Spin-to-charge conversion in lateral and vertical topological-insulator/ferromagnet heterostructures with microwave-driven precessing magnetization

Using the charge-conserving Floquet-Green function approach to open quantum systems driven by an external time-periodic potential, we analyze how spin current pumped by the precessing magnetization of a ferromagnetic (F) layer is injected laterally into the interface with strong spin-orbit coupling (SOC) and converted into charge current flowing in the same direction. In the case of a metallic interface with the Rashba SOC used in recent experiments [J. C. R. Sánchez, L. Vila, G. Desfonds, S. Gambarelli, J. P. Attané, J. M. De Teresa, C. Magén, and A. Fert, Nat. Commun. 4, 2944 (2013)], both spin $I^{S_{\alpha }}$ and charge $I$ current flow within the interface where $I/I^{S_{\alpha }}\simeq$ 2–8% (depending on the precession cone angle), while for a F/topological-insulator (F/TI) interface employed in related experiments [Y. Shiomi, K. Nomura, Y. Kajiwara, K. Eto, M. Novak, K. Segawa, Y. Ando, and E. Saitoh, arXiv:1312.7091] the conversion efficiency is greatly enhanced $(I/I^{S_{\alpha }}\simeq$ 40–60%) due to perfect spin-momentum locking on the surface of a TI. The spin-to-charge conversion occurs also when spin current is pumped vertically through the F/TI interface with smaller efficiency $\left(I/I^{S_{\alpha }}\sim 0.001\%\right)$, but with the charge current signal being sensitive to whether the Dirac fermions at the interface are massive or massless.

Figure 1

Schematic of (a) lateral and (b) vertical F/TI heterostructures whose precessing magnetization, driven by the absorption of microwaves of frequency $\omega$ under the FMR conditions, pumps pure spin current along the $z$ or $x$ axis, respectively, in the absence of any dc bias voltage between the semi-infinite N leads. The strong interfacial SOC converts pumped spins into charge current flowing along the $x$ axis in both setups, which is measured as the voltage signal $V_{\mathrm{pump}}$ in an open circuit. Besides lateral heterostructures of an F/TI type in panel (a), we also study lateral F/2DEG heterostructures with conventional Rashba SOC at the interface for comparison. The unit vector ${\mathbf{n}}_M$ specifies the axis around which magnetization $\mathbf{m}\left(t\right)$ is precessing, while the vector ${\mathbf{n}}_{\mathrm{TI}}$ is perpendicular to quintuple layers (QLs) comprising the 3D TI slab [e.g., panel (b) illustrates the case ${\mathbf{n}}_{\mathrm{TI}}=(1,1,0)]$.

Figure 2

(a), (c) The angular dependence of the dc pumping voltage in lateral F/TI and F/2DEG heterostructures illustrated in Fig. 1 for different orientations of the axis ${\mathbf{n}}_M$ around which magnetization is precessing with a cone angle $\theta$. Panels (b) and (d) show the spin current $I^{S_{\alpha }}$ which accompanies the charge current $I=V_{\mathrm{pump}}G$ in (a) and (c), respectively, where both $I^{S_{\alpha }}$ and $I$ flow along the $x$ axis in Fig. 1. Note that charge current in panels (a) and (c) is nonzero only when magnetization is precessing around the $y$ axis in Fig. 1.

Figure 3

Efficiency of spin-to-charge conversion in lateral F/TI and F/2DEG heterostructures is quantified by the ratio of charge $I$ to spin $I^{S_{\alpha }}$ currents from Fig. 2 for magnetization precessing around the $y$ axis $[{\mathbf{n}}_M=(0,1,0)]$ in Fig. 1.

Figure 4

(a), (c) The angular dependence of the dc pumping voltage in F/TI vertical heterostructures illustrated in Fig. 1 whose magnetization is precessing around the $z$ axis, ${\mathbf{n}}_M=(0,0,1)$. Panels (b) and (d) show the spin current $I^{S_{\alpha }}$ which accompanies the charge current $I=V_{\mathrm{pump}}G$ in (a) and (c), respectively, where both $I^{S_{\alpha }}$ and $I$ flow along the $x$ axis in Fig. 1. The QLs of the 3D TI slab are oriented perpendicular to ${\mathbf{n}}_{\mathrm{TI}}=(1,0,0)$ in panels (a) and (b), or perpendicular to ${\mathbf{n}}_{\mathrm{TI}}=(0,1,1)$ in panels (c) and (d).

Figure 5

The TAMR of F/TI vertical heterostructures illustrated in Fig. 1 for a gapless $({\Delta }_{\mathrm{surf}}\equiv 0$ for dotted lines) or gapped $({\Delta }_{\mathrm{surf}}\ne 0$ for solid lines) surface of the TI that is in direct contact with the F layer. The QLs of the 3D TI slab are oriented perpendicular to ${\mathbf{n}}_{\mathrm{TI}}=(1,0,0)$ for curves with the left ordinate, or perpendicular to ${\mathbf{n}}_{\mathrm{TI}}=(0,1,1)$ for curves with the right ordinate.

Figure 6

The local DOS $g(E,k_y,k_z=0)$ on the surface ML of the TI slab in contact with the F layer within the vertical heterostructure illustrated in Fig. 1, as well as on the adjacent second and third MLs. In the top row of panels, the TI slab is weakly (F/TI hopping is $J=0.25$ eV) coupled to the F layer, and in the bottom row of panels the coupling (F/TI hopping is $J=0.4$ eV) is stronger. The static magnetization of the F layer is perpendicular to the F/TI interface, and it is assumed to induce an energy gap on the surface ML of TI via the magnetic proximity effect. The QLs of the TI slab are oriented perpendicular to ${\mathbf{n}}_{\mathrm{TI}}=(1,0,0)$.