{\vec{S}}_1|+|{\vec{S}}_2|$ fixed.