All-electrical generation of spin-polarized currents in quantum spin Hall insulators

The control and generation of spin-polarized current (SPC) without magnetic materials and an external magnetic field is a big challenge in spintronics and normally requires a spin-flip mechanism. In this Rapid Communication, we show the theoretical discovery of all-electrical generation of SPC without relying on spin-flip spin-orbit coupling (SOC). We find that the SPC can be produced as long as an energy-dependent phase difference between the spin up and down electrons can be established. We verify this through quantum transport calculations on a gated stanene zigzag nanoribbon, which is a quantum spin Hall (QSH) insulator. Our calculations indicate that the transient current as well as ac conductance are significantly spin polarized, which results from the genetic phase difference between spin up and down electrons after traversing the system. Our results are robust against edge imperfections and generally valid for other QSH insulators, such as silicene and germanene, etc. These findings establish a different route for generating SPCs by purely electrical means and open the door for interesting applications of semiconductor spintronics.

Figure 1

(a) Schematic plot of a zigzag nanoribbon device with different views. Here, $E_z$ is the external electric field generated by the back-gate voltage. The source and drain electrodes extend to $x=\mp \infty$, as shown by the left/right pointing arrows. $V_{\text{in}}$ is the input pulse voltage. $a$ is the lattice constant. $l$ is the buckling height. $W$ (unit $\sqrt{3}a$) and $L$ (unit $a$) denote the width and the length of the gated region, respectively. (b)–(d) Band structures for the nanoribbon with $W=12$. (b) QSH insulator phase with $E_z=0$. Trivial BI phase with (c) $l{E}_z=0.1h_0$ ($E_z>0$) and (d) $l{E}_z=-0.1h_0$ ($E_z<0$). $K$ (0.71) and $K^{\prime }$ (1.29) denote the position of the conduction band minimum (or valence band maximum) for different spins. ${\mathrm{Gap}}_{\uparrow }$ (${\mathrm{gap}}_{\downarrow }$) denotes the band gap for spin up (spin down).

Figure 2

The transient current $I_{\sigma }\left(t\right)$ ($\sigma =\uparrow$ for spin up, $\downarrow$ for spin down) through the nanoribbon in response to a upward step pulse. $W=12$, $L=16$, and $V_b\left(t\right)=0.2\theta \left(t\right)$ V. (a) $E_z=0$, (b) $l{E}_z=0.1h_0$ ($E_z>0$), and (c) $l{E}_z=-0.1h_0$ ($E_z<0$). Inset: The spin-polarized current defined as $P\left(t\right)=I_{\uparrow }\left(t\right)-I_{\downarrow }\left(t\right)$. ${\tau }_{\uparrow }$ (${\tau }_{\downarrow }$) denotes roughly the characteristic time for the spin up (down) electron to reach the steady state. The short-dashed lines that denote the dc current for spin up/down are calculated by using the Landauer-Büttiker formula.

Figure 3

Dynamic conductance (in units of $e^2/h$) as a function of frequency $\omega$. (a), (b) The total dynamic conductance $G_{LR}$. (c), (d) The dynamic conductance $G_{LR}^c$ contributed by the particle current. (e), (f) The dynamic conductance $G_{LR}^d$ contributed by the displacement current. (a), (c), (e) and (b), (d), (f) show the results for the nanoribbon without and with $E_z$ in the central region, respectively. Re: real part; Im: imaginary part. Parameters $W$, $L$, and $E_z>0$ are the same as those given in the caption of Fig. 2.

Figure 4

The scattering matrix $s_{LR}$ as a function of energy for (a) QSH insulator phase and (b) BI phase in the central region. (c) The phase difference $\mathrm{\Delta }\theta ={\theta }_{\downarrow }-{\theta }_{\uparrow }$ as a function of energy when the central region is in the BI phase. $S_{LR}$: $\mathrm{\Delta }\theta \left(E\right)$ is determined from the scattering matrix. Model: $\mathrm{\Delta }\theta \left(E\right)$ is estimated from $\mathrm{\Delta }\theta \left(E\right)=[q_{\downarrow }\left(E\right)-q_{\uparrow }\left(E\right)]d$. Parameters $W$, $L$, and $E_z>0$ are the same as those given in the caption of Fig. 2. (d) The complex band structure for the nanoribbon in the BI phase. $q$ and $\kappa$ (inset) correspond to the real and imaginary components of wave vector $k_x=q+i\kappa$, respectively.