Pair correlations and the survival of superconductivity in and around a   super-conducting impurity
  The problem of the survival of superconductivity in a small super-conducting grain placed in a metal substrate is addressed. For this aim the pair correlations and super-conducting gap around and inside a negative-U impurity in one and two dimensions is calculated, in a discrete tight-binding model and a continuous model. Using the Hartree-Fock-Gorkov mean-field decomposition, it is found that pairing inside the gap occurs when the system has a degeneracy between successive number of electron pairs, and is highly sensitive to the chemical potential. For finite pairing in the island, pair correlations in the normal part may either decay exponentially or be long-ranged, depending on the strength of interaction. In addition, it is shown analytically that there is a minimal island size under-which pairing vanishes, and that it scales as a power-law, rather then exponentially as in isolated grains.   These results are interpreted in terms of screening of the negative-U impurity by the electron gas. 
Yonatan Dubi