Local density of states at zigzag edges of carbon nanotubes and graphene

The electron-phonon matrix element for edge states of carbon nanotubes and graphene at zigzag edges is calculated for obtaining renormalized energy dispersion of the edge states. Self-energy correction by electron-phonon interaction contributes to the energy dispersion of edge states whose energy bandwidth is similar to phonon energy. Since the energy uncertainty of the edge state is larger than temperature, we conclude that the single-particle picture of the edge state is not appropriate when the electron-phonon interaction is taken into account. The longitudinal acoustic phonon mode contributes to the matrix element through the on-site deformation potential because the wave function of the edge state has an amplitude only on one of the two sublattices. The on-site deformation potentials for the longitudinal and in-plane tangential optical phonon modes are enhanced at the boundary. The results of local density of states are compared with the recent experimental data of scanning tunneling spectroscopy.