Single-Walled Carbon Nanotubes: Structures and Symmetries
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Single-walled carbon nanotubes (SWNTs) represent a novel class of low-dimensional materials exhibiting exceptional electronic properties. Many of these properties were predicted by theory, and then confirmed by experiment after the successful synthesis of single-walled carbon nanotubes. In this work, we provide an introduction to the structure and symmetries of SWNTs with emphasis on the use of their helical symmetry. We start by showing how all extended SWNTs can be envisioned by rolling up a single sheet of graphite. We then show that all such SWNTs have translational symmetry along the nanotube axis, but often with hundreds, if not thousands, of atoms in the translational unit cell. Next, we show how all SWNTs defined by rolling up a single sheet of graphite have both rotational and helical symmetries that can be used to reduce the number of atoms necessary to generate any SWNT to two. (A Fortran program that may be used to calculate the x, y, z coordinates for any SWNT defined by rolling up the graphene sheet is provided in Appendix A. This program is based on the helical and rotational symmetries of SWNTs.) We then illustrate some of the uses of helical and rotational symmetries in the calculation of the band structures of SWNTs. Finally, we conclude with brief comments.