Abstract
A mesoscopic superconductor in the shape of a circular cone, and size of the order of the coherence length, is investigated theoretically. The external magnetic field is applied perpendicular to its circular basis, and vortex states are obtained in the framework of the Ginzburg-Landau theory. We find patterns made of giant vortex states (GVS), curved multivortex states (MVS), and a combination of both of them. The results are summarized in phase diagrams, where the GVS and MVS regimes are determined according to the applied field and geometric parameters. We find that superconductivity persists up to fields much larger than the upper critical field $({H}_{c2})$ in case of a very small apex angle. The results are relevant to understand the properties of superconducting tips, which are currently used in scanning tunneling microscopy, in the presence of an external applied field.
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