Page 46 - Carbon Nanotubes

ELECTRONIC AND STRUCTURAL PROPERTIES
OF CARBON NANOTUBES

J. W. MINTMIRE and C. T. WHITE
Chemistry Division, Naval Research Laboratory, Washington, DC 20375-5342, U.S.A.
(Received 12 October 1994; accepted in revised form 15 February 1995)

Abstract-Recent developments using synthetic methods typical of fullerene production have been used
to generate graphitic nanotubes with diameters on the order of fullerene diameters: “carbon nanotubes.”
The individual hollow concentric graphitic nanotubes that comprise these fibers can be visualized as con-
structed from rolled-up single sheets of graphite. We discuss the use of helical symmetry for the electronic
structure of these nanotubes, and the resulting trends we observe in both band gap and strain energy ver-
sus nanotube radius, using both empirical and first-principles techniques. With potential electronic and
structural applications, these materials appear to be appropriate synthetic targets for the current decade.
Key Words-Carbon nanotube, electronic properties, structural properties, strain energy, band gap, band
structure, electronic structure.

1. INTRODUCTION ture used in the rest of the manuscript, and present an
analysis of the rotational and helical symmetries of the
Less than four years ago Iijima[l] reported the novel
synthesis based on the techniques used for fullerene nanotube. Then, we will discuss the electronic struc-
synthesis[2,3] of substantial quantities of multiple-shell ture of the nanotubes in terms of applying Born-von
graphitic nanotubes with diameters of nanometer di- Karman boundary conditions to the two-dimensional
mensions. These nanotube diameters were more than graphene sheet. We will then discuss changes intro-
duced by treating the nanotube realistically as a three-
an order of magnitude smaller than those typically ob-
tained using routine synthetic methods for graphite fi- dimensional system with helicity, including results
bers[4,5]. This work has been widely confirmed in the both from all-valence empirical tight-binding results
literature, with subsequent work by Ebbesen and and first-principles local-density functional (LDF)
Ajayan[6] demonstrating the synthesis of bulk quan- results.
tities of these materials. More recent work has further
demonstrated the synthesis of abundant amounts of 2. NANOTUBE STRUCTURE AND SYMMETRY
single-shell graphitic nanotubes with diameters on the
order of one nanometer[7-9]. Concurrent with these Each single-walled nanotube can be viewed as a
experimental studies, there have been many theoreti- conformal mapping of the two-dimensional lattice of
cal studies of the mechanical and electronic properties a single sheet of graphite (graphene), depicted as the
of these novel fibers[lO-30]. Already, theoretical stud- honeycomb lattice of a single layer of graphite in Fig. 1,
ies of the individual hollow concentric graphitic nano- onto the surface of a cylinder. As pointed out by
tubes, which comprise these fibers, predict that these Iijima[ 11, the proper boundary conditions around the
nanometer-scale diameter nanotubes will exhibit con- cylinder can only be satisfied if one of the Bravais lat-
ducting properties ranging from metals to moderate tice vectors of the graphite sheet maps to a circumfer-
bandgap semiconductors, depending on their radii and ence around the cylinder. Thus, each real lattice vector
helical structure[lO-221. Other theoretical studies have of the two-dimensional hexagonal lattice (the Bravais
focused on structural properties and have suggested lattice for the honeycomb) defines a different way of
that these nanotubes could have high strengths and rolling up the sheet into a nanotube. Each such lattice
rigidity resulting from their graphitic and tubular vector, E, can be defined in terms of the two primi-
structure[23-30]. The metallic nanotubes- termed ser- tive lattice vectors RI and R2 and a pair of integer in-
pentine[23] -have also been predicted to be stable dices [n,,nz], such that B =nlR1 + n2R2, with Fig. 2
against a Peierls distortion to temperatures far below depicting an example for a [4,3] nanotube. The point
room temperaturejl01. The fullerene nanotubes show group symmetry of the honeycomb lattice will make
the promise of an array of all-carbon structures that many of these equivalent, however, so truly unique
exhibits a broad range of electronic and structural nanotubes are only generated using a one-twelfth ir-
properties, making these materials an important syn- reducible wedge of the Bravais lattice. Within this
thetic target for the current decade. wedge, only a finite number of nanotubes can be con-
Herein, we summarize some of the basic electronic structed with a circumference below any given value.
and structural properties expected of these nanotubes The construction of the nanotube from a confor-
from theoretical grounds. First we will discuss the ba- mal mapping of the graphite sheet shows that each
sic structures of the nanotubes, define the nomencla- nanotube can have up to three inequivalent (by point
37