Lattice dynamics of single-walled achiral ${\mathrm{BC}}_3$ nanotubes

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 $\omega =C∕R^{\alpha }$ with tube radius $R$, where the scaling exponent $\alpha =1$ and the proportional constant $C=939.6{\mathrm{cm}}^{-1}\mathrm{\AA }$ in the former and $\alpha =1.1$ and $C=321.5{\mathrm{cm}}^{-1}{\mathrm{\AA }}^{1.1}$ 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.

Figure 1

Calculated phonon dispersion relation of the ${\mathrm{BC}}_3$ sheet (solid lines) together with the experimental points (open circles).

Figure 2

(a) PDR and (b) DOS of zigzag $(4,0)$ BCNT; (c) PDR and (d) DOS of armchair $(4,4)$ BCNT.

Figure 3

Frequencies of (a) RBM and (b) $E_{2\mathrm{g}}$ of achiral BCNTs as a function of radius. The solid lines are a linear fit to the data for BCNTs. In the insets, the open and solid circles represent the vibrational patterns of carbon and boron atoms, respectively.

Figure 4

(a) Specific heat of armchair $(4,4)$, $(6,6)$, and $(8,8)$ BCNTs and ${\mathrm{BC}}_3$ sheet; (b) specific heat of BCNTs at $T=300\mathrm{K}$ as a function of the index $n$. For a comparison, the inset shows the result of CNTs.

Figure 5

(a) The 2D Brillouin zones of $(5,5)$ and $(10,10)$ CNTs; (b) PDR of armchair $(5,5)$ CNT (open circles), and $(10,10)$ CNT (solid lines).