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The Structural and Electronic Properties of Carbon Nanotubes

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Book cover Conduction in Carbon Nanotube Networks

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

Carbon has a rich and varied chemistry, forming compounds with almost every other known element of the periodic table. It is also the compounds that it forms with itself, however, that are of great technological interest.

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Notes

  1. 1.

    Note that double-walled CNTs are often referred to as multi-walled CNTs.

  2. 2.

    Specific conductivity, defined as the conductivity normalised by the density, is used to allow comparison of CNT networks with different morphologies and densities.

  3. 3.

    This estimate is obtained by assuming that conductance within a CNT can be considered ballistic over the electron mean free path of carbon nanotubes \(\ell _{\mathrm {m.f.p.}}\sim 1~\mathrm {\mu m}\) (see Chap. 3 for a discussion of ballistic conductance and mean free paths). This defines a conductivity of \(\sigma \sim G\ell _{\mathrm {m.f.p.}}/A\) where the conductance G is of order the quantum conductance \(G_0=7.75\times 10^{-5}~\mathrm {S}\) and A is the CNT cross-sectional area. Assuming that each CNT within the network contributes independently to the conductivity, the specific conductivity of the network as a whole is equal to the specific conductivity of an individual CNT. The mass density of an individual CNT is obtained as \(\rho =N m_C / A d\) where d is the unit length of the tube, \(m_C\sim 10^{-26}~\mathrm {kg}\) is the atomic mass of a carbon atom and \(N=2\pi D d / 3\sqrt{3} a_{CC}^2\) is the number of carbon atoms per unit length d with \(D\sim 1~\mathrm {nm}\) the CNT diameter and \(a_{CC}=1.4\,{\AA }\) the carbon-carbon bond length. The resulting expression for the specific conductivity is

    $$\begin{aligned} \sigma ^{\prime } = \frac{\sigma }{\rho } = \frac{3\sqrt{3}G_0\ell _\mathrm{m.f.p.}a_{CC}^2}{2\pi D m_C}. \end{aligned}$$
    (2.1)
  4. 4.

    For clarity, we omit the normalisation of the wavefunction.

  5. 5.

    We assume the graphene lattice vectors to be

    $$\begin{aligned} \mathbf {a}_1 = (a_{1,x},a_{1,y}) = \left( \frac{\sqrt{3}}{2}, \frac{1}{2}\right) a, \quad \quad \mathbf {a}_2 = \left( \frac{\sqrt{3}}{2},-\frac{1}{2}\right) a. \end{aligned}$$
  6. 6.

    Placed in each spin channel.

  7. 7.

    These points are equivalent under time-reversal symmetry.

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  1. Cavendish Laboratory, Theory of Condensed Matter Group, University of Cambridge, Cambridge, UK

    Robert A. Bell

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Bell, R.A. (2015). The Structural and Electronic Properties of Carbon Nanotubes. In: Conduction in Carbon Nanotube Networks. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-19965-8_2

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