Double-resonant Raman scattering with optical and acoustic phonons in carbon nanotubes

We analyze and assign two low-intensity, double-resonant Raman bands at around 1950 ${\mathrm{cm}}^{-1}$ and 2450 ${\mathrm{cm}}^{-1}$ in carbon nanotube ensembles. We present Raman spectra of a high-pressure carbon monoxide sample for laser energies in the visible range. By using density functional theory plus zone folding, we simulate both combination modes for various excitation energies with respect to the chiral-index distribution of carbon nanotubes in the sample. For a comparison, double-resonant Raman spectra of single carbon nanotubes are also modeled within a symmetry-based nonorthogonal tight-binding approach, resulting in a different assignment of the 1950-${\mathrm{cm}}^{-1}$ band. Apparent discrepancies in the assignments are discussed in the article.

Figure 1

Second-order Raman spectrum of an HiPCO-produced buckypaper carbon nanotube sample. The laser excitation energy is $2.41\text{eV}$. The insets show a magnification of the analyzed double-resonant Raman modes.

Figure 2

Experimental Raman spectra of the 1950-${\mathrm{cm}}^{-1}$ and the 2450-${\mathrm{cm}}^{-1}$ bands from a HiPCO sample for many excitation energies given next to the spectra.

Figure 3

(a), (b) Experimentally obtained dispersions for the 1950-${\mathrm{cm}}^{-1}$ and 2450-${\mathrm{cm}}^{-1}$ Raman bands as a function of the excitation energy. Lines are linear fits of the two components in the Raman modes, as shown in Fig. 2. The FWHM of the two components in the Raman modes are as shown in Fig. 2 (top: 2450-${\mathrm{cm}}^{-1}$ band, bottom: 1950-${\mathrm{cm}}^{-1}$ band). Arrows indicate the range of the experimentally obtained optical resonances of carbon nanotubes tubes belonging to a family $(2{\mathrm{n}}_1$+${\mathrm{n}}_2$) as given under the arrows. (d)–(g) Calculated Raman spectra in a DFT + zone-folding approach. The dot diameter represents the intensity of single peaks. In (d) and (e) the intensities of the LO+LA and TO+LA combination modes are separately normalized to 1 and then composed to a spectrum. Instead, in (f) and (g) the intensities of the LO+LA and TO+LA combination modes are summarized and then normalized to 1.

Figure 4

(left) Phonon dispersion of graphene along the high-symmetry directions of the first Brillouin zone. Highlighted are parts of the LO/LA and TO/ZO phonon branches that are used to calculate Raman spectra for various excitation energies on the right side. The calculations are done to visualize the likewise dispersions of the two different processes where all matrix elements are set to 1 [12]. In graphene, a TO+ZO process is forbidden by symmetry [28].

Figure 5

Second-order Raman spectra for the (7,7), (16,0), (12,0), (9,9), (15,0), and (11,0) carbon nanotubes excited in the range of their E11 (metallic) and E22 (semiconducting) transitions calculated within a nonorthogonal tight-binding approach. Equivalent to the zone-folding approach, we find a DR TO+LA @ K process responsible for the 2450-${\mathrm{cm}}^{-1}$ band. The 1950-${\mathrm{cm}}^{-1}$ band originates from a TO+ZO @K  process. The Raman intensity of each nanotube, normalized per atom, is summed over the considered nanotube types equally.