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 non-linear 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 2D 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 super-current 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. 
Hoi-Yin Hui, Alejandro M. Lobos, Jay D. Sau and S. Das Sarma
cond-mat.supr-con cond-mat.mes-hall