Eur. Phys. J. B 22, 307–320 (2001)

THE EUROPEAN
PHYSICAL JOURNAL B
EDP Sciencesc Società Italiana di Fisica
Springer-Verlag 2001

Determination of SWCNT diameters from the Raman responseof the radial breathing mode
H. Kuzmany1,a , W. Plank1 , M. Hulman1 , Ch. Kramberger1, A. Grüneis1 , Th. Pichler1,2 , H. Peterlik1 , H. Kataura3 ,and Y. Achiba4
1

Universität Wien, Institut für Materialphysik, Strudlhofgasse 4, 1090 Wien, Austria
Institut für Festkoerperphysik und Werkstofforschung, Dresden, Germany
3
Department of Physics, Tokyo Metropolitan University, Tokyo, Japan
4
Department of Chemistry, Tokyo Metropolitan University, Tokyo, Japan
2

Received 18 February 2001 and Received in final form 3 April 2001
Abstract. We report on the evaluation of the distribution of diameters for nanotube samples with a wide
variation of mean diameters. Such results were obtained from a detailed analysis of the radial breathingmode Raman response and compared to results obtained from an evaluation of optical spectra and X-ray
diffraction pattern. The evaluation of the Raman data needs a well refined analysis as the experimental
analysis exhibits a rather complicated and oscillating relation between response and exciting laser. Both,an exact calculation where the density of states was considered explicitly and an approximate calculation
were applied. Both models used for the analysis are able to explain several unexpected results from theexperiment such as the oscillating behavior of the spectral moments, unusual discontinuities in the first
moments of the Raman response for excitation in the IR, a fine structure for the response in optics andRaman, and an up shift of the RBM frequency as compared to qualified ab initio calculations. In detail thefirst moment and the variance of the spectra were used for the evaluation of the diameter distribution. To
obtain good results between experimental and theoretical oscillation pattern the transition energy between
the first two van Hove singularities had to be scaled up which is considered as a result from coulombinteraction of the electrons in the tubular material. On the other hand the analysis does not only allow
to determine the mean value and the width of the diameter distribution but yields also a value for the
average bundle diameters or, alternatively, the strength of the tube-tube interaction. The model used forthe analysis of the Raman data is also appropriate to analyze the optical response, at least for the spectral
range from 0.5 eV to 3.5 eV. The fine structure in the response for the transitions between the three lowest
van Hove singularities is well reproduced and the mean tube diameters and their distribution is obtainedin very good agreement with the results from the Raman analysis. From the X-ray analysis the same meanvalues and comparable distributions for the tube diameters were received whereas the bundle diameters
could not be retained with high precision in this case.
PACS. 61.46.+w Nanoscale materials: clusters, nanoparticles, nanotubes, and nanocrystals –63.20.Dj Phonon states and bands, normal modes, and phonon dispersion – 78.30.-j Infrared and Raman
spectra

1 Introduction
Carbon nanotubes as discovered in 1991 by Iijima et al. [1]exhibit a multiwall structure consisting of several concentric rolled up graphene planes. Structure and properties
of such tubes are difficult to analyze since contributions of
the various shells mix and can hardly be separated. Only
after a successful preparation of large quantities of singlewall carbon nanotubes (SWCNTs) by Thess et al. [2] andlater on by Journet et al. [3] and other groups, research
expanded dramatically [4]. The single, rolled up grapheneae-mail: kuzman@ap.univie.ac.atplane can be considered as an one dimensional (1D) electronic systems. Each tube is described by a lattice vector(n, m) in the graphene plane which turns into the circumference of the tube on rolling up the sheet. n, m are integers and are called the components of the Hamada vector
or folding vector. Dividing the length of the Hamada vector by π yields the diameter of the tube. Likewise, n and mdetermine the helicity of the rolled up sheet. Within a certain diameter range a rather large number of tubes withdifferent helicities is geometrically allowed which yields
an almost continuous distribution of possible diameters.
There is good evidence and wide agreement that for certain preparation conditions all geometrically allowed tubes

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are grown and any selection comes only from the tube diameter [5–8]. This is agreed even though Raman spectra
and optical absorption exhibit a fine structure which can
be fitted with a rather small number of individual oscillators. These oscillators were ascribed in some early work
to a selective collection of certain tube diameters [9–11]
and could thus indicate selective tube growth. By varying growth temperature and catalyst tube material with amean diameter between 0.9 nm 1.5 nm could be grown so
far [12,13]. In extreme cases where the tubes were grown inthe cavities of a zeolite tube diameters as small as 0.4 nm
have been reported [14].
The electronic properties of the nanotubes are obtained from a zone folding procedure in the graphene
plane. Electrons can accommodate quasicontinuous kvectors along the tube axis but are trapped into well
separated k-vectors perpendicular to this axis. The ideal,
defect-less SWCNT thus exhibits typical 1D van Hove singularities in the density of electronic states (DOS). For optical absorption only symmetric transitions between singularities from the valence band to singularities in the
conduction band are allowed. The number of singularities
in the DOS scales up with the diameter of the tubes but it
is independent from their helicity. Electronic transition energies scale with the π-π overlap integral γ0 . The value forthe latter is 2.5 eV as calculated for the graphene plane ina tight binding approximation where only π-orbitals are
considered. For the nanotubes best agreement with experiments has been obtained for a value between 2.9 and
3 eV [8,9,15]. For such values the first two allowed opticaltransitions for tubes with a standard diameter of 1.36 nm
will occur at 0.7 and 1.2 eV in semiconducting tubes andat 1.9 eV for the first allowed transition in metallic tubes.
The positions of the van Hove singularities and thus thetransition energies scale up linearly with the inverse of the
tube diameter [12,16,17].
Experimental values for the tube diameters were firstobtained from X-ray analysis of nanotube bundles [2].
Subtracting a van der Waals distance of 0.3 nm from theevaluated center to center separation of the tubes a tube
diameter in reasonable good agreement with statisticalvalues from a TEM analysis was obtained [2]. Also, theaverage position of the absorption bands from thin films
have been used successfully to obtain a good estimate forthe average tube diameter [12].
The Raman spectrum of SWCNTs has two importantregions. The first one around 1580 cm−1 is strongly related
to the tangential Raman mode in graphite. The second one
between 140 and 220 cm−1 originates from a radial breathing mode (RBM) and is unique for SWCNTs. Its positionand shape depends strongly on the exciting laser wavelength as demonstrated for the first time by Rao et al. [18].
This was explained by a photo-selective resonance scattering of the modes. Each laser selects resonantly those tubes
for which the electronic transitions match best with thelaser energy. Like the transition energies, the RBM frequencies also scale inverse with the tube diameter. Since
the original work of Rao et al. the unusual response of thisline in the Raman spectra has been discussed in several

