Josephson effect in topologically confined channels

In recent times the fractional Josephson effect has been used as a signature of Majorana fermions forming in Josephson junctions mediated by the topologically protected spin helical edge states of a quantum spin Hall insulator. Here we propose a setup based on silicene in which either a single spin helical Kramers pair of edge states or two sets of Kramers pair edge states can exist at domain walls depending on an externally applied inhomogeneous electrical field. We furthermore investigate a Josephson junction mediated by these edge states through the Bogoliubov de Gennes formalism. We show that the energy phase relation is 4 π periodic when the system is in the quantum spin Hall regime but changes to a 2 π periodicity due to backscattering effects when two sets of Kramers pair edge states are present.