  Graphene antidot lattices have recently been proposed as a new breed of
graphene-based superlattice structures. We study electronic properties of
triangular antidot lattices, with emphasis on the occurrence of dispersionless
(flat) bands and the ensuing electron localization. Apart from strictly flat
bands at zero-energy (Fermi level), whose existence is closely related to the
bipartite lattice structure, we also find quasi-flat bands at low energies. We
predict the real-space electron density profiles due to these localized states
for a number of representative antidot lattices. We point out that the studied
low-energy, localized states compete with states induced by defects on the
superlattice scale in this system which have been proposed as hosts for
electron spin qubits. We furthermore suggest that local moments formed in these
midgap zero-energy states may be at the origin of a surprising saturation of
the electron dephasing length observed in recent weak localization measurements
in graphene antidot lattices.
