A global prediction of the Kataura plot for chiral carbon nanotubes: Topological family effect revealed in the natural helical crystal lattice scheme

The optical transition energies of chiral carbon nanotubes (CNTs) are intimately related to their geometrical structures, and are frequently applied in the recognition and isolation of single nanotubes. Conventional translational crystal lattice (TCL) scheme failed to offer an accurate description of the optical properties of the chiral CNTs due to the formidable unit cell size constraint as well as the complexity of the band structures. Most fitting equations for the Kataura plot, usually based on the empirical correlations with the chiral indices of the CNTs, lack a clear physical interpretation. In this study, we apply the natural helical crystal lattice (NHCL) scheme to systematically analyze the band gap variation of arbitrary chiral CNTs. We derive a global analytical formula that takes into account the comprehensive geometrical connection topologies for the band gap of all (n, m)-CNTs with a diameter larger than 1.2 nm. With the inclusion of the many-body effect, we are able to provide a global Kataura plot prediction with a minimal set of only two fitting parameters, and the results are consistent with the experimental Kataura plot within 0.67% deviation for over 1300 data points. Our formula precisely captures the observed experimental branching features and presents a clear physical rationalization for the topological family effect on the optical excitation energies of chiral CNTs.