Calculation of phonon dispersion in carbon nanotubes using a continuum-atomistic finite element approach

Dispersion calculations are presented for cylindrical carbon nanotubes using a manifold-based continuum-atomistic finite element formulation combined with Bloch analysis. The formulated finite elements allow any (n,m) chiral nanotube, or mixed tubes formed by periodically-repeating heterojunctions, to be examined quickly and accurately using only three input parameters (radius, chiral angle, and unit cell length) and a trivial structured mesh, thus avoiding the tedious geometry generation and energy minimization tasks associated with ab initio and lattice dynamics-based techniques. A critical assessment of the technique is pursued to determine the validity range of the resulting dispersion calculations, and to identify any dispersion anomalies. Two small anomalies in the dispersion curves are documented, which can be easily identified and therefore rectified. They include difficulty in achieving a zero energy point for the acoustic twisting phonon, and a branch veering in nanotubes with nonzero chiral angle. The twisting mode quickly restores its correct group velocity as wavenumber increases, while the branch veering is associated with a rapid exchange of eigenvectors at the veering point, which also lessens its impact. By taking into account the two noted anomalies, accurate predictions of acoustic and low-frequency optical branches can be achieved out to the midpoint of the first Brillouin zone.