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We present a thorough tight-binding analysis of the band structure of a widevariety of lattices belonging to the class of honeycomb and Kagome systemsincluding several mixed forms combining both lattices. The band structure ofthese systems are made of a combination of dispersive and flat bands. Thedispersive bands possess Dirac cones (linear dispersion) at the six corners (Kpoints) of the Brillouin zone although in peculiar cases Dirac cones at thecenter of the zone $(\Gamma$ point) appear. The flat bands can be of differentnature. Most of them are tangent to the dispersive bands at the center of thezone but some, for symmetry reasons, do not hybridize with other states. Theobjective of our work is to provide an analysis of a wide class of so-calledligand-decorated honeycomb Kagome lattices that are observed in 2Dmetal-organic framework (MOF) where the ligand occupy honeycomb sites and themetallic atoms the Kagome sites. We show that the $p_x$-$p_y$ graphene model isrelevant in these systems and there exists four types of flat bands: Kagomeflat (singly degenerate) bands, two kinds of ligand-centered flat bands (A$_2$like and E like, respectively doubly and singly degenerate) and metal-centered(three fold degenerate) flat bands.

journal of physics. condensed matter

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