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

Abstract

As a function of the magnetic flux threading the object, the Little-Parks oscillation in the critical temperature of a large-radius, thin-walled superconducting ring or hollow cylinder has a period given by $h/2e$, due to the binding of electrons into Cooper pairs. On the other hand, the single-electron Aharonov-Bohm oscillation in the resistance or persistent current for a clean (i.e. ballistic) normal-state system having the same topological structure has a period given by $h/e$. A basic question is whether the Little-Parks oscillation changes its character, as the radius of the superconducting structure becomes smaller, and even comparable to the zero-temperature coherence length. We supplement the argument that the $h/e$ oscillations should also be exhibited via a microscopic analysis of this regime, formulated in terms of the Gor'kov approach of BCS theory. We see that as the radius of the ring becomes smaller, an oscillation of critical temperature in the period of $h/e$ emerges in addition to the usual Little-Parks dependence. We argue that in the clean limit there is a superconductor-normal transition, at nonzero flux, as the ring radius becomes sufficiently small, and that the transition can be either continuous or discontinuous, depending on the radius and the external flux. In the dirty limit, we argue that the transition is rendered continuous, which results in continuous quantum phase transitions tuned by flux and radius.