Spin-thermoelectric transport induced by interactions and spin-flip processes in two-dimensional topological insulators

We consider thermoelectric transport properties of the edge states of a two-dimensional topological insulator in a double quantum point contact geometry coupled to two thermally biased reservoirs. Both spin-preserving and spin-flipping tunneling processes between opposite edges are analyzed in the presence of electron-electron interactions. We demonstrate that the simultaneous presence of spin-flipping processes and interactions gives rise to a finite longitudinal spin current. Moreover, its sign and amplitude can be tuned by means of gate voltages with the possibility to generate a pure spin current, with a vanishing charge current.

Figure 1

Scheme of the setup. A 2D TI in a two-terminal configuration with fixed chemical potential ${\mu }_R={\mu }_L=\mu$ and a finite thermal gradient $T_L\ne T_R$ is considered. On each edge, electrons with spin $\uparrow$ (solid line) and spin $\downarrow$ (dashed line) counterpropagate. Two quantum point contacts pinch the two edges, allowing for electron-tunneling events. Gate voltages $V_t$ and $V_b$ locally modify electron density in the region between the two QPCs.

Figure 2

Dispersion relation of edge states shifted by gate voltages inside the tunneling region. (a) $p$ tunneling: two $R\uparrow$ electrons move to the $L\uparrow$ branch, changing their chirality; the transferred momentum depends on energy. (b) $f$ tunneling: two $R\uparrow$ electrons move to the $R\downarrow$ branch, changing their spin; the transferred momentum does not depend on energy.

Figure 3

Two possible paths for a $p$-tunneling event. A $R\uparrow$ electron (solid red line) can tunnel to $L\uparrow$ (solid blue line) through left (black solid line) or right (black dashed line) QPC.

Figure 4

Average spin current $\langle I_s\rangle$ in units of ${10}^{-6}{\left|{\lambda }_f\right|}^2/\left(8\pi \right)$ as a function of the parameter $\mathcal{K}$. Different curves correspond to the different values $T_R=0.4,0.6,0.8$, and $1.0$ K, while $T_L$ is fixed at $T_L=0.1$ K. Parameters $\eta$ and $\alpha$ are set to $\eta =8\pi$ and $\alpha =0.4$. The chemical potential and the high-energy cutoff are $\mu =100$ K and ${\omega }_c=200$ K.

Figure 5

Density plot of (a) charge current $〈I_c〉$ and (b) spin current $〈I_s〉$ as a function of $\eta$ $(y$ axis) and $\mathcal{K}$ $(x$ axis) for a double QPC geometry, in units of ${10}^{-4}e{\left|{\lambda }_p\right|}^2/\left(2\pi \right)$ and ${10}^{-6}{\left|{\lambda }_f\right|}^2/\left(8\pi \right)$, respectively. Parameter $\alpha$ is set to $\alpha =0.4$. The two temperatures are $T_L=0.1$ K and $T_R=1$ K. The chemical potential and the high-energy cutoff are $\mu =100$ K and ${\omega }_c=200$ K.

Figure 6

The plot shows the ratio $\langle I_s\rangle /\langle I_c\rangle$ as a function of $\eta$, in units of $|{\lambda }_f{|}^2/\left(4e\right|{\lambda }_p{|}^2)$. Different curves refer to different values of $\alpha =0.1$, 0.25, and 0.4. Temperatures are set to $T_L=0.1$ K and $T_R=1$ K, while the interaction strength is $\mathcal{K}=0.6$. The chemical potential and the high-energy cutoff are $\mu =100$ K and ${\omega }_c=200$ K.

Figure 7

Average spin current $\langle I_s\rangle$ in units of ${10}^{-6}{\left|{\lambda }_f\right|}^2/\left(8\pi \right)$ as a function of the parameter $\alpha$. Different curves refer to different values of $\eta =3\pi /2,3\pi$, and $9\pi /2$. In all cases $\langle I_c\rangle =0$ with a finite pure spin current. Other parameters are the same as in Fig. 6.