hundred papers. Ab initio calculations confirmed the 1/ddependence of the mode frequency [19,20] and yielded an
appropriate factor of proportionality. This has motivatedseveral authors to use Raman results from the RBM forthe analysis of tube diameters as well. A problem for suchanalyzes arises certainly from the large differences in the
average and peak line positions if different lasers are used
for excitation. Only very recently a better understanding for the response of the RBM was obtained from anextended experimental and theoretical analysis of the response [8,17]. The response of the peak position and of
the first and second moments of the spectra were foundin reference [17] to oscillate with the energy of the exciting laser. This oscillation was found to be due to themacroscopic quantization of the electronic levels as a consequence of the finite size of the tubes in direction perpendicular to the tube axis and due to the distribution of thestates along the tube axis into van Hove singularities. Furthermore, in the latter work a useful approximation wasdeveloped which allowed to evaluate the spectral moments
for the Raman response of the RBM without explicit useof the joined density of states. This simplified model wasused to determine the mean and the width of the diameter
distribution of a SWCNT sample. The knowledge of thefirst and second moment of the spectral distribution for
excitation with only one laser line was demonstrated tobe sufficient for this.
On account of the above described results it was certainly demanding to check the range of validity for thetwo developed models and to apply them to samples withdifferent mean diameters. In this paper we present therefore a detailed version of the mathematical backgroundfor the two models and the pitfalls which must be known
for the evaluation of diameters from the RBM. The exact model which uses explicitly the DOS for all geometrically allowed tubes is related to the approximate evaluation where only the resonance positions are considered asa continuous distribution. For the evaluation of the firstand second spectral moments the different models yield
almost the same result for standard tubes. Discrepancies
are found for very thin tubes with diameters below 1 nm.
The consequences of the oscillating behavior of the moments with respect to the energy of the exciting laser are
discussed and the unusual response observed for several
laser lines is explained. The model for the analysis is applied to nanotubes with 6 different mean diameters. Goodagreement with the experiment is only obtained if the energies for the two lowest transitions are scaled up by about
10% from their tight binding value. Both, the model whichuses the full DOS and the approximate model are also applied to interpret spectra for optical absorption and to
derive the mean diameters and their distribution. Almost
the same type of up scale is requested in this case as for
the Raman spectra. The fine structure observed in the optical spectra is well reproduced by the model. Results forthe diameters are found in very good agreement with theresults from the evaluation of the Raman data and froma detailed X-ray analysis but thin tube material needs a
special treatment. Average diameters of carbon nanotube

H. Kuzmany et al.: Determination of SWCNT diameters from Raman scattering 2n+m  √3kam
2n + m+qπ
n,m (k) = ± γ0 1 + 2 (−1) dR cos
2 (n2 + m2 + nm)1/2
n2 + m2 + nm
√
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bundles could only be determined from the Raman data
but their quantitative value relies on details of the theory
used for the evaluation. The fine structure in both, theRaman response and the optical response is found to be
due to a clustering of diameters as a consequence of theirsmall value.

309

(2)

for the RBM frequency in this work. C2 is a function todescribe the tube-tube interaction as discussed in detailin Section 6.3.
The density of states was evaluated for all geometrically allowed tubes from zone folding the tight binding
band structure
See equation (2) above

2 Evaluation of Raman intensities
and spectral moments, basic relations
The evaluation of the Raman spectra was performed inseveral steps. First the frequency of the RBM was calculated using an ab initio program package. Then the densityof states was evaluated for all geometrically allowed modes
between 0.5 nm and 2.0 nm diameter within a tight binding approximation. With these results a theory for resonance Raman scattering was applied to evaluate spectral
intensities and their moments. Finally, a continuum approximation is used to evaluate the spectral moments andto determine the parameters of the distribution function.
2.1 Evaluation of mode frequencies and densityof states
For the evaluation of the mode frequencies νRBM the results obtained from the Vienna ab initio simulation package were used as reported previously [20]. The package allows only to calculate armchair tubes (n = m)and zigzag tubes (n or m = 0) in the requested diameter range. From an interpolation of the form νRBM =[239 − 5(n − m)]/d(n, m) one can obtain safely the RBM
frequencies for arbitrary helicities. Within the limits of theaccuracy of the calculation and to be consistent with thevarious analyzes performed in the present work we haveused a constant value of 234 cm−1 nm for the factor of proportionality between the RBM frequency of a single tube
and 1/d.
The RBM frequency is known to be up shifted by intertube interaction in the bundles. Several recent calculations estimate this up shift to be between 8% and 12% fortypically (10,10) tube diameters [21,23]. From additional
theoretical analyzes this up shift was found to depend on
the diameter of the tubes d and on the number N of tubes
in the bundles [22]. Hence, to obtain best agreement withdetails of the analysis we have used
νRBM (n, m) =234
+ C2 (N, d)d(n, m)(1)

where γ0 ≈ 2.9 eV and a = 2.46 Å are the carboncarbon interaction energy and the lattice constant of thegraphene sheet, respectively. dR is the greatest common
divisor of (2n + m) and (n + 2m) and q is the subband index extending from {−(n2 + m2 + nm)/dR } to{(n2 + m2 + nm)/dR − 1}. For each subband k extends
from −π/T to π/T , where T is the length of the unit cell.
The density of sates was evaluated for every 10 meV.
To avoid instabilities in the calculations the resulting divergencies were cut off at a factor of 100 above their average value and subsequently smoothed by a 6 point gliding
average. This procedure considers the finite effective widthof the electronic states due to life time effects and spectral
resolution.
2.2 Calculation of Raman intensities and spectral
moments by using the density of states explicitly
For the evaluation of the resonance Raman intensities thetheory of Martin and Falicov [24] is used. The Feynman
diagram yields for the normalized matrix elements of aone phonon Raman process
K2f,10 =X

Mf j Mji Mi0
,
(~ω
−
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−
i~α)(~ω2 − l − i~α)
1
j
ji

(3)

where 0 and f assign the initial state with a photon
ω1 = ωi and the final state with a photon ω2 = ωs anda phonon ±ν = ω2 − ω1 , respectively. The first and thethird matrix element in the equation describe the electronphoton coupling. The matrix element in the center originates from the electron-phonon interaction where one
phonon is created, and 1/α is the lifetime of the excitedstate. i and j assign intermediate states with and without a phonon. The denominator expresses the resonances
for the incoming and for the outgoing photon. The scattering cross section σ(ωi ) is proportional to the absolutesquare of the expression in equation (3). In a solid there
is a continuous possibility for transitions from the valenceband to the conduction band which means the sum in

