  Serving as a new two-dimensional plasmonic material, graphene has stimulated
an intensive study of its optical properties which benefit from the unique
electronic band structure of the underlying honeycomb lattice of carbon atoms.
In classical electrodynamics, nanostructured graphene is commonly modeled by
the computationally demanding problem of a three-dimensional conducting film of
atomic-scale thickness. Here, we propose an efficient alternative
two-dimensional electrostatic approach where all the calculation procedures are
restricted to the plane of the graphene sheet. To explore possible quantum
effects, we perform tight-binding calculations, adopting a random-phase
approximation. We investigate the multiple plasmon modes in triangles of
graphene, treating the optical response classically as well as quantum
mechanically in the case of both armchair and zigzag edge termination of the
underlying atomic lattice. Compared to the classical plasmonic spectrum which
is "blind" to the edge termination, we find that the quantum plasmon
frequencies exhibit blueshifts in the case of armchair edge termination, while
redshifts are found for zigzag edges. Furthermore, we find spectral features in
the zigzag case which are associated with electronic edge states not present
for armchair termination. Merging pairs of such triangles into dimers, the
plasmon hybridization leads to energy splitting in accordance with
plasmon-hybridization theory, with a lower energy for the antisymmetric modes
and a smaller splitting for modes with less confinement to the gap region. The
hybridization appears strongest in classical calculations while the splitting
is lower for armchair edges and even more reduced for zigzag edges. Our various
results illustrate a surprising phenomenon: Even 20 nm large graphene
structures clearly exhibit quantum plasmonic features due to atomic-scale
details in the edge termination.
