Strong enhancement of superconductivity in a nanosized Pb bridge

In recent experiments with a superconducting nanosized Pb bridge formed between a scanning tunneling microscope tip and a substrate, superconductivity has been detected at magnetic fields, that are a few times larger than the third (surface) critical field. We describe the observed phenomenon on the basis of a numerical solution of the Ginzburg-Landau equations in a model structure consisting of six conoids. The spatial distribution of the superconducting phase is shown to be strongly inhomogeneous, with a concentration of the superconducting phase near the narrowest parr (the "neck") of the bridge. We show that suppression of superconductivity in the bridge by applied magnetic field or by temperature first occurs near the bases and then in the neck region, what leads to a continuous superconducting-to-normal resistive transition. A position of the transition midpoint depends on temperature and, typically, is by one order of magnitude higher than the second critical field H-c2. We find that the vortex states can be realized in the bridge at low temperatures T/T-c less than or equal to 0.6. The vortex states lead to a fine structure of the superconducting-to-normal resistive transition. We also analyze vortex states in the bridge that are characterized by a varying vorticity as a function of the bridge's height.