Bogoliubov-de Gennes study of nanoscale Hubbard superconductors

The effects of quantum confinement on the superconducting ground state are studied within the Bogoliubov-de Gennes (BdG) formalism and an attractive Hubbard model. We consider a periodic arrangement of two-dimensional superconducting grains composed by N x N atoms surrounded by insulating, metallic or superconducting stripes with a thickness of s atoms, leading to 2(N + s)² coupled self-consistent BdG equations for a supercell of (N + s) x (N + s) atoms. These equations determine the spatial variation of superconducting gap as functions of temperature, electron-electron interaction, and hopping integrals analyzing three types of boundary stripes. The results show a clear enhancement of the superconducting gap and critical temperature induced by the electron confinement in the grain, being larger for the insulating boundary case. Finally, the numerical solutions of BdG equations are compared with those obtained by applying the BCS theory to each grain site. (copyright 2016 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)