Josephson effect and critical currents in trivial and topological full-shell hybrid nanowires

We perform microscopic numerical simulations of the Josephson effect through short junctions between two full-shell hybrid nanowires, comprised of a semiconductor core fully wrapped by a thin superconductor shell, both in the trivial and topological regimes. We explore the behavior of the current-phase relation and the critical current $I_c$ as a function of a threading flux for different models of the semiconductor core and different transparencies of the weak link. We find that $I_c$ is modulated with flux due to the Little-Parks (LP) effect and displays a characteristic skewness towards large fluxes within non-zero LP lobes, which is inherited from the skewness of a peculiar kind of subgap states known as Caroli-de Gennes-Matricon (CdGM) analogs. The appearance of Majorana zero modes at the junction in the topological phase is revealed in $I_c$ as fin-shaped peaks that stand out from the background at low junction transparencies. The competition between CdGMs of opposite electron- and hole-like character produces steps and dips in $I_c$. A rich phenomenology results, which includes 0-, $π$- and $φ$-junction behaviors depending on the charge distribution across the wire core and the junction transparency.