We study theoretically proximity-induced superconductivity and its inverse effect in dice lattice flat band model by considering Josephson junction with an s-wave pairing in the superconducting leads. Using self-consistent tight-binding Bogoliubov-de Gennes method, we show that there is a critical value for chemical potential of the superconductors depending on paring interaction strength over which for undoped normal region the proximity effect is enhanced. Whereas if the superconductor chemical potential is less than the critical one the proximity effect decreases regardless of normal region doping and in the meanwhile, the pairing amplitude of superconducting region increases significantly. Furthermore, we unveil that the supercurrent passing through the junction is large (vanishingly small) when the superconductor chemical potential is smaller (larger) than the critical value which increases as a function of normal region chemical potential.

A multiscale modeling and simulation approach, including first-principles calculations, ab initio molecular dynamics simulations, and a tight binding approach, is employed to study band flattening of the electronic band structure of oxidized monolayer graphene. The width of flat bands can be tuned by strain, the external electric field, and the density of functional groups and their distribution. A transition to a conducting state is found for monolayer graphene with impurities when it is subjected to an electric field of ∼1.0 V/Å. Several parallel impurity-induced flat bands appear in the low-energy spectrum of monolayer graphene when the number of epoxy groups is changed. The width of the flat band decreases with an increase in tensile strain but is independent of the electric field strength. Here an alternative and easy route for obtaining band flattening in thermodynamically stable functionalized monolayer graphene is introduced. Our work discloses a new avenue for research on band flattening in monolayer graphene.

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