Trions in semiconducting single-walled carbon nanotubes

We study trions (charged excitons), a complex of an electron-hole pair and an additional electron or hole, in semiconducting single-walled carbon nanotubes (s-SWCNT), by means of the exact diagonalization of the realistic Hamiltonian based on the k . p scheme and on the screened Hartree-Fock approximation. By comparing different classes of models that partially or fully include the band nonparabolicity, the form factors in the density operators, the screening effects on interaction, and the self-energy correction in the energy bands, we succeed in capturing the essential features of s-SWCNT. It turns out that the trion binding energy is significantly suppressed by the form factor as well as by the screening effect. Further, an unconventional feature of s-SWCNT is found: the trion has a larger binding energy than the biexciton, since the biexcitons are more strongly affected by screening than trions. We also consider the effects of the short-range part of the Coulomb interaction, and clarify the fine structures in the trion energy levels. It is shown that the bright (optically allowed) trion with the lowest energy can be interpreted as a bound state of a dark (optically forbidden) exciton and an extra electron or hole.