Abstract
The first reports on synthesis of carbon [1] and transition metal chalcogenide [2] nanotubes triggered extensive research on both organic and inorganic nanostructures which proved to have potential of becoming a key nanotechnological material due to the outstanding physical properties. It was found that many compounds which crystallize in a bulk or/and in a layered form can grow into the cylindrical structures, under specific conditions. After the discoveries of nanotubes made of carbon, transition metal chalcogenides and oxides [3], boron nitride [4], silicon [5] and metal (e.g., Au [6]), recent discovery of the functional semiconducting oxide nanostructures [7] paved the way for synthesis of diverse nanosized forms of zinc oxide as well. Diameters of the synthesized nanotubes (or lateral dimensions of the other nanostructures) vary from few Angstroms to few micrometers. In this chapter we derive symmetry of arbitrary nanotubes and discuss their common symmetry-based properties. Then we focus on carbon nanotubes: deriving easily many of their famous properties, we show that symmetry is the most profound way of understanding them.
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Notes
- 1.
Recall that \(\tilde{x}=x/n\) (division by the line group parameter n), while \(\overline{x},\underline{x}\) denote numerator and denominator of the rational \(x=\overline{x}/\underline{x}\).
- 2.
However, if the hamiltonian is real, then the time reversal must be taken into account to predict band degeneracy and topology, Sect. 8.1.2.
- 3.
As in the considered case there is a single orbit and single orbital per atom, the term \({\mid~\!\!\!{{A,\psi_A}}~\!\!\rangle}{{\boldsymbol r}}a{B,\psi_B}\) determining block \(H^{\downarrow{\lambda}}_{AB}\) in the pulled down hamiltonian matrix (8.22) reduces to the superfluous factor, projector \({\mid~\!\!\!{{000}}~\!\!\rangle}{{\boldsymbol r}}a{000}\).
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Damnjanović, M., Milošsević, I. (2010). Nanotubes. In: Line Groups in Physics. Lecture Notes in Physics, vol 801. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11172-3_9
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