Fig. 19.51. Phonon dispersion relations for a (5,5) carbon nanotubule (corresponding to the armchair tubule that fits on to a CM hemispherical cap) [19.84],

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Fig. 19.51. Phonon dispersion relations for a (5,5) carbon nanotubule (corresponding to the armchair tubule that fits on to a CM hemispherical cap) [19.84], a (5,5) carbon nanotube (corresponding to the armchair tubule that fits on to a C60 hemispherical cap) [19.84], This figure shows that many phonon branches result from the zone-folding procedure described by Eq. (19.41). We show that the phonon dispersion relations for a carbon nanotube depend on the (n, m) indices for the nanotube, or equivalently on the tubule diameter and chirality.

The structure of zigzag (n, 0) tubules can also be described by groups with Dnh or Dnd symmetry for even or odd n, respectively. All zigzag tubules have, among others, 3Alg, 3A2u, 6Elg, 6Eu, and 6E2g modes, irrespective of whether they have Dnh or Dnd symmetries. This gives 15 Raman-active modes and 7 infrared-active modes with distinct nonzero frequencies for zigzag tubules. As an example, consider the (9,0) zigzag tubule which has N = 18 and 108 degrees of freedom and 60 distinct mode frequencies from Eq. (19.41). Of these distinct mode frequencies, 15 are Raman active, 7 are infrared active, two have zero frequency, and 36 are silent. For zigzag tubules (n odd, Dnd symmetry)

IT" = 3Alg + 3Alu + 3A2g + 3 A2u + 6 Elg + 6 Eu + 6 E2g + 6 Elu + ■ • • + 6%~i)/2]g + 6£l(n_1)/2]H. (19.42)

For zigzag tubules (n even, Dnh symmetry)

Tf = 3Alg + 3A,U + 3^42g + 3Y42„ + 3 Blg + 3 BXu + 3 B2g + 3Blu + 6 Elg + 6Elu + 6 E2g + 6 E2u + • ■ • + 6£[(„-2)/2]g + 6£|(„-2)/2]U- (19-43)

We conclude this section with a summary of the phonon dispersion relations for chiral nanotubes. For the case where n and m are nonzero and have no common divisor, the point group of symmetry operations for chiral carbon nanotubes is given by the nonsymmorphic Abelian groups CN/il discussed in §19.4.3. Here ft is an integer [see Eq. (19.13)] which has no common divisor with N, and il denotes the number of 277 cycles traversed by the vector R before reaching a lattice point (see Fig. 19.24). Here N is the number of hexagons per ID unit cell in the tubule and d is the highest common divisor of (n, m) [19.84], Every tubule thus has 6N vibrational degrees of freedom, which are conveniently expressed in terms of their symmetry types [19.84]

r;b = 6 A + 6B + 6£, + 6£2 + • • ■ + 6EN/2__V (19.44)

Of these modes, the Raman-active modes are those that transform as A, £,, and E2 and the infrared-active modes are those that transform as A or Ex. This gives 15 nonzero Raman-active mode frequencies and 9 nonzero infrared-active mode frequencies, after subtracting the modes associated with acoustic translations ( A + £,) and with rotation of the cylinder (^4). Zone-folding techniques can also be used to yield the phonon dispersion relations w(k) for specific (n, m) chiral nanotubes, using Eq. (19.39). As an example of a chiral tubule, consider the (n, m) = (7,4) tubule. In this case N — 62, so that the ID unit cell has 372 degrees of freedom. The phonon modes include branches with 6A + 6B + 6Ex + 6E2 -I-----1- 6E30 symmetries,

192 branches in all. Of these, 15 are Raman active at k = 0, while 9 are infrared active, 3 corresponding to zero-frequency modes at k — 0, and 162 are silent.

A major difference between the symmorphic armchair and zigzag tubules, on the one hand, and the nonsymmorphic chiral tubules, on the other hand, is that for symmorphic tubules, the M point of the 2D graphene sheet Brillouin zone is folded into the T point, while for the nonsymmorphic tubules the M point does not map into the T point. This difference in behavior with regard to zone folding causes a larger spread in the values of the Raman and infrared frequencies in armchair and zigzag tubules as compared with general chiral tubules. This feature is further discussed in §19.7.2.

Phonon dispersion relations, similar to those shown in Fig. 19.51 for the («, m) — (5,5) tubule, have also been calculated for various zigzag and chiral tubules [19.83]. In general, the dispersion relations will have many more branches than are shown in Fig. 19.51 because of the larger size of the ID unit cell for a general (n, m) nanotube. For many experiments involving phonon dispersion relations for carbon nanotubes, the measurements are made on a multitude of nanotubes, each having its own (n, m) indices. If the measurements involve spectroscopy, then the observed spectra will emphasize those mode frequencies which are common to many nanotubes. In §19.7.2, we will consider the Raman- and infrared-active frequencies for the various types of carbon nanotubes as a function of tubule diameter, to assist with the interpretation of spectroscopic measurements on multiple carbon nanotubes, arising both from multiwall nanotubes and from nanotubes with different tubule axes.

19.7.2. Calculated Raman- and Infrared-Active Modes

Of particular use for the interpretation of spectroscopic data is the dependence of the phonon frequencies on tubule diameter. Explicit experimental results are presented in §19.7.3.

The general method used to obtain the calculated dependence of the tubule mode frequencies on tubule diameter and chirality considers all tubules of a given chirality and then considers the effect of the tubule diameters on the mode frequencies. Results are now available for armchair, zigzag, and some representative chiral nanotubes [19.83,84,87],

In Figs. 19.52 and 19.53, we see the plots for the dependence of the Raman-active and infrared-active tubule modes, respectively, as a function of tubule diameter for the zigzag («,0) tubules up to n = 40. In making these plots we make use of the fact that the number and symmetries of the optically active modes are independent of tubule diameter dt, thereby allowing us to follow the evolution of the various optically active modes as a function of d,. In these plots we see that many of the modes are strongly dependent on d„ so that these modes do not contribute constructively to a single sharp Raman or infrared spectral line for an ensemble of tubules of varying diameters. Some features in Fig. 19.52 that are candidates for observed Raman lines include features at 840,1350, and 1590 cm-1. In addition, the Raman cross sections decrease strongly as the diameter increases,

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