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where y is in units of cm-1 and d is the nanotube diameter in nm [95].

This correlation has been widely used to study samples with different diameter distributions [75, 78, 96-98], confirming the diameter dependence of the electronic properties of SWNTs. As a result, the study of the nanotube's diameter by Raman scattering has provided complementary information to the direct diameter measurements conducted by TEM, STM, and other techniques [99-105].

As mentioned above, the information obtained by Raman spectroscopy is the result of the resonant Raman effect which is particularly strong in SWNTs. Therefore the nanotubes that contribute most strongly to the Raman scattering for a given excitation wavelength are those which are in resonance with the incident or scattered light. Consequently, the spectral shape and position of the Raman RBM do not directly reflect the true diameter distribution in the SWNT sample, but rather the subset of nanotubes that are in resonance with the incident or scattered photon. So the relative intensities of the individual peaks that compose the RBM band are not only associated with the relative concentrations of the different nanotubes in the sample but rather reflect their electronic properties, which in their turn modulate the Raman cross section [106, 107]. This fact has been clearly illustrated by Dresselhaus et al. in a series of publications [73, 75, 108-111]. Figure 9 illustrates the Raman band associated with the radial breathing mode of SWNTs obtained by laser vaporization of a carbon target containing 1 to 2 at% of Ni/Co. In this case, the Raman spectra was obtained with six different laser excitation energies ranging from 1.17 to 2.71 eV [108]. Notice that the band shapes are completely different from one excitation laser to another. This result is consistent with a sample containing

Figure 9. Raman spectra of the radial breathing mode of single-wall carbon nanotubes collected with six different laser lines: 1.17, 1.59, 1.92, 2.41, 2.54, and 2.71 eV. The dotted curves represent individual Lorentzian curves used by the authors to fit the experimental data. Reprinted with permission from [108], M. A. Pimenta et al., J. Mater. Res. 13, 2396 (1998). © 1998, Materials Research Society.

Figure 9. Raman spectra of the radial breathing mode of single-wall carbon nanotubes collected with six different laser lines: 1.17, 1.59, 1.92, 2.41, 2.54, and 2.71 eV. The dotted curves represent individual Lorentzian curves used by the authors to fit the experimental data. Reprinted with permission from [108], M. A. Pimenta et al., J. Mater. Res. 13, 2396 (1998). © 1998, Materials Research Society.

carbon nanotubes with different diameters, which resonate at different energies. In this case, the average diameter as determined by TEM of the SWNT was 1.24 nm, but the RBM bands indicate that they range from 1.09 to 1.44 nm.

Figure 10 shows a similar result, on a set of samples produced by catalytic disproportionation of CO over a supported CoMo catalyst at 700 °C [88]. Under these particular conditions the diameter distribution is centered at around 0.9 nm as shown by TEM analysis. Once again the RBM bands show a strong dependence on the laser energy, which confirms the relation among the energy separation on DOS singularities with the nanotube diameter.

The differences in the average peak positions when different lasers are used complicate the analysis. Recently a better understanding of the response of the RBM has been attempted on the basis of an extended experimental and theoretical analysis [82, 112, 113]. The responses of the peak position and of the first and second moments of the spectra were found to oscillate with the energy of the exciting laser. This oscillation was found to be due to the microscopy quantization of the electronic levels as a consequence of the finite size of the nanotubes in the direction perpendicular to the nanotube axis and, as mentioned previously, to the distribution of the states along the nanotube axis into the van Hove singularities. Furthermore, an approximation has been successfully used to evaluate the spectral moments of the RBM Raman response without explicit use of the joined density of states. This simplified model has been used to determine the mean and the width of the diameter distribution of a SWNT sample. A remarkable conclusion is that just a single laser energy could be used to evaluate the first and second moments of the spectra and, therefore, the mean and the width of the diameter distribution of the sample [113].

In principle, the radial breathing mode frequency could provide the identity (n, m) of the individual nanotubes participating in the Raman scattering since the RBM frequency is proportional to the diameter, and the value of the SWNT

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wavenumber (cm-1)

Figure 10. Radial breathing mode resonant Raman spectra of SWNTs grown at 750 °C by CO disproportionation over a Co:Mo catalyst. The laser excitation energies were 2.55 eV, 2.4 eV, and 2.0 eV. Reprinted with permission from [88], J. E. Herrera et al., J. Nanotech., in press. © American Scientific Publishers.

diameter depends on the integers (n, m) [39, 114]. However several distinct (n, m) nanotubes can exhibit sufficiently similar diameters so that their RBM frequency differs only by 1-2 cm-1. This prevents one from using the RBM frequencies to determine (n, m) in a sample of SWNTs, since the natural linewidth of the RBM band is 6 cm-1 and the individual contributions to the band from nanotubes with similar diameters would be very difficult to resolve [84, 93, 99].

Besides the radial breathing mode band, the Raman spectra of SWNTs also comprise a disorder-induced Raman band (D band), which is a feature common to all sp2 hybridized disordered carbon materials. This D band appears in the Raman spectra between 1250 and 1450 cm-1 and is associated with phonons close to the K point of the graphite Brillouin zone and it becomes Raman active when translational symmetry is lost [110, 111, 115]. The D band also shows a strong dependence on the excitation laser energy, although in this case the dependence is highly linear. Pimenta et al. [116]. studied the D band behavior on isolated SWNTs and observed that its intensity appeared to be random from one nanotube to another; although the linear dependence on the excitation laser energy was observed, this was attributed to be more a consequence of the same general trend that is observed for all sp2 carbon materials. The authors suggested that this phenomenon is linked to phonons that are not at the center of the 1D Brillouin zone of SWNTs and concluded that they become Raman active due to the finite size of the SWNTs or to the presence of defects, which would break the translational symmetry along the nanotube axis [111, 117]. Recently, it has been proposed that the excitation energy dependence of the D band is an intrinsic property of the SWNT, and it has been suggested that the metallic character of an individual nanotube can be inferred from the shape and position of its D band [118].

The D band has been related not just to defects on SWNTs but also to the presence of other forms of disordered carbon such as carbon nanoparticles and amorphous carbon [119]. Its relative intensity has been used as a semiquantitative indicator of the presence of undesired forms of carbon (i.e., microcrystalline graphite, amorphous carbon, MWNTs, carbon nanofibers). Figure 11 shows an example in which the relative intensity of the D band can be related to the presence of carbon impurities in a SWNT sample. The Raman spectra of SWNTs obtained by a catalytic method over a selective solid catalyst are presented as function of the time on the reaction stream. It is observed that at longer times there is an increase in the intensity of all the Raman bands as a result of the increasing carbon deposition. However, the relative intensity of the D band clearly increases with reaction time, which indicates that undesired forms of carbon start being formed at longer reaction times. The TEM observations on these same samples confirm this conclusion. A higher density of nanofibers and graphite is observed after long reaction times than after short reaction periods [32].

The Raman spectra of SWNTs also contain the typical G band associated with the tangential C-C stretching modes of SWNTs, which consist of two A, two Er, and two E2 phonon modes for chiral nanotubes and one A1g, one E1g, and one E2g mode for armchair or zigzag nanotubes, as predicted by group theory and phonon frequency calculations

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wavenumber (cm-1)

Figure 10. Radial breathing mode resonant Raman spectra of SWNTs grown at 750 °C by CO disproportionation over a Co:Mo catalyst. The laser excitation energies were 2.55 eV, 2.4 eV, and 2.0 eV. Reprinted with permission from [88], J. E. Herrera et al., J. Nanotech., in press. © American Scientific Publishers.

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