We analyze the crossover from an hc/e-periodicity of the persistent current in flux threaded clean metallic rings towards an hc/2e-flux periodicity of the supercurrent upon entering the superconducting state. On the basis of a model calculation for a
one-dimensional ring we identify the underlying mechanism, which balances the hc/e versus the hc/2e periodic components of the current density. When the ring circumference exceeds the coherence length of the superconductor, the flux dependence is strictly hc/2e periodic. Further, we develop a multi-channel model which reduces the Bogoliubov - de Gennes equations to a one-dimensional differential equation for the radial component of the wave function. The discretization of this differential equation introduces transverse channels, whose number scales with the thickness of the ring. The periodicity crossover is analyzed close the critical temperature.
The magnetic flux periodicity of superconducting loops as well as flux quantization itself are a manifestation of macroscopic quantum phenomena with far reaching implications. They provide the key to the understanding of many fundamental properties o
f superconductors and are the basis for most bulk and device applications of these materials. In superconducting rings the electrical current has been known to periodically respond to a magnetic flux with a periodicity of $bm{h/2e}$. Here, the ratio of Plancks constant and the elementary charge defines the magnetic flux quantum $bm{h/e}$. The well-known $bm{h/2e}$ periodicity is viewed to be a hallmark for electronic pairing in superconductors and is considered evidence for the existence of Cooper pairs. Here we show that in contrast to this long-term belief, rings of many superconductor bear an $bm{h/e}$ periodicity. These superconductors include the high-$bm{T_c}$ cuprates, Sr$_2$RuO$_4$, the heavy-fermion superconductors, as well as all other unconventional superconductors with nodes in the energy gap functions, and s-wave superconductors with small gaps or states in the gap. As we show, the 50-year-old Bardeen--Cooper--Schrieffer theory of superconductivity implies that for multiply connected paths of such superconductors the ground-state energies and consequently also the supercurrents are generically $bm{h/e}$ periodic. The origin of this periodicity is a magnetic-field driven reconstruction of the condensate and a concomitant Doppler-shifted energy spectrum. The robust, flux induced reconstruction of the condensate will be an important aspect to understand the magnetic properties of mesoscopic unconventional superconductors.
