Elastic Continuum Models of Phonons in Carbon Nanotubes

Abstract

In this chapter, elastic continuum models are used to describe phonons in carbon nanotubes and vibrational modes in biological structures. Based on elastic continuum theory, acoustic vibrational modes are modeled for both zigzag and armchair nanotubes of finite length using a variational solution of Donnell's equation. The acoustic phonon modes in these calculations are determined for both even and odd modes of the acoustic displacement. The dispersion relations vary with the length of the tube. The displacement field of the nanotube is used to calculate the deformation potential interaction Hamiltonian. In addition, the optical vibrational modes are derived for finite length nanotubes in the elastic continuum approximation. A quantum mechanical normalization prescription is applied to facilitate the determination of the optical phonon modes. The dispersion relation is calculated based on the continuum approach and the quantum normalized amplitude is used to calculate the optical deformation potential. These fully three-dimensional elastic continuum models are compared with other approaches to modeling phonons in carbon nanotubes such as the popular zone-folding technique. In a related underlying topic, the applicability of continuum models for the analysis of nanoscale structures is demonstrated for the case fullerenes. It is shown that the b₂ elongation mode of C₆₀ may be described within the continuum approximation. Indeed, for these fullerene structures, the frequencies of selected vibrational modes are predicted to within a few percent.