Theory of Superconductivity in Mesoscopic Systems*

By using Bogoliubov-de Gennes (BdG) equations, we study superconducting (SC) states in a quasi 2-dimensional system of radius $R$. It is shown that no vortices exist in $s$-wave SC samples with $R < R_c\sim\xi(0)$, the $T=0$ coherence length. We predict that chiral $p$-wave states exhibit superconductivity for $R < R_c$ only in the presence of a vortex with opposite chirality. This {\it reentrant} SC phase is a consequence of non-zero chirality of the pairing order parameter and implies the presence of chiral edge currents. Our study may be applied to sharply probing the pairing symmetry of unconventional superconductors.