Low-frequency phonons in carbon nanotubes: A continuum approach

Abstract

Low-frequency phonons in carbon nanotubes are studied using a continuum model which allows consideration of an arbitrary wall thickness for the nanotube. Phonon dispersion relations are calculated for two archetypal examples of carbon nanotubes, the ͑5,5͒ and ͑10, 10͒ tubes. The dependence of the radial breathing mode frequency at ⌫ on the inverse nanotube diameter is verified within this model; furthermore, we prove it to hold for all pure modes within the thin-shell approximation. The effect of the nanotube wall thickness on the eigenfrequencies of carbon nanotubes is also analyzed, and a criterion to fix this parameter within a continuum model is presented. We compare our results to other continuum approaches, and show that by choosing the appropriate parameters, excellent agreement with recent first-principles calculations can be achieved.

Theory of coherent phonons in carbon nanotubes and graphene nanoribbons

“…Phonons in graphene, SWNTs and GNRs have been studied by a number of techniques including elastic continuum models [93][94][95][96][97][98][99], valence force field models [34,[100][101][102][103] bond charge models [104], and ab initio methods [105][106][107][108][109][110].…”

Carbon Nanostructures as an Electromechanical Bicontinuum

“…Vibrations in carbon nanostructures such as tubes, fullerenes, or graphene sheets [1,2,3] have a ubiquitous influence on electronic, optical and thermal response: scattering from optical phonons limits charge transport in otherwise ballistic nanotube conductors [4,5]; twist deformations gap metallic tubes [6,7]; ballistic phonons transport heat in nanotubes with great efficiency [8,9,10]; resonant Raman spectroscopy can unambiguously identify a tube's wrapping indices (n,m) [11,12,13,14]; electron-phonon interactions may ultimately limit the electrical performance of graphene [15,16]. Computationally intensive atomistic models of lattice dynamics often lack simplified model descriptions that can facilitate insight, yet traditional analytical continuum models [1,2,17,18], while very useful and important, cannot describe atomistic phenomena without phenomenological extensions [19,20,21]. Although continuum models are restricted to long-wavelength physics, they have been used to describe atomic-scale phenomena in bulk binary compounds by incorporating a separate continuum field for each sublattice [23]: in graphene, two fields are necessary.…”

Internal resonance of vibrational modes in single-walled carbon nanotubes

“…Continuum thin-shell model, which neglects non-local atomic interactions, can successfully predict the RBM frequency of SWCNTs (Mahan 2002;Wang & Hu 2005;Chico et al 2006). When flexural waves are involved, non-local continuum model may take the long-range forces of nanotube atoms into account (Zhang et al 2004(Zhang et al , 2005Wang & Hu 2005;Hu et al 2008).…”