| The progress of nanofabrication technologies from the 1990s resulted in an increase of interest in the study of superconducting properties of mesoscopic samples. A mesoscopic sample is such that its size is comparable to the coherence lengthξand penetration depthλ. The properties of mesoscopic systems are considerably influenced by confinement effects. Therefore, the vortex state will depend on the size and the geometry of the sample.The vortex state and vortex charge of a thin mesoscopic ring are investigated theoretically in the framework of the phenomenological Ginzburg-Landau theory within the de Gennes boundary condition ( 2)SSn ieA i? ? ?? cψ= bψ. The surfaceextrapolation lengths b on the inner surface and the outer surface can own different value and sign, and are denoted as bi and bo , respectively. b < 0 and b > 0 correspond to surface superconductivity of enhancement and suppression. The main properties of the vortex state are listed as follows: (1) In mesoscopic samples two kinds of vortex states can exist, that is, the giant vortex state and the multivortex state. The giant vortex state is circular symmetric with a fixed value of angular momentum L . For sufficiently large rings the giant vortex can break up into multivortices. The multivortex state ( L1 , L2 ) represents that one vortex with angular momentum L1 situates in the center of the ring with L2 ? L1 single vortices on a single ring. The phase transition between the giant vortex and the multivortex state is of second order. There are saddle points of the free energy representing the surface barrier which has to be overcome for transition between the different vortex states. Due to the saddle state, the transition L→L+ 1 does not necessarily coincide with the magnetic field H tr where the ground state changes from L to L + 1 state. For increasing applied field, the L state remains stable up the penetration field H p > Htr and transits then to the L + 1 state. For decreasing field, the L + 1 state...
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