Josephson junctions of Weyl and multi-Weyl semimetals

We study a Josephson junction involving a Weyl and a multi-Weyl semimetal separated by a barrier region of width $d$ created by putting a gate voltage $U_0$ over the Weyl semimetal. The topological winding number of such a junction changes across the barrier. We show that $I_c R_N$ for such junctions, where $I_c$ is the critical current and $R_N$ the normal state resistance, in the thin barrier limit, has a universal value independent of the barrier potential. We provide an analytical expression of the Andreev bound states and use it to demonstrate that the universal value of $I_c R_N$ is a consequence of change in topological winding number across the junction. We also study AC Josephson effect in such a junction in the presence of an external microwave radiation, chart out its current-voltage characteristics, and show that the change in the winding number across the junction shapes the properties of its Shapiro steps. We discuss the effect of increasing barrier thickness $d$ on the above-mentioned properties and chart out experiments which may test our theory.