  In recent experiments, superconductivity and correlated insulating states
were observed in twisted bilayer graphene (TBG) with small magic angles, which
highlights the importance of the flat bands near Fermi energy. However, the
moir\'e pattern of TBG consists of more than ten thousand carbon atoms that is
not easy to handle with conventional methods. By density functional theory
calculations, we obtain a flat band at E$_F$ in a novel carbon monolayer coined
as cyclicgraphdiyne with the unit cell of eighteen atoms. By doping holes into
cyclicgraphdiyne to make the flat band partially occupied, we find that
cyclicgraphdiyne with 1/8, 1/4, 3/8 and 1/2 hole doping concentration shows
ferromagnetism (half-metal) while the case without doping is nonmagnetic,
indicating a hole-induced nonmagnetic-ferromagnetic transition. The calculated
conductivity of cyclicgraphdiyne with 1/8, 1/4 and 3/8 hole doping
concentration is much higher than that without doping or with 1/2 hole doping.
These results make cyclicgraphdiyne really attractive. By studying several
carbon monolayers, we find that a perfect flat band may occur in the lattices
with both separated or corner-connected triangular motifs with only including
nearest-neighboring hopping of electrons, and the dispersion of flat band can
be tuned by next-nearest-neighboring hopping. Our results shed insightful light
on the formation of flat band in TBG. The present study also poses an
alternative way to manipulate magnetism through doping flat band in carbon
materials.
