Theoretical study of the vibrational edge modes in graphene nanoribbons

We investigate the phonon normal modes in hydrogen-terminated graphene nanoribbons (GNRs) using the second-generation reactive empirical bond order (REBOII) potential and density-functional theory calculations. We show that specific modes, absent in pristine graphene and localized at the GNR edges, are intrinsic signatures of the vibrational density of states of the GNRs. Three particular modes are described in details: a transverse phonon mode related to armchair GNRs, a hydrogen out-of-plane mode present in both armchair and zigzag GNRs, and the Raman radial-breathing-like mode. The good agreement between the frequencies of selected edge modes obtained using REBOII and first-principles methods shows the reliability of this empirical potential for the calculation and the assignment of phonon modes in carbon nanostructures where carbon atoms present a $s{p}^2$ hybridization.