Page 145 - Carbon Nanotubes

Vibrational modes of carbon nanotubes 135
spectroscopic measurements, which yield the phonon der the assumption that the hybridization between the
dispersion curves along the M direction in the BZ. The sp2 and pz orbitals is small. For example, in the arm-
dispersion curves are somewhat different near M, and chair nanotube based on CG0, with a diameter of ap-
along M-K, than the 2D calculations shown in Fig. Ib. proximately 0.7 nm, the three bond angles are readily
The lattice dynamical model for 3D graphite produces calculated and they are found to be 120.00", 118.35',
dispersion curves q(q) that are in good agreement and 118.35'. Because the deviation of these angles
with experimental results from inelastic neutron scat- from 120" is very small, the effect of curvature on the
tering, Raman scattering, and IR spectroscopy. force constants might be expected to be small. Based
The zone-folding scheme has two shortcomings. on a calculation using the semi-empirical interatomic
First, in a 2D graphene sheet, there are three modes Tersoff potential, Bacsa et al. [26,36] estimate consid-
with vanishing frequencies as q + 0; they correspond erable mode softening with decreasing diameter. For
to two translational modes with in-plane C-atom dis- tubes of diameter greater than -10 nm, however, they
placements and one mode with out-of-plane C-atom predict tube wall curvature has negligible effect on the
displacements. Upon rolling the sheet into a cylinder, mode frequencies.
the translational mode in which atoms move perpen-
dicular to the plane will now correspond to the breath- 3.3 Raman- and infrared-active modes
ing mode of the cylinder for which the atoms vibrate The frequencies of the tubule phonon modes at the
along the radial direction. This breathing mode has a r-point, or BZ center, are obtained from eqn (17) by
nonzero frequency, but the value cannot be obtained setting k = 0. At this point, we can classify the modes
by zone folding; rather, it must be calculated analyt- according to the irreducible representations of the
ically. The frequency of the breathing mode w,,d is symmetry group that describes the nanotube. We be-
readily calculated and is found to be[l,2] gin by showing how the classification works in the case
of chiral tubules. The nanotube modes obtained from
the zone-folding eqn by setting p = 0 correspond to
t-he I'-point modes of the 2D graphene sheet. For these
modes, atoms connected by any lattice vector of the
2D sheet have the same displacement. Such atoms, un-
der the symmetry operations of the nanotubes, trans-
where a = 2.46 A is the lattice constant of a graphene form into each other; consequently, the nanotubes
sheet, ro is the tubule radius, mc is the mass of a car- modes obtained by setting 1.1 = 0 are completely sym-
bon atom, and +:) is the bond stretching force con- metric and they transform according to the A irreduc-
stant between an atom and its ith nearest neighbor. It ible representation.
should be noted that the breathing mode frequency is Next, we consider the r-point nanotube modes ob-
found to be independent of n and m, and that it is in- tained by setting k = 0 and p = N/2 in eqn (17). The
versely proportional to the tubule radius. The value modes correspond to 2D graphene sheet modes at the
of = 300 cmp' for r, = 3.5 A, the radius that cor- point k = (Mr/C)e in the hexagonal BZ. We consider
responds to a nanotube capped by a C60 hemisphere. how such modes transform under the symmetry op-
Second, the zone-folding scheme cannot give rise erations of the groups ed and C3hd/,. Under the ac-
to the two zero-frequency tubule modes that corre- tion of the symmetry element C,, an atom in the 2D
spond to the translational motion of the atoms in the graphene sheet is carried into another atom separated
two directions perpendicular to the tubule axis. That from it by the vector
is to say, there are no normal modes in the 2D graph-
ene sheet for which the atomic displacements are such
that if the sheet is rolled into a cylinder, these displace-
ments would then correspond to either of the rigid tu- The displacements of two such atoms at the point
bule translations in the directions perpendicular to the k = (Nr/C)C have a phase difference given by
cylinder axis. To convert these two translational modes
into eigenvectors of the tubule dynamical matrix, a
perturbation matrix must be added to the dynamical N (20)
- k.rl = 27r(n2 + m2 + nm)/(dciR)
matrix. As will be discussed later, these translational 2
modes transform according to the El irreducible rep-
resentation; consequently, the perturbation should be which is an integral multiple of 2n. Thus, the displace-
constructed so that it will cause a mixing of the El ments of the two atoms are equal and it follows that
modes, but should have no effect in first order on
modes with other symmetries. The perturbation ma-
trix turns out to cause the frequencies of the El
modes with lowest frequency to vanish, affecting the
other El modes only slightly. The symmetry operation RNd,, carries an atom
Finally, it should be noted that in the zone-folding into another one separated from it by the vector
scheme, the effect of curvature on the force constants
has been neglected. We make this approximation un-