Symmetry of the carbon nanotube modes and their origin from the phonon branches of graphene

A group-theory method to relate the spectra of tubular and parent planar structures is elaborated. The spectra correlation is obtained using a virtual intermediate planar periodic structure, which has a local symmetry coinciding with that of the tube surface. The method is applied to deduce the zone-center phonon modes of carbon nanotubes from the dispersion branches of the graphene monolayer. The results obtained open an additional way for experimental indexing of the carbon nanotubes by means of Raman spectroscopy, and the origin of the Raman-active modes in the achiral nanotubes is determined. We demonstrate that the number of modes in the $G$ band and the relation between their frequencies are governed by the nanotube type. The proposed theory can be easily explored to understand the phonon and electron spectra of many other tubular structures (boron nitride, transition metal dichalcogenides) which have emerged in the past decade.

Figure 1

Structures of the single-walled carbon nanotubes are presented on the cylindrical surfaces. Panels (a) and (b) show achiral zigzag ($\mathit{P}=7{\mathit{a}}_1$) and armchair ($\mathit{P}=7{\mathit{a}}_1+7{\mathit{a}}_2$) nanotubes respectively, while panel (c) demonstrates the chiral ($\mathit{P}=8{\mathit{a}}_1+2{\mathit{a}}_2$) nanotube. Mirror planes ${\sigma }_h$ and ${\sigma }_v$ are shown in achiral structures (a, b). In all three panels bold arrows show the characteristic translations ${\mathit{a}}_{\phi }$. The nanotube unit cells are located between two dotted circles.

Figure 2

Formation of the SWCNT first Brillouin zone from the nonequivalent lines cutting the reciprocal space of the GL is shown. Panels (a–c) demonstrate zigzag ($\mathit{P}=5{\mathit{a}}_2$), armchair ($\mathit{P}=10{\mathit{a}}_1-5{\mathit{a}}_2$), and chiral ($\mathit{P}=-2{\mathit{a}}_1+7{\mathit{a}}_2$) nanotubes, respectively. Circles denote the nodes of the reciprocal lattice with the basis vectors defined by (14-15). Nodes with the same numbers are equivalent with respect to the GL reciprocal lattice translations. In both achiral cases $N=10$ and the number of equally spaced cutting lines is also 10, while for chiral nanotube $N=26$. The point $\mu =N/2$ is the $M$ point of the GL reciprocal space, which is translationally equivalent to the point $\frac{1}{2}{\mathit{b}}_1$of the first Brillouin zone.

Figure 3

The origin of Raman-active modes in the zigzag, armchair, and chiral SWCNTs is shown in panels (a–c), respectively. Filled circles correspond to the Raman-active modes. The $\mathit{\Gamma }$, $\mathbf{M}$, and $\mathbf{X}$ points belong to the boundary the first Brillouin zone. The exact location of the $\mathbf{X}$ point is determined by the fact that the ${\mathit{b}}_{\phi }$ vector (14) is directed to it. Phonon dispersion curves of the GL are adapted from Ref. 1.

Figure 4

Atomic shifts corresponding to the functions $p_j$ (shown for one hexagon in each case) in the achiral SWCNTs. Functions of the iLO and iTO type are presented in panels (a, b) and (c, d), respectively. Functions shown in (b, c) are even with respect to ${\sigma }_h$, and the ones in (a, d) are odd.