Fluxoid valve effect in full-shell nanowire Josephson junctions

We introduce a new type of supercurrent valve based on full-shell nanowires. These hybrid wires consist of a semiconductor core fully wrapped in a thin superconductor shell and subjected to an axial magnetic field. Due to the tubular shape of the shell, the superconductor phase acquires an integer number $n$ of $2\ensuremath{\pi}$ twists or fluxoids that increases in steps with applied flux. By connecting two such hybrid wires, forming a Josephson junction (JJ), a flux-modulated supercurrent develops. If the two superconducting sections of the JJ have different radii ${R}_{1}$ and ${R}_{2}$, they can develop equal or different fluxoid numbers ${n}_{1},{n}_{2}$ depending on the field. If ${n}_{1}\ensuremath{\ne}{n}_{2}$, the supercurrent is blocked, while it remains finite for ${n}_{1}={n}_{2}$. This gives rise to a fluxoid valve effect controlled by the applied magnetic field or a gate voltage at the junction. We define a fluxoid-valve quality factor that is perfect for cylindrically symmetric systems and decreases as this symmetry is reduced. We further discuss the role of Majorana zero modes at the junction when the full shell nanowires are in the topological superconducting regime.