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The European Physical Journal B

equation (3) can be replaced by an integral over the joint
density of states gjds () and the scattering cross section
becomesZ
2
Mf Mph M0 gjds (n, m, )dσ(ωi ) ∝(4)(~ωi −  − i~α)(~ωs −  − i~α)where the three symbols Mf , Mph , and M0 represent thethree matrix elements from equation (3). For matrix elements which depend weakly on the energy as comparedto the denominators in equation (3) or equation (4) the
triple product of the matrix elements can be removed fromthe integral. In this case the spectral intensity for a set oftubes (n, m) with Gaussian distributed diameters dn,m isobtained from(X
(dn,m − d0 )2
I0 (ν) =L(ν, ν(n, m), Γ )exp −
2σ2
n,m
)
Z
2
gjds (n, m, ) d
×· (5)(~ωi −  − i~α)(~ωs −  − i~α)L(ν, ν(n, m), Γ ) is a line shape function for the phononmode of tube (n, m) with width Γ . For the discussion performed below ~α = 0.01 eV was assumed and a Lorentzianline was used for L. d0 and σ2 are the mean and the variance of the Gaussian function describing the diameter distribution of the sample [8,10,17]. The variance is related
to the FWHM as σ2 = F W HM 2 /8ln2. gjds (n, m, ) is the
DOS from Section 2.1 with  replaced by 2. Equation (5)allows immediately to calculate the first moment fromR
νI0 (ν)dνhν()i = R(6)
I0 (ν)dνand the spectral variance from∆(ν) = hν 2 i − hνi2 ,(7)both as function of the excitation energy ~ωi . The evaluation of the spectral moments according to the above model
will be assigned in the following as full DOS calculation.
2.3 Continuum approximation of spectral moments
Since the van Hove singularities dominate the spectral response one can substitute the explicit DOS function by aconstant value for each tube and thus drop the integration
over the energy in equation (5). Considering only the first
8 transitions we obtain for the first momenthν()i =8 aRj+P[νRBM + C2 (N, d)] ρ(d) exp[−(d − d0 )2 /2σ2 ] dd
j=1 aj−
8 aRj+
P·
ρ(d) exp[−(d − d0 )2 /2σ2 ] dd

j=1 aj−(8)

The integration extends over a range determined by theupper and lower border lines for each van Hove singularity
which limit the relation between 1/d and . For the jth
singularity these borderlines can be evaluated from thetheory of [16] and have the formaj± () = (2jgj aCC γ0 /d) ± jaCC /6,
where aCC = 1.44 Å is the C-C separation in the graphenesheet.
ρ(d) = ∆Z(d)/∆d =d(n, m)π3√
9 3a2CC(9)is the continuum description for the density of tube diameters as derived from the total number Z(d(n, m)) oftubes in a diameter range from 0 to d. gj = 1.1 for j = 1, 2
and 1 otherwise. It is considered as a small scaling factor
to tune the oscillation position and amplitude to the values observed in the experiment.
The second moments (variances) are evaluated from asimilar relation according to the definition in equation (7).
The evaluation of the spectral moments according to theabove model will be assigned in the following as continuum
approximation.
The described model holds only for not too small values of σ. A reasonable limit is σ ≥ aCC /3 which corresponds to the smallest width of the integration interval(±aCC /6 for j = 1).
2.4 Comparison between full DOS and continuum
calculation
To check the difference between full DOS and continuum
calculation an evaluation of the spectral moments was performed for a typical tube with mean diameter of 1.36 nm
and variance σ2 = 0.01. The full DOS calculation yields
a damped oscillation versus excitation energy for the firstand for the second moment. This is demonstrated in Figure 1a. The noise in the result originates from the singularities in the density of states. Up to 4 oscillations canbe detected before the latter are damped out for excitation in the blue and ultraviolet spectral range. To obtain
the results as plotted the energy scale was up-shifted by
10% and then continuously chirped to obtain un-scaledenergies from 1.6 eV upwards. This process is necessary
to compensate for the influence of gj in the approximate
evaluation of the moments according to equation (8). As it
will be discussed below this rescaling of the tight bindingtransition energies is an intrinsic property of the response.
The dashed line in the figure represents the result fromthe continuum approximation according to equation (8).
For the first moment the agreement between full DOS andapproximate calculation is very good except that the latter yields a too large amplitude for the first oscillation.
For the variance the agreement is almost as good exceptthat for the approximation the first oscillations appear too
sharp and too strong. In both cases the response evaluated for the variance is phase shifted by π/4 as comparedH. Kuzmany et al.: Determination of SWCNT diameters from Raman scattering
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Fig. 2. Reduced first moment hνi = hν(d, σ)i − hν(d0 , σ0 )i(a) and reduced variance ∆ν = ∆ν(d, σ) − ∆ν(d0 , σ0 ) (b) as afunction of d and σ for a distribution with d0 = 1.36 nm andσ = 0.16 nm.

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good, particularly for low energies. The peaks from thecontinuum approximation appear up shifted as comparedto the full DOS calculation. Only for energies higher thanabout 2.3 eV both models yield the same result within an
expected experimental error. Thus, as a rough estimate,for diameters down to 1 nm the approximate model isreliable for the whole spectral range whereas for smaller
tubes analysis must be limited either to energies larger
2.3 eV or the full DOS evaluation must be used.

