Phonon and thermal properties of achiral single wall carbon nanotubes
ABSTRACT A detailed theoretical study of the phonon and thermal properties of achiral single wall carbon nanotubes has been carried
out using force constant model considering up to third nearestneighbor interactions. We have calculated the phonon dispersions,
density of states, radial breathing modes (RBM) and the specific heats for various zigzag and armchair nanotubes, with radii
ranging from 2.8 Å to 11.0 Å. A comparative study of phonon spectrum with measured Raman data reveals that the number of Raman
active modes for a tube does not depend on the number of atoms present in the unit cell but on its chirality. Calculated phonon
modes at the zone center more or less accurately predicted the Raman active modes. The radial breathing mode is of particular
interest as for a specific radius of a nanotube it is found to be independent of its chirality. We have also calculated the
variation of RBM and Gband modes for tubes of different radii. RBM shows an inverse dependence on the radius of the tube.
Finally, the values of specific heat are calculated for various nanotubes at room temperature and it was found that the specific
heat shows an exponential dependence on the diameter of the tube.
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 rPhonon and thermal properties of achiral SWCNTs. S Ijima, C Barbec, A Maiti, J Bernholc .
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PRAMANA
— journal of
physics
c ? Indian Academy of SciencesVol. 67, No. 2
August 2006
pp. 305–317
Phonon and thermal properties of achiral single
wall carbon nanotubes
PRAPTI SAXENA1,∗and SANKAR P SANYAL1,2
1Condensed Matter Laboratory, Department of Physics, Barkatullah University,
Bhopal 462 026, India
2Department of Electronics, Barkatullah University, Bhopal 462 026, India
∗Corresponding author. Email: prapti194@rediffmail.com; spsanyal@sancharnet.in
MS received 11 August 2005; revised 17 April 2006; accepted 23 June 2006
Abstract.
single wall carbon nanotubes has been carried out using force constant model considering
up to third nearestneighbor interactions. We have calculated the phonon dispersions,
density of states, radial breathing modes (RBM) and the specific heats for various zigzag
and armchair nanotubes, with radii ranging from 2.8˚ A to 11.0˚ A. A comparative study of
phonon spectrum with measured Raman data reveals that the number of Raman active
modes for a tube does not depend on the number of atoms present in the unit cell but
on its chirality. Calculated phonon modes at the zone center more or less accurately
predicted the Raman active modes. The radial breathing mode is of particular interest
as for a specific radius of a nanotube it is found to be independent of its chirality. We
have also calculated the variation of RBM and Gband modes for tubes of different radii.
RBM shows an inverse dependence on the radius of the tube. Finally, the values of specific
heat are calculated for various nanotubes at room temperature and it was found that the
specific heat shows an exponential dependence on the diameter of the tube.
A detailed theoretical study of the phonon and thermal properties of achiral
Keywords. Nanotubes; thermal properties; zone folding; phonons.
PACS Nos63.32.+m; 65.80.+n; 61.48.+c
1. Introduction
Carbon nanotubes are quasionedimensional structures and are projected as one
of the most promising candidates of future nanoelectronics owing to their extra
ordinary electronic, thermal, mechanical, elastic and electrical properties [1–10].
Although a satisfactory amount of theoretical work has already been published on
the lattice dynamical properties of carbon nanotubes, there is a great need for sim
pler models that can explain accurately a satisfactory amount of experimental data
available on single wall carbon nanotubes (SWCNTs). So far, many theoretical
models have been proposed for the study of the vibrational spectra and phonon
density of state of SWCNTs. In prior efforts by Jishi et al [11], the force constant
305
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Prapti Saxena and Sankar P Sanyal
model using zonefolding approach was employed for the calculation of phonon dis
persion relations of the nanotubes. In this method the graphite force constants are
modified to include the effect of curvature. However, only a few calculations of
phonon dispersions in nanotubes have been accomplished by zonefolding method
with the correction in the dynamical matrix to obtain two acoustic modes. Later,
Saito et al [12] and others have reported the phonon properties calculated using the
force constant model. In these approaches the authors have modified the force con
stants in order to fulfill the rotational sum rule and to obtain the torsional acoustic
branch. Lately, SanchezPortal et al [13] have reported the ab initio and Yu et al
[14] have reported the tight binding calculations of vibrational properties of nan
otubes. The latter approaches are free from the deficiencies of the first two models.
