Cutting lines near the Fermi energy of single-wall carbon nanotubes

Analytic expressions for the cutting line number $\mu$ of a single-wall carbon nanotube (SWNT) which gives van Hove singularities for optical transitions are found to exist around $\mu =N∕3$ for most SWNTs, where $N$ is the number of cutting lines of the SWNT. However, for metal-2 SWNTs, the cutting line number shows an irregular dependence on $(n,m)$. In this paper a classification is proposed for metal-2 nanotubes which are classified as either metal-$2p$ or metal-$2m$ SWNTs. The appearance of metal-1, metal-$2p$, and metal-$2m$ SWNTs in a two-dimensional $(n,m)$ plot is not regular.

Figure 1

(a) Cutting lines of (6,3) metal-1 nanotubes on a hexagonal 2D BZ, where $N=42,p=q=1$, and $\mu =14$,28. (b) The cutting lines of (9,3) where $N=78,p=4,q=3$, and $\mu =26$, 52. Starting from the $\Gamma$ point, we meet first $K^{\prime }$ point prior to $K$ point.

Figure 2

Three cutting lines around the $K$ points for (a) metal-1 (6,3), (b) metal-$2p$ (4,1), (c) metal-$2m$ (5,2) SWNTs. While the $K$ point is located at the center of the cutting line for metal-1, the $K$ points are located at a 1:5 (or 5:1) position of the cutting line for metal-$2p$ (or metal-$2m$).

Figure 3

(Color online) Classification of $(n,m)$ single-wall carbon nanotubes. Small open symbols denote type I (red circle) and type II (blue square) semiconducting nanotubes. Solid symbols denote metal-1 (black circle), metal-$2p$ (yellow triangle), metal-$2m$ (green square). The appearance of the solid symbols is not periodic except for zigzag and armchair SWNTs.