Kondo interaction of quantum spin Hall edge channels with charge puddles

Under time-reversal symmetry, quantum spin Hall edge channels are protected against elastic backscattering. However, even for samples which exhibit conductance quantization due to the quantum spin Hall effect, reproducible fluctuations shape the quantization plateau when the chemical potential is tuned through the bulk gap. Here, we examine those fluctuations in micron-sized HgTe quantum well devices. By performing temperature- and gate-dependent measurements, we conclude that “charge puddles” in the narrow-gap material have a Kondo-type interaction with the edge channels resulting in the observed conductance fluctuations. Our results provide insight into the underlying mechanisms of scattering in quantum spin Hall edge channels.

Figure 1

Conductance fluctuations in the quantum spin Hall regime. (a) The four-terminal conductance $G_{\text{4t}}=I_{A,D}/V_{B,C}$ of Device 1 is plotted as a function of gate voltage $V_G$ at 17 mK. The dashed line indicates the expected quantization conductance ($4e^2/h$) for a four-terminal device. The inset shows a schematic of the device. The letters $A,B,C,\mathrm{and}D$ indicate Ohmic contacts. The distance between the contacts $B$ and $C$ is $\sim 1.7 \mu \mathrm{m}$. (b) The two-terminal conductance $G_{\text{2t}}=I_{A,C}/V_{A,C}$ of Device 2 is plotted as a function of $V_G$ at 27 mK. The dashed line indicates the expected quantization conductance ($3/2e^2/h$) for a three-terminal device. The inset shows a schematic of the device along with the molecular beam epitaxy grown layer stack. The distance between the contacts $A$ and $C$ is $\sim 3 \mu \mathrm{m}$. The width is $\sim 2 \mu \mathrm{m}$ for both devices. The CdTe, (Hg,Cd)Te bottom and top barriers have a thickness of 50, 150, and 15 nm, respectively. The green boxes indicate the regions of interest for the study of fluctuations in these devices.

Figure 2

Odd-even patterns in the temperature dependence of the conductance fluctuations. (a) $G_{\text{4t}}$ as a function of $V_G$ from 17 mK (blue) to 1.7 K (red) for Device 1. (b) $G_{\text{2t}}$ as a function of $V_G$ from 27 mK (blue) to 2 K (red) for Device 2. The various conductance peaks are labeled as $P_{\text{ev}}$ and $P_{\text{odd}}$ (indicated by vertical lines) while the valleys are labeled as $V$.

Figure 3

Temperature evolution of odd and even peaks. The temperature-dependent change in conductance $\mathrm{\Delta }G\left(T\right)=G\left(T\right)-G\left(T_{\text{High}}\right)$ of the peaks marked in Fig. 2 for (a) Device 1 ($T_{\text{High}}=1.7\text{K}$) and (b) Device 2 ($T_{\text{High}}=2.0\text{K}$). The color code corresponds to the color map of the labels for $P$ and $V$ introduced in Fig. 2. The dashed lines indicate fits to Eq. (1). In Device 2, the temperature dependence saturates at $\sim 200$ mK, which we attribute to higher electron temperature during the measurement compared to Device 1 ($\sim 50$ mK). The error bars are based on a maximum error estimation in 0.2 mV (Device 1) and 0.6 mV (Device 2) windows around the vertical lines in Fig. 2.

Figure 4

Fluctuations based on the interaction between edge channels and a charge puddle. (a) A schematic of the helical edge channels and the Kondo quantum dot formed by a charge puddle in the vicinity of the edge channels. The inset indicates the wave function overlap of edge channels and quantum dot states. The out-of-plane arrows indicate the orientation of the spin eigenstates of helical edge electrons (spin momentum locked) and the quantum dot system which are not necessarily parallel to each other, as further discussed in the Appendix. (b) Temperature dependence of the conductance of a Kondo-correlated quantum dot as a function of occupation number, where red indicates high ($T>T_K$) and blue low ($T\sim T_K$) temperatures. (c) An illustration of the corresponding quantum spin Hall conductance due to the interaction between helical edge channels and a Kondo quantum dot as described in the main text.