Shape resonances and the T ( c ) dependence on film thickness of Ni/Bi systems

We report on the experimentally observed variation of the superconducting critical temperature (T ( c )) of Ni/Bi systems as a function of the total deposited film thickness and on its explanation using a theoretical model. Two series of Ni/Bi systems have been analyzed which were obtained by depositions of Ni onto Bi in the proportions Ni3Bix (3 nm of Ni onto x nm of Bi) and NiyBi6y (y nm Ni onto 6 y nm of Bi). As shown recently, the formation of the superconducting compound NiBi3 at Ni/Bi interfaces in the resulting NiBi3-Bi films is thermodynamically favored by a volume contraction. Here we corroborate this result and estimate the thickness of the resulting NiBi3 and of the remaining Bi layers for the Ni3Bix and NiyBi6y series using the laws of mass and conservation of number of atoms. We consider the resulting film as being made up of two homogeneous and uniform layers of NiBi3 and Bi, respectively, and study this idealizing model using the Bogoliubov de Gennes (BdG) equations. It is assumed that superconductivity originates in the NiBi3 layer and penetrates the Bi layers via a potential barrier. Our theoretical calculations predict the dependence of T ( c ) with respect to the thicknesses of the NiBi3 and Bi layers, and also with the strength of the potential barrier that blocks the migration of electrons from the NiBi3 to the Bi layer. The calculations show that the superconducting gap also exists in Bi, although much weaker than in the NiBi3 layer. We compare the predicted T ( c ) values with the experimental data and find sufficient agreement to suggest that our model can explain the experimentally observed variation of T ( c ) with thickness. We interpret this dependence as shape resonance oscillations which are derived from the BdG theory applied to thin superconducting films.