Anomalous Josephson current via Majorana bound states in topological insulators

We propose a setup involving Majorana bound states (MBS) hosted by a vortex on a superconducting surface of a 3D topological insulator (TI). We consider a narrow channel drilled across a TI slab with both sides covered by s-wave superconductor. In the presence of a vortex pinned to such a channel, it acts as a ballistic nanowire connecting the superconducting surfaces, with a pair of MBS localized in it. The energies of the MBS possess a 4π-periodic dependence on the superconductive phase difference φ between the surfaces. It results in the appearance of an anomalous term in the current-phase relation I(a)(φ) for the supercurrent flowing along the channel between the superconductive surfaces. We have calculated the shape of the 4π-periodic function I(a)(φ), as well as the dependence of its amplitude on temperature and system parameters.