Chiral symmetry analysis and rigid rotational invariance for the lattice dynamics of single-wall carbon nanotubes

We provide a detailed expression of the vibrational potential for the lattice dynamics of single-wall carbon nanotubes (SWCNT’s) satisfying the requirements of the exact rigid translational as well as rotational symmetries, which is a nontrivial generalization of the valence force model for the planar graphene sheet. With the model, the low-frequency behavior of the dispersion of the acoustic modes as well as the flexure mode can be precisely calculated. Based upon a comprehensive chiral symmetry analysis, the calculated mode frequencies (including all the Raman- and infrared-active modes), velocities of acoustic modes, and the polarization vectors are systematically fitted in terms of the chiral angle and radius, where the restrictions of various symmetry operations of SWCNT’s are fulfilled.

Figure 1

Sketch of carbon atoms on a cylindric surface.

Figure 2

The projection of ${\vec{e}}_1^{rc}$ on ${\vec{e}}_1^r$ for tubes $(2n,n)$ with $n∊[1,15]$. It shows that ${\vec{e}}_1^{rc}$ only deviates about 2% from ${\vec{e}}_1^r$ even in the small-radius (2, 1) tube.

Figure 3

The effect of bond lengths on the TW mode in tube (5, 2). (a) All bonds are assumed to be the same. The frequency of the TW mode is nonzero. (b) The differences between bonds are considered. The frequency of the TW mode is precisely zero $({\omega }_{\mathrm{TW}}<{10}^{-4}{\mathrm{cm}}^{-1})$.

Figure 4

As functions of $r$ and $\theta$, we show in (a) the frequency of the ${\vec{e}}_z$ OP mode at $(\kappa ,n)=(0,0)$, (b) $C_{\mathrm{TW}}$ for small-radius tubes, and (c) the $z$ component of the eigenvector of the ${\vec{e}}_r$ AC mode.