Even-odd flux quanta effect in the Fraunhofer oscillations of an edge-channel Josephson junction

We calculate the beating of h/2e and h/e periodic oscillations of the flux-dependent critical supercurrent through a quantum spin-Hall insulator between two superconducting electrodes. A conducting pathway along the superconductor connects the helical edge channels via a non-helical channel, allowing an electron incident on the superconductor along one edge to be Andreev reflected along the opposite edge. We find the appearance of Fraunhofer oscillations with an even-odd effect: Large peaks in the critical current at even multiples of h/2e alternate with smaller peaks at odd multiples. This even-odd effect has been recently observed in one series of experiments by Pribiag et al.