However, their work is limited to tubes with radius less than 7˚ A. Maultzsch et al
[15] have used the force constant approach but emphasized mainly on the behavior
of highenergy Raman active modes. In another approach, Popov et al [16] have
used the valence force field model for the calculations of phonon dispersion rela
tions. This model takes into account the screw symmetry of the nanotubes, which
results in a reduction of the size of the dynamical matrix to six for all tube types
and allows one to study the vibrations of all observable nanotubes. Also they have
considered up to second nearestneighbor interactions and focused mainly on the
breathing like modes in carbon nanotubes. All these models discussed above are
successful in some way or other but have some inherent deficiencies in explaining
phonon modes at lower frequencies. The torsional mode remained unexplained in
these studies.
On the other hand, very few theoretical approaches are reported so far on the
thermal properties of carbon nanotubes [5,6]. Berber et al [6] have used the mole
cular dynamic simulation to predict the thermal conductivity of nanotubes. They
showed that the thermal conductivity of an isolated (10,10) carbon nanotube could
be as high as 6600 W/mK at room temperature. This exceptionally high value of
thermal conductivity reflects the onedimensional conduction behavior of nanotubes
with least phonon scattering. Recent experimental results have, however, proved
the authenticity of these results [7–10]. Few authors [17,18] have also reported the
negative thermal expansion coefficient for nanotubes at room temperature. In view
of these facts there is a great need for an appropriate theoretical model, which can
be useful in the characterization of the nanotubes on the basis of these thermal
properties, as well.
Since carbon nanotubes are huge structures, it is unphysical to depend on mod
els, which have large number of parameters. On the other hand, ab initio methods
are much timeconsuming. Therefore, in order to simplify the problem and to get
reasonable quantitative results and understanding of the phonon dynamics and
thermal properties, we have employed the force constant approach for the calcu
lation of phonon modes in carbon nanotubes. This model is based on Born–von
Karman theory [19] of lattice dynamics. In this model interatomic interactions are
represented in terms of sum of radial and angular bending forces extended up to
third nearest neighbors. The Brillouin zone of nanotube is defined by applying zone
folding technique to the graphene layer. We have calculated the density of states
and compared them with the experimental Raman modes. Also, we have correctly
predicted the diameter dependence of the radial breathing mode (RBM) for all the
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Phonon and thermal properties of achiral SWCNTs
Figure 1. Various possible rollup directions of graphene layer to generate
different types of nanotubes. a1 and a2 represent the primitive vectors of
graphene monolayer.
nanotubes. The variation of RBM with the radius of the tube further allows us to
predict the RBM for tubes of arbitrary chirality. We presented the results of radius
dependence of Gmode and discussed the overbending of highenergy longitudinal
optic modes in single wall nanotubes. Lastly, by using the phonon spectrum, we
have calculated the specific heat of both zigzag and armchair nanotubes and found
that their values show an exponential diameter dependence. We organize the paper
as follows: In §2 we give a brief description of the structure of achiral nanotubes
and the phenomena of zone folding. Section 3 deals with the theoretical model. In
§4 we present our results, followed by discussion.
2. Structure of nanotube
Nanotubes are generally classified on the basis of their angle of chirality (θ), which
is defined as the angle between the circumference and the nearest zigzag carbon
atoms. Accordingly they are classified into two groups, namely, achiral (armchair
(θ = 30◦) and zigzag (θ = 0◦)) and chiral (0◦< θ < 30◦) tubes. Conventionally,
they are characterized by the indices n and m of graphene lattice vectors a1and
a2, used in the definition of chiral vector (Ch) of nanotube. The chiral vector is
one of the unit vectors of onedimensional carbon nanotubes and defines the rollup
direction of graphene monolayer, defined as [20] (figure 1)
Ch= na1+ ma2≈ (n,m).
