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

We calculate the beating of $h\mathrm{/}2e$ and $h\mathrm{/}e$ periodic oscillations of the flux-dependent critical supercurrent $I_c(\mathrm{\Phi })$ 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. In the limit of small Andreev reflection probability the resulting even-odd effect is described by $I_c\propto \mathrm{\mid }\cos (e\mathrm{\Phi }\mathrm{/}\mathrm{\hslash })+f\mathrm{\mid }$, with $\mathrm{\mid }f\mathrm{\mid }\ll 1$ proportional to the probability for phase-coherent inter-edge transmission. Because the sign of $f$ depends on microscopic details, a sample-dependent inversion of the alternation of large and small peaks is a distinctive feature of the beating mechanism for the even-odd effect.