## Phonon Properties

A general approach for obtaining the phonon dispersion relations of carbon nanotubes is given by tight binding molecular dynamics (TBMD) calculations adopted for the nanotube geometry, in which the atomic force potential for general carbon materials is used [25,34]. Here we use the scaled force constants from those of 2D graphite [2,14], and we construct a force constant tensor for a constituent atom of the SWNT so as to satisfy the rotational sum rule for the force constants [35,36]. Since we have 2N carbon atoms in the unit cell, the dynamical matrix to be solved becomes a 6N x 6N matrix [35,37].

In Fig. 9 we show the results thus obtained for (a) the phonon dispersion relations u(k) and (b) the corresponding phonon density of states for 2D graphite (left) and for a (10,10) armchair nanotube (right). For the 2N = 40 carbon atoms per circumferential strip for the (10,10) nanotube, we have 120 vibrational degrees of freedom, but because of mode degeneracies, there are only 66 distinct phonon branches, for which 12 modes are non-degenerate and 54 are doubly degenerate. The phonon density of states for the (10,10) nanotube is close to that for 2D graphite, reflecting the zone-folded nanotube phonon dispersion. The same discussion as is used for the electronic structure can be applied to the van Hove singularity peaks in the phonon density of states of carbon nanotubes below a frequency of 400 cm-1 which can be observed in neutron scattering experiments for rope samples.

xt/it slates'1C-atorr.'anr~1 kt/h states/1 C-Btarn/Errf1

Fig. 9. (a) Phonon dispersion relations and (b) phonon DOS for 2D graphite (left) and for a (10,10) nanotube (right) [35]

xt/it slates'1C-atorr.'anr~1 kt/h states/1 C-Btarn/Errf1

Fig. 9. (a) Phonon dispersion relations and (b) phonon DOS for 2D graphite (left) and for a (10,10) nanotube (right) [35]

There are four acoustic modes in a nanotube. The lowest energy acoustic modes are the Transverse Acoustic (TA) modes, which are doubly degenerate, and have x and y displacements perpendicular to the nanotube z axis. The next acoustic mode is the "twisting" acoustic mode (TW), which has ^-dependent displacements along the nanotube surface. The highest energy mode is the Longitudinal Acoustic (LA) mode whose displacements occur in the z direction. The sound velocities of the TA, TW, and LA phonons for a (10,10) carbon nanotube, vTa'10', wtw10) and «La'10', are estimated as v^a'10' =9.42km/s, v^w10' = 15-00 km/s, and «La'10' =20.35km/s, respectively. The calculated phase velocity of the in-plane TA and LA modes of 2D graphite are «¡a=15.00 km/s and «¡.=21.11 km/s, respectively. Since the TA mode of the nanotube has both an 'in-plane' and an 'out-of-plane' component, the nanotube TA modes are softer than the in-plane TA modes of 2D-graphite. The calculated phase velocity of the out-of-plane TA mode for 2D graphite is almost 0km/s because of its k2 dependence. The sound velocities that have been calculated for 2D graphite are similar to those observed in 3D graphite [10], for which «¡3° = 12-3km/s and «¡¡3° = 21.0km/s. The discrepancy between the vTa velocity of sound for 2D and 3D graphite comes from the interlayer interaction between the adjacent graphene sheets.

The strongest low frequency Raman mode for carbon nanotubes is the Radial Breathing A1g mode (RBM) whose frequency is calculated to be 165 cm-1 for the (10,10) nanotube. Since this frequency is in the silent region for graphite and other carbon materials, this A1g mode provides a good marker for specifying the carbon nanotube geometry. When we plot the A1g frequency as a function of nanotube diameter for (n, m) in the range 8 < n < 10, 0 < m < n, the frequencies are inversely proportional to dt [5,35], within only a small deviation due to nanotube-nanotube interaction in a nanotube bundle. Here ^(1010) and d(1010) are, respectively, the frequency and diameter dt of the (10,10) armchair nanotube, with values of W(10'10)=165cm~1 and d(10 10)=1.357nm, respectively. However, when we adopt 70 = 2-90 eV, the resonant spectra becomes consistent when we take W(ioiio) = 177cm-1. As for the higher frequency Raman modes around 1590 cm-1 (G-band), we see some dependence on dt, since the frequencies of the higher optical modes can be obtained from the zone-folded k values in the phonon dispersion relation of 2D graphite [26].

Using the calculated phonon modes of a SWNT, the Raman intensities of the modes are calculated within the non-resonant bond polarization theory, in which empirical bond polarization parameters are used. [38] The bond parameters that we used in this chapter are a|| — a± = 0.04 A3, a'^ + 2a'L = 4.7 A2, and a'n — a= 4.0 A2, where a and a' denote the polarizability parameters and their derivatives with respect to bond length, respectively. [35] The eigen-functions for the various vibrational modes are calculated numerically at the r point (k = 0). When some symmetry-lowering effects, such as defects and finite size effects occur, phonon modes away from the r point are observed in the Raman spectra. For example, the DOS peaks at 1620 cm-1 related to the highest energy of the DOS, and some DOS peaks related to M point phonons can be strong. In general, the lower dimensionality causes a broadening in the DOS, but the peak positions do not change much. The 1350 cm-1 peaks (D-band) are known to be defect-related Raman peaks which originate from K point phonons, and exhibit a resonant behavior [39].

## Post a comment