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Fig. 1. Comparison between calculated first and second moments using the DOS explicitly (full squares) and continuum approximation (dashed line). C2 = 9.5 was assumed inboth cases. For the full DOS calculation a phonon line widthΓ = 2 cm−1 was used. (a) for a distribution with d0 =1.36, σ0 = 0.1, (b) for a distribution with d0 = 0.9, σ0 = 0.1.

to the first moment. Also, it is important to note that
both, the first moments and the variances are characterized by their absolute value, their oscillation frequency andshape, their oscillation amplitude, their oscillation phaseand by their damping. It means all these quantities are
relevant parameters to be determined in an experiment.
Figure 1b presents the same comparison but for tubes
with 0.9 nm diameter. Obviously the agreement is not as

2.5 Evaluation of diameters and diameter distributions
For the evaluation of d0 and σ equations (7) and (8) must
be solved simultaneously. Since these equations are highly
nonlinear solubility can be expected if the two equations
depend strongly on d and σ. This dependence can bechecked by studying the two quantities as a function of dand σ for a known distribution with parameters d0 and σ0 .
Figure 2 depicts this response for a particular laser excitation of 1.92 eV in a two-dimensional graph for a distribution with d0 = 1.36 nm and σ0 = 0.16. The values on thez-axis represent the evaluated response reduced by the
moments valid for the distribution under consideration.
Thus, the ditch for zero or minimum response represents
the correct solution for d and σ in both cases. Since theditch is well expressed a unique solution can be expected

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The European Physical Journal B
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Fig. 3. First moments and variance for the Raman response of the radial breathing mode versus energy of the exciting laser
for different parameters of the distribution. (a) different values of d0 as indicated; σ = 0.1, C2 = 9.5 cm−1 . (b) different valuesof σ as indicated; d0 = 1.36 nm. C2 = 9.5 cm−1 . (c) different values of C2 as indicated. d0 = 1.36, σ = 0.1.

for this sample. Similar graphs can be obtained for otherlaser energies. The ditch becomes rather flat for excitation
with low energy lasers.

3 Response to the parametersof the diameter distribution

As a summary from the analyzes carried out on thevarious samples described below in 90% of all trials thesystem of the two equation was soluble by Mathematica.
Lack of convergence for the solution was only encountered
for the tube material with the smallest diameter and for
excitation with deep red laser light.

In order to use the above model for the analysis of SWCNT
material it is necessary to study its response to a wide
range of parameters. Results for excitation with a continuous distribution of laser energies are depicted in Figure 3.
Part (a) exhibits the oscillations of the first moment fordifferent diameters between 1 and 1.7 nm. As expected theH. Kuzmany et al.: Determination of SWCNT diameters from Raman scattering
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mean value shifts upwards with decreasing mean diameter d0 . In addition there is a phase shift of the oscillations.
The positions of the first, second, third, etc. maxima shift
to higher laser energies with decreasing tube diameter.
For very small tubes a new peak enters on the low energy
side. From the position of the peaks for the tube with diameter 1.36 nm (dotted line) we can formally correlate
the first, second and third peak with the second and third
transition of semiconducting tubes and with the first transition of metallic tubes, respectively. Interestingly the oscillations become rather flat for excitation energies above
2.5 eV, almost independent from the tube diameter. For
excitation with the Nd:YAG laser (1.17 eV) the responseeven from the first transitions in semiconducting tubes
becomes relevant for small tubes and exhibits a giant oscillation of 120 cm−1 . Similarly, the variance becomes verylarge for this laser. On the other hand the discrimination
between tubes with different diameters in the range of 1.2
to 1.7 nm becomes difficult whereas the discrimination oftubes with smaller diameter is easy. In fact from the graphone expects a discontinuity of the mean value of the RBM
frequency for tubes around than 1 and 1.2 nm. This isindeed observed in the experiments as shown in Section 5.
Figure 3b depicts a similar set of response curves forconstant tube diameters but varying second moment. As
one might have expected the oscillations smear out forbroad distributions but they also decrease for very narrow
distributions. In the latter case the peaks in the variancesplit into two components.
Finally, Figure 3c exhibits the response to C2 whichcharacterizes the intertube interaction. The up shift of the
response with increasing intertube interaction is natural
but it occurs without phase shift. This is in contrast to
the shift originating from decreasing diameter. It meansthat the up shift of the RBM frequencies as a consequenceof decreasing tube diameter and as a consequence of increasing intertube interaction can be discriminated in theevaluation.
The quenching of the oscillations for narrow distributions becomes only significant for values of σ ≤ 0.05 nm.
The evaluation of this behavior can not be obtained
straight forwardly from equation (8) since this equation
holds only for σ ≥ aCC /3 as already mentioned above.
For smaller values of σ contributions from the energy region between the van Hove singularities can not be neglected. A formal extension of the formula as depicted in
the Appendix allows, however, to include very narrow distributions in the analysis. The resulting amplitudes of theoscillations are shown in Figure 4 for two different meanvalues of the tube diameters.
The above analysis indicates that the determinationof the mean tube diameter from the first moments needssome care. The 1/d law may not be valid under certain
conditions. Figure 5 gives an example. The dashed line
represents the 1/d law. The two full drawn lines represent the connection between the observed first moments
for a particular distribution characterized by 1/d0 and two
different excitation energies. For the IR excitation the deviations from the 1/d law are dramatic but even for the

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Fig. 4. Values for the second and the third peak to minimum
amplitude of the oscillations in Figure 3 versus width of the
distribution. (a) mean diameter d0 = 0.9 nm, (b) mean diameter d0 = 1.7 nm. C2 = 9.5 cm−1 was used in both cases. The
third peak in the oscillations in Figure 3 originates from thefirst van Hove singularities in metallic tubes.

green laser excitation measurable deviations occur. A difference between 1/d0 = 0.65 and 1/d0 = 0.7 is significant
in the experiment.

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The European Physical Journal B
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Fig. 5. Inverse of average tube diameters in nm−1 versus
first spectral moment for a Gaussian distributions with σ =0.15 nm. Excitation energies are as indicated. The dashed line
represents 1/d = hνi/234.

4 Experimental
SWCNTs with 6 different mean diameters were preparedby arc discharge and laser ablation as described previously in detail [12,13]. The diameters estimated from previous Raman and X-ray analyzes extended from 0.9 nm
to 1.7 nm. The samples were suspended in toluene andthen drop-coated to form a thick film on a hot silicon substrate (for the Raman and X-ray experiments) or dropcoated on a hot quartz substrate to form a thin film forthe optical transmission experiments. The obtained films
were annealed at 1000◦ in high vacuum for 12 hours andthen slowly cooled to room temperature. Raman spectra were recorded in the spectral range of the RBM with
7 different laser lines extending from 454 nm to 1064 nm(1.17–2.73 eV) for all samples. For two selected samples(smallest and largest diameter) a full set of 30 laser lines
was used for the excitation. The spectral resolution was
2 cm−1 and the measurements were taken at a room temperature. Analysis and detection of the scattered light was
performed with a Dilor xy spectrometer and a liquid nitrogen cooled CCD detector. The system was calibrated
for intensities by recording the resonance cross section of
the F1g mode of Si.
Optical transmission was studied for the thin films onquartz with a Hitachi U3410 spectrometer in the spectral
range between 0.7 and 5 eV with a spectral resolution of
20 meV.
In order to control the analysis from Raman and optical experiments X-ray diffraction was recorded for all
samples. CuKα from a rotating anode was used together
with a pinhole camera and a 2D position sensitive detector [25] for scattering vectors between 2 and 18 nm−1 .

