![Calculated RPA loss function for the carbon (3,3) distant tubes (continuous thin line) and solid (continuous thick line) of tubes. q is vanishing with an orientation parallel to the tube axis. The dashed curves denote the results for graphite (thick) and graphene (thin) for small in-layer q [35]. The loss function for the (5,0) tube is shown in the inset as a dotted curve. LFE are negligible for this q orientation.](https://www.researchgate.net/profile/Angel-Rubio-5/publication/226341393/figure/fig3/AS:670016894668801@1536755884139/Calculated-RPA-loss-function-for-the-carbon-3-3-distant-tubes-continuous-thin-line_Q320.jpg)
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arXiv:cond-mat/0308126v1 [cond-mat.mtrl-sci] 7 Aug 2003
Applied Physics A manuscript No.
(will be inserted by the editor)
Optical absorption and electron energy loss spectra of carbon and
boron nitride nanotubes: a first principles approach.
A.G. Marinopoulos1, Ludger Wirtz2, Andrea Marini2, Valerio Olevano1, Angel Rubio2, and Lucia
Reining1
1Laboratoire des Solides Irradi´es, UMR 7642 CNRS/CEA, ´
Ecole Polytechnique, F-91128 Palaiseau, France
2Department of Material Physics, University of the Basque Country, Centro Mixto CSIC-UPV/EHU, and Donostia Interna-
tional Physics Center (DIPC), Paseo Manuel de Lardizabal 4, 20018 San Sebasti´an, Spain
Received: date / Revised version: date
Abstract We present results for the optical absorp-
tion spectra of small-diameter single-wall carbon and
boron nitride nanotubes obtained by ab initio calcula-
tions in the framework of time-dependent density func-
tional theory. We compare the results with those ob-
tained for the corresponding layered structures, i.e. the
graphene and hexagonal BN sheets. In particular, we
focus on the role of depolarization effects, anisotropies
and interactions in the excited states. We show that al-
ready the random phase approximation reproduces well
the main features of the spectra when crystal local field
effects are correctly included, and discuss to which extent
the calculations can be further simplified by extrapolat-
ing results obtained for the layered systems to results ex-
pected for the tubes. The present results are relevant for
the interpretation of data obtained by recent experimen-
tal tools for nanotube characterization such as optical
and fluorescence spectroscopies as well as polarized res-
onant Raman scattering spectroscopy. We also address
electron energy loss spectra in the small-q momentum
transfer limit. In this case, the interlayer and intertube
interactions play an enhanced role with respect to opti-
cal spectroscopy.
Pacs71.45.Gm,77.22.Ej,78.20.Bh,78.67.Ch
1 Introduction
The science of nanostructures is one of the fields of grow-
ing interest in materials science. Nanotubes, in partic-
ular, are of both fundamental and technological impor-
tance; being quasi-one-dimensional (1D) structures, they
possess a number of exceptional properties. While the
peculiar electronic structure – metallic versus semicon-
ducting behavior – of carbon nanotubes depends sen-
sitively on the diameter and the chirality, i.e., on the
way the graphene sheet is wrapped up into a tube [1],
boron nitride (BN) tubes display a more uniform behav-
ior with a wide band-gap (larger than 4 eV), almost in-
dependent of diameter and chirality [2,3]. The potential
applications of nanotubes in nanodevices are numerous
[4]: super-tough nanotube fibres [5], gas sensors [6], field
effect displays [7], and electromechanical devices [8], to
name only a few. One of the most spectacular exam-
ples is the realization of field effect transistors both with
carbon nanotubes [9, 10] and with BN-nanotubes [11].
Since carbon and BN nanotubes are routinely pro-
duced in gram quantities, the challenge now consists in
having fast experimental tools for characterization of
nanotube samples and, if possible, single isolated nan-
otubes. Optical spectroscopic techniques provide us with
this tool [12, 13,14,15, 16, 17]. The final goal is to find a
unique mapping of the measured electronic and vibra-
tional properties onto the tube indices (n, m). To that
end, the electronic structure and dielectric properties of
the tubes are two key areas to study. One possible spec-
troscopic method is optical absorption spectroscopy with
direct excitations from occupied to unoccupied states.
