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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?

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    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.Subbands created by transverse wavevector quantization in carbon nanotubes

    Source: my lecture notes from 'Graphene and graphene-based materials' course.

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

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