One-Dimensional Character of Combination Modes in the Resonance Raman Scattering of Carbon Nanotubes

Resonance Raman spectroscopy with an energy tunable system is used to analyze the $600–1100{\mathrm{c}\mathrm{m}}^{-1}$ spectral region in single-wall carbon nanotubes. Sharp peaks are associated with the combination of zone folded optic and acoustic branches from 2D graphite. These combination modes exhibit a peculiar dependence on the excitation laser energy that is explained on the basis of a highly selective resonance process that considers phonons and electrons in low dimensional materials.

Figure 1

(a) Two-dimensional plot for the $E_{\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{r}}$ dependence of the Raman spectra of SWNT bundles in the intermediate frequency mode (IFM) range. The light areas indicate high intensity. Arrows point to five well-defined ${\omega }_{\mathrm{I}\mathrm{F}\mathrm{M}}^{-}$ features. (b) Raman spectra in the IFM range taken with $E_{\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{r}}=2.05$, 2.20, 2.34, and 2.54 eV.

Figure 2

(a) van Hove singularity (vHS) $E_{ii}$ transition energies (from the tight binding model with ${\gamma }_0=2.9\mathrm{e}\mathrm{V}$ and $a_{\mathrm{C}\mathrm{-}\mathrm{C}}=0.142\mathrm{n}\mathrm{m}$) as a function of $d_t$. The symbols “×,” “+,” and “○” stand for semiconducting $\mathrm{m}\mathrm{o}\mathrm{d}(2n+m,3)=1$, $\mathrm{m}\mathrm{o}\mathrm{d}(2n+m,3)=2$, and metallic SWNTs, respectively. Gray curves connect $E_{ii}^S$ for SWNTs with $2n+m=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$. The perpendicular broad gray lines are guides to the eye showing the connection between the $E_{33}^S$ and $E_{44}^S$ vHSs for selected $(n,m)$ assigned SWNTs. (b) Theoretical prediction for the ${\omega }_{\mathrm{I}\mathrm{F}\mathrm{M}}^{+}$ and ${\omega }_{\mathrm{I}\mathrm{F}\mathrm{M}}^{-}$ for special $(n,m)$ SWNTs assigned in (a). Considering only the $E_{33}^S$ and $E_{44}^S$ vHSs for the SWNTs assigned in (a), the plot in (b) is constructed by considering $E_{\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{r}}=E_{ii}$ and ${\omega }_{\mathrm{I}\mathrm{F}\mathrm{M}}^{\pm }={\omega }_0^{\pm }\pm v^{\pm }(6/d_t)$, ${\omega }_0^{+}=540{\mathrm{c}\mathrm{m}}^{-1}$, and ${\omega }_0^{-}=1195{\mathrm{c}\mathrm{m}}^{-1}$ (see text).

Figure 3

(a) Schema for the 2D phonon dispersion of one acoustic and one optical phonon branch close to the $\Gamma$ point. The cutting lines represent the allowed 1D wave vectors for a given $(n,m)$ SWNT. The symmetry for each cutting line is indicated close to the optical modes. (b) Phonon density of states showing the confinement of phonon states at the 1D vHSs, that occurs at $q_{\mathrm{1}\mathrm{D}}=0$.