Lattice dynamics of single-walled achiralBC3nanotubes

Abstract

The phonon dispersion relation and specific heat of single-walled ${\mathrm{BC}}_{3}$ nanotubes have been investigated using a force constant model. The obtained phonon dispersion relation of ${\mathrm{BC}}_{3}$ sheet reproduces well the experimental data. The tube-diameter dependent frequencies of both the radial breathing mode and the lowest phonon mode ${E}_{2\mathrm{g}}$ can be well fitted by a power law $\ensuremath{\omega}=C∕{R}^{\ensuremath{\alpha}}$ with tube radius $R$, where the scaling exponent $\ensuremath{\alpha}=1$ and the proportional constant $C=939.6\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}\phantom{\rule{0.2em}{0ex}}\mathrm{\AA{}}$ in the former and $\ensuremath{\alpha}=1.1$ and $C=321.5\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}\phantom{\rule{0.2em}{0ex}}{\mathrm{\AA{}}}^{1.1}$ in the latter, at variance with carbon nanotubes and BN nanotubes. The specific heat of ${\mathrm{BC}}_{3}$ nanotubes are also calculated, less than that of the ${\mathrm{BC}}_{3}$ sheet, in which several crossings are observed at low temperature due to the first optical phonon mode excited at different temperature. By virtue of the simple zone-folding model, in addition, a universal formula is derived to describe the tube diameter dependence of specific heat for various types of nanotube systems. The results provide an alternative way to characterize the ${\mathrm{BC}}_{3}$ nanotubes and suggest the underlying quantized phonon structures in one-dimensional nanotube systems.