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The problem of the survival of superconductivity in a small super-conductinggrain placed in a metal substrate is addressed. For this aim the paircorrelations and super-conducting gap around and inside a negative-U impurityin one and two dimensions is calculated, in a discrete tight-binding model anda continuous model. Using the Hartree-Fock-Gorkov mean-field decomposition, itis found that pairing inside the gap occurs when the system has a degeneracybetween successive number of electron pairs, and is highly sensitive to thechemical potential. For finite pairing in the island, pair correlations in thenormal part may either decay exponentially or be long-ranged, depending on thestrength of interaction. In addition, it is shown analytically that there is aminimal island size under-which pairing vanishes, and that it scales as apower-law, rather then exponentially as in isolated grains.  These results are interpreted in terms of screening of the negative-Uimpurity by the electron gas.Comment: Accepted for publication in Phys.Rev.

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