Tunneling between helical edge states through extended contacts

We consider a quantum spin Hall system in a two-terminal setup, with an extended tunneling contact connecting upper and lower edges. We analyze the effects of this geometry on the backscattering current as a function of voltage, temperature, and strength of the electron interactions. We find that this configuration may be useful to confirm the helical nature of the edge states and to extract their propagation velocity. By comparing with the usual quantum point-contact geometry, we observe that the power-law behaviors predicted for the backscattering current and the linear conductance are recovered for low enough energies, while different power laws also emerge at higher energies.

Figure 1

Extended contact geometry for a quantum spin Hall system with one Kramers doublet of helical edge states. The full (dashed) lines represent helical edge states carrying electrons with spin up (down). Right- (left)-moving electrons are in equilibrium with the left (right) contact at chemical potential ${\mu }_L$ (${\mu }_R$). The black arrow represents a possible spin-conserving electron tunneling event through the extended region.

Figure 2

Modulating function as a function of (a) bias $V$ (in units of ${\epsilon }_F/e$) at low temperature ($k_{B}T={10}^{-2}{\epsilon }_F$) and (b) temperature $T$ (in units of ${\epsilon }_F/k_B$) at low bias ($\mathrm{eV}={10}^{-2}{\epsilon }_F$), for different lengths of the contact: ${\alpha }_l=1$ (long dashed red line), 2 (dashed green line), and 5 (short dashed blue line). Note that the behavior at low temperature in (a) is indistinguishable from the $T=0$ case. This comment holds as well for (b) between low $V$ and $V=0$. Other parameters: $K=0.75$.

Figure 3

(a) Differential conductance as a function of bias $V$ (in units of ${\epsilon }_F/e$) at low temperature ($k_{B}T={10}^{-2}{\epsilon }_F$) and (b) linear conductance as a function of the temperature $T$ (in units of ${\epsilon }_F/k_B$), for different lengths of the contact: ${\alpha }_l=1$ (long dashed red line), 2 (dashed green line), and 5 (short dashed blue line). Units of the conductance: $G_0=\frac{2e^2}{{\epsilon }_F^2}\frac{{\left|{\Lambda }_0\right|}^2}{{\left(2\pi a\right)}^2}{\left(k_{F}a\right)}^{2d_K}$. Other parameters: $K=0.75$.

Figure 4

Differential conductance as a function of bias $V$ (in units of ${\epsilon }_F/e$) for different interaction strengths: $K=1$ (long dashed red line), $0.75$ (dashed green line), and $0.5$ (short dashed blue line). Note that the conductance is plotted in unity of $G_0$ as in Fig. 3, which depends on $K$ and thus does not allow for a direct comparison of the size between the different curves. Other parameters: ${\alpha }_l=5$; $k_{B}T={10}^{-2}{\epsilon }_F$.

Figure 5

Differential conductance as a function of bias $V$ (in units of ${\epsilon }_F/e$) for different temperatures (in units of ${\epsilon }_F/k_B$): $T=0.1$ (long dashed red line), 0.5 (dashed green line), and 2 (short dashed blue line). Units of $G_0$ as in Fig. 3. Other parameters: ${\alpha }_l=5$; $K=0.75$.

Figure 6

Log-log plot (a) of the backscattering current [in units of $I_0\equiv ({\epsilon }_F/e)G_0$] as a function of the bias voltage $V$ (in units of ${\epsilon }_F/e$) at low temperature ($k_{B}T={10}^{-2}{\epsilon }_F$) (long dashed red curve) and (b) of the linear conductance (in units of $G_0$) as a function of temperature (in units of ${\epsilon }_F/k_B$) (long dashed red curve). Other parameters: ${\alpha }_l=1$, $K=0.75$. Straight lines represent the asymptotic power-law behavior with exponent (a) $2d_K-1=13/12$ (dashed green line) and $2d_K-2=1/12$ (short dashed blue line) and (b) $2d_K-2=1/12$ (dashed green line) and $2d_K-3=-11/12$ (short dashed blue line).