Symmetry properties of vibrational modes in graphene nanoribbons

We present symmetry properties of the lattice vibrations of graphene nanoribbons with pure armchair (AGNRs) and zigzag edges (ZGNRs). In nonsymmorphic nanoribbons, the phonon modes at the edge of the Brillouin zone are twofold degenerate whereas the phonon modes in symmorphic nanoribbons are nondegenerate. We identified the Raman-active and infrared-active modes. We predict 3N and 3 (N+1) Raman-active modes for N-ZGNRs and N-AGNRs, respectively (N is the number of dimers per unit cell). These modes can be used for the experimental characterization of graphene nanoribbons. Calculations based on density-functional theory suggest that the frequency splitting of the LO and TO modes in AGNRs (corresponding to the E-2g mode in graphene) exhibits characteristic width and family dependence. Further, all graphene nanoribbons have a Raman-active breathing like mode, the frequency of which is inversely proportional to the nanoribbon width and thus might be used for experimental determination of the width of graphene nanoribbons.