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 $4\pi$-periodic dependence on the superconductive phase difference $\phi$ between the surfaces. It results in the appearance of an anomalous term in the current-phase relation $I_a(\phi )$ for the supercurrent flowing along the channel between the superconductive surfaces. We have calculated the shape of the $4\pi$-periodic function $I_a(\phi )$, as well as the dependence of its amplitude on temperature and system parameters.

Figure 1

The system. A slab of TI has both surfaces covered by superconductor. A hole channel in the layer hosting a vortex forms an $S\mathrm{\text{−}}N\mathrm{\text{−}}S$ junction between the surfaces. The superconducting surfaces are connected away from the hole, completing an $S\mathrm{\text{−}}N\mathrm{\text{−}}S$ circuit with supercurrent flowing through the hole.

Figure 2

The anomalous current $I_a(\phi )$ is computed for $p_{F}R=2$ (dashed line) and $p_{F}R=3$ (solid blue line). Other parameters are fixed as $p_F\xi =10$, $p_F{\xi }_{\mathrm{sc}}=5$, $p_{F}L=6$, $T=0.05\Delta$. The extrema of $I_a(\phi )$ occur whenever a pair of conjugated levels crosses at $ϵ=0$.

Figure 3

Temperature dependence of $I_a(\phi =\pi )$ for the same sets of parameters as in Fig. 2. Solid blue line and dashed line correspond to $p_{F}R=3$ and $p_{F}R=2$, respectively.