We present a detailed study of the vibrational properties of single wall carbon nanotubes (SWNTs). The phonon dispersions of SWNTs are strongly shaped by the effects of electron-phonon coupling. We analyze the separate contributions of curvature and confinement. Confinement plays a major role in modifying SWNT phonons and is often more relevant than curvature. Due to their one-dimensional character, metallic tubes are expected to undergo Peierls distortions (PD) at $T=0\mathrm{K}$. At finite temperature, PD are no longer present, but phonons with atomic displacements similar to those of the PD are affected by strong Kohn anomalies (KA). We investigate by density functional theory (DFT) KA and PD in metallic SWNTs with diameters up to $3\mathrm{nm}$, in the electronic temperature range from $4\mathrm{K}$ to $3000\mathrm{K}$. We then derive a set of simple formulas accounting for all the DFT results. Finally, we prove that the static approach, commonly used for the evaluation of phonon frequencies in solids, fails because of the SWNTs reduced dimensionality. The correct description of KA in metallic SWNTs can be obtained only by using a dynamical approach, beyond the adiabatic Born-Oppenheimer approximation, by taking into account nonadiabatic contributions. Dynamic effects induce significant changes in the occurrence and shape of Kohn anomalies. We show that the SWNT Raman $G$ peak can only be interpreted considering the combined dynamic, curvature and confinement effects. We assign the $G^{+}$ and $G^{-}$ peaks of metallic SWNTs to TO (circumferential) and LO (axial) modes, respectively, the opposite of semiconducting SWNTs.

• Received 23 September 2006

DOI:https://doi.org/10.1103/PhysRevB.75.035427