Ultrathin superconductors of different materials are becoming a powerful platform to find mechanisms for enhancement of superconductivity, exploiting shape resonances in different superconducting properties. Since 2004, the observation of shape resonances in superconducting nanofilms of Pb and first evidences of shape resonances in the superconducting critical temperature in metallic nanowires of Sn and Al [1–2] clearly established the importance of the interplay between quantum size effects, leading to multiple bands, and superconductivity, when the lateral dimensions of the system are reduced to the order of the interparticle distance or the pair correlation length.
Moreover, superconductivity in iron-based, magnesium diborides, and other high-Tc superconductors has a strong multi-band and multi-gap character [3,4]. Recent experiments support the possibility for a BCS-BEC crossover induced by the proximity of the chemical potential to the band edge of one of the bands, with evidences for Lifshitz transitions associated with changes in the Fermi surface topology [5, 6].
Here we present the simplest theoretical model which accounts for superconducting shape resonances and the BCS-BEC crossover in a multi-band / multi-gap superconductor, considering tunable interactions and nanostructured geometries. When the gap is of the order of the local chemical potential, superconductivity is in the crossover regime of the BCS-BEC crossover and the Fermi surface of the small band is completely smeared by the gap opening. In this situation, small and large Cooper pairs coexist in the total condensate, which is the optimal condition for high-Tc or even for room temperature superconductivity [7,8]. As a realizable example of enhancement of superconductivity in nanostructured materials, we consider here superconducting stripes organized in a parallel pattern (Superstripes), in which shape resonances and multigap physics at the band edge play a cooperative role in enhancing superconductivity in the crossover regime of pairing, while allowing for a sizable screening of the detrimental superconducting fluctuations [7,8,9]. Finally, we focus on a key prediction of the above discussed physics: the isotope effect of the superconducting critical temperature in the vicinity of a Lifshitz transition, which has a unique dependence on the energy distance (or density) between the chemical potential and the Lifshitz transition point [10].