Abstract
The Raman-active radial breathing modes ͑RBM͒ and tangential modes ͑TM͒ of single wall carbon nano-tubes ͑SWNT͒ are studied at fixed laser excitation energy 2.41 eV ͑514.5 nm͒. We focus on the striking diameter dependence of the relative intensity of the TM and RBM between 0.9 and 2.1 nm, which displays a series of plateaux separated by well-defined minima around 1.35 and 0.9 nm. This relates to the diameter dependence of allowed optical transitions ͑AOT͒ in SWNT. Diameters in the range 1-1.3 nm ͑above 1.4 nm͒ correspond to metallic ͑semiconducting͒ SWNT in resonance at 2.41 eV. The minima correspond to out-of-resonance conditions for TM. The measurement of the diameter dependence of the TM intensity for fixed laser energies is an alternative experimental way to plot the envelopes of the domains of AOT in SWNT. Raman spectroscopy of single wall carbon nanotubes ͑SWNT͒ is well known to be a resonant process associated with optical transition between spikes in the one-dimensional ͑1D͒ electronic density of states which fall in the visible and near-infrared range. 1-6 The energy of these allowed optical transitions ͑AOT͒ depend both on the diameter and on the metallic or semiconducting character of the tubes, as illustrated in Fig. 1 ͑from Ref. 7͒ where the ranges of energies of the AOT have been calculated for semiconducting ͑black ar-eas͒ and metallic ͑dashed areas͒ tubes. Raman spectroscopy allows us to study the diameter distribution from the analysis of the RBM range (below 300 cm Ϫ1) and the electronic properties from the line profiles in the TM range ͑1400-1700 cm Ϫ1). 2-10 As far as bundles of SWNT are concerned, inter-tube coupling must be considered to derive properly the relation between tube diameter and RBM frequency. This was achieved recently by considering a Lennard-Jones potential in addition to a force constant model in order to account for van der Waals intertube interactions. 11 A significant upshift of the RBM is found for tubes in bundles with respect to isolated tubes ͓of about 16 cm Ϫ1 for a ͑10,10͒ SWNT͔. The whole calculated data were best fitted by the following non-linear phenomenologic relation between the RBM frequency and the tube diameter: RBM ͑ cm Ϫ1 ͒ϭ238/d͑ nm͒ 0.93. ͑1͒ Equation ͑1͒ has been proposed as a useful tool to estimate tube diameters in SWNT. This was achieved on various samples and a good agreement was evidenced with TEM or neutron diffraction results on the same samples. 11,12 In Fig. 1 the top scale indicates the RBM frequency calculated from Eq. 1. As far as the TM range ͑1400-1700 cm Ϫ1) is concerned , two distinct profiles corresponding to the specific responses of semiconducting and metallic nanotubes were measured. For semiconducting tubes, the TM essentially displays a symmetric profile with a dominant peak around 1590 cm Ϫ1 and two other structures around 1560 and 1550 cm Ϫ1. For metallic tubes, the TM displays an intense, broad and asymmetric band around 1540 cm Ϫ1 , a line at 1560 cm Ϫ1 , and a sharp peak around 1580 cm Ϫ1. 2,4-6,8,9 On the other hand, the TM profile was used as a probe to study photose-lectively semiconducting or metallic tubes of selected diameter and a good agreement was found with calculations of Fig. 1. 5,6 Because resonance in Raman can occur via incident or scattered photons, Stokes and anti-Stokes TM spectra can display very distinct profiles when (laser ϩ TM) and (laser-TM) correspond to AOT for tubes of different elec-FIG. 1. Allowed optical transitions for SWNT of various diameters and helicities ͑from Ref. 7͒. Black and dashed areas correspond to semiconducting and metallic tubes, respectively. The grey frame corresponds to the 2.41 eV ͑514.5 nm͒ laser excitation used in this work: resonance is expected for RBM and TM at the top and bottom of the frame, respectively. The arrows indicate the diameters corresponding to expected minima in the TM intensity ͑see text͒.