Edge effect and significant increase of the superconducting transition onset temperature of 2D superconductors in flat and curved geometries

In this paper, we present a simple method to model the curvature activated phonon softening in a 2D superconducting layer. The superconducting transition temperature Tc in the case of a 2D rectangular sheet, a hollow cylinder and a hollow sphere of one coherence length thickness is calculated by the quantum mechanical electron-phonon scattering matrix, and a series of collective lattice vibrations in the surface state. We will show that being extremely thin in a flat rectangular shape is not enough to significantly enhance the Tc through phonon softening. However, if a curvature is added, Tc can be strongly enhanced. The increase in Tc with respect to the bulk is greatest in a hollow sphere, intermediate in a hollow cylinder and weakest for the rectangular sheet, when systems of identical length scale are considered. In addition, we find that the edge effect of such a 2D sheet has a strong broadening effect on Tc in addition to the effect of order parameter phase fluctuations.