Bloch theory for single-walled carbon nanotubes

Characterization of electronic structure of SWCNTs is a challenging problem. Traditionally, a single unit cell of armchair, zigzag, and chiral nanotubes defined as a part of the unrolled nanotube is a rectangle formed by the chiral vector C=(n,m) and the translation vector T and comprises N hexagons, with N being 28 for C=(4,2) and 194 for C=(8,3) (Saito at al, Physical Properties of Carbon Nanotubers, 1998). This approach also employs a flat reciprocal lattice and defines the first Brillouin zone as a single line segment. The tight-binding approximation method for recovering electronic structure of SWCNTs available in the literature is based either on the graphene model (Hamada et al, Phys. Rev. Lett. 1992) or its modifications by including some effects due to curvature (Blaise et al, Phys. Rev. Lett. 1994). The goal of this work is to develop an analogue of the Bloch theory for crystals with translational symmetry to SWCNTs of any chirality (n,m). A unit cell employed is a single hexagon rolled over the nanotube surface. Characteristic three-dimensional vectors of the real nanotube and its reciprocal tube in the three-dimensional reciprocal space are introduced. The first Brillouin zone of the reciprocal tube is, in general, an irregular hexagon rolled around the reciprocal tube, and its sides are elliptical arcs. Its symmetric properties depend on the chirality vector of the SWCNT. Eigenfunctions of the Hamiltonian on a SWCNT subjected to cyclic boundary conditions and their properties are discussed. An analogue of the Bloch theorem for nanotubes is stated and proved. A modification of the tight-binding method for SWCNTs is proposed.