For carbon tubes, the energy difference Eii between cor-
responding occupied and unoccupied van Hove singular-
ities (VHSs) in the 1-dimensional electronic density of
states (DOS) is approximately inversely proportional to
the tube diameter d. In resonant Raman spectroscopy [14]
and scanning tunneling spectroscopy [13], this scaling is
employed for the determination of tube diameters (and
(n, m) indexes in resonant Raman [18]). A recent ex-
ample is the fluorescence spectroscopy of single carbon
tubes in aqueous solution, where E22 is probed through
the frequency of the excitation laser and E11 is probed
simultaneously through the frequency of the emitted flu-
orescent light [15]. Also optical absorption spectroscopy
of nanotubes in aqueous solution [16] allows for the spec-
tral resolution of peaks that can be associated with dis-
tinct types of nanotubes. For the distance between the
first VHSs in semiconducting carbon tubes, a simple
π-electron tight-binding fit yields the relation E11 =
2 A.G. Marinopoulos, L. Wirtz, et al
E22/2 = 2aC−Cγ0/d, where aC−Cis the distance be-
tween nearest-neighbor carbon atoms. The value for the
hopping matrix element γ0varies between 2.4 eV and 2.9
eV, depending on the experimental context in which it is
used. This fact is a clear indication that the above rela-
tion gives only qualitative and not quantitative informa-
tion on the tube diameter and/or chirality. Furthermore,
for small-diameter tubes the band structure completely
changes with respect to the band structure of large di-
ameter tubes, including a reordering of the VHSs in the
density of states and displaying fine structure beyond
the first and second VHSs [19, 20]. This structure sensi-
tively depends on the tube indices and may be probed
by optical absorption spectroscopy over a wider energy
range, possibly extending into the UV regime.
The spectroscopical characterization of macroscopic
tube samples is made difficult by the fact that in the
bulk solid and even in bundles the tubes are not per-
fectly aligned and do not have a well defined diameter
and helicity, rendering, in the case of carbon tubes, a
random mixture of semiconducting and metallic tubes
[21]. Additionally, the role of intertube interaction in the
spectra needs to be addressed since the tubes are close
packed and could interact with each other via long-range
forces induced by the excitations. Only very recently op-
tical absorption spectra were reported for aligned single-
wall carbon tubes of a very narrow diameter distribution
(4±0.2 ˚
A) grown inside the channels of a zeolite matrix
[22]. Geometric arguments predict three possible tubu-
lar helicities for the range of diameters around 4 ˚
A: the
armchair (3,3), the zig-zag (5,0) and the chiral (4,2) con-
figuration. Therefore, this particular case serves as an
important case study where a direct comparison between
experimental data and theoretical calculations becomes
possible. Indeed in Ref. [19] we have shown the relevance
of a first-principles calculation for these small-diameter
carbon nanotubes by reproducing the polarization de-
pendence of the measured optical spectra.
At present, the dielectric response of tubes in the
frequency range of the electronic interband transitions
and the collective excitations (plasmons) of the valence
electrons (up to 50 eV) is not well understood since
previous studies focused on the low-energy regime and
excitations [23,24]. For higher frequencies, up to now,
the majority of the theoretical studies of the response,
besides model calculations [25], are mostly limited to
summing over independent band-to-band transitions ob-
tained within the semi-empirical tight-binding method
[26] or the density-functional theory (DFT) framework
[22,27,28], or to calculations of the joint density of states
(JDOS) [29] which use the bandstructure with no ex-
plicit evaluation of the transition matrix elements. As
we will show below, these approximations are not suf-
ficient for a full interpretation of the experiments. This
shortcoming is not due to the quality of the bandstruc-
ture calculation itself but instead due to the neglect
of the induced microscopic components in the response
to the external field, the local field effects (LFE) [19].
