Proximity-induced superconductivity and Josephson critical current in quantum spin Hall systems

We consider recent experiments on wide superconductor–quantum spin Hall insulator (QSHI)–superconductor Josephson junctions, which have shown preliminary evidence of proximity-induced superconductivity at the edge modes of the QSHI system based on an approximate analysis of the observed Fraunhofer spectra of the Josephson critical current as a function of the applied magnetic field. Using a completely independent exact numerical method involving a nonlinear constrained numerical optimization, we calculate the supercurrent profiles, comparing our results quantitatively with the experimental Fraunhofer patterns in both HgCdTe and InAs-GaSb based QSHI Josephson junctions. Our results show good qualitative agreement with the experiments, verifying that the current distribution in the two-dimensional sample indeed has peaks at the sample edges when the system is in the QSHI phase, thus supporting the interpretation that superconductivity has indeed been induced in the QSHI edge modes. On the other hand, our numerical work clearly demonstrates that it will be very difficult, if not impossible, to obtain detailed quantitative information about the supercurrent distribution just from the analysis of the Josephson Fraunhofer spectra, and therefore, conclusions regarding the precise width of the edge modes or their topological nature are most likely premature at this stage.

Figure 1

Consistency check of the Dynes-Fulton method. (a) Simulated $J\left(x\right)$ (solid black line) and reconstructed $J_{\text{DF}}\left(x\right)$ (dashed blue line) density profiles. (b) The corresponding Fraunhofer patterns. Although the original $J\left(x\right)$ and reconstructed $J_{\text{DF}}\left(x\right)$ profiles are qualitatively similar, there is no quantitative agreement.

Figure 2

(a) The chosen current density. (b) The critical current against $\beta$ with $D=0$. (c) The critical current against $\beta$ with $D=0.5$. (d) The critical current against $\beta$ with $D=1$.

Figure 3

(a) The black line shows the Fraunhofer pattern obtained from digitalization of the experimental results in Fig. 2(c) of Ref. [24], in which the sample is believed to be in the SQH insulating regime. The red, green, and blue lines correspond to the Fraunhofer patterns obtained from the profiles below, found with our numerical minimization, for $D=0,D=0.5$, and $D=1$, respectively. In (b)–(d) we show the corresponding profiles for transparencies $D=0,D=0.5$, and $D=1.0$, respectively. Slightly different profiles are obtained in each case (here we show only three for each value of $D)$ as a consequence of the above-mentioned nonuniqueness of the procedure.

Figure 4

(a) The black line shows the Fraunhofer pattern obtained from direct digitalization of the experimental results in Fig. 5(a) of Ref. [25] when the sample is in the SQH insulating regime. The red, green, and blue lines correspond to the Fraunhofer patterns obtained from the profiles below, found with our numerical minimization, for $D=0,D=0.5$, and $D=1$, respectively. In (b)–(d) we show the corresponding profiles for transparencies $D=0,D=0.5$, and $D=1.0$, respectively. Slightly different profiles are obtained in each case (here we show only three for each value of $D)$ as a consequence of the above-mentioned nonuniqueness of the procedure.