5 Results
5.1 Raman response of the RBM for selected lasers
As expected from many previous experiments the response
for excitation with different lasers revealed very rich and1&./



1L<+.

1L<+.














5DPDQ VKLIW FP














5DPDQ VKLIW FP

Fig. 6. Raman response of the RBM for 6 different samples;(a) as excited with a yellow laser at 568 nm and (b) with an
IR laser at 1064 nm (b).

strongly differing spectra for each sample. Figure 6 depicts
two examples, one for excitation with a yellow laser and
one for excitation with the IR laser. In the latter casethe discontinuous jump of the position of the RBM withchanging diameter for small tubes is obvious. In contrast
to this there is very little change on the mean line position
for larger tube diameters. This is exactly as predicted from
Figure 3a. Also, since the shape and width of the spectrain part (a) of the figure exhibit strong variations they can
not reflect directly the width of the diameter distribution.
A straight forward analysis of the average positionsof the RBM for the 7 laser lines yields a fluctuation up
to 30 cm−1 for the first moment evaluated for one andthe same sample and a corresponding high uncertainty inthe average tube diameter of 15%. In contrast, applying
the above described formalism the diameter of the tubes
could be determined with a much smaller experimental
error which eventually turned out to be of the order of 2%.
The analysis was performed in general by determining d0 and σ for a constant value of C2 = 9.5 in a first
approximation. A fine tuning was obtained by calculating
the moments for the obtained values of d0 and σ and fitting the oscillations to the experimental points with C2
as a fitting parameter. The resulting curves touched theexperimental points very well. A comparison between experiment and calculation is depicted in Figures 7a, b for
two different samples. The figure clearly demonstrates theimportance of using the oscillatory model. With the newvalues for C2 new values for d0 and σ were retained. In
general the applied fitting cycle had only little influence on
the results for d0 and σ but revealed variations in C2 . The
final values for d0 , σ, and C2 are collected in Table 1 forthe 6 samples, together with the root mean square error
for d0 .

H. Kuzmany et al.: Determination of SWCNT diameters from Raman scattering

315

Table 1. Diameters and diameter distributions for single wall carbon nanotubes. N is the number of tubes per bundle asdetermined from equation (13) in combination with equation (14) and for c = 0.7. D is the bundle diameter in nm.
Sample

RP249
RHF3
RHF2
NCKL167
RPTL33NiYHK33

Tube
diameter

Distribution
width σ(Raman)[nm]0.11±0.04
0.14 ±0.03
0.14 ±0.02
0.15±0.03
0.14 ±0.02
0.14 ±0.04

Intertube
shift C2and (N, D)(Raman)[cm−1 ]
14 (8.2,5)9 (3)11 (4)
16 (10,7)
18 (20,10)
20 (44,15)(Raman)[nm]0.94±0.02
1.17±0.04
1.30 ±0.02
1.39 ±0.01
1.47 ±0.02
1.53±0.02(X-ray)
[nm]
0.97
1.16
1.24
1.35
1.40
1.48

Coherence
length(width σ)(X-ray)[nm]([nm])20 (-)
25 (0.2)
25 (0.18)
50 (0.07)
10 (0.18)
25 (0.16)

D

537/

E



1L<+.







Tube
diameter

Distribution
width σ(optics)
[nm]
0.94
1.18
1.29
1.38
1.41
1.47

(optics)
[nm]
0.13
0.16
0.11
0.11
0.12
0.12



E











∆ν FP













( H9




















∆ν FP

53

D


ν! FPν! FP5+)

Tube
diameter







( H9

Fig. 7. Experimentally evaluated first moments and variances
for two different nanotube samples; (a) RHF2 with mean diameter d0 = 1.30 nm, σ = 0.14 nm, C2 = 11 cm−1 and (b)RPTL33 with mean diameter d0 = 1.47 nm, σ = 0.14 nm
C2 = 18 cm−1 . Full drawn lines are from the continuum approximation.

5.2 Samples with very large and very small diameter
For the samples with largest (NiYHK33) and smallest(RP249) diameter a full set of 30 laser lines was used todetermine the spectral moments. Results are depicted in
Figure 8. For the sample with large diameter four oscillations waves can be seen. The continuum approximationfits the four oscillations from the experiment well eventhough the oscillation frequency appears a bit too large.
Since in this case as well as for the sample RP249 a large
enough number of experimental data was available a direct fit of the experimental results to the first moments
was performed in addition. Fitting parameters were simultaneously d0 , and C2 . In the case of NiYHK33 results ford0 were about 4% smaller than those from solving the two
equations (7) and (8). With the smaller values for d0 theagreement between experiment and calculation improves.
For sample RP249 the smallest diameter of only
0.94 nm was observed but only laser energies larger than
2 eV could be used for the evaluation. Also, plotting thefirst moments versus laser energy for the continuum approximation as well as for the full DOS calculation re-












( H9







( H9

Fig. 8. Same presentation as in Figure 7 but for 30 different laser lines: (a) sample NiYHK33 with mean diameterd0 = 1.53 nm, σ = 0.14 nm, and (b) sample RP249 with mean
diameter d0 = 0.94 nm, σ = 0.1 nm. The full drawn line in (b)is obtained from the full DOS calculation with γ0 = 2.55 eV.

vealed noticeable disagreement between experiment andcalculation. This discrepancy could only be avoided by
re-fitting the experimental data and including γ0 as a fitting parameter. Very good agreement was then obtainedwith a value for γ0 = 2.55, particularly for the full DOS
calculation. Data from this fit are depicted in Table 1.
A comparison between experimental results and calculation is depicted in Figure 8 for the two samples. Notethat in the case of the sample with the small tube diameters the full DOS calculation was applied to evaluate thetheoretical curve.

5.3 Results from optical absorption
Optical absorption has been used as an alternative tool
to determine SWCNT diameters [11,12]. It was therefore
important to compare results form this technique with theresults from Raman experiments. After recording the spectra for all 6 samples the overall background was subtracted
and the well known absorption peaks for the first and second transition for semiconducting tubes and the first transition for metallic tubes were observed. These spectra are316

The European Physical Journal B

NCKL167

Intensity (a. u.)

