Spin-Orbit Coupling Induced Spin-Transfer Torque and Current Polarization in Topological-Insulator/Ferromagnet Vertical Heterostructures

We predict an unconventional spin-transfer torque (STT) acting on the magnetization of a free ferromagnetic (F) layer within N/TI/F vertical heterostructures, which originates from strong spin-orbit coupling on the surface of a three-dimensional topological insulator (TI), as well as from charge current becoming spin polarized in the direction of transport as it flows perpendicularly from the normal metal (N) across the bulk of the TI layer. The STT vector has both in-plane and perpendicular components that are comparable in magnitude to conventional torque in ${\mathrm{F}}^{\prime }/\mathrm{I}/\mathrm{F}$ (where I stands for insulator) magnetic tunnel junctions, while not requiring additional spin-polarizing ${\mathrm{F}}^{\prime }$ layer with fixed magnetization, which makes it advantageous for spintronics applications. Our principal formal result is a derivation of the nonequilibrium Green function-based formula and the corresponding gauge-invariant nonequilibrium density matrix, which makes it possible to analyze the components of the STT vector in the presence of arbitrary strong spin-orbit coupling either in the bulk or at the interface of the free F layer.

Figure 1

Schematic view of the topological insulator-based vertical heterostructure operated by spin-transfer torque. The junction contains a single F layer of finite thickness with free magnetization $\mathbf{m}$, and the N leads are semi-infinite. We assume that each layer is composed of atomic monolayers (modeled on an infinite square tight-binding lattice).

Figure 2

(a) The angular dependence of torque components, ${\mathbf{T}}_{\parallel }={\tau }_{\parallel }\mathbf{m}\times (\mathbf{m}\times {\mathbf{e}}_z)$ and ${\mathbf{T}}_{\perp }={\tau }_{\perp }\mathbf{m}\times {\mathbf{e}}_z$, acting on the free magnetization $\mathbf{m}$ within N/TI/F heterostructure in Fig. 1. (b) The torque components, ${\mathbf{T}}_{\parallel }={\tau }_{\parallel }\mathbf{m}\times (\mathbf{m}\times {\mathbf{m}}^{\prime })$ and ${\mathbf{T}}_{\perp }={\tau }_{\perp }\mathbf{m}\times {\mathbf{m}}^{\prime }$, acting on the free-layer magnetization $\mathbf{m}=(\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta )$ in conventional ${\mathrm{F}}^{\prime }/\mathrm{I}/\mathrm{F}$ symmetric MTJ where magnetization of the reference layer ${\mathrm{F}}^{\prime }$ is fixed at ${\mathbf{m}}^{\prime }={\mathbf{e}}_z$. (c) The torque components in N/I/F junction, defined in the same fashion as in panel (a), with the Rashba SOC of strength ${\alpha }_R/2a=0.1\mathrm{eV}$ located on the last monolayer of F which is in contact with I barrier. (d) The angular dependence of conductances for N/TI/F, ${\mathrm{F}}^{\prime }/\mathrm{I}/\mathrm{F}$ and N/I/F junctions. The bias voltage $V_b$ in all panels is sufficiently small to ensure the linear-response regime.

Figure 3

The spin-polarization vector ${\mathbf{P}}^{\mathrm{out}}=(0,0,P_z^{\mathrm{out}})$ of current [28] in the right N lead of $\mathrm{N}/\mathrm{TI}/\mathrm{N}$ junction as a function of the thickness $d_{\mathrm{TI}}$ of the 3D TI layer after unpolarized charge current is injected from the left N lead.