These effects strongly modify the total response for cer-
tain polarizations of the external perturbation. Also, in-
duced exchange and correlation (XC) components ob-
tained beyond the random-phase approximation (RPA)
might contribute [23,30]. Therefore, important questions
concerning the electron interaction, excitations and screen-
ing still remain unanswered. Our approach here is to
determine the spectra by ab-initio calculations incorpo-
rating important ingredients of the electron response as
in previous works [19, 20].
The present paper is organized as follows: after an
exposition of the theoretical framework in section 2, we
present in section 3 ab initio calculations of optical spec-
tra of small single-wall carbon and boron nitride nan-
otubes. We identify the influence of crystal LFE, XC
effects and intertube interaction in the spectra. We also
point out the similarities and differences between carbon
and BN structures which are related to similarities and
differences in the respective electronic band structures.
A certain number of comparisons with experiment allows
us to verify that the chosen ab initio approach is an im-
portant improvement and well suited for the description
of the spectra of these systems. In section 4, we anal-
yse the spectra of the building blocks of the tubes, i.e.
the graphene and hexagonal BN sheets (layers) including
the optical spectrum of graphite. This gives information
about the the interaction between objects (sheet-sheet)
in the excited state, even when these objects can be con-
sidered to be isolated in the ground state. We also dis-
cuss to which extent the dielectric response of the tubes
can be understood in terms of the response of the sheets.
This will help to understand whether some aspects of the
response of the tubes are inherent to the sheets, and if
so, they could also be observed in other systems of more
practical interest, e.g. samples comprising a mixture of
tubes with different diameters and orientation or multi-
wall tubes. In section 5, we present results for electron
energy loss in order to have some additional validation
from existing experimental data, and because a compari-
son between optical and loss spectra allows a deeper dis-
cussion and understanding of interaction effects. Finally,
we conclude with section 6, with an overall discussion of
the results.
2 Theoretical Framework
Ab initio calculations in the framework of Density Func-
tional Theory (DFT) have yielded high-quality results
for a large variety of systems: from molecules to pe-
riodic solids and structural defects [31]. These results
are however mostly limited to quantities related to the
electronic ground state, whereas additional phenomena
that occur in the excited state are not correctly de-
scribed [32]. In particular, the self-consistency between
the total perturbing potential and the charge response
Optical properties of C and BN nanotubes 3
induces Hartree and exchange-correlation potentials that
have to be dealt with. The former give rise to the so-
called local field effects, whereas the latter can lead, for
example, to excitonic effects. Today, in the solid state ab
initio framework two main approaches can include both
LFE and XC effects [30] and can be therefore suitable
for the description of electronic excitations in nanostruc-
tures. First, Green’s functions approaches within many-
body perturbation theory: here one adds self-energy cor-
rections to the DFT Kohn-Sham bandstructure and the
electron-hole interaction is included via the solution of
the Bethe-Salpeter equation [30]. This approach has given
excellent results for various bulk and finite systems, but
it is computationally very cumbersome and not ready yet
to be applied systematically to more complex systems.
The time-dependent Hartree-Fock (TDHF) method rep-
resents a certain level of approximation within this ap-
proach, where correlation (i.e. screening) in the self en-
ergy is neglected. LFE, on the other hand, are still re-
tained. TDHF has been employed, up to now in a semi-
empirical way, to calculate optical spectra, including ex-
citons, of carbon nanotubes [23] and also for the 4 ˚
A-
diameter ones [33]. However, the obtained absorption
peak assignments in the latter case were in disagreement
with predictions based on the operative dipole selection
rules for the specific tubular space groups [22, 27]. This
suggests that in such systems with important metallic
character a neglect of screening may lead to problems.
The second way to calculate spectra is given by the
time-dependent DFT (TDDFT) [34], where all many-
body effects are embodied in the exchange-correlation
potential and kernel. It is in practice an approximate way
but always treats the variations of the Hartree poten-
tial exactly (right as in both the Green’s functions and
TDHF approaches). This approach, using the (static)
adiabatic local density approximation (TDLDA) for the
XC effects, or even completely neglecting the density
variations of the XC potential (RPA, on the basis of
an LDA bandstructure) has been applied successfully to
many finite and infinite systems [30]. In particular, ex-
cellent results for the loss spectra of graphite have been
obtained using this approximation [35]. This is the ap-
proach we follow in the present work.
