The magnetic flux dependence of the vortex state for small mesoscopic superconducting rings surrounded by a medium is investigated by the phenomenological Ginzburg-Landau theory. The influences of the ring size and the surface superconductivity on the free energy and total supercurrent are studied. For narrow rings, the persistent current evolves towards a periodic behaviour with magnetic flux. The complete paramagnetic or diamagnetic state, corresponding to positive or negative current flowing in the whole ring, can occur. A remarkable intermittent superconducting behaviour for the ground-state transition is found when the strength of surface-suppressed superconductivity is enlarged or the ring size is decreased. Consequently, a pure superconducting state with positive total current can be obtained.
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