[Paper Review] Topology, connectivity and electronic structure of C and B cages and the corresponding nanotubes

Frederik Leys, C. Amovilli+1 more|arXiv (Cornell University)|Feb 3, 2003

Boron and Carbon Nanomaterials Research4 citations

TL;DR

This paper establishes a one-to-one topological correspondence between boron (B) and carbon (C) cages, showing that B₃₂ corresponds to C₆₀, and uses Hartree-Fock and free-electron network models to analyze their electronic structures. It derives the dispersion relations and density of states (DOS) for planar boron lattices and boron nanotubes via zone folding, revealing a direct link between connectivity, curvature, and electronic band structure, with key differences arising from electronegativity in BN and fluorinated fullerenes.

ABSTRACT

After a brief discussion of the structural trends which appear with increasing number of atoms in B cages, a one-to one correspondence between the connectivity of B cages and C cage structures will be proposed. The electronic level spectra of both systems from Hartree-Fock calculations is given and discussed. The relation of curvature introduced into an originally planar graphitic fragment to pentagonal 'defects' such as are present in buckminsterfullerene is also briefly treated. A study of the structure and electronic properties of B nanotubes will then be introduced. We start by presenting a solution of the free-electron network approach for a 'model boron' planar lattice with local coordination number 6. In particular the dispersion relation E(k) for the pi-electron bands, together with the corresponding electronic Density Of States (DOS), will be exhibited. This is then used within the zone folding scheme to obtain information about the electronic DOS of different nanotubes obtained by folding this model boron sheet. To obtain the self-consistent potential in which the valence electrons move in a nanotube, 'the March model' in its original form was invoked and results are reported for a carbon nanotube. Finally, heterostructures, such as BN cages and fluorinated buckminsterfullerene, will be briefly treated, the new feature here being electronegativity difference.

Motivation & Objective

• To establish a one-to-one correspondence between the connectivity of B and C cages based on Euler’s theorem, showing that B₃₂ corresponds to C₆₀.
• To analyze the electronic structure of B and C cages using Hartree-Fock calculations, highlighting similarities in their level spectra.
• To investigate how curvature in planar graphitic fragments leads to pentagonal defects, as seen in fullerenes.
• To model the electronic properties of boron nanotubes by folding a 2D model boron lattice using the zone folding scheme.
• To extend the analysis to heterostructures such as BN cages and fluorinated fullerenes, emphasizing the role of electronegativity differences.

Proposed method

• Uses Euler’s theorem to establish a topological correspondence between B and C cages, where the number of pentagonal faces and coordination numbers (5- and 6-fold) are conserved.
• Applies Hartree-Fock calculations to Bₙ cages (n = 30–54) with nuclei constrained on a spherical surface, revealing a dominance of triangular faces and ~12 five-coordinated atoms in large cages.
• Employs a free-electron network model akin to Kirchhoff’s circuit laws, where π-electrons move along bonds with local coordination number 6, to derive the dispersion relation E(k) and DOS for a 2D model boron lattice.
• Uses the zone folding scheme to map the 2D band structure onto one-dimensional energy bands and DOS for boron nanotubes of various chiralities.
• Applies the March model (spherical surface charge approximation) to self-consistently determine the potential in which valence electrons move in infinite nanotubes, using a generalized form of the original model.
• Analyzes heterostructures like BN and fluorinated C₆₀ by introducing electronegativity differences via modified Coulomb integrals (α_B and α_N), leading to energy gap formation in BN layers.

Experimental results

Research questions

• RQ1 Is there a one-to-one topological correspondence between B and C cages based on their connectivity and face structure?
• RQ2 How does curvature in planar graphitic fragments lead to pentagonal defects, and how is this reflected in the electronic structure of fullerenes?
• RQ3 What is the electronic band structure of boron nanotubes, and how does it emerge from the zone folding of a 2D boron lattice?
• RQ4 How do electronegativity differences between B and N atoms affect the electronic structure of BN layers and related heterostructures?
• RQ5 To what extent can the March model and free-electron network approach predict the electronic properties of nanotubes and fullerenes?

Key findings

• A one-to-one correspondence is established between B and C cages: for example, C₆₀ corresponds to B₃₂, with both systems showing similar electronic level spectra from Hartree-Fock calculations.
• In large Bₙ cages (n ≥ 30), the number of triangular faces dominates, and the number of five-coordinated B atoms approaches ~12, consistent with the icosahedral symmetry of fullerenes.
• The dispersion relation E(k) for the 2D model boron lattice is derived, and the resulting DOS is compared to that of graphene, showing analogous band broadening from molecular orbitals.
• Using the zone folding scheme, the 1D energy bands and DOS of boron nanotubes are obtained from the 2D band structure, enabling prediction of electronic behavior in different chiralities.
• For BN layers, the electronegativity difference between B and N leads to a finite energy gap of ~4 eV, with the π-bands split into two sub-bands separated by 2δ, where δ = ½(α_B − α_N).
• In contrast to BN, graphene’s π-bands touch at the Dirac point due to α_B = α_N, resulting in δ → 0 and no band gap, highlighting the critical role of electronegativity in electronic structure.