  The global phase coherence in doubly-connected superconductors leads to
fluxoid quantization, allowing the superfluid velocity vs to be controlled by
an applied magnetic flux. In ultrasmall samples this quantization requirement
leads, surprisingly, to the destruction of the phase coherence itself around
half-integer flux quanta, because of the sample-size-induced growth in vs, as
predicted by de Gennes. We report observations of the predicted phenomenon in
ultrathin Al and Au0.7In0.3 cylinders, and the corresponding phase diagram for
ultrathin superconducting cylinders. The new phase diagram features
disconnected superconducting regions, as opposed to the single one seen in the
conventional Little-Parks experiment.