NIYHK33

NCKL167

to line position but also with respect to line width and intensity. Similar good agreement was obtained for the othersamples. For the smallest tube PR249 again the value forγ0 had to be reduced to yield best agreement with theexperiment.
The fine structure in the second and third peak of thecalculated spectra is retained with a very similar shape as
observed in the experiment. Thus, similar to the Raman
spectra it is not smeared out by the overlap from the large
number of tubes considered.
The alternate possibility for the analysis of the optical
spectra is to apply an approximation where only the density of geometrically allowed tubes is considered weighted
with the Gaussian probability for each tube.
(dn,m − d0 )2α() ∝
exp −
2σ2
m,n,j
X

NIYHK33

0,5

1,0

1,5

2,0

2,5

3,0

Energy (eV)Fig. 9. Optical absorption for two different SWCNT samples
as indicated and after subtraction of the background (full line).
Calculated absorption spectra are shown as dashed lines. The
upper two dashed spectra hold for a calculation using the fullset of DOS functions. The lower two dashed spectra hold forthe approximation according to equation (11).

very similar to the absorption spectra reported by Kataura
et al. in [12]. The two higher absorption peaks exhibited afine structure as it was also reported before [11]. Results
for two selected samples are depicted in Figure 9 as fulldrawn lines.
To obtain the parameters for the diameter distribution
from the experiment one can proceed as in the case of theRaman experiments. The full set of DOS functions can besupper imposed to represent the quasi-continuous distribution of nanotubes. Each tube is again weighted with aGaussian factor. Assuming constant matrix elements likein the case of the Raman analysis the absorption is obtained fromX
(dn,m − d0 )2 2α() ∝
exp −gn,m ().
(10)
2σ2
m,n
Subtracting again the background renders characteristic
peaks for the optical transitions between the van Hove
singularities. Using the same value of γ0 = 2.9 eV for theπ-π overlap and a similar up scaling for the first two transitions as it was applied in the case of the Raman analysis
one can fit d0 and σ to the experiments. Corresponding
results are depicted for the two tube materials as dashedlines in the upper part of Figure 9. The calculated spectra
were obtained for d0 = 1.38 nm, σ = 0.11 nm (NCKL167)
and d0 = 1.47 nm, σ = 0.12 nm (NiYHK33), respectively,in very good agreement with the results from the Raman
experiments. The agreement is not only good with respect×δ·( − j (n, m))2 + (δ/2)2(11)

δ is a small value of the order of 10 meV describing thefinite resolution of the spectrometer and the width of
the resonant electronic states j due to lifetime effects.
j (n, m) = 2jgj aCC γ0 /d(n, m) is the same quantity as
used in equation (8). This means the evaluation is performed in direct analogy to the continuum approximationof the Raman data. Experimental results can be fitted
again by varying d0 and σ. In the lower part of Figure 9
results are depicted for the same two samples as selectedin the upper part. Agreement with the experiment is evenbetter than for the full DOS calculation. Similar good
agreement was obtained for the other samples. Resultsfor d0 and σ are listed in Table 1.
5.4 Results from X-ray diffraction
The traditional method to determine diameters of SWCNTs is X-ray diffraction. The diffraction pattern of large
bundles provides insight into the lattice constant of thehexagonal lattice and can thus give information on the
tube diameter, coherence length (coherent bundle diameter) and diameter distribution. Evaluation of all thesedata is only possible if samples are of high quality as itneeds to consider several of the small side bands accompanying the main diffraction peak from the hexagonal lattice
at a scattering vector q = 4.3 nm−1 . All samples revealed
reasonably good peaks which enabled a detailed analysis,
except for the material with the smallest tube diameters.
For this material, due to the high background, only one
peak could be identified. The position of the maximum
was used to calculate the tube-tube distance and the tube
diameter by subtracting the Van der Waals distance. In
the raw data a strong increase in scattering intensity canalways be observed for small q. This small-angle scattering
originates from large objects and has to be subtracted by
an appropriate procedure [5,7]. Figure 10 depicts an example for sample NCKL167 after this background correction. Details of the subtraction procedure as well as details

H. Kuzmany et al.: Determination of SWCNT diameters from Raman scattering,QWHQVLW\ DUEX

1&./     
T>QP@Fig. 10. X-ray diffraction intensity after appropriate subtraction of the background (circles). The line is a fit from thediffraction model. The disagreement between experiment andfit for the fourth peak is due to model simplifications. The fifth
peak at 18 nm−1 originates from graphitic carbon.

for the following analysis will be described elsewhere [26].
The four diffraction peaks between 4 and 16 nm−1 originate from the nanotubes in the bundles.
The model used for the evaluation of the diffraction
spectra is based on the following assumptions: The tubes
exhibit an outer radius r1 , an inner radius r2 and thus awall thickness w = r1 − r2 . They are of infinite length,arranged in a perfect hexagonal lattice and diameters are
subjected to a Gaussian distribution. The mean tube diameter d0 = r1 + r2 . The intensity of the Bragg reflections was calculated by using the model for random layer
lattices [28]. The average intensity of the Debye-Scherrer
rings is then proportional to the following equation, excluding constants but including the polarization factor
and the Lorentz factor for 2D layer lattices.
I(θ) ∝cosθ(1 + cos2 2θ)
√
sinθ
 √

2L π
×G
(sinθ − sinθ0 ) F 2 (q(θ)),
λ

(12)

with
Z ∞
G(a) =

dx exp[−(x2 − a)2 ].

0

F is the form factor, L the particle dimension and θ0 therespective Bragg reflection angle. A Debye-Waller factor
was neglected, as for carbon the intensity reduction in
the whole observed q-range is below 2%. The form factor
for a cylinder is given by the Bessel-function of the firstorder [29] and has the form F 2 (q) ∝ [r1 J1 (q, r1 )/(qr1 )]2 /q.
The diffraction from the tube can be evaluated from theabsolute value of the subtraction of two cylinders, one withthe outer and the other with the inner radius of the tube.
The model is similar to the one described by Rols et al. [7],but is based on the model of layer lattices [27]. Thus, allpeaks can be used for a quantitative description, whereas

