Flat band in twisted graphene bilayer has garnered widespread attention, and whether flat bands can be realized in carbon monolayers is an interesting topic worth exploring in condensed matter physics. In this work, we demonstrate that based on the theory of compact localized states, a series of two-dimensional carbon allotropes with flat bands can be achieved. Two of them, named 191-8-66-C-r567x-1 and 191-10-90-C-r567x-1, are evaluated to be dynamically stable carbon phases with isolated or weakly overlapped flat bands at the Fermi level. The maximum Fermi velocities of the flat-band electrons are evaluated to be $1\times {10}^4$ and $0.786\times {10}^4$ m/s, respectively, both of which are lower than the Fermi velocity of the flat-band electrons in magic-angle graphene ($4\times {10}^4$ m/s). Furthermore, 191-8-66-C-r567x-1 has been predicted to be a flat-band related magnetic half metal with a magnetic moment of 1.854 µB per cell, while 191-10-90-C-r567x-1 is a flat-band related magnetic normal metal with a magnetic moment of 1.663 µB per cell. These results not only show that flat bands can be constructed in carbon monolayers, but also indicate the potential for achieving metal-free magnetic materials with light elements based on flat-band theory.

• Flat bands
• Graphene
• Density functional theory
• Lattice models in condensed matter
• Tight-binding model