Page 41 - Carbon Nanotubes

et
32 M. S. DRESSZLHAUS a/.
same. This is a symmetry-imposed result that is gen- search directions. The multi-wall calculations have
erally valid for all carbon nanotubes. been predominantly done for double-wall tubules, al-
Regarding the electronic structure, the number of though some calculations have been done for a four-
energy bands for (n,O) zigzag carbon nanotubes is walled tubule[16-18] and also for nanotube arrays
2n, the number of carbon atoms per unit cell, with [ 16,171.
symmetries The first calculation for a double-wall carbon
nanotube[l5] was done using the tight binding tech-
nique, which sensitively includes all symmetry con-
straints in a simplified Hamiltonian. The specific
geometrical arrangement that was considered is the
most commensurate case possible for a double-layer
nanotube, for which the ratio of the chiral vectors for
A symmetry-imposed band degeneracy occurs for the the two layers is 1 :2, and in the direction of transla-
Ef+3)/21g and EE(~-~,~~~ Fermi level, tional vectors, the ratio of the lengths is 1 : 1. Because
at
the
bands
when n = 3r, r being an integer, thereby giving rise the C60-derived tubule has a radius of 3.4 A, which is
to zero gap tubules with metallic conduction. On the close to the interlayer distance for turbostratic graph-
other hand, when n # 3r, a bandgap and semicon- ite, this geometry corresponds to the minimum diam-
ducting behavior results. Independent of whether the eter for a double-layer tubule. This geometry has
tubules are conducting or semiconducting, each of many similarities to the AB stacking of graphite. In
the [4 + 2(n -1)J energy bands is expected to show a the double-layer tubule with the diameter ratio 1:2, the
(E - Eo)-1’2 type singularity in the density of states interlayer interaction y1 involves only half the num-
at its band extremum energy Eo [ 101. ber of carbon atoms as in graphite, because of the
The most promising present technique for carrying smaller number of atoms on the inner tubule. Even
out sensitive measurements of the electronic proper- though the geometry was chosen to give rise to the
ties of individual tubules is scanning tunneling spec- most commensurate interlayer stacking, the energy
troscopy (STS) becaise of the ability of the tunneling dispersion relations are only weakly perturbed by the
tip to probe most sensitively the electronic density of interlayer interaction.
states of either a single-wall nanotube[l2], or the out- More specifically, the calculated energy band struc-
ermost cylinder of a multi-wall tubule or, more gen- ture showed that two coaxial zigzag nanotubes that
erally, a bundle of tubules. With this technique, it is would each be metallic as single-wall nanotubes yield
further possible to carry out both STS and scanning a metallic double-wall nanotube when a weak inter-
tunneling microscopy (STM) measurements at the layer coupling between the concentric nanotubes is
same location on the same tubule and, therefore, to introduced. Similarly, two coaxial semiconducting tu-
measure the tubule diameter concurrently with the STS bules remain semiconducting when the weak interlayer
spectrum. coupling is introduced[l5]. More interesting is the case
Although still preliminary, the study that provides of coaxial metal-semiconductor and semiconductor-
the most detailed test of the theory for the electronic metal nanotubes, which also retain their individual
properties of the ID carbon nanotubes, thus far, is the metallic and semiconducting identities when the weak
combined STMISTS study by Olk and Heremans[ 131. interlayer interaction is turned on. On the basis of this
In this STM/STS study, more than nine individual result, we conclude that it might be possible to prepare
.multilayer tubules with diameters ranging from 1.7 to metal-insulator device structures in the coaxial geom-
9.5 nm were examined. The I- Vplots provide evidence etry without introducing any doping impurities[20], as
for both metallic and semiconducting tubules[ 13,141. has already been suggested in the literature[10,20,21].
Plots of dl/dVindicate maxima in the 1D density of A second calculation was done for a two-layer tu-
states, suggestive of predicted singularities in the 1D bule using density functional theory in the local den-
density of states for carbon nanotubes. This STM/ sity approximation to establish the optimum interlayer
STS study further shows that the energy gap for the distance between an inner (53) armchair tubule and
semiconducting tubules is proportional to the inverse an outer armchair (10,lO) tubule. The result of this
tubule diameter lid,, and is independent of the tubule calculation yielded a 3.39 A interlayer separation
chirality. [16,17], with an energy stabilization of 48 meV/car-
bon atom. The fact that the interlayer separation is
about halfway between the graphite value of 3.35 A
4. MULTI-WALL NANOTUBES AND ARRAYS
and the 3.44 A separation expected for turbostratic
Much of the experimental observations on carbon graphite may be explained by interlayer correlation be-
nanotubes thus far have been made on multi-wall tu- tween the carbon atom sites both along the tubule axis
bules[15-19]. This has inspired a number of theoretical direction and circumferentially. A similar calculation
calculations to extend the theoretical results initially for double-layered hyper-fullerenes has also been car-
obtained for single-wall nanotubes to observations in ried out, yielding an interlayer spacing of 3.524 A for
multilayer tubules. These calculations for multi-wall C60@C240 with an energy stabilization of 14 meV/C
tubules have been informative for the interpretation atom for this case[22]. In the case of the double-
of experiments, and influential for suggesting new re- layered hyper-fullerene, there is a greatly reduced pos-