A bird's eye view on the flat and conic band world of the honeycomb and Kagome lattices: towards an understanding of 2D metal-organic frameworks electronic structure

We present a thorough tight-binding analysis of the band structure of a wide variety of lattices belonging to the class of honeycomb and Kagome systems including several mixed forms combining both lattices. The band structure of these systems are made of a combination of dispersive and flat bands. The dispersive bands possess Dirac cones (linear dispersion) at the six corners (K points) of the Brillouin zone although in peculiar cases Dirac cones at the center of the zone (Gamma point) appear. The flat bands can be of different nature. Most of them are tangent to the dispersive bands at the center of the zone but some, for symmetry reasons, do not hybridize with other states. The objective of our work is to provide an analysis of a wide class of so-called ligand-decorated honeycomb Kagome lattices that are observed in a 2D metal-organic framework where the ligand occupy honeycomb sites and the metallic atoms the Kagome sites. We show that the p(x)-p(y) graphene model is relevant in these systems and there exists four types of flat bands: Kagome flat (singly degenerate) bands, two kinds of ligandcentered flat bands (A(2) like and E like, respectively doubly and singly degenerate) and metalcentered (three fold degenerate) flat bands.