Detection of second-order topological superconductors by Josephson junctions

We study Josephson junctions based on second-order topological superconductors (SOTSs) which can be realized in quantum spin Hall insulators with large inverted gap in proximity to unconventional superconductors. We find that tuning the chemical potential in the superconductor strongly modifies the induced pairing of the helical edge states, resulting in topological phase transitions. In a corresponding Josephson junction, a $0$-$\pi$ transition is realized by tuning the chemical potentials in the superconducting leads. This striking feature is stable in junctions with respect to different sizes, doping the normal region, and the presence of disorder. Our transport results can serve as novel experimental signatures of SOTSs. Moreover, the $0$-$\pi$ transition constitutes a fully electric way to create or annihilate Majorana bound states in the junction without any magnetic manipulation.