317

the intensities for the higher order peaks were weaker inthe experiments than predicted by the model of Rols [7].
The tube thickness was fixed to 0.08 nm to reduce thenumber of free parameters. This value is consistent witha nanotube of only one wall. The resolution function of
the spectrometer was measured by crystalline powder andused for modeling the width of the peaks. An additional
broadening was simulated by assuming a distribution of
tube diameters which leads to an increased peak width
for the higher Bragg reflections and is similar to a lattice
distortion. The model with the tube diameter, the tube
distance, the diameter distribution and the crystal size as
four parameters is then fitted to the experimental results.
The first peak determines the center to center distance of the tubes and can be measured with high precision. The absence of the (2 0), (3 0) and (4 0) reflections
along the q-axis is the most remarkable point and gives astrong restriction to the form factor, i.e. it determines thetube diameter. The width of the higher reflection peaks
increases which is attributed to the size distribution of
tube diameters. In the model, due to geometric restrictions larger tube diameters cause larger tube distances,i.e. the Van der Waals distance is assumed to be a constant. This suggests that within the bundles regions withlarger and smaller tube diameters coexist. A description
by using a mean tube diameter and a mean tube distance,
which would be the consequence if tubes of different sizeare completely mixed, does not adequately reproduce theexperimental results. This interpretation is particularly reliable for specimen NCKL167, where the narrow diameter
distribution permits such a refined evaluation.
The peak at 18 nm−1 originates from the (002) reflection of graphite. Its small width which is even smaller
than the width of the first nanotube reflection, and the
pronounced asymmetry towards small q is typically forhighly oriented (turbostratic) graphitic carbon [28]. As
the area of the graphitic peak is significantly smaller thanthe one of the nanotubes, it can be concluded that theamount of graphitic carbon is significantly smaller thanthe one which formed the nanotubes. There exists an additional part of amorphous carbon in the specimen, which
could not be determined, since the background from smallangle scattering and from amorphous carbon cannot be
distinguished with sufficient precision.
The results obtained from the evaluation are shownin Table 1, columns 5 and 6. Precise values are obtained
for the diameter and the distance (diameter plus Van der
Waals distance) of the tubes. The differences between fit
and experimental results in the wings of the first peak may
be attributed to a non-Gaussian distribution of the tube
diameters. This could lead to a small overestimation of
the fit parameter for σ of the tube diameter distribution.
As the parameter L only slightly influences the fit thedetermination of the crystal size is not very accurate.

6 Discussion
The two models developed in this work are the basis forthe analysis of the tube material. Both models have their

The European Physical Journal B



F



ν

merits. The continuum model is easier to handle and allows a direct analysis of tube diameters, even if only asmall number of laser lines have been used for the excitation of the spectra. In the extreme case of small tubes
or very narrow diameter distributions it suffers from over
simplification. The full DOS model is more elaborate but
holds also for thin tubes. The reduced value of γ0 for small
tubes is really surprising. From hand waving arguments
one would rather expect a further increase of the π-π overlap with increasing curvature of the tubes. On the other
hand it is known that σ orbitals increasingly contributeto the electronic structure of the π system [30] which may
indeed change the electronic response and has some influence on the van Hove pattern.
The need for the increase of the transition energies between the first two van Hove singularities must be considered as intrinsic as it holds for the evaluation of the Raman
data as well as for the evaluation of the optical spectra.
It is most probably due to the shortcomings of the tightbinding approximation which was used for the evaluationof the band structure (Eq. (2)). In a recent theoreticalanalysis by Ando et al. [31] it was demonstrated that the
Coulomb interaction between the electrons in the π-band
leads to exciton levels and to an up shift of the transitionenergies between the van Hove singularities. This up shift
is particularly strong for the lowest energy transition andcan reach 20% for a Coulomb interaction of the order of
0.1 eV. This upshift was observed recently from a mismatch between experimentally observed absorption andresults from tight binding calculations [32].

GQG

318


E


D













ν












FP

Fig. 11. Clustering in the distribution of tube diameters forsmall SWCNTs and origin of fine structure in the Raman line
for the RBM; a density of RBM frequencies versus RBM frequency (or tube diameter); a sampling width of 3 cm−1 wasapplied; b same plot as a but for large tubes. (The left handside of the line corresponds to 10 nm diameter or 23.4 cm−1 .)The whole curve was shifted onto the top of curve a by scaling
the x-axis and y-axis to provide immediate comparison. c as
curve a but each tube was weighted with its probability in a
Gaussian distribution of diameters.

Both, the Raman results and the optical spectra reveal
a fine structure which needs some more discussion.
6.2 Origin of the fine structure

6.1 Quantum oscillation and diameter distribution
The presented models can explain several unexpected experimental results. One of them is the oscillation of thespectral moments which are identified as a quantum size
effect originating from the small dimension in radial direction. The laser energy is swept across the van Hove
singularities and excites oscillating tube diameters to resonance. Other unexpected results are the discontinuous
switch of line position for the RBM mode with changing
tube diameter if the excitation is performed with red or IR
light. Oscillations are well expressed for such excitations
whereas for the determination of diameter distributions
blue or green lasers are recommended.
For the Raman as well as for the optical and X-ray
analysis the size of the probe extends over several thousand bundles. Therefore the obtained diameter distributions should be representative in all three cases. Indeed,
the results from the Raman experiments are in excellent
agreement with results from optics and X-rays. The experimental error for d0 given in Table 1 is a real root meansquare error obtained for the excitation with the various
lasers. The error for σ is of the same relative magnitude.
X-ray analysis yields very reliable data for tube diameters but more details suffer from the strong backgroundwhich has to be subtracted. Better samples and even morerefined analyzes might help.

In previous work and in the current presentation we havedemonstrated that the full DOS model as well as the continuum approximation provide a good representation for
the fine structure in the Raman spectra of the RBM and in
the optical absorption spectra. This is still surprising since
the response originates from an overlap of a rather large
number of tubes with only slightly varying properties. The
physical reason for the fine structure can be traced backin fact to the nanoscopic geometry of the tubes. For suchsmall entities the distribution of tube diameters is not really quasicontinuous but exhibits clusters. This is demonstrated in Figure 11. The figure shows the density of diameters or, more precisely, the density of RBM frequencies
for a certain frequency interval for two selected ranges oftube diameters. For the small tube as they are usually prepared in nanotube research clustering is evident (curve a).
For large tubes (which have not been prepared so far as
single wall species) the density versus frequency relationis completely smooth (curve b). If we weight curve a witha Gaussian distribution we obtain curve c which is almost
a representative for the fine structure in the Raman response of the RBM.
The clustering of the RBM frequencies is related tothe clustering of the diameters and the latter correlates
immediately to the clustering of resonance transitions.
This means there is a clustering of resonance cross sections which certainly further enhances the development ofa fine structure.

