Mapping the Topological Phase Diagram of Multiband Semiconductors with Supercurrents

We show that Josephson junctions made of multiband semiconductors with strong spin-orbit coupling carry a critical supercurrent I-c that contains information about the nontrivial topology of the system. In particular, we find that the emergence and annihilation of Majorana bound states in the junction is reflected in strong even-odd effects in I-c at small junction transparency. This effect allows for a mapping between I-c and the topological phase diagram of the junction, thus providing a dc measurement of its topology.