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In this paper, we present a simple method to model the curvature activatedphonon softening in a 2D superconducting layer. The superconducting transitiontemperature Tc in the case of a 2D rectangular sheet, a hollow cylinder and ahollow sphere of one coherence length thickness is calculated by the quantummechanical electron-phonon scattering matrix, and a series of collectivelattice vibrations in the surface state. We will show that being extremely thinin a flat rectangular shape is not enough to significantly enhance the Tcthrough phonon softening. However, if a curvature is added, Tc can be stronglyenhanced. The increase in Tc with respect to the bulk is greatest in a hollowsphere, intermediate in a hollow cylinder and weakest for the rectangularsheet, when systems of identical length scale are considered. In addition, wefind that the edge effect of such a 2D sheet has a strong broadening effect onTc in addition to the effect of order parameter phase fluctuations.

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