Geometry-controlled engineering of the low-temperature proximity effect in normal metal-superconductor junctions

In the ballistic regime at finite temperatures, the proximity effect diminishes following an exponential pattern; however, at low or zero temperatures, this transition alters to a decay characterized by a power law with a dimensionality-dependent exponent. Here, we extend the current understanding of the proximity effect by exploring the role of normal metal-superconductor (NS) junction geometry in altering the spatial propagation of the superconducting order. Specifically, we demonstrate that geometric factors, such as interface curvature, significantly affect the decay exponent of the Cooper pair wave function, with negative curvature increasing the proximity range exponent and positive curvature shortening it. Furthermore, we discuss how the geometry of the NS interface governs the transparency of the clean NS junction and thus influences the proximity effect. These results deepen our understanding of how geometry and the proximity effect interact, which is important for the design and optimization of superconducting hybrid devices.