Electron-phonon effects in graphene and armchair (10,10) single-wall carbon nanotubes

The electron-phonon interaction in low-dimensional tight-binding systems is discussed. A sheet of graphite, which is two-dimensional, and an armchair single-wall carbon nanotube (SWNT), which is quasi-one-dimensional, are taken as examples. For the modulated hopping the matrix elements for both systems are derived in the context of a two parameter model for the phonon vibrational spectrum. It is found that they (for both structures) display a deformation type of potential, and are reduced by a factor of $\left(1\mathrm{}-R\right),$ where R depends on the phonon parameters. It is also shown that the ordinary electron-phonon coupling displays a deformation type of approximation for both systems. Next, a different type of interaction is considered—the phonon modulated electron-electron interaction. It gives two contributions—random phase approximation with one phonon line and exchange interaction with one phonon line. We find that for the two-dimensional (2D) graphene and for the quasi-1D (10,10) SWNT, the modulated hopping and exchange coupling govern the electron transport at room temperatures.