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Real-space superconducting properties are increasingly important tocharacterize low-dimensional, layered, and nanostructured materials. Here, wepresent a method to extract the real-space superconducting order parameter fromthe superconducting gap spectrum obtained via anisotropic Migdal-Eliashbergcalculations, using the Bloch wave functions of the Fermi states. We apply thisapproach to a selection of atomically thin material systems. Our analysis ofgallenene, a monolayer of gallium atoms, shows that its planar and buckledphases exhibit distinct superconducting order parameter behaviors, shaped bytheir structural and electronic properties. Furthermore, we demonstrate thatour real-space approach is exceptionally suited to identify and characterizeJosephson junctions made from van der Waals materials. Our examination of abilayer of NbSe$_2$ reveals that the van der Waals gap acts as an intrinsicweak link between the superconducting NbSe$_2$ layers. Therefore, a bilayer ofNbSe$_2$ represents one of the thinnest and most tunable Josephson junctionarchitectures, with potential applications in quantum devices. Our findingsunderscore the utility of transformation into real-space in understandingsuperconducting properties through ab initio calculations.

Real-space superconducting properties in the atomically-thin limit: Ab initio approach and its application to Josephson junctions