Armchair configuration is represented by the indices (n,n), while (n,0) and (n,m)
represent zigzag and chiral tubes respectively. Another unit vector, which is called
translational vector T, is parallel to the nanotube axis and is directed normal to the
chiral vector Chin the unrolled graphene lattice. The reciprocal lattice of nano
tubes is defined by two vectors, q1and q2, such that vector q1is directed along the
circumference and is quantized to give N discrete wavevectors while q2is directed
parallel to the translational vector, along the axis of the tube and is continuous.
Quantization of the reciprocal lattice vector q1around the circumference is known
as zone folding and is the result of boundary condition, which states that the
(1)
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Prapti Saxena and Sankar P Sanyal
wavelength of the mode multiplied by an integer should not be greater than the
circumference of the tube. In our calculations of phonon dispersion curves for the
carbon nanotubes are plotted along q2direction. Number of atomic pairs in the
unit cell of a nanotube is given by N = 2(n2+ m2+ nm)/dR where dR is the
greatest common divisor (gcd) of (2m + n) and (2n + m).
3. Theoretical model
The present model has been worked out in the framework of Born–von Karman
theory of lattice dynamics [19,21]. Two types of forces are assumed to exist between
the atoms, the central force and the angular force. Central forces act along the line
joining the two atoms while angular forces are proportional to the angle, which
the line joining the displaced atoms makes with the line joining their equilibrium
position. Components of total radial and angular forces, which are also called
noncentral forces, are given by
Fk(??) = −α?
p[uk(?) − uk(??)]
+(αp− α?
p)λk(??)
?
j
λj(??)[uj(??) − uj(?)],
(2)
where, uk(?) and uk(??) are the components of instantaneous displacement of ?th
and ??th atoms respectively for k = 1,2,3. αpand α?
force constants respectively where, p = 1,2,3 for the first, second and third nearest
neighbors. λj(??) are the direction cosines of ??th atom with respect to ?th atom,
with j = 1,2,3.
Thus F1(??) denotes the xcomponent (k = 1) of the force acting on the ?th par
ticle arising from the displacement of the ??th atom. After Fourier transformation
of the atomic displacements and the force constant matrix, the dynamical problem
is reduced to the solution of the system of 6N equations for the tube having 2N
atoms in its unit cell.
pare the central and angular
Dkk?(q) − mω2δkk? = 0.
Equation (3) is the eigenvalue equation and describes adequately the phonon dis
persion relation. One can define the frequency distribution function, g(ω), as the
phonon density of states obtained from the solution of eq. (3), such that g(ω)dω
represents the number of vibrational modes of the lattice in the frequency range ω
and ω + dω.
?
where Ω is the volume of the unit cell of the nanotube, dSqis the surface element
of constant frequency surface in qspace, ∇ωqis the group velocity.
We have used Lorentzian distribution function and performed the multiplepeak
fitting to obtain the exact frequency distribution function, which is given by
(3)
g(ω)dω =
Ω
(2π)3
dSq
∇ωq,
(4)
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Phonon and thermal properties of achiral SWCNTs
g(ω)dω =
Ω
(2π)3
?
q
δ
(ω − ωq)2+ δ2,
(5)
where ωq is the central frequency of the Lorentzian for a given value of phonon
wave vector q while δ is the fullwidth at halfmaximum for a single fitted peak.
Ω is the volume of the unit cell of the nanotube. Once the frequency distribution
function is framed one can calculate the specific heat from the relation given by
?ωmax
where
??ω
where Cq(ω) is the heat capacity of lattice wave with wave vector q and angu
lar frequency ω. kB and ? are the Boltzmann’s constant and Planck’s constant
respectively.
CV(T) =
0
Cq(ω)g(ω)dω,
(6)
Cq(ω) = kB
kBT
?2
exp(?ω/kBT)
[exp(?ω/kBT) − 1]2,
(7)
4. Result and discussions
In the present paper we have reported the phonon dispersion curves for a number of
zigzag and armchair single wall carbon nanotubes with their radius ranging from 2.8
˚ A to 11.0˚ A. The values of model parameters are derived by associating them to the
known experimental results on Raman scattering [22] and those reported by Bond
polarizability model of SWCNT [20]. The bestfit model parameters for (10,10)
armchair single wall carbon nanotube are summarized in table 1. It can be seen
Table 1.
wall carbon nanotube.
