Electronic and Optical Properties of Narrow-Gap Carbon Nanotubes

The sp³ tight-binding model is used to calculate electronic structures of narrow-gap carbon nanotubes. The curvature effects cause a small energy gap, a one-dimensional parabolic bands, and a shift in the Fermi wave vector. The energy gap is described by the approximate relation Eg ≃ 5|V_ppπ|b² cos(3θ)/16R_d². It is almost identical to the first-principles result. Due to the curvature effects, the optical spectra exhibit single-particle absorption peaks and a plasmon structure. They are consistent with the experimental results, the absorption peaks in the optical conductivity and the plasmon edge in the reflectance spectrum, thereby verifing the relation between Eg and (R _d, θ).