As a first step in our TDDFT approach, we deter-
mined the electronic ground state of the systems. The
Kohn-Sham single-particle equations were solved self-
consistently in the LDA for exchange and correlation [36].
For the description of the valence-core interaction we
have used norm-conserving pseudopotentials which were
generated from free-atom all-electron calculations [37].
For the “isolated” geometries of the tubes and of the
sheets we calculated periodic arrays with a large distance
between the building blocks in a supercell geometry, in
order to minimize the mutual interaction. The crystalline
valence-electron wavefunctions were expanded using a
plane-wave basis set (with an energy cutoff of 62 Ryd-
berg for carbon systems, and 50 Rydberg for BN sys-
tems). A part of the calculations was carried out using
the ABINIT code [38].
The next step is the linear response calculation of
the independent particle polarizability χ0[39,40]. It in-
volves a sum over excitations from occupied bands to
unoccupied bands:
χ0
G,G′(q, ω) = 2 Rdk3
(2π)3Pocc.
nPunocc.
m
hn,k|e−i(q+G)r|m,k+qihm,k+q|ei(q+G′)r|n,ki
ǫn,k−ǫm,k+q−ω−iη −(m↔n),(1)
where (m↔n) means that the indices mand nof the
first term are exchanged. The result is checked for con-
vergence with respect to the number of bands [41] and
the discrete sampling of k-points within the first Bril-
louin zone.
Within TDLDA, the full polarizability χis connected
to χ0via[42]
χ=χ0+χ0(VC+fxc)χ, (2)
where VCis the bare Coulomb interaction and fxc , the
so-called exchange-correlation kernel, is the functional
derivative of the LDA exchange-correlation potential with
respect to the electron density. By setting fxc to zero,
exchange and correlation effects in the electron response
are neglected and one obtains the Random-Phase Ap-
proximation (RPA). We have carried out our calcula-
tions in the RPA and also in the TDLDA for certain
cases. The inverse dielectric function for a periodic sys-
tem and momentum transfer qis obtained from:
ε−1
G,G′(q) = δG,G′+VC(q+G)χG,G′(q) (3)
with qin the first Brillouin zone and G,G′are reciprocal
lattice vectors. The absorption spectrum is obtained as
the imaginary part of the macroscopic dielectric function
εM(ω) = 1/ε−1
G=G′=0(q→0; ω) (4)
whereas the loss function for a transferred momentum
(q+G) is given by −Im[ε−1
G,G(q)]. In inhomogeneous sys-
tems, e.g. periodic solids, clusters and structural imper-
fections the inhomogeneity in the electron response gives
rise to local field effects (LFE) and the εG,G′cannot
be considered as being purely a diagonal matrix. There-
fore, its off-diagonal elements have to be included in the
matrix inversion. Making the approximation ε−1
G,G(q)≈
1/εG,G(q) corresponds to neglecting the inhomogeneity
of the response, i.e. to neglecting the LFE. When these
effects are neglected and, moreover, all transition matrix
elements in χ0are supposed to be constant, one arrives
at the widely used approximation that the absorption
spectrum is proportional to the JDOS, i.e. proportional
to the sum over interband transitions from occupied (v)
to empty (c) states over the Brillouin zone points k,
Pv,c,kδ(ǫck−ǫvk−ω).
4 A.G. Marinopoulos, L. Wirtz, et al
3 Optical Absorption Spectrum for carbon and
BN tubes
Let us first look at the optical absorption spectra of
small-diameter carbon nanotubes (all three possible he-
licities), for which experimental results have recently
become available. The calculations of the spectra [43]
were done for the tubes arranged in a hexagonal lat-
tice with an intertube distance (distance between tube
walls), equal to Dt= 5.5 ˚
A which leads to nearly iso-
lated tubes. Additionally, for the (3,3) armchair ones we
repeated the calculations for a solid with a smaller inter-
tube distance, Dt= 3.2 ˚
A, which is close to the interlayer
distance in graphite (∼3.4 ˚
A).
