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As a function of the magnetic flux threading the object, the Little-Parksoscillation in the critical temperature of a large-radius, thin-walledsuperconducting ring or hollow cylinder has a period given by $h/2e$, due tothe binding of electrons into Cooper pairs. On the other hand, thesingle-electron Aharonov-Bohm oscillation in the resistance or persistentcurrent for a clean (i.e. ballistic) normal-state system having the sametopological structure has a period given by $h/e$. A basic question is whetherthe Little-Parks oscillation changes its character, as the radius of thesuperconducting structure becomes smaller, and even comparable to thezero-temperature coherence length. We supplement the argument that the $h/e$oscillations should also be exhibited via a microscopic analysis of thisregime, formulated in terms of the Gor'kov approach of BCS theory. We see thatas the radius of the ring becomes smaller, an oscillation of criticaltemperature in the period of $h/e$ emerges in addition to the usualLittle-Parks dependence. We argue that in the clean limit there is asuperconductor-normal transition, at nonzero flux, as the ring radius becomessufficiently small, and that the transition can be either continuous ordiscontinuous, depending on the radius and the external flux. In the dirtylimit, we argue that the transition is rendered continuous, which results incontinuous quantum phase transitions tuned by flux and radius.

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