Page 139 - Carbon Nanotubes

VIBRATIONAL MODES OF CARBON NANOTUBES;
SPECTROSCOPY AND THEORY

P. C. EKLUND,’ J. M. HOLDEN,’ and R. A. JISHI*
’Department of Physics and Astronomy and Center for Applied Energy Research,
University of Kentucky, Lexington, KY 40506, U.S.A.
’Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.;
Department of Physics, California State University, Los Angeles, CA 90032, USA.

(Received 9 February 1995; accepted in revised form 21 February 1995)

Abstract-Experimental and theoretical studies of the vibrational modes of carbon nanotubes are reviewed.
The closing of a 2D graphene sheet into a tubule is found to lead to several new infrared (1R)- and Raman-
active modes. The number of these modes is found to depend on the tubule symmetry and not on the di-
ameter. Their diameter-dependent frequencies are calculated using a zone-folding model. Results of Raman
scattering studies on arc-derived carbons containing nested or single-wall nanotubes are discussed. They
are compared to theory and to that observed for other sp2 carbons also present in the sample.
Key Words-Vibrations, infrared, Raman, disordered carbons, carbon nanotubes, normal modes.

1. INTRODUCTION 2. OVERVIEW OF RAMAN SCATTERING
FROM SP2 CARBONS
In this paper, we review progress in the experimental
detection and theoretical modeling of the normal modes Because a single carbon nanotube may be thought
of vibration of carbon nanotubes. Insofar as the theo- of as a graphene sheet rolled up to form a tube, car-
retical calculations are concerned, a carbon nanotube bon nanotubes should be expected to have many prop-
is assumed to be an infinitely long cylinder with a mono- erties derived from the energy bands and lattice
layer of hexagonally ordered carbon atoms in the tube dynamics of graphite. For the very smallest tubule di-
wall. A carbon nanotube is, therefore, a one-dimensional ameters, however, one might anticipate new effects
system in which the cyclic boundary condition around stemming from the curvature of the tube wall and the
the tube wall, as well as the periodic structure along closing of the graphene sheet into a cylinder. A natu-
the tube axis, determine the degeneracies and symmetry ral starting point for the discussion of the vibrational
classes of the one-dimensional vibrational branches modes of carbon nanotubes is, therefore, an overview
i1-31 and the eIectronic energy bands[4-12]. of the vibrational properties of sp2 carbons, includ-
Nanotube samples synthesized in the laboratory are ing carbon nanoparticles, disordered sp2 carbon, and
typically not this perfect, which has led to some confu- graphite. This is also important because these forms
sion in the interpretation of the experimental vibrational of carbon are also often present in tubule samples as
spectra. Unfortunately, other carbonaceous material “impurity phases.”
(e.g., graphitic carbons, carbon nanoparticles, and In Fig. la, the phonon dispersion relations for 3D
amorphous carbon coatings on the tubules) are also graphite calculated from a Born-von Karman lattice-
generally present in the samples, and this material dynamical model are plotted along the high symmetry
may contribute artifacts to the vibrational spectrum. directions of the Brillouin zone (BZ). For comparison,
Defects in ithe wall (e.g., the inclusion of pentagons we show, in Fig. lb, the results of a similar calcula-
and heptagons) should also lead to disorder-induced tion[29] for a 2D infinite graphene sheet. Interactions
features in the spectra. Samples containing concentric, up to fourth nearest neighbors were considered, and
coaxial, “nested” nanotubes with inner diameters the force constants were adjusted to fit relevant exper-
-8 nm and outer diameters -80 nm have been syn- imental data in both of these calculations. Note that
thesized using carbon arc methods[l3,14], combustion there is little dispersion in the k, (I? to A) direction
fllames[l5], and using small Ni or Co catalytic parti- due to the weak interplanar interaction in 3D graphite
cles in hydrocarbon vapors[lb-201. Single-wall nano- (Fig. IC). To the right of each dispersion plot is the
tubes (diameter 1-2 nm) have been synthesized by calculated one-phonon density of states. On the energy
adding metal catalysts to the carbon electrodes in a dc scale of these plots, very little difference is detected
arc[21,22] To date, several Raman scattering stud- between the structure of the 2D and 3D one-phonon
~
ies[23-281 of nested and single-wall carbon nanotube density of states. This is due to the weak interplanar
samples have appeared. coupling in graphite. The eigenvectors for the optically
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