The magnetic flux dependence of energy and supercurrent in mesoscopic $d$-wave symmetric and asymmetric superconducting loops is investigated by numerically solving the Bogoliubov-de Gennes equations self-consistently. For square loops, we find an $hc/e$-flux periodicity in energy and supercurrent and demonstrate that the flux periodicity is sensitive to the hole size and the superconducting pairing strength as well as temperature. The $hc/2e$-periodic behavior can be restored almost entirely when we displace the central hole sufficiently out of the center of the sample. In rectangular loops, the discrete current-carrying low-energy spectrum can exist for an odd winding number of the order parameter.

• Received 16 June 2009

DOI:https://doi.org/10.1103/PhysRevB.80.144501