Page 143 - Carbon Nanotubes

Vibrational modes of carbon nanotubes 133
images show that this carbon nanosoot consists of dis- tube specified by (n, m) is obtained. If n = m, the re-
ordered sp2 carbon particles with an average particle sulting nanotube is referred to as an “armchair”
diameter of -200 A. The Raman spectrum (Fig. 2d) tubule, while if n = 0 or m = 0, it is referred to as a
of the “as synthesized” carbon black is very similar to “zigzag” tubule; otherwise (n # m # 0) it is known as
that of glassy carbon (Fig. 2e) and has broad disorder- a “chiral” tubule. There is no loss of generality if it is
induced peaks in the first-order Raman spectrum at assumed that n > m.
1359 and 1600 cm-’, and a broad second-order fea- The electronic properties of single-walled carbon
ture near 2950 cm-’. Additional weak features are nanotubes have been studied theoretically using dif-
observed in the second-order spectrum at 2711 and ferent methods[4-121. It is found that if n - m is a
3200 cm-’ , similar to values in HOPG, but appear- multiple of 3, the nanotube will be metallic; otherwise,
ing closer to 2(1359 CII-’) = 2718 cm-’ and 2(1600 it will exhibit a semiconducting behavior. Calculations
cm-I) = 3200 cm-’ , indicative of somewhat weaker on a 2D array of identical armchair nanotubes with
3D phonon dispersion, perhaps due to weaker cou- parallel tube axes within the local density approxi-
pling between planes in the nanoparticles than found mation framework indicate that a crystal with a hex-
in HQPG. TEM images[34] show that the heat treat- agonal packing of the tubes is most stable, and that
ment of the laser pyrolysis-derived carbon nanosoot intertubule interactions render the system semicon-
to a temperature THT = 2820°C graphitizes the nano- ducting with a zero energy gap[35].
particles (Le., carbon layers spaced by -3.5 A are
aligned parallel to facets on hollow polygonal parti- 3.1 Symmetry groups of nanotubes
cles). As indicated in Fig. 2c, the Raman spectrum of A cylindrical carbon nanotube, specified by (n,m),
this heat-treated carbon black is much more “gra- can be considered a one-dimensional crystal with a
phitic” (similar to Fig. 2a) and, therefore, a decrease fundamental lattice vector T, along the direction of the
in the integrated intensity of the disorder-induced band tube axis, of length given by[1,3]
at 1360 cm-’ and a narrowing of the 1580 cm-’ band
is observed. Note that heat treatment allows a shoul-
der associated with a maximum in the mid-BZ density
of states to be resolved at 1620 cm-I, and dramati-
cally enhances and sharpens the second-order features. where

dR = d if n - rn # 3dr
3. THEORY OF VIBRATIONS IN
CARBON NANOTUBFS = 3d if n - m = 3dr (3)
A single-,wall carbon nanotube can be visualized by
referring to Fig. 3, which shows a 2D graphene sheet where Cis the length of the vector in eqn (l), d is the
with lattice vectors a1 and a2, and a vector C given by greatest common divisor of n and m, and r is any in-
teger. The number of atoms per unit cell is 2N such
that

N = 2(n2 + m2 + nm)/dR. (4)
where n and m are integers. By rolling the sheet such
that the tip and tail of C coincide, a cylindrical nano-
For a chiral nanotube specified by (n, m), the cylin-
der is divided into d identical sections; consequently
a rotation about the tube axis by the angle 2u/d con-
stitutes a symmetry operation. Another symmetry op-
eration, R = ($, 7) consists of a rotation by an angle
$ given by

s2
$=27r-
Nd

followed by a translation 7, along the direction of the
tube axis, given by

d
s=T-.
N
The quantity s2 that appears in eqn (5) is expressed in
terms of n and m by the relation
Fig. 3. Translation vectors used to define the symmetry of
a carbon nanotube (see text). The vectors a, and a2 define
the 2D primitive cell. s2 = (p(m + 2n) + q(n + 2m)I (d/dR) (7)