Page 36 - Carbon Nanotubes

PHYSICS OF CARBON NANOTUBES

M. S. DRESSELHAUS,’ DRESSELHAUS,* R. SAITO~
and
G.
‘Department of Electrical Engineering and Computer Science and Department of Physics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A.
’Francis Bitter National Magnet Laboratory, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, U.S.A.
‘Department of Electronics-Engineering, University of Electro-Communications,
Tokyo 182, Japan
(Received 26 October 1994; accepted 10 February 1995)

Abstract-The fundamental relations governing the geometry of carbon nanotubes are reviewed, and ex-
plicit examples are presented. A framework is given for the symmetry properties of carbon nanotubes for
both symmorphic and non-symmorphic tubules which have screw-axis symmetry. The implications of sym-
metry on the vibrational and electronic structure of ID carbon nanotube systems are considered. The cor-
responding properties of double-wall nanotubes and arrays of nanotubes are also discussed.
Key Words-Single-wall, multi-wall, vibrational modes, chiral nanotubes, electronic bands, tubule arrays.
1. INTRODUCTION governing these parameters, and list typical numeri-
Carbon nanotube research was greatly stimulated by cal values for these parameters.
the initial report of observation of carbon tubules of In the theoretical carbon nanotube literature, the
nanometer dimensions[l] and the subsequent report focus is on single-wall tubules, cylindrical in shape
on the observation of conditions for the synthesis of with caps at each end, such that the two caps can be
large quantities of nanotubes[2,3]. Since these early re- joined together to form a fullerene. The cylindrical
ports, much work has been done, and the results show portions of the tubules consist of a single graphene
basically that carbon nanotubes behave like rolled-up sheet that is shaped to form the cylinder. With the re-
cylinders of graphene sheets of sp2 bonded carbon cent discovery of methods to prepare single-walled
atoms, except that the tubule diameters in some cases nanotubes[4,5], it is now possible to test the predic-
are small enough to exhibit the effects of one-dimen- tions of the theoretical calculations.
sional(1D) periodicity. In this article, we review sim- It is convenient to specify a general carbon nano-
ple aspects of the symmetry of carbon nanotubules tubule in terms of the tubule diameter d, and the chi-
(both monolayer and multilayer) and comment on the ral angle 0, which are shown in Fig. 1. The chiral
significance of symmetry for the unique properties vector Ch is defined in Table 1 in terms of the integers
predicted for carbon nanotubes because of their 1D (n,rn) and the basis vectors a, and a2 of the honey-
periodicity. comb lattice, which are also given in the table in
Of particular importance to carbon nanotube phys- terms of rectangular coordinates. The integers (n, m)
ics are the many possible symmetries or geometries uniquely determine dr and 0. The length L of the chi-
that can be realized on a cylindrical surface in carbon ral vector c, (see Table 1) is directly related to the tu-
nanotubes without the introduction of strain. For 1D bule diameter &. The chiral angle 0 between the Ch
systems on a cylindrical surface, translational sym- direction and the zigzag direction of the honeycomb
metry with a screw axis could affect the electronic lattice (n,O) (see Fig. 1) is related in Table 1 to the
structure and related properties. The exotic electronic integers (n,m).
properties of 1D carbon nanotubes are seen to arise We can specify a single-wall C,,-derived carbon
predominately from intralayer interactions, rather nanotube by bisecting a Cm molecule at the equator
than from interlayer interactions between multilayers and joining the two resulting hemispheres with a cy-
within a single carbon nanotube or between two dif- lindrical tube having the same diameter as the C60
ferent nanotubes. Since the symmetry of a single nano- molecule, and consisting of the honeycomb structure
tube is essential for understanding the basic physics of of a single layer of graphite (a graphene layer). If the
carbon nanotubes, most of this article focuses on the C6, molecule is bisected normal to a five-fold axis,
symmetry properties of single layer nanotubes, with the “armchair” tubule shown in Fig. 2 (a) is formed,
a brief discussion also provided for two-layer nano- and if the C,, molecule is bisected normal to a 3-fold
tubes and an ordered array of similar nanotubes. axis, the “zigzag” tubule in Fig. 2(b) is formed[6].
Armchair and zigzag carbon nanotubules of larger di-
ameter, and having correspondingly larger caps, can
2. FUNDAMENTAL PARAMETERS AND likewise be defined, and these nanotubules have the
RELATIONS FOR CARBON NANOTUBES general appearance shown in Figs. 2(a) and (b). In ad-
In this sect.ion, we summarize the fundamental pa- dition, a large number of chiral carbon nanotubes can
rameters for carbon nanotubes, give the basic relations be formed for 0 < 10 1 < 30°, with a screw axis along

27