Acoustic and optical phonons in nanotubes

The elastic continuum model is applied for single wall nanotubes (SWNTs) and multiple wall nanotube (MWNTs) with the nanotube approximated as an elastic cylindrical membrane of finite thickness. Donnell´s equations of motion for a cylindrical membrane are solved using the Rayleigh-Ritz method to give the acoustic vibrational modes of SWNTs. The displacement modes are used to calculate the deformation potential and the dispersion relations are compared for various diameter nanotubes. Results indicate that phonon-bottleneck effects should be important for the short nanotubes being considered for nanotube-based devices exhibiting quasi-ballistic transport.