Exciton-photon, exciton-phonon matrix elements, and resonant Raman intensity of single-wall carbon nanotubes

Within the framework of the tight-binding model, we have developed exciton-photon and exciton-phonon matrix elements for single-wall carbon nanotubes. The formulas for first-order resonance and double-resonance Raman processes are discussed in detail. The lowest-energy excitonic state possesses an especially large exciton-photon matrix element compared to other excitonic states and continuum band states because of its localized wave function with no node. Unlike the free-particle picture, the photon matrix element in the exciton picture shows an inverse diameter dependence but no tube type or chirality dependences. As a result, the optical absorption intensity shows a strong diameter dependence but no tube type or chirality dependences. Moreover, the continuum band edge can be determined from the wave function or exciton-photon matrix element. For the radial breathing mode (RBM) and G-band modes, the phonon matrix elements in the exciton and free-particle pictures are almost the same. As a result, the intensity for the Kataura plots for the RBM or G-band modes by the exciton and free-particle pictures show similar family patterns. However, the excitonic effect has greatly increased the diameter dependence and magnitude of the intensities for the RBM and G band by enhancing the diameter dependence and magnitude of the photon matrix element. Therefore, excitons have to be considered in order to explain the strong diameter dependence of the Raman signal observed experimentally.