Page 42 - Carbon Nanotubes

Physics of carbon nanotubes 33
sibility for interlayer correlations, even if C60 and tion is zero in the limit of 2D graphite, this result is
CZm take the same I), axes. Further, in the case of consistent with the limiting case of N = 00. This very
C240, the molecule deviates from a spherical shape to small Peierls gap is, thus, negligible at finite temper-
an icosahedron shape. Because of the curvature, it is atures and in the presence of fluctuations arising from
expected that the spherically averaged interlayer spac- 1D conductors. Very recently, Viet et al. showed[29]
ing between the double-layered hyper-fullerenes is that the in-plane and out-of-plane distortions do not
greater than that for turbostratic graphite. occur simultaneously, but their conclusions regarding
In addition, for two coaxial armchair tubules, es- the Peierls gap for carbon nanotubes are essentially as
timates for the translational and rotational energy discussed above.
barriers (of 0.23 meV/atom and 0.52 meV/atom, re- The band structure of four concentric armchair tu-
spectively) were obtained, suggesting significant trans- bules with 10, 20, 30, and 40 carbon atoms around
lational and rotational interlayer mobility of ideal their circumferences (external diameter 27.12 A) was
tubules at room temperature[l6,17]. Of course, con- calculated, where the tubules were positioned to min-
straints associated with the cap structure and with de- imize the energy for all bilayered pairs[l7]. In this
fects on the tubules would be expected to restrict these case, the four-layered tubule remains metallic, simi-
motions. The detailed band calculations for various lar to the behavior of two double-layered armchair
interplanar geometries for the two coaxial armchair tu- nanotubes, except that tiny band splittings form.
bules basically confirm the tight binding results men- Inspired by experimental observations on bundles
tioned above[ 16,171. of carbon nanotubes, calculations of the electronic
Further calculations are needed to determine whether structure have also been carried out on arrays of (6,6)
or not a Peierls distortion might remove the coaxial armchair nanotubes to determine the crystalline struc-
nesting of carbon nanotubes. Generally 1D metallic ture of the arrays, the relative orientation of adjacent
bands are unstable against weak perturbations which nanotubes, and the optimal spacing between them.
open an energy gap at EF and consequently lower the Figure 5 shows one tetragonal and two hexagonal ar-
total energy, which is known as the Peierls instabil- rays that were considered, with space group symme-
ity[23]. In the case of carbon nanotubes, both in-plane tries P4,/mmc (DZh)h), P6/mmm (Dih), and P6/mcc
and out-of-plane lattice distortions may couple with (D,‘,), respectively[16,17,30]. The calculation shows
the electrons at the Fermi energy. Mintmire and White
have discussed the case of in-plane distortion and have
concluded that carbon nanotubes are stable against a
Peierls distortion in-plane at room temperature[24],
though the in-plane distortion, like a KekulC pattern,
will be at least 3 times as large a unit cell as that of
graphite. The corresponding chiral vectors satisfy the
condition for metallic conduction (n - m = 3r,r:in-
teger). However, if we consider the direction of the
translational vector T, a symmetry-lowering distortion
is not always possible, consistent with the boundary
conditions for the general tubules[25]. On the other
hand, out-of-plane vibrations do not change the size
of the unit cell, but result in a different site energy for
carbon atoms on A and B sites for carbon nanotube
structures[26]. This situation is applicable, too, if the
dimerization is of the “quinone” or chain-like type,
where out-of-plane distortions lead to a perturbation
approaching the limit of 2D graphite. Further, Hari-
gaya and Fujita[27,28] showed that an in-plane alter-
nating double-single bond pattern for the carbon
atoms within the 1D unit cell is possible only for sev-
eral choices of chiral vectors.
Solving the self-consistent calculation for these
types of distortion, an energy gap is always opened by
the Peierls instability. However, the energy gap is very
small compared with that of normal 1D cosine energy
bands. The reason why the energy gap for 1D tubules
is so small is that the energy gain comes from only one
of the many 1D energy bands, while the energy loss Fig. 5. Schematic representation of arrays of carbon nano-
due to the distortion affects all the 1D energy bands. tubes with a common tubule axial direction in the (a) tetrag-
Thus, the Peierls energy gap decreases exponentially onal, (b) hexagonal I, and (c) hexagonal I1 arrangements. The
reference nanotube is generated using a planar ring of twelve
with increasing number of energy bands N[24,26-281. carbon atoms arranged in six pairs with the Dsh symmetry
Because the energy change due to the Peierls distor- [16,17,30].