Kohn Anomalies in Graphene Nanoribbons

The quantum corrections to the energies of the $\Gamma$ point optical phonon modes (Kohn anomalies) in graphene nanoribbons are investigated. We show theoretically that the longitudinal optical modes undergo a Kohn anomaly effect, while the transverse optical modes do not. In relation to Raman spectroscopy, we show that the longitudinal modes are not Raman active near the zigzag edge, while the transverse optical modes are not Raman active near the armchair edge. These results are useful for identifying the orientation of the edge of graphene nanoribbons by G band Raman spectroscopy, as is demonstrated experimentally. The differences in the Kohn anomalies for nanoribbons and for metallic single wall nanotubes are pointed out, and our results are explained in terms of pseudospin effects.