Controlling the flow of spin and charge in nanoscopic topological insulators

Controlling the flow of spin and charge currents in topological insulators (TIs) is a crucial requirement for applications in quantum computation and spin electronics. We demonstrate that such control can be established in nanoscopic two-dimensional TIs by breaking their time-reversal symmetry via magnetic defects. This allows for the creation of nearly fully spin-polarized charge currents, and the design of highly tunable spin diodes. Similar effects can also be realized in mesoscale hybrid structures in which TIs interface with ferro- or antiferromagnets.

Figure 1

(a) Schematic representation of the spin-resolved spatial current patterns in a 2D TI with ${\mathrm{\Lambda }}_{SO}=0.1, e\mathrm{\Delta }V=0.01t$, and $k_{B}T={10}^{-7}t$. (b) Energy dependence of the LDOS $N_{\sigma }$ at $\mathbf{L}$ in a clean TI with $N_{\sigma }(-E)=N_{\sigma }\left(E\right)$. Spatial pattern of $I_{\mathbf{r}{\mathbf{r}}^{\prime }}^{\uparrow }$ for $t_l=0.1t$ carried by (c) the lowest energy edge state at $E_1=0.0342t$ [blue dashed arrow in (b)], and (d) the lowest energy edge state at $E_1=0.031t$ in the presence of two potential defects (red circles) with scattering strength $U_0=10t$. Spatial pattern of (e) $I_{\mathbf{r}{\mathbf{r}}^{\prime }}^{\uparrow }$ and (f) $I_{\mathbf{r}{\mathbf{r}}^{\prime }}^{\downarrow }$ carried by the edge state at $E_1=0.0335t$ for $t_l=0.5t$.

Figure 2

(a) LDOS $N_{\sigma }(\mathbf{L},E)$ in a TI without and with a magnetic defect [red circle in (b)] with Ising symmetry, $J_{z}S=5t$ and $t_l=0.1t$. (b) Spatial pattern of the charge current, $I_{\mathbf{r}{\mathbf{r}}^{\prime }}^c$, carried by the lowest energy edge state at $E_1=0.0142t$ [see blue dashed arrow in (a)]. (c) LDOS $N_{\sigma }(\mathbf{L},E)$ for a TI containing two magnetic defects [red circles in (d)] with $xy$ symmetry, $J_{\pm }S=5t$ and $t_l=0.5t$. Spatial pattern of (d) $I_{\mathbf{r}{\mathbf{r}}^{\prime }}^c$, (e) $I_{\mathbf{r}{\mathbf{r}}^{\prime }}^{\uparrow }$, and (f) $I_{\mathbf{r}{\mathbf{r}}^{\prime }}^{\downarrow }$ carried by the edge state at $E_1=0.0175t$ [see blue dashed arrow in (c)] for $t_l=0.5t$.

Figure 3

TI containing two magnetic defects of Heisenberg symmetry with $J_{z}S=J_{\pm }S=5t$ [red circles in (c)] and $t_l=0.275t$: (a) ${\eta }_{\uparrow ,\downarrow }$ as a function of $V_g$ for forward, $\mathrm{\Delta }V$ (${\eta }_{\uparrow }$: black line, ${\eta }_{\downarrow }$: red line), and backward bias, $-\mathrm{\Delta }V$ (${\eta }_{\uparrow }$: blue dashed line, ${\eta }_{\downarrow }$: green dashed line). (b) $N_{\sigma }(E=e{V}_g)$ at $\mathbf{L}$. Spatial pattern of $I_{\mathbf{r}{\mathbf{r}}^{\prime }}^c$, for (c) forward bias $\mathrm{\Delta }V$, and (d) backward bias $-\mathrm{\Delta }V$ at $V_{g,1}$. (e) Total normalized charge current $I_{\mathrm{out}}^c\left(V_g\right)/I_c^{\max }$ with $I_c^{\max }={\text{max}}_{V_g}\left(I_c^{\mathrm{out}}\right)$.

Figure 4

TI with edge disorder, $t_l=0.5t$ and two magnetic defects (red circles) with $J_{\pm }S=5t, J_{z}S=0$. Spatial pattern of (a) $I_{\mathbf{r}{\mathbf{r}}^{\prime }}^{\uparrow }$ and (b) $I_{\mathbf{r}{\mathbf{r}}^{\prime }}^{\downarrow }$ carried by the edge state at $E_1=0.04t$.

Figure 5

Hybrid structure of TI ($N_a=14, N_z=15, t_l=0.5t$) and (a),(b) a disordered ferromagnet with $J_{\pm }S=5t$; the sites with $J_{\pm }S=0$ are indicated by white circles. (c),(d) An antiferromagnet with $J_{\pm }S=\pm 5t$ with an alternating sign between neighboring sites. Spatial pattern of (a),(c) $I_{\mathbf{r}{\mathbf{r}}^{\prime }}^{\uparrow }$, and (b),(d) $I_{\mathbf{r}{\mathbf{r}}^{\prime }}^{\downarrow }$ carried by the edge state at $E_1=0.1t$