The sensitive structural dependence of the optical properties of single-walled carbon nanotubes, which are dominated by excitons and tunable by changing diameter and chirality, makes them excellent candidates for optical devices. Because of strong many-electron interaction effects, the detailed dependence of the optical oscillator strength f(s) of excitons on nanotube diameter d, chiral angle θ, and electronic subband index P (the so-called family behavior), however, has been unclear. In this study, based on results from an extended Hubbard Hamiltonian with parameters derived from ab initio GW plus Bethe-Salpeter equation (GW-BSE) calculations, we have obtained an explicit formula for the family behavior of the oscillator strengths of excitons in semiconducting single-walled carbon nanotubes (SWCNTs), incorporating environmental screening. The formula explains recent measurements well and is expected to be useful in the understanding and design of possible SWCNT optical and optoelectronic devices.
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