Flux-Periodicity Crossover from hc/e in Normal Metallic to hc/2e in Superconducting Loops

The periodic response of a metallic or a superconducting ring to an external magnetic flux is one of the most evident manifestations of quantum mechanics. It is generally understood that the oscillation period hc/2e in the superconducting state is half the period hc/e in the metallic state, because the supercurrent is carried by Cooper pairs with a charge 2e. On the basis of the Bardeen-Cooper-Schrieffer theory we discuss, in which cases this simple interpretation is valid and when a more careful analysis is needed. In fact, the knowledge of the oscillation period of the current in the ring provides information on the electron interactions. In particular, we analyze the crossover from the hc/e periodic normal current to the hc/2e periodic supercurrent upon turning on a pairing interaction in a metal ring. Further, we elaborate on the periodicity crossover when cooling a metallic loop through the superconducting transition temperature Tc.