Structural and model parameters for single
Structural parameters
C–C bond length (aC–C)
Length of primitive vectors of
graphene (a1 = a2 = a)
1.44˚ A
√3 × aC–C
Model parameters
(central and angular
force constants in
dynes/cm)
α1
α2
α3
α?
α?
α?
9705.0
4070.0
−2775.0
3975.0
−247.5
−635.0
1
2
3
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Prapti Saxena and Sankar P Sanyal
Table 2. Experimental and theoretical Raman mode vibrational frequencies
for (10,10) single wall carbon nanotube.
Raman frequencies (cm−1)
Bond polarizability model
Ref. [20]
Rao et al [22]
Exp. (Theoret.)Mode Present model
E2g
E1g
A1g
E2g
E1g
A1g
E2g
A1g
E2g
E1g
E2g
16.45*
92.19*
165.56*
373.50*
1569*
1587.28*
1592.77*
687.07
859.17
1352.85
1528.6
17
118
165
368
1585
1587
1591
–
–
–
–
(22)
116(117)
186(165)
377(368)
1567(1584)
1593(1585)
1609(1590)
673(670)
855(866)
1347(1374)
1550(1543)
*Represents the frequencies used for fitting force constants.
from table 1 that the values of central force constants are significantly larger than
their angular counterparts for the first and second nearestneighbor interactions,
while for the third nearestneighbor interactions angular forces dominate over the
central one. This shows that the effect of curvature is more prominent in the case
of third nearest neighbor. This is quite justified when one considers the cylindrical
structures, like nanotubes, as emphasized by Saito et al [20]. The Raman modes,
which are used for fitting, together with the calculated frequencies from the present
model are listed in table 2.
The phonon band structure, in general, reflects the symmetry as well as the num
ber of atoms present in the unit cell and dynamical response of atomic species that
constitutes the nanotubes. However, the order and characteristics of the resulting
symmetries are ascertained from the way the twodimensional hexagonal sheet’s
boundaries are connected to each other to form the cylindrical structure. In arm
chair and zigzag type of single wall carbon nanotubes, the translational periodicity
is reflected, and hence gives rise to maximum symmetry. Due to the optimum sym
metry properties they possess maximal degeneracy [23]. Therefore, for (10,10) and
(10,0) nanotubes, instead of 120 modes one gets only 66 phonon modes at the zone
center, out of which 54 are doubly degenerate while 12 are nondegenerate. Low
frequency E2gmode for (10,10) carbon nanotube obtained from our calculations is
16.45 cm−1, which is also reported as 17 cm−1theoretically [20]. The dispersion
curves for (10,10) and (10,0) nanotubes are shown in figures 2a and 3a, respec
tively. Both of them contain 40 atoms in their unit cells, and hence 120 phonon
modes are obtained for each of them. At the zone center we have obtained four
acoustic modes, for both the tubes, out of which the lowest two acoustic modes are
doubly degenerate and transverse in nature. The highest one is the longitudinal
acoustic mode. The fourth mode is a special feature of nanotubes and comes into
existence due to the movement of atoms around the axis of the tube. Since we
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Phonon and thermal properties of achiral SWCNTs
Figure 2. (a) Dispersion curves of (10,0) nanotubes plotted in axial qvector.
(b) Calculated phonon density of states.
reported in ref. [5].
(c) Phonon density of states as
Figure
qvector. (b) Calculated phonon density of states. (c) Phonon density of
states as reported in ref. [20].
3. (a) Dispersion curves of (10,10) nanotubes plotted in axial
consider the nanotubes as onedimensional continuous cylindrical structures, this
type of mode is possible and is referred to as ‘twisting or torsional mode’. One
important finding of our calculations is that we have obtained four acoustic modes
for all the tube structures having radius greater than 5˚ A while for nanotubes
Pramana – J. Phys., Vol. 67, No. 2, August 2006
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with lower radius only three acoustic modes are observed. This can be explained
from the fact that for lower radius the radial dimensions become comparable to
the interatomic dimensions. Therefore, this type of atomic displacements around
the circumference become less probable and the structure behaves truly nanoscopic
contrary to the bulk behavior. However, the possibility of cross circumference
interactions for lower radius cannot be ruled out completely. The total phonon
density of states for both the tubes are plotted and compared with the results
reported by Cao et al [5] and Saito et al [20] in figures 2b–2c and 3b–3c. It can
be noticed from these figures that various high and lowfrequency peak positions
of SWCNT agree well with the observed Raman peaks. However, there is some
discrepancy, which is obvious in the two plots at the medium frequency range. The
reason of this deviation is the values of Raman frequencies that we have taken for
fitting the model parameters, which cover only high and low frequency modes (table
2). Also in refs [5] and [20], they have used the force constant directly obtained
for graphene sheet while in our case we have used the force constants obtained for
nanotubes, by fitting the Raman modes of (10,10) nanotube only. This could be
another reason for inconsistency between the two plots in the intermediate energy
range.
