  We explore the consequences of electron-phonon (e-ph) coupling in graphene
antidot lattices (graphene nanomeshes), i.e., triangular superlattices of
circular holes (antidots) in a graphene sheet. They display a direct band gap
whose magnitude can be controlled via the antidot size and density. The
relevant coupling mechanism in these semiconducting counterparts of graphene is
the modulation of the nearest-neighbor electronic hopping integrals due to
lattice distortions (Peierls-type e-ph coupling). We compute the full momentum
dependence of the e-ph vertex functions for a number of representative antidot
lattices. Based on the latter, we discuss the origins of the previously found
large conduction-band quasiparticle spectral weight due to e-ph coupling. In
addition, we study the nonzero-momentum quasiparticle properties with the aid
of the self-consistent Born approximation, yielding results that can be
compared with future angle-resolved photoemission spectroscopy measurements.
Our principal finding is a significant e-ph mass enhancement, an indication of
polaronic behavior. This can be ascribed to the peculiar momentum dependence of
the e-ph interaction in these narrow-band systems, which favors small phonon
momentum scattering. We also discuss implications of our study for recently
fabricated large-period graphene antidot lattices.
