Phonon dispersions in graphene sheet and single-walled carbon nanotubes

In the present research paper, phonons in graphene sheet have been calculated by constructing a dynamical matrix using the force constants derived from the second-generation reactive empirical bond order potential by Brenner and co-workers. Our results are comparable to inelastic X-ray scattering as well as first principle calculations. At I" point, for graphene, the optical modes (degenerate) lie near 1685 cm (-aEuro parts per thousand 1). The frequency regimes are easily distinguishable. The low-frequency (omega -> 0) modes are derived from acoustic branches of the sheet. The radial modes can be identified with omega -> 584 cm (-aEuro parts per thousand 1). High-frequency regime is above 1200 cm (-aEuro parts per thousand 1) (i.e. ZO mode) and consists of TO and LO modes. The phonons in a nanotube can be derived from zone folding method using phonons of a single layer of the hexagonal sheet. The present work aims to explore the agreement between theory and experiment. A better knowledge of the phonon dispersion of graphene is highly desirable to model and understand the properties of carbon nanotubes. The development and production of carbon nanotubes (CNTs) for possible applications need reliable and quick analytical characterization. Our results may serve as an accurate tool for the spectroscopic determination of the tube radii and chiralities.