H. Kuzmany et al.: Determination of SWCNT diameters from Raman scattering

6.3 Bundle diameters and intertube interaction(10.3d − 2.3)2.56
·
d

(14)

c is a scaling factor describing the tube-tube interaction in
the bundles (c = 1 for the work of Henrard et al.) explicitly
and a(N ) is a function describing the behavior for small
numbers of tubes in the bundle. a(N ) was determinedfrom a best fit to the relation calculated in reference [22].
The fit and a set of relations according to equation(14) are depicted in Figure 12 for different tube diameters. Using the values for the intertube interaction for
infinitely large tubes from the calculation in reference [23]the resulting shifts are much larger than the observed up
shifts. This would require very few tubes per bundle incontrast to various other measurements. The intertube interaction calculated in reference [21] is about 30% smaller
and yields more realistic results in the present case. Evenfor this reduced interaction potential the limiting case forthe stiffening is not reached. Therefore equation (14) could
be solved for N for all tubes under investigation using theobserved up shifts. The resulting values for C2 and N are
depicted in Table 1 in column four. From the values obtained for N the bundle diameter can be evaluated froms
D = (d + 0.6)

√
2 3N
π

 QP QP(13)

In a recent paper by Henrard et al. [22] it was demonstrated that C2 depends not only on the tube diameterbut also on the number of tubes in the bundle, at least as
long as the bundles are not too big. Up to about 100 tubes
per bundle can give a selective response. In this case thevalue of C2 can provide information on the bundle diameter. The final shift of the RBM mode for infinite large
bundles increases about linearly with the tube diameter.
Considering this findings one can use the following refined
formula for the evaluation of the number of tubes in thebundle.
C20 (N, d) = cC2 (d)a(N )
c(10.3d − 2.3)2.56
1=
1 − 0.46 ·dN QP

8SVKLIW 

The stiffening of the RBM frequency by the intertube
interaction was demonstrated by us already in previous
work [8,17]. This effect is fully confirmed here for tubes
with different diameters. The requested up shift of thefrequencies calculated for individual tubes is determinedby the function C2 . This function was found by Henrard
et al. [23] to depend on the tube diameter. Evaluating
their data explicitly yields a relation of the form
C2 (d) =

319

(15)

for a hexagonal arrangement of the tubes. This yields forthe four largest bundles from Table 1: 15, 10, 7, and 5 nm,
respectively. The evaluated diameters for the bundles areof course only as good as the calculation. A more rapid
saturation for the up shift of the RBM frequency with















1XPEHU RI 7XEHVFig. 12. Relative shift of the RBM frequency in SWCNT bundles as a consequence of intertube interaction. Bullets are as
calculated from reference [22]. The dashed line is a best fit using equation (14). The full drawn lines hold for d0 as indicated.

bundle diameter was found by Popov et al. [33]. In thiscase bundle diameters would be even smaller. A weaker
interaction would yield larger bundles.
Certainly also the constant of 234 cm−1 nm is crucial for the evaluation. A smaller value would result in astronger up shift from the tube-tube interaction. Higher
values would reduce the number of tubes in the bundles.
However, as the ab initio calculations from reference [20]must be considered as very accurate there is little room
for adjustment from this side.
The coherence lengths from X-ray might be a comparative quantity to the evaluated bundle diameters. From
Table 1 they turn out much larger than the diameters evaluated from Raman. This is not really surprising. X-rays
sees only the large bundles but never an individual tube or
bundles with two or three tubes. In contrast, Raman scattering records all tubes, no matter how small the bundles
are to which they belong. This makes certainly a difference.

7 Summary
Raman scattering and in particular the response from the
RBM has again proven to be an excellent tool for the
analysis of SWCNTs. However, to draw the full information buried in this response a careful analysis is requested.
The oscillations in the frequency response could be traced
back to the size quantization of the tubes in transversal
direction. The evaluation of the first and second spectral
moment of the radial breathing mode response allows todetermine tube diameters and, if the bundle size is not in
the limit of infinite large bundles, also bundle diameters.
The latter evaluation of diameters relies, however, on thevalidity of results from theoretical models. In fact, turning

320

The European Physical Journal B

P8 (I + 10−7 ∗ a R −[νj+1jhν()i =j=1

2
2
RBM + C2 (N, d)] ρ(d) exp[−(d − d0 ) /2σ ] dd

aj+

P8 (I + 10−7 ∗ a R − ρ(d) exp[−(d − d )2 /2σ2 ] dd
j+1

j

j=1

,

(16)

0

aj+

the problem around, from known bundle diameters intertube interaction can be evaluated. The fine structure in
the Raman response and in the optical response is due
to a clustering of geometrically allowed diameters which
again is a consequence of the nanoscopic sizes of the tubes.

For the work carried out at the Universität Wien the authors acknowledge the Austrian Fonds zur Förderung derWissenschaflichen Forschung project 12924-TPH and the European network HPRN-CT-1999-00011 for financial support.
T.P. acknowledges the ÖAW for an APART grant. For thework carried out at the Tokyo Metropolitan University support
is acknowledged from the Japan Society for the Promotion of
Science, Research for the Future Program and from a Grantin-Aid for Scientific Research on the Priority Area “Fullerenes
and Nanotubes” by the Ministry of Education, Science, andCulture of Japan.

Appendix
The evaluation of the oscillation for very narrow diameter
distributions can be performed fromSee equation (16) above
with
Zaj+


[νRBM + C2 (N, d)] ρ(d) exp −(d − d0 )2 /2σ2 dd.
Ij =
aj−

(17)
Note that the integration in equation (16) extends over
the regions in between the van Hove singularities. The
factor 10−7 is empirical. For lower values oscillations are
not quenched for small values of σ and for larger values
the calculated diameters disagree with the experiment if
σ is specified in the experimental range.

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2. A. Thess et al., Science 273, 483 (1996).
3. C. Journet et al., Nature 388, 756 (1997).
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Saito, G.Dresselhaus, M.S. Dresselhaus, Physical Properties of Carbon Nanotubes (Imperial College Press, London, 1998); M. Dresselhaus, P. Eklund, Adv. Phys. 49,
705 (2000).
5. A. Rinzler et al., Appl. Phys. A 67, 29 (1998).
6. J.W.G. Wildöer et al., Nature 381, 59 (1998).
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9. M.A. Pimenta et al., Phys. Rev. B 58, R16016 (1998).
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11. O. Jost et al., Appl. Phys. Lett. 75, 2217 (1999).
12. H. Kataura et al., Synth. Metals 103, 2555 (1999).
13. H. Kataura et al., Carbon 38, 1691 (2000).
14. Z.K. Tang et al., Appl. Phys. Lett. 73, 2287 (1998).
15. C. Thomsen et al., Phys. Rev. B 61, 16179 (2000).
16. R. Saito et al., Phys. Rev. B 61, 2981 (2000).
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