Excitons and exciton-phonon coupling in metallic single-walled carbon nanotubes: Resonance Raman spectroscopy

Abstract

The first four transitions (upper and lower branches of ${E}_{11}^{M}$ and ${E}_{22}^{M}$) for a broad diameter range (0.7--4 nm) of metallic single-walled carbon nanotubes are studied in the 1.26--2.71 eV energy range using resonance Raman spectroscopy of their radial breathing modes (RBMs). A scaling-law analysis of transition energies from 77 spectral features suggests that the transitions are excitonic in nature and that relative scaling of electron self-energies and exciton binding energies in metallic nanotubes closely matches that found in semiconductors. The previously elusive upper-branch signatures are observed at large diameters $(g1.3\text{ }\text{nm})$ for several chiralities for both ${E}_{11}^{M}$ and ${E}_{22}^{M}$ excitation. These results are discussed as a consequence of the nodal behavior of exciton-phonon coupling. Additionally, while theoretical calculations for the $(n,m)$-dependent matrix elements predict that the RBM intensity should decrease with increasing diameter, the opposite behavior is observed experimentally. We show that this is a consequence of an increase in the resonance Raman broadening factor $\ensuremath{\Gamma}$ as diameter decreases.