Intensity of the resonance Raman excitation spectra of single-wall carbon nanotubes

Abstract

The electron-phonon matrix elements are calculated for the radial breathing mode (RBM) and the $G$-band $A$ symmetry mode of single-wall carbon nanotubes. The RBM intensity decreases with increasing nanotube diameter and chiral angle. The RBM intensity at van Hove singular $k$ points is larger outside the two-dimensional Brillouin zone around the $K$ point than inside the Brillouin zone. For the $G$ band $A$ symmetry mode, the matrix element shows that all semiconducting nanotubes have nonzero LO mode intensity, and the LO mode generally has a larger intensity than the TO mode, while the ratio of the intensity of the LO mode to that of the TO mode decreases with increasing chiral angle. In particular, zigzag nanotubes have zero intensity for the TO mode, and armchair nanotubes have zero intensity for the LO mode. Using the matrix elements thus obtained, the resonance Raman excitation profiles are calculated for nanotube samples under different broadening factor $\gamma$ regimes. For semiconducting nanotubes, the excitation profiles for the RBM are consistent with experiments. For metallic nanotubes, a quantum interference effect in the Raman intensity is found for both the RBM and LO modes. For the RBM and LO modes, different kinds of excitation profiles are discussed for nanotube samples in the large and small $\gamma$ regimes by considering the electron-phonon matrix element and the trigonal warping effect. For nanotube samples in the large $\gamma$ regime, a shift in the energy of the peak in the RBM intensity relative to the corresponding peak in the joint density of states is found.