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

Real-space superconducting properties are increasingly important to characterize low-dimensional, layered, and nanostructured materials. Here, we present a method to extract the real-space superconducting order parameter from the superconducting gap spectrum obtained via anisotropic Migdal-Eliashberg calculations, using the Bloch wave functions of the Fermi states. We apply this approach to a selection of atomically thin material systems. Our analysis of gallenene, a monolayer of gallium atoms, shows that its planar and buckled phases exhibit distinct superconducting order parameter behaviors, shaped by their structural and electronic properties. Furthermore, we demonstrate that our real-space approach is exceptionally suited to identify and characterize Josephson junctions made from van der Waals materials. Our examination of a bilayer of ${\mathrm{NbSe}}_{2}$ reveals that the van der Waals gap acts as an intrinsic weak link between the superconducting ${\mathrm{NbSe}}_{2}$ layers. Therefore, a bilayer of ${\mathrm{NbSe}}_{2}$ represents one of the thinnest and most tunable Josephson junction architectures, with potential applications in quantum devices. Our findings underscore the utility of transformation into real space in understanding superconducting properties through atomistic simulations.