  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.