The optical absorption spectra for the small diameter
tubes are diplayed in Fig. 1 (as well as JDOS curves for
two cases). In the upper panel of the Figure, the JDOS
divided by the square of the excitation energy, for the
(3,3) nearly isolated (thin line) and interacting (in the
solid) (thick line) tubes are shown. It can be seen that af-
ter an initial very steep decrease (<1 eV), in both JDOS
curves there is a gradual increase starting from 2 eV. In
the case of isolated tubes a sequence of pronounced peak
structures up to 5 eV is observed: this can be explained
from the occurrence of direct interband transitions be-
tween the van Hove singularities of the density of states
(DOS). These peaks are smeared out in the JDOS of the
solid where the tubes are strongly interacting [44]. In the
lower panels of Fig. 1 the calculated absorption spectra
for light polarizations perpendicular and parallel to the
tube axis are displayed (as well as the experimental data
plotted in the inset). The dashed lines denote results
in the RPA neglecting LFE, continuous lines including
LFE. For parallel polarization both LFE and LDA-XC
effects were found to be negligible and, therefore, in this
case only the RPA results without LFE are presented.
The fact that both these effects turned out to be negli-
gible can explain why the peak positions (A, B and C)
predicted here for the parallel polarization match very
well the ones found in two recent DFT-RPA studies [27],
which completely neglected LFE and XC effects in the
response. Our calculated peak positions (A, B and C) in
Fig. 1 are in good agreement (to within 0.2 eV) with the
experimentally observed peak structures for the parallel
polarization (inset).
It is also important to note that the present results
support the same peak-to-helicity assignment as in pre-
vious works [22,27]. Namely, the three observed peaks –
A, B and C – are due to the (5,0), (4,2) and (3,3) tubes,
respectively. Such an assignment is also in accordance
with the dipole selection rules for these tubular space
groups [22,27].
Concerning the effect of intertube interaction in the
absorption we found that it is rather small for this po-
larization: tube-tube interaction leads only to a broad-
ening of the main absorption peak (thick solid curve for
the (3,3) tubes). This can be explained from an increase
of the energy range of the possible interband transitions
brought about by the interaction.
All the discussion of the results up to now seems
to indicate that LFE may not be needed. Nonetheless,
for light polarizations perpendicular to the tube axis –
due to the presence of depolarization – LFE play an
important role which cannot be ignored. More specif-
ically, for this polarization the experimental spectrum
displays vanishing intensity for frequencies up to 4 eV
[22] (see inset in Fig. 1, dotted curve). Clearly, neither
the JDOS calculation nor the calculated spectrum with-
out LFE can capture this effect. Instead, they both pre-
dict a series of peaks from 2 to 5 eV. The reason that
RPA-without-LFE fails here is due to the depolarization
effect [19,23, 45] which is created by the induced polar-
ization charges. The depolarization is only accounted for
if LFE are included: as it can be seen in Fig. 1 (second
panel; continuous curve) LFE suppress the low-energy
absorption peaks and render the tubes almost transpar-
ent below 5 eV in agreement with the experiment. It
should not come, therefore, as a surprise that the recent
DFT-RPA studies [27] (which did not consider LFE) did
not reproduce this transparency for the perpendicular
polarization.
The TDLDA result, displayed as the dotted curve in
the second panel, turned out to be qualitatively similar
to the RPA-with-LFE result: again the low-energy ab-
sorption peaks are suppressed. This shows that the main
effect comes from fluctuations of the Hartree-, and not
from those of the XC-potential (as also obtained for the
case of graphite [35]). LDA-XC effects only cause a small
(0.3 eV) redshift of the remaining absorption peak to 5.5
eV, with respect to RPA-with-LFE. This is a character-
istic behavior of finite systems [30]. Still there is an open
question about the role of electron-hole interaction in the
case of the small band-gap (4,2) tube [46], however the
good agreement with experiment indicates that its con-
tribution should be small for the parallel polarization.
