Single-walled carbon nanotubes: the Bloch theory, reciprocal tubes, and a tight-binding approximation

Abstract: The electronic structure of a graphene sheet is altered when it is rolled up to form a single-walled carbon nanotube (SWCNT), and the curvature effects for small radius nanotubes become significant. In the paper, an analogue of the Bloch theory of crystals with translational symmetry to armchair, zigzag, and chiral SWCNTs, cylindrical lattices with rotation-translational symmetry, is proposed. It is based on the use of cylindrical coordinates, three-dimensional characteristic vectors of the lattice, a reciprocal tube, and rotation-translation transformations in the real and reciprocal 3d spaces. The Brillouin zone on the reciprocal tube and the domain of the wave-vector are determined. An analogue of the Bloch theorem for SWCNTs is stated and proved. A tight-binding approximation scheme for orbitals orthogonal to the nanotube surface is described, and the Hamiltonian and overlap matrices associated with the first and second nearest-neighbor tight-binding approximations are derived.