Reciprocal tube tight-binding approximation method for carbon nanotubes

Technical Summary: Reciprocal Tube Tight-Binding Approximation Method for Carbon Nanotubes

Problem Statement
The electronic structure of single-walled carbon nanotubes (SWCNTs) is fundamentally altered when a graphene sheet is rolled into a cylinder, particularly for nanotubes with small radii where curvature effects become significant. While the zone-folding method and standard tight-binding approximations based on graphene are widely used, they exhibit limitations. Comparisons with density functional theory (DFT) have shown that for small-radius nanotubes, the band gaps calculated by traditional tight-binding methods can deviate by more than 50% from DFT results. Furthermore, existing techniques often rely on one-dimensional wave-vectors and treat the Brillouin zone as a segment, which may not fully capture the complexity of chiral, armchair, and zigzag structures, especially when curvature is non-negligible. The goal of this work is to advance the tight-binding approximation technique specifically for narrow and intermediate-radius nanotubes where graphene-based approximations are inefficient.

Methodology
The paper proposes a modified Bloch theory and tight-binding approximation framework that operates simultaneously in real and reciprocal three-dimensional (3D) spaces. The core of the methodology involves the following components:

1. Reciprocal Nanotubes: The author introduces the concept of a "reciprocal nanotube" described in reciprocal cylindrical coordinates. Just as the real nanotube is formed by rolling a graphene sheet, the reciprocal tube is constructed by rolling a portion of the reciprocal lattice around the reciprocal axis.
2. Unit Cells and Transformations: The unit cell for the real nanotube is defined as a hexagon rolled around the tube, while the reciprocal unit cell is a parallelogram rolled around the reciprocal tube. The method employs simultaneous rotation-translation transformations ($T^\pm_j$ in real space and $\tilde{T}^\pm_j$ in reciprocal space) to map these unit cells across the entire structure.
3. 2D Bloch Sums and Brillouin Zones: Unlike traditional methods that use 1D wave-vectors, this approach constructs the Bloch sum for tight-binding approximation as a double sum along two intersecting helices. This structure naturally leads to a two-dimensional first Brillouin zone.
   • For chiral nanotubes $(n, m)$, the first Brillouin zone is a rectangle.
   • For armchair $(n, n)$ and zigzag $(n, 0)$ nanotubes, the first Brillouin zone is a square.
4. Bloch Theorem Analogue: The author proves an analogue of the Bloch theorem for these 2D cylindrical quasi-hexagonal lattices. They demonstrate that the Bloch state $\psi(r; k)$ satisfies $\psi(T^\pm_j r; \tilde{T}^\pm_j k) = e^{\pm ij\kappa_\pm c_\pm} \psi(r; k)$, where $u(r; k)$ is invariant under the simultaneous application of real and reciprocal transformations.
5. Hamiltonian and Overlap Integrals: The study derives the secular equation for $2p_z$-orbitals. It computes Hamiltonian and overlap matrices by establishing relations between the cylindrical coordinates of the nanotube axis and the local spherical coordinates of the atoms. Calculations are performed for both first and second nearest-neighbor approximations.

Key Contributions

• Conceptual Framework: The introduction of the reciprocal tube and the associated rotation-translation transformations in both real and reciprocal spaces provides a unified geometric description for chiral, armchair, and zigzag SWCNTs.
• 2D Brillouin Zone: The construction of a 2D first Brillouin zone (rectangular or square) distinguishes this method from zone-folding techniques and helical structure methods that utilize 1D wave-vectors.
• Theoretical Proof: The paper provides a rigorous proof of the Bloch theorem analogue for SWCNTs, establishing the specific phase factors and invariance properties of the wave functions under the defined transformations.
• Numerical Implementation: The author implements the method to calculate electronic band structures for sample nanotubes, handling the macroscopic number of terms in the double sums efficiently.

Results
Numerical results based on the first nearest-neighbor tight-binding approximation were reported for various chiral, armchair, and zigzag nanotubes:

• Chiral Nanotubes: For narrow chiral tubes, the upper and lower energy dispersion curves touch along two symmetric curves within the 2D Brillouin zone. As the radius increases, these curves degenerate into single points, eventually leading to a small, visible band gap for certain chiralities (e.g., $A(15, m)$ with specific $m$ values). The study found no correlation between the metallic nature of the tube and the condition that $n-m$ is a multiple of 3, contrary to some previous predictions.
• Armchair Nanotubes: The electronic structure of armchair tubes is closest to that of graphene, often forming Dirac cones. However, for narrow tubes ($n \le 5$), the curves are smooth at the touching points. For intermediate radii ($6 \le n \le 10$), a small band gap appears, which vanishes for $n \ge 11$, where the touching points become Dirac points.
• Zigzag Nanotubes:
  • Small radius tubes ($n=4, 5, 6$) are metallic, exhibiting a 2D zero-band-gap domain where the upper and lower dispersion surfaces overlap.
  • Intermediate radius tubes ($7 \le n \le 13$) are semi-metallic, with the 2D domains degenerating into two symmetric short curves.
  • Large radius tubes ($n \ge 14$) behave like graphene, with curves touching at two points to form Dirac cones, indicating that curvature effects become insignificant.
• Second Nearest-Neighbor Approximation: The inclusion of second nearest-neighbor terms did not yield visible improvements, as the additional integrals were of the order $10^{-5}$, comparable to the imaginary parts of eigenvalues in the first nearest-neighbor approximation.

Significance and Claims
The paper claims that the proposed reciprocal tube tight-binding approximation is efficient for the entire range of nanotubes, including narrow, intermediate-radius, and "megatubes." The method addresses the inefficiency of graphene-based approximations for small-radius nanotubes by explicitly accounting for curvature through the geometry of the reciprocal tube and the 2D Brillouin zone. The author asserts that their numerical results are consistent with DFT methods regarding the metallic nature of small-radius zigzag nanotubes. The work establishes a theoretical foundation for analyzing SWCNTs that treats the real and reciprocal spaces symmetrically, offering a distinct alternative to zone-folding and helical symmetry methods.