Emergence of $h/e$-period oscillations in the critical temperature of small superconducting rings enclosing magnetic flux

The Little-Parks critical-temperature oscillations, with magnetic flux, of a large-radius hollow cylindrical superconductor have a period $h/2e$. This oscillation period reflects the binding of electrons into Cooper pairs. On the other hand, the single-electron Aharonov-Bohm oscillations in the resistance or persistent current in a clean metallic ring have period $h/e$. By using the Gor'kov approach to BCS theory, we investigate oscillations in the critical temperature of a superconducting ring, for radii that are comparable to the superconducting coherence length. In this regime, oscillations in the critical temperature of period $h/e$ emerge, in addition to the usual Little-Parks-period oscillations. We argue that in the clean limit there is a superconductor-normal phase transition at nonzero flux, as the ring radius becomes sufficiently small, and that this transition can be either second- or first-order, depending on the ring radius and the external flux. In the dirty limit, we argue that the transition is rendered second-order, which results in continuous quantum phase transitions tuned by flux and radius.