Approaching Quantization in Macroscopic Quantum Spin Hall Devices through Gate Training

Quantum spin Hall edge channels hold great promise as dissipationless one-dimensional conductors. However, the ideal quantized conductance of $2e^2/h$ is only found in very short channels-in contradiction with the expected protection against backscattering of the topological insulator state. In this Letter we show that enhancing the band gap does not improve quantization. When we instead alter the potential landscape by charging trap states in the gate dielectric using gate training, we approach conductance quantization for macroscopically long channels. Effectively, the scattering length increases to $175\mu \mathrm{m}$, more than 1 order of magnitude longer than in previous works for HgTe-based quantum wells. Our experiments show that the distortion of the potential landscape by impurities, leading to puddle formation in the narrow gap material, is the major obstacle for observing undisturbed quantum spin Hall edge channel transport.

Figure 1

(a) Schematic sketch of the potential landscape along the edge channel for small (left) and large fluctuations (center), and an enlarged band gap (right). CB and VB denote the conduction and valence band, respectively. (b) Calculated band dispersions around the $\mathrm{\Gamma }$ point of samples $A$ through $E$. The arrow indicates the band gaps. (c) Longitudinal resistance as a function of the gate voltage for samples $A$ through $E$ for large (red, $l_{\text{edge}}=620\mu \mathrm{m}$) and small devices (blue, $l_{\text{edge}}=58\mu \mathrm{m}$) at $T\approx 2\mathrm{K}$. The gate voltages $V_G$ are shifted by $V_0$ corresponding to the resistance maximum.

Figure 2

Resistance ratio ${\gamma }_G=R_{xx58}/R_{xx620}$ as a function of gate voltage. Dashed lines indicate ideal values for pure sheet conductance (${\gamma }_G=1$) and pure edge conductance (${\gamma }_G=0.1$).

Figure 3

Gate training. Longitudinal resistance $R_{xx}$ as a function of gate voltage at $T=2\mathrm{K}$ for (a) the large device ($l_{\text{edge}}=620\mu \mathrm{m}$), (b) the small ($l_{\text{edge}}=58\mu \mathrm{m}$), and (c) the microdevice ($l_{\text{edge}}=13\mu \mathrm{m}$). The left and right panel represent different sweep directions, from zero to $V_{\max }$ (indicated by a colored bar) and $V_{\max }$ back to zero, respectively. Black dots mark the resistance maximum for each training sequence.

Figure 4

(a) Minimum conductance values of the corresponding maximum resistance (dots in Fig. 3) as a function of $V_{\max }$ for all three channel lengths. Open and closed dots represent sample $F$ and triangles sample $D$. Open dots are obtained from a second cool-down. The dashed line marks the expected quantized conductance value of $2e^2/h$. (b) shows the conductance closest to the expected value of $2e^2/h$ as a function of channel length for initial (red) and optimized (blue) minimum conductance (extracted from (a)). The solid line corresponds to a fit of the average scattering length within the edge channel, which yields $\lambda =8$ (red) and $175\mu \mathrm{m}$ (blue). The dashed line represents the Ohmic behavior with $G_{\min }\propto 1/l_{\text{edge}}$.