Effect of quantized electronic states on the dispersive Raman features in individual single-wall carbon nanotubes

This work reports how resonance Raman experiments are used to study details of the electronic structure of individual single-wall carbon nanotubes (SWNTs) by measuring the phonon spectra and how the quantized electronic structure affects the dispersive Raman features of SWNTs. We focus our analysis on the dispersive D and $G^{\prime }$ bands observed in the Raman spectra of isolated semiconducting nanotubes. By using a laser excitation energy of 2.41 eV, we show that both the D-band and $G^{\prime }$-band frequencies are dependent on the wave vector $k_{\mathrm{ii}}$ where the electrons are confined in the one-dimensional subband i of the electronic structure of SWNTs. By making use of the $\mathrm{}(n,m)\mathrm{}$ assignment for each tube, we theoretically correlate the observed frequency dependences for the D- and $G^{\prime }$-band modes with the electronic structure predicted for each $\mathrm{}(n,m)\mathrm{}$ pair and we determine the dependence of ${\omega }_D$ and ${\omega }_{G^{\prime }}$ on the diameter and chirality for individual electronic transitions $E_{\mathrm{ii}}$ for nanotube bundles. We use the D- and $G^{\prime }$-band dependence on electron wave vector $k_{\mathrm{ii}}$ to predict the dominant phonon wave vector q selected by the quantum-confined electronic state $k_{\mathrm{ii}}$ and to explain the anomalous dispersion observed for ${\omega }_D$ and ${\omega }_{G^{\prime }}$ in SWNT bundles as a function of laser excitation energy, yielding excellent agreement between experiment and theory.