From ${\mathrm{C}}_{60}$ to a fullerene tube: Systematic analysis of lattice and electronic structures by the extended Su-Schrieffer-Heeger model

The development from ${\mathrm{C}}_{60}$ and ${\mathrm{C}}_{70}$ to an infinitely long tube is studied by changing the carbon number N. The extended Su-Schrieffer-Heeger Hamiltonian is applied to various geometrical structures and solved for the half-filling case of π electrons. For finite N (∼100), appreciable dimerizations (∼0.01 Å) exist, and a fairly large gap (∼0.1–1 eV) remains. The solution, which includes the perfect Kekulé structure, always gives the lowest energy. Other solutions, where there are deviations from the Kekulé structure, have higher energies. When N goes to infinity, the strength of the unique dimerization pattern, i.e., the perfect Kekulé structure, becomes too small to be observed, but the gap width (≃0.02 eV) is comparable to room temperature and can be measured. Therefore, the infinitely long tube will have properties like those of semiconductors with a very narrow gap. We would not expect perfect metallic properties, but peculiar properties due to the small gap could be observed in experiments.