Summary

In summary, the spectra of the DOS for SWNTs have a strong chirality dependence. Especially for metallic nanotubes, the DOS peaks are found to be split into two peaks because of the trigonal warping effect, while semiconducting nanotubes do not show a splitting. The width of the splitting becomes a maximum for the metallic zigzag nanotubes (3n, 0), and is zero for armchair nanotubes (n,n), which are always metallic. In the case of semiconducting nanotubes, the upper and lower bounds of the peak positions of E^ (dt) on the Kataura chart shown in Fig. 5 are determined by the values of Ef1 (dt) for the (3n + 1, 0) or (3n — 1,0) zigzag nanotubes. The upper and lower bounds of the widths of the Efi (dt) curves alternate with increasing i between the (3n +1, 0) and (3n — 1,0) zigzag nanotubes.

The existence of a splitting of the DOS spectra for metallic nanotubes should depend on the chirality which should be observable by STS/STM experiments, consistent with the experiments of Kim et al. [22]. The width of the metallic window can be observed in resonant Raman experiments, especially through the differences between the analysis for the Stokes and the anti-Stokes spectra. Some magnetic effects should be observable in the resonant Raman spectra because an applied magnetic field should perturb the 1D DOS for the nanotubes, since the magnetic field will break the symmetry between the K and K' points. The magnetic susceptibility, which has been important for the determination of 70 for 3D graphite [77,78], could also provide interesting results regarding a determination of Epp (dt) for SWNTs, including the dependence of Epp (dt) on dt.

Purification of SWNTs to provide SWNTs with a known diameter and chirality should be given high priority for future research on carbon nanotube physics. Furthermore, we can anticipate future experiments on SWNTs which could illuminate phenomena showing differences in the E(k) relations for the conduction and valence bands of SWNTs. Such information would be of particular interest for the experimental determination of the overlap integral s as a function of nanotube diameter. The discussion presented in this article for the experimental determination of Epp(dt) depends on assuming s = 0, in order to make direct contact with the tight-binding calculations. However, if s =0, then the determination of Epp (dt) would depend on the physical experiment that is used for this determination, because different experiments emphasize different k points in the Brillouin zone. The results of this article suggest that theoretical tight binding calculations for nanotubes should also be refined to include the effect of s = 0. Higher order (more distant neighbor) interactions should yield corrections to the lowest order theory discussed here.

The 1540 cm-1 feature appears only in the Raman spectra for a metallic bundle, but not for semiconducting SWNTs nor for individual metallic SWNTs. The inter-tube interaction in MWNTs gives 5% higher RBM mode frequencies than in SWNT bundles, and the intertube-interaction effect between the MWNT innermost shell and its adjacent outer shell is important for splitting the RBM peaks of a MWNT sample.

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