Metallic and Semiconducting Nanotubes, symmetry discussion I'm interested in band gaps of Single-walled Carbon Nanotubes (SWNTs).
I know that there are three kinds of SWNTs:

* Zigzag : $(n,0)$

* Armchair : $(n,n)$

* Chiral : $(n,m)$
Electical properties of SWNTs depend on indices $(n,m)$:
 Because of the symmetry and unique electronic structure of graphene, the structure of a nanotube strongly affects its electrical properties. For a given (n,m) nanotube, if n = m, the nanotube is metallic; if n − m is a multiple of 3, then the nanotube is semiconducting with a very small band gap, otherwise the nanotube is a moderate semiconductor.

However, this rule has some exceptions.
Where I can find proof of this statement or how I can prove it using symmetry of SWNTs?  
 A: I'll give the explanation that helped me.
First, the diameter of the nanotube depends on the indices as $$d = \frac{a}{\pi}\sqrt{n^2+nm+m^2}.$$
Electrons in the nanotube will have a momentum vector $k$. The electronic properties will then depend on the orientation of this vector with respect to the Brillouin zone.
The momentum perpendicular to the nanotube axis, $k_\perp$, is quantized (as in 'electrons don't leave the nanotube'): $$k_\perp = \frac{2\pi \ell}{nd}.$$
This quantization creates subbands separated by $\Delta k \sim \frac{1}{d}$. Two possible scenarios are:


*

*If the subband does not pass the Dirac point ($K$ in the picture below), the intersection of the subband and the energy surface of a graphene sheet is a gapped energy dispersion curve. The material will be semiconducting.

*If the subband passes the Dirac point precisely, the intersection of the subband and the energy surface of a graphene sheet is a Dirac-like linear spectrum. The material will be metallic.


I'm illustrating this below.
Source: my lecture notes from 'Graphene and graphene-based materials' course.
