  We investigate higher-order plasmons in graphene nanoribbons, and present how
electronic edge states and wavefunction fine structure influence the graphene
plasmons. Based on nearest-neighbor tight-binding calculations, we find that a
standing-wave model based on nonlocal bulk plasmon dispersion is surprisingly
accurate for armchair ribbons of widths even down to a few nanometers, and we
determine the corresponding phase shift upon edge reflection and an effective
ribbon width. Wider zigzag ribbons exhibit a similar phase shift, whereas the
standing-wave model describes few-nanometer zigzag ribbons less satisfactorily,
to a large extent because of their edge states. We directly confirm that also
the larger broadening of plasmons for zigzag ribbons is due to their edge
states. Furthermore, we report a prominent fine structure in the induced
charges of the ribbon plasmons, which for armchair ribbons follows the
electronic wavefunction oscillations induced by inter-valley coupling.
Interestingly, the wavefunction fine structure is also found in our analogous
density-functional theory calculations, and both these and tight-binding
numerical calculations are explained quite well with analytical Dirac theory
for graphene ribbons.
