Nonequilibrium Josephson Effect through Helical Edge States

We study Josephson junctions between superconductors connected through the helical edge states of a two-dimensional topological insulator in the presence of a magnetic barrier. As the equilibrium Andreev bound states of the junction are $4\pi$ periodic in the superconducting phase difference, it was speculated that, at finite dc bias voltage, the junction exhibits a fractional Josephson effect with half the Josephson frequency. Using the scattering matrix formalism, we show that his effect is absent in the average current. However, clear signatures can be seen in the finite-frequency current noise. Furthermore, we discuss other manifestations of the Majorana bound states forming at the edges of the superconductors.

Figure 1

(a) Schematic view of the S-TI-S junction. (b) dc current, (c) real part, and (d) imaginary part of the first harmonic of the ac current as a function of applied bias for various transparencies. Here $I_{\Delta }=G_N\Delta /e$ with $G_N=D{e}^2/h$.

Figure 2

Zero-frequency current noise for the same transparencies as in Fig. 1. The inset shows the effective charge $q^{*}$ as a function of applied bias for $D=0.4$, 0.7, and 0.9.

Figure 3

Finite-frequency noise of a junction with transparency $D=0.4$ for various values of the applied bias.