# 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?

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.

• Thank you for your interest in the question. I'm little busy right now, so I'll take a deeper look later on. @svavil
– VlS
Dec 8 '15 at 9:17
• You are totally right, thank you for your answer, really helped me a lot! Just one question, where can I get the notes? @svavil
– VlS
Dec 20 '15 at 23:09
• The team behind the course is in the process of publishing. I'll post back here as soon as we release them. Dec 20 '15 at 23:30
• @VlS added a link to the full lecture notes. Mar 26 '16 at 19:41