Supercurrent determined from the Aharonov-Bohm effect in mesoscopic superconducting rings

We have solved the Bogoliubov-de Gennes equation for a clean, one-dimensional mesoscopic superconducting ring threaded by a magnetic flux Φ. We show that the superfluid velocity is driven directly by Φ while the relative motion of the pair of electrons is independent of Φ. Meanwhile, the fluxoid quantization is obtained straightforwardly. More importantly, we have also calculated the supercurrent numerically and self-consistently and find it is periodic in Φ with the period Φshc/2e for Φs≤Φd=(mvdL/ Latin small letter h with strokeπ)Φs and with the period Φ0hc/e for Φd<Φs, which arises from mesoscopic effects.