Amongst the 6N phonon modes obtained from our calculations for the nanotube
having 2N atoms in its unit cell, only a few are Raman and infrared active [22,24].
The Raman active vibrational modes are mainly symmetry dependent.
calculations we have obtained a higher frequency band around 1600 cm−1(Gband),
which arises from six tangential bondstretching vibrational modes of symmetry
2A1, 2E1 and 2E2 for all the tubes [25]. It can be argued that these modes are
only approximately longitudinal or transverse [26]. However, the order of the three
pairs of frequencies depends on the particular model [27]. The variation of Gband
modes with radius for various armchair and zigzag tubes are presented in figures
4a and 4b, respectively.
Overbending [28] is one of the special features of graphene that persists even
when the graphene layer is rolledup to form a tube. When the longitudinal high
energy optic mode has its local minima at Γ point, we refer to it as overbending.
The magnitude of overbending is the difference of energy of phonon branch local
minima at q = 0 and the maxima at any nonzero value of q. Due to this softening
of LO phonon, a peak appears at the lowenergy side of the highenergy band
of Raman spectra of nanotubes. In this way one can interpret the double peak
structure of the highenergy subband of carbon nanotubes. In the case of achiral
tubes it was found that the magnitude of overbending is higher for zigzag tubes
(∼12.7 cm−1) than for the armchair tubes (∼10.9 cm−1).
The seventh and the most important lowfrequency (A1g) Raman active mode
of nanotubes is called the radial breathing mode (RBM). It’s value is definite and
different for each tube and also varies with tube radius. When all the atoms move
in a phase perpendicular to the tube axis changing the radius of the tube, one gets
the RBM [29]. It involves the collective movement of atoms towards and away from
the central axis of SWCNTs (see figure 5). The radial breathing mode emerges from
the periodicity of the graphene lattice, which is imposed on nanotubes on account
of wrapping. Therefore, the frequency of this mode is directly influenced by the
perimeter of the tube. We have calculated the radial breathing modes for a family
In our
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Phonon and thermal properties of achiral SWCNTs
of zigzag and armchair nanotubes. For (10,10) nanotube, which has a larger radius,
the value of RBM is obtained as 165 cm−1, while for (10,0) nanotube, which has
smaller radius, it’s value is 282 cm−1. Both the values are in good agreement with
the values reported by various authors [22,29–31].
The variation of RBM with the radius of the tubes is given in figure 6. By fitting
power trend line to these results we have obtained an inverse relationship between
the radius and the RBM for both the tubes. A general representation of the relation
between RBM and radius of the tube can be given as
ω(RBM)= AR−1.
(8)
We have calculated different values of proportionality constants A for the two types
of nanotubes such that, A = 1134 cm−1˚ A for armchair nanotubes and A = 1092
cm−1˚ A for zigzag nanotubes. On averaging the values of A for all achiral tubes
one gets A = 1107.8 cm−1˚ A, which is in the range reported by various authors
[29,31].
Finally we have calculated the specific heats for these sets of nanotubes. For this
purpose, we have formulated the expression of vibrational density of state (g(ω)) by
using Lorentzian function (eq. (5)). Since lots of Lorentzian peaks are possible for
a given value of wave vector q we have used the method of averaging. The variation
of specific heat at 300 K with the radius of the nanotube is shown in figure 7. The
values of specific heats of armchair nanotubes are comparatively larger than those
for the zigzag nanotubes having the same value of n. This variation comes from a
basic structural detail, which is the difference in volumes of the unit cells of these
tubules in spite of them possessing the same number of carbon atoms in their unit
cells. We have calculated the dependence of specific heat on the radius of tubes for
both nanotubes.
