Short-range correlation effects on the plasmons in cylindrical tubules
B. Tanatar
Department of Physics, Bilkent University, Bilkent, 06533 Ankara, Turkey
Received 21 August 1996
We study the collective excitation of single and coaxial cylindrical tubules in the very low density regime.
Exchange-correlation effects neglected in the random-phase approximation are shown to be important for
plasmon modes. We also calculate the spectral weight function Im 1/ ( q , ) to investigate the collective
modes as measured in electron energy-loss spectroscopy experiments. S0163-18299706804-5
The recent developments in the synthesis techniques have
led to the fabrication of hollow tubules of graphitic
materials.
1
A graphene tubule is essentially a graphitic sheet
rolled up in the cylindrical form. Charge carriers may be
introduced onto the tubules in a controlled manner through
intercalation
2
as in the case of graphite intercalated
compounds.
3
The typical radius of such cylindrical tubules is
several nanometers, which results in a one-dimensional char-
acter of the motion of these carriers. Experimentally it has
been found that the synthesized tubules can be open ended.
4
Metallic or semiconducting behavior of single tubules is
predicted.
5
From a theoretical point of view carbon-based
microstructures offer opportunities to study the effects of di-
mensionality on the collective excitations.
6
Moreover, the
carbon nanotubules have caused considerable excitement and
activity because of possible device applications.
7
The charge carriers on a tubule may be modeled by a
quasi-one-dimensional Q1D electron-gas as a first step to-
wards the full understanding of complicated excitation spec-
tra of graphitic materials. The elementary excitations in cy-
lindrical tubules were studied by Lin and Shung
8
using the
random-phase approximation RPA. Longe and Bose
9
have
presented semiclassical calculations for metallic graphene tu-
bules. A more systematic study of the excitation spectrum of
multishell fullerenes and coaxial nanotubules appeared
recently.
6
A microscopic approach utilizing the symmetry
properties is also reported for carbon fibers.
10
Band-structure
effects have recently been considered.
11
The plasma modes
in coaxial carbon tubules have been observed experimentally
by means of electron energy-loss spectroscopy.
12
Our aim in this paper is to examine the collective excita-
tions of very low density electrons on single and coaxial
tubules. We include the short-range correlation effects
through the local-field corrections. A similar approach was
taken by Miesenbo
¨
ck and Tosi
13
in their study of layered
metal intercalated graphite. The previous works on the col-
lective excitations of graphene tubules have used the RPA,
which takes into account dynamic screening in the electron-
gas but does not include the corrections due to exchange-
correlation xc effects. The local-field theory
14
of Singwi
et al. STLS uses the static pair-distribution function to ap-
proximate the short-range electronic correlations. It has been
noted that corrections to the RPA become more important in
one- and two-dimensional systems than that in the bulk. Sys-
tems with finite number of particles also show strong depen-
dence on the many-body effects.
15
It has been found, for
instance, that the xc effect on the plasmon dispersion de-
creases with increasing cluster size.
A free-electron gas model for cylindrical tubules was de-
veloped by several researchers.
8,16,17
The one-dimensional
1D nature of the system has similarities to other quasi-one-
dimensional electron-gas models commonly used to describe
the semiconducting quantum wires.
18
A notable feature of
the hollow tubule model is that the angular momentum quan-
tum number L completely decouples the intrasubband
( L =0) and intersubband ( L 0) excitations from each
other.
The collective excitations of a single tubule are deter-
mined from the zeros of the dielectric function
( q , ; L ) =1 -V ( q ; L ) 1 -G ( q ; L )
( q , ; L ), where the
Coulomb interaction between electrons on a tubule is
19
V ( q ; L ) =(4 e
2
/
0
) I
L
( qR ) K
L
( qR ), in which I
L
( x ) and
K
L
( x ) are the modified Bessel functions of the first and sec-
ond kind, respectively. The local-field factor G ( q ; L ) modi-
fies the bare Coulomb interaction by describing the xc effects
neglected in the RPA. The density response function
( q , ; L ) appropriate for electrons on a tubule is derived by
several authors.
8,16,17
It has contributions from occupied sub-
bands ( l =0,1,2, . . . ), thus giving rise to a rich excita-
tion spectrum for the collective modes. In the following we
shall concentrate on the most energetic plasmon mode, since
the oscillator strength is dominated by this mode.
8
The local-
field factor G ( q ; L ) is calculated using the 1D version of the
STLS scheme
14
self-consistently with the static structure fac-
tor S ( q ) as obtained from the frequency integral of the
fluctuation-disspation theorem.
We first discuss the results of our calculations for a single
tubule. For illustration purposes we take the radius of the
tubule R =11 Å, the effective mass of the electrons
m * =m
e
/4, and the background dielectric constant
0
=2.4,
so that a graphene tubule is closely described. We assume
that the electrons are introduced through intercalation. The
linear electron density is chosen to be n
1D
a
B
* =1/(2 r
s
)
=0.5, where a
B
* is the effective Bohr radius and r
s
is the
dimensionless electron-gas coupling parameter. This corre-
sponds to a sheet density of n
2D
=n
1D
/(2 R )
1.4510
13
cm
-2
, low enough to make the short-range cor-
relation effects appreciable. For the above parameters we
find the Fermi energy to be E
F
=0.14 eV, so that only the
lowest three subbands are occupied, viz., l =0,1.
PHYSICAL REVIEW B 15 JANUARY 1997-I VOLUME 55, NUMBER 3
55 0163-1829/97/553/13613/$10.00 1361 © 1997 The American Physical Society