Phonon Dispersion Relation of Carbon Nanotube

Some researchers obtained the phonon dispersion relation of carbon nanotube by considering harmonic oscillation of carbon atoms. Recently, we could find an alternative method for determining carbon nanotube band energy by modifying orthogonalized-plane-wave method and combining it with tight binding method. Here, we did some calculations based on these methods and solved the dynamical matrix of carbon nanotube by referring the above methods and using force constant model. Some necessary adjustments are made in order to take into account the effect of the second nearest neighbor atoms which are not harmonic vibrations.