Pair correlations and the survival of superconductivity in and around a superconducting impurity

The problem of the survival of superconductivity in a small superconducting grain placed in a metal substrate is addressed. For this aim the pair correlations and superconducting gap around and inside a negative-U impurity in one and two dimensions are calculated in a discrete tight-binding model and a continuous model. Using a mean-field decomposition, it is shown that finite pairing in the grain develops when the system has a degeneracy between a successive number of electron pairs, and thus may oscillate as a function of the chemical potential. For finite pairing in the island, pair correlations in the normal part exhibit a crossover from being long ranged to exponentially decaying, depending on the strength of interaction in the grain. It is shown analytically that there is a minimal island size under which pairing vanishes, which is different from that given by Anderson's criterion [J. Chem. Phys. Solids 11, 26 (1959)], and that it scales as a power law with island size, rather than exponentially as in isolated grains.