LFE also have a similar drastic impact for the system
of isolated zig-zag (5,0) tubes (not shown). The manner
according to which LFE operate may be understood as
follows: for perpendicular polarization the tubes form an
assembly of almost isolated, but highly polarizable, ob-
jects. An applied external field induces hence a local, i.e.
microscopic, response - the LFE - which strongly weak-
ens the total perturbation (i.e. it is a “depolarization”).
The macroscopic response to this weak perturbation is
only very moderate, because the electrons are localized
on the tubes. This is totally different from the screening
in a bulk metal or small-gap semiconductor, where even
a very weak total perturbation still leads to a strong
response at low frequencies. For polarization parallel to
the tube axis, the situation resembles rather this latter
case, something which explains the absence of LFE for
this polarization. On the other hand – for the perpendic-
ular polarization – when the tubes are interacting in the
solid (third panel) the depolarization is much weaker:
Optical properties of C and BN nanotubes 5
1 2 3 4 567
energy (eV)
0
10
optical absorption
perpendicular
JDOS /
1 2 3 4
0
2
4
6
polarization
polarization
parallel
(3,3)
(4,2)
(5,0)
(3,3)
(3,3) solid
isolated
(3,3)
solid
isolated
ω2
(x 2)
(x 2)
AB
C
A
BC
Fig. 1 Calculated optical spectra for the (3,3), (5,0) and
(4,2) nearly isolated and solid (only for (3,3) case; thick lines)
of carbon tubes with (continuous lines) and without (dashed
lines) LFE in the RPA. For comparison the TDLDA result
for the isolated (3,3) tubes is given by the dotted line in the
second panel as well as the JDOS curves. The results are pre-
sented for both perpendicular and parallel to the tube-axis
polarization. For the (4,2) tube only the result for the par-
allel polarization without LFE is given, which should be a
sufficiently good approximation for this polarization, accord-
ing to our results for the (3,3) and (5,0). The experimental
data [22] are displayed in the inset.
the tubes start to absorb (they are no longer transpar-
ent) because the electronic states start to delocalize and
the system is now more similar to a bulk metal. This
different behavior of the response depending on the in-
tertube distance leads to a very important consequence:
it suggests that the inter-tube interaction can be detected
experimentally in a qualitative study of the absorption
spectra for perpendicular light polarizations.
0
2
4
6
0510 15
energy (eV)
0
2
4
6
8
10
12
optical absorption
0
2
4
6⊥
||
aver.
JDOS/ω2
a)
b)
c)
BN(6,0)
BN(5,5)
BN(3,3)
BN(3,3)
BN(3,3)
Fig. 2 Calculated optical absorption spectra for quasi-
isolated BN nanotubes [47] with (solid line) and without
(dashed line) LFE in the RPA approximation: a) Compari-
son of the spatially averaged spectrum of the (3,3) tube with
the joint density of states divided by the square of the ex-
citation energy (dotted line); b) spectrum with polarization
perpendicular to the tube axis for the (3,3) tube; c) spectra
with polarization parallel to the tube axis for the (3,3), (5,5)
and (6,0) tubes.
In Fig. 2 we present optical absorption spectra for
three different BN nanotubes. First, we compare for the
case of the BN(3,3) tube the spatially averaged spec-
trum with the joint density of states (JDOS), divided
by the square of the transition energy. If local field eff-
fects are neglected, most peaks of the JDOS are visible
in the averaged absorption spectra while some peaks are
suppressed due to small or vanishing oscillator strength
in Eq. (1). This demonstrates that the fine structure
in the spectra is not an artifact of low k-point sam-
pling but is due to the presence of van-Hove singular-
ities in the 1-dimensional density of states of the tubes.
Proper inclusion of LFE leads to a smoothing of the spec-
trum. However, some fine structure survives and may
be discernible in high-resolution optical absorption ex-
periments. The spectrum with perpendicular polariza-
tion demonstrates that for BN nanotubes, depolariza-
tion effects play a similarly important role as for carbon
tubes: Neglecting LFE, the onset of absorption would be
around 4.5 eV. LFE, however, lead to a redistribution of
6 A.G. Marinopoulos, L. Wirtz, et al
the oscillator strength to higher energies and render the
tