Phonon dispersion of carbon nanotubes

We present the phonon dispersion relations of single-wall carbon nanotubes calculated within a force-constants approach. By using the full symmetry group of the tubes, we are able to calculate the dispersion relations for any chirality starting from one single carbon atom. We find an overbending in the highest optical branch between 6 and 12 cm−1 independent of the tube diameter. The order of the high-energy modes at the Γ-point differs from the results derived from simple zone folding. The splitting between the two Raman active optical modes with A1 symmetry at the Γ-point of chiral tubes is ≈4 cm−1 for typical diameters; it increases with decreasing tube diameter.