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

TL;DR

This work develops a Bloch-theorem–like framework tailored to single-walled carbon nanotubes by exploiting cylindrical rotation-translational symmetry. It introduces a reciprocal tube concept, constructs corresponding Brillouin zones, and derives a differential equation for the periodic component of the wavefunction, all within a cylindrical geometry. A tight-binding scheme for 2p_z orbitals is formulated, yielding explicit secular equations and Hamiltonian/overlap matrices for both first and second nearest neighbors. The approach extends zone-folding ideas beyond flat graphene, enabling systematic analysis of armchair, zigzag, and chiral SWCNTs with curvature effects accounted for through symmetry-adapted transformations. The methodology lays groundwork for computational electronic-structure studies and potential DFT-based refinements in SWCNTs.

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.

Figures (9)

• Figure 1: (a): Hexagonal unit cell. (b): Image of the segment $A_1M$ on the armchair tube ${\cal A}$.
• Figure 2: Unrolled lattice of the tube $\tilde{{\cal A}}$, the reciprocal of an armchair nanotube ${\cal A}$, and the Brillouin zone (the deep blue hexagon).
• Figure 3: Domain (the deep blue rectangle) of the transformations $\tilde{{\cal T}}_j^\pm{\bf k}$ associated with an armchair nanotube ${\cal A}$.
• Figure 4: (a): Graphene sheet and the chiral and translation vectors ${\bf C}_h$ and ${\bf T}$. (b): Projections of the vectors $j{\bf a}_\pm$ ($j=1,2$) on the $x$ and $z$-axes.
• Figure 5: Chiral symmetry: the unrolled parallelogram of periods $B_1B_2B_3B_4$ of the reciprocal tube $\tilde{{\cal A}}$, and the Brillouin hexagon $S_1S_2\ldots S_6$.