Proximity-induced minimum radius of superconducting thin rings closed by   the Josephson 0 or pi junction
  Superconductivity is shown to be completely destroyed in thin mesoscopic or nanoscopic rings closed by the junction with a noticeable interfacial pair breaking and/or a Josephson coupling, if a ring's radius r is less than the minimum radius r_min. The quantity r_min depends on the phase difference across the junction, or on the magnetic flux that controls the phase difference in the flux-biased ring. It also depends on the Josephson and interfacial effective coupling constants, and in particular, on whether the ring is closed by 0 or pi junction. The current-phase relation is substantially modified when the ring's radius exceeds r_min for some of the phase difference values, or slightly goes beyond its maximum. The modified critical temperature Tc, as well as the temperature dependent supercurrent near Tc are identified here as functions of the ring's radius and the magnetic flux. 
Yu. S. Barash
cond-mat.supr-con