  The Dirac bands and flat bands are difficult to coexist because they
represent two extreme ends of electronic properties. However, in this paper, we
propose a carbon-Kagome-lattice (CKL) family based on first-principles
calculations, and the coexistence of Dirac and flat bands are observed in this
series of three-dimensional carbon structures. The flat bands are originated
from the orbital interactions of the Kagome lattices, while the Dirac bands are
related to the carbon zigzag chains. A tight-binding model is used to explain
the various band structures in different CKLs. The coexistence of flat and
Dirac bands around the Fermi level implies that CKL structures maybe can serve
as superconductors. In addition, electronic properties of the thinnest CKL
slabs, only consisting of benzene rings, are studied. Flat bands are found in
the band spectra of the two-dimensional structures, and split into spin-up and
spin-down bands because of strong correlated effect in the case of hole doping.
