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The electron-phonon matrix element for edge states of carbon nanotubes andgraphene at zigzag edges is calculated for obtaining renormalized energydispersion of the edge states. Self-energy correction by electron-phononinteraction contributes to the energy dispersion of edge states whose energybandwidth is similar to phonon energy. Since the energy-uncertainty of the edgestate is larger than temperature, we conclude that the single-particle pictureof the edge state in not appropriate when the electron-phonon interaction istaken into account. The longitudinal acoustic phonon mode contributes to thematrix element through the on-site deformation potential because thewavefunction of the edge state has an amplitude only on one of the twosublattices. The on-site deformation potentials for the longitudinal andin-plane tangential optical phonon modes are enhanced at the boundary. Theresults of local density of states are compared with the recent experimentaldata of scanning tunneling spectroscopy.Comment: 8 pages, 5 figure

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