Figure 4. Variation of Gband modes with the chiral index n of (a) armchair
nanotubes and (b) zigzag nanotubes.
Pramana – J. Phys., Vol. 67, No. 2, August 2006
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Prapti Saxena and Sankar P Sanyal
Figure 5. RBM mode (unique to SWNTs).
Figure 6. Variation of radial breathing mode with the radius of (n,n) nano
tubes (line with triangles) and (n,0) nanotubes (line with circles).
CV = CV0+ AVe(−R/R0),
(9)
where CV0, AV and R0are the constants such that for armchair nanotubes CV0=
749.93 J/KgK, AV = −3600 J/KgK and R0= 2˚ A while for zigzag nanotubes
CV0= 778.47 J/KgK, AV = −3200 J/KgK and R0= 1.5˚ A.
The temperature dependence of specific heat for (10,10) (armchair) and (10,0)
(zigzag) SWCNT is shown in figure 8. It is observed that specific heat exhibits
almost the same behavior with temperature for both tube types. However, the
magnitude of CV is greater for (10,10) carbon nanotube as compared to (10,0)
nanotube. It is also useful to note that at very low temperatures, up to 8 K,
the specific heat increases almost linearly with temperature. At low temperatures,
CV probes only the lowest energy phonons, i.e. only the lowenergy modes get
activated. These are the acoustic modes, whose dispersion can often be expressed
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Phonon and thermal properties of achiral SWCNTs
Figure 7. The variation of specific heat with the chiral index n of nanotubes
at 300 K.
Figure 8. Temperature dependence of specific heat of (10,10) and (10,0)
carbon nanotubes. Inset shows the variation of specific heats at low temper
atures.
Pramana – J. Phys., Vol. 67, No. 2, August 2006
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as a power law,
g(ω) ∝ kα.
(10)
For a single such mode, eq. (7) simplifies to
CV ∝ Td/α,
(11)
where d is the dimensionality of the system. For a linearly dispersing mode (α = 1)
(in eq. (10)) the specific heat is linear in T for a 1D system, and shows the
familiar Debye T3behavior for a 3D system. Therefore, the linear behavior at low
temperature is a direct consequence of the 1D quantized nature of the nanotube
phonon band structure. Above 8 K, the slope of CV increases gradually, as the
optical subbands start to contribute. Around 30 K there is a sharp rise in CV,
which indicates that at this temperature the first highenergy optical subband gets
activated and the specific heat increases drastically with temperature. Hone et al [9]
and Dobardzic et al [28] have also depicted this type of lowtemperature behavior
(below 30 K) for thermal conductivity. As the temperature increases further, the
value of specific heat increases to finally reach its classical limit.
In conclusion, we have developed a lattice vibrational theory to explain the
phonon dispersion in achiral single wall carbon nanotubes using a phenomeno
logical force constant model. Its extension for chiral tubes is in progress and will
be reported subsequently. Also, we have successfully predicted the behavior of ra
dial breathing mode and its dependence on the radius of the nanotube. We have
obtained all the Raman modes in close proximity to those reported earlier from
experiments. We have also developed the relation between specific heat and the
radius of the tube that could be further employed for thermal characterization of
nanotubes. Finally, we have provided an approach, which could be further used to
predict the phonon behavior of noncarbon [21,31] as well as filled nanotubes.
Acknowledgement
The authors are grateful to the Department of Science and Technology (through
Project No. SR/S5/NM12/2005 Nanoscience and Technology Initiative) for finan
cial support to this work. PS is also thankful to DST for SRF and to Prof. Stefano
Bellucci (INFN, Roma), Dr Sandro Scandolo and Dr Ralph Gabeur (Condensed
Matter Division, ICTP, Italy) for valuable discussions. One of us (SPS) is thankful
to Prof. G P Shrivastava and Steven Hepplstone of the University of Exeter, UK
and Prof. Peter Entel, University of Duisburg, Germany for useful discussions and
Royal Society, London and DAAD, Bonn, for financial support.
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