Electron-phonon and electron-photon interactions and resonant Raman scattering from the radial-breathing mode of single-walled carbon nanotubes

The resonance Raman profile of the radial-breathing mode is calculated for all 300 single-walled carbon nanotubes in the radius range from $2\mathrm{\AA }\text{to}12\mathrm{\AA }$ and for all optical transitions up to $3.5\mathrm{eV}$ using a symmetry-adapted nonorthogonal tight-binding model [V. N. Popov, New J. Phys. 6, 17 (2004)]. The influence of the electron-phonon and electron-photon interactions on the Raman intensity is studied using an approximate expression for the intensity in the vicinity of each optical transition as the product of the electron-phonon coupling matrix element, the momentum matrix element, and the effective mass raised to different powers. The dependence of the latter three quantities and the maximum Raman intensity on the nanotube radius, the chiral angle, and the optical transition energy is discussed in detail. In particular, the points of the corresponding plots exhibit family behavior of three different types. It is shown that the widespread practice to neglect the electron-photon and electron-phonon interactions in the estimation of the intensity can lead to incorrect prediction of the Raman spectra.