Chirality Dependence of the $K$-Momentum Dark Excitons in Carbon Nanotubes

TL;DR

This study investigates how the energy of $K$-momentum dark excitons in semiconducting carbon nanotubes varies with chirality, revealing systematic dependencies and comparing experimental results with theoretical models.

Contribution

It provides experimental measurements of the $K$-momentum dark exciton energies across different chiralities and offers theoretical insights into their chiral angle dependence.

Findings

Dark exciton energy is 20-38 meV above bright exciton.

Energy separation depends systematically on diameter, family, and semiconducting type.

Observed chiral dependence exceeds theoretical predictions, suggesting larger phonon dispersion near the $K$ point.

Abstract

Using a collection of twelve semiconducting carbon nanotube samples, each highly enriched in a single chirality, we study the chirality dependence of the $K$-momentum dark singlet exciton using phonon sideband optical spectroscopy. Measurements of bright absorptive and emissive sidebands of this finite momentum exciton identify its energy as 20 - 38 meV above the bright singlet exciton, a separation that exhibits systematic dependencies on tube diameter, $2n+m$ family, and semiconducting type. We present calculations that explain how chiral angle dependence in this energy separation relates to the Coulomb exchange interaction, and elaborate the dominance of the $K_{A_1'}$ phonon sidebands over the zone-center phonon sidebands over a wide range of chiralities. The Kataura plot arising from these data is qualitatively well described by theory, but the energy separation between the sidebands shows a larger chiral dependence than predicted. This latter observation may indicate a larger dispersion for the associated phonon near the $K$ point than expected from finite distance force modeling.