| Summary: | Superconductivity is a remarkable quantum phenomenon in which a macroscopic number of electrons form a condensate of Cooper pairs that can be described by a single quantum wave function. According to the celebrated Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity, there is a minimum length scale (the coherence length) below which the condensate has a rigid quantum phase. The fate of superconductivity in a system with spatial dimensions smaller than [the coherence length] has been the subject of intense interest for decades and recent studies of superconductivity in ultra-thin epitaxial metal films have revealed some surprising behaviors in light of BCS theory. Notably, it was found that superconductivity remains robust in thin lead films with thicknesses orders of magnitude smaller than the coherence length (i.e. in the extreme two dimensional limit). Such studies raise the critical question: what happens to superconductivity as all dimensions are reduced toward the zero dimensional limit? By controlling the lateral size of ultra thin 2D islands, we systematically address this fundamental question with a detailed scanning tunneling microscopy/spectroscopy study. We show that as the lateral dimension is reduced, the strength of the superconducting order parameter is also reduced, at first slowly for dimensions larger than the bulk coherence length, and then dramatically at a critical length scale of ~ 40nm. We find this length scale corresponds to the lateral decay length of the order parameter in an island containing regions of different heights and different superconducting strength. Overall, our results suggest that fluctuation corrections to the BCS theory are important in our samples and may need to be systematically addressed by theory. === text
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