Accurate density functional calculations for the phonon dispersion relations of graphite layer and carbon nanotubes

Accurate calculations for the phonon dispersion relations of single-wall armchair and zigzag nanotubes are presented. The calculations are performed using a plane-wave basis set and density functional theory. To ensure the accuracy of the presented calculations, the phonon dispersion relation of an isolated graphite layer is calculated and the results are compared to experiment. Errors are small, but some notable discrepancies between experiment and theory are observed and discussed. For armchair and zigzag nanotubes the dependence of Raman-active and infrared-active modes on the radius is investigated in detail concentrating on the modes in the G band. The results are compared to those predicted by the zone-folding method using the calculated force constants for graphite. We find a general softening of most high-frequency modes and a substantial lowering of one particular longitudinal $A_1$ mode in metallic tubes. We associate this mode with the Breit-Wigner-Fano lines observed usually in metallic tubes. The precise electronic mechanism leading to the softening of the longitudinal $A_1$ mode is discussed in detail.