Page 146 - Carbon Nanotubes

136 P. c. EKI :UND et al.

wherep and q are the integers uniquely determined by tically active modes with nonzero frequencies; thus
eqn (8). The atoms in the 2D graphene sheet have dis- there are 15 Raman- and 7 IR-active modes.
placements, at the point k = (Nr/C)&, that are com-
pletely out of phase. This follows from the observation 3.4 Mode frequency dependence
that on tubule diameter
In Figs. 4-6, we display the calculated tubule fre-
quencies as a function of tubule diameter. The results
are based on the zone-folding model of a 2D graph-
ene sheet, discussed above. IR-active (a) and Raman-
and that Wd is an odd integer; consequently active (b) modes appear separately for chiral tubules
(Fig. 4), armchair tubules (Fig. 5) and zig-zag tubules
(Fig. 6). For the chiral tubules, results for the repre-
sentative (n, m), indicated to the left in the figure, are
displaced vertically according to their calculated diam-
From the above, we therefore conclude that the nano- eter, which is indicated on the right. Similar to modes
tube modes obtained by setting p = N/2, transform in a Ca molecule, the lower and higher frequency
according to the B irreducible representation of the modes are expected, respectively, to have radial and
chiral symmetry group e. tangential character. By comparison of the model cal-
Similarly, it can be shown that the nanotube modes culation results in Figs. 4-6 for the three tube types
at the I?-point obtained from the zone-folding eqn by (armchair, chiral, and zig-zag) a common general be-
setting p = 9, where 0 < 9 < N/2, transform accord- havior is observed for both the IR-active (a) and
ing to the Ev irreducible representation of the symme- Raman-active (b) modes. The highest frequency
try group e. Thus, the vibrational modes at the modes exhibit much less frequency dependence on di-
F-point of a chiral nanotube can be decomposed ac- ameter than the lowest frequency modes. Taking the
cording to the following eqn large-diameter tube frequencies as our reference, we
see that the four lowest modes stiffen dramatically
(150-400 cm-') as the tube diameter approaches -1
nm. Conversely, the modes above -800 cm-' in the
Modes with A, E,, or E2 symmetry are Raman ac- large-diameter tubules are seen to be relatively less sen-
tive, while only A and El modes are infrared active. sitive to tube diameter: one Raman-active mode stiff-
The A modes are nondegenerate and the E modes are ens with increasing tubule diameter (armchair), and a
doubly degenerate. According to the discussion in the few modes in all the three tube types soften (100-200
previous section, two A modes and one of the E, cm-'), with decreasing tube diameter. It should also
modes have vanishing frequencies; consequently, for be noted that, in contrast to armchair and zig-zag tu-
a chiral nanotube there are 15 Raman- and 9 IR-active bules, the mode frequencies in chiral tubules are
modes, the IR-active modes being also Raman-active. grouped near 850 cm-' and 1590 cm-'.
It should be noted that the number of Raman- and IR- All carbon nanotube samples studied to date have
active modes is independent of the nanotube diameter. been undoubtedly composed of tubules with a distri-
For a given chirality, as the diameter of the nanotube bution of diameters and chiralities. Therefore, whether
increases, the number of phonon modes at the BZ cen- one is referring to nanotube samples comprised of
ter also increases. Nevertheless, the number of the single-wall tubules or nested tubules, the results in
modes that transform according to the A, E,, or E2 Figs. 4-6 indicate one should expect inhomogeneous
irreducible representations does not change. Since only broadening of the IR- and Raman-active bands, par-
modes with these symmetries will exhibit optical activ- ticularly if the range of tube diameter encompasses the
ity, the number of Raman or IR modes does not in- 1-2 nm range. Nested tubule samples must have a
crease with increasing diameter. This, perhaps unantic- broad diameter distribution and, so, they should ex-
ipated, result greatly simplifies the data analysis. The hibit broader spectral features due to inhomogeneous
symmetry classification of the phonon modes in arm- broadening.
chair and zigzag tubules have been studied in ref. [2,3]
under the assumption that the symmetry group of
these tubules is isomorphic with either Dnd or Bnh, 4. SYNTHESIS AND RAMAN SPECTROSCOPY
depending on whether n is odd or even. As noted ear- OF CARBON NANOTUBES
lier, if one considers an infinite tubule with no caps, We next address selected Raman scattering data
the relevant symmetry group for armchair and zigzag collected on nanotubes, both in our laboratory and
tubules would be the group 6)2nh. For armchair tu- elsewhere. The particular method of tubule synthesis
bules described by the Dnd group there are, among may also produce other carbonceous matter that is
others, 3A1,, 6E1,, 6E2,, 2A2,, and SEI, optically both difficult to separate from the tubules and also ex-
active modes with nonzero frequencies; consequently, hibits potentially interfering spectral features. With
there are 15 Raman- and 7 IR-active modes. All zig- this in mind, we first digress briefly to discuss synthe-
zag tubules, under Dnd or Bnh symmetry group have, sis and purification techniques used to prepare nano-
among others, 3A1,, 6E,,, 6E2,, 2A2,, and 5E,, op- tube samples.