Planar Josephson junctions in germanium: Effect of cubic spin-orbit interaction

Planar Josephson junctions comprising semiconductors with strong spin-orbit interaction (SOI) are promising platforms to host Majorana bound states (MBSs). Previous works on MBSs in planar Josephson junctions have focused on electron gases, where SOI is linear in momentum. In contrast, a two-dimensional hole gas in planar germanium (Ge) exhibits SOI that is cubic in momentum. Nevertheless, we show here that due to the particularly large SOI, Ge is a favorable material. Using a discretized model, we numerically simulate a Ge planar Josephson junction and demonstrate that also cubic SOI can lead to the emergence of MBSs. Interestingly, we find that the cubic SOI yields an asymmetric phase diagram as a function of the superconducting phase difference across the junction. We also find that trivial Andreev bound states can imitate the signatures of MBSs in a Ge planar Josephson junction, therefore making the experimental detection of MBSs difficult. We use experimentally realistic parameters to assess if the topological phase is accessible within experimental limitations. Our analysis shows that two-dimensional Ge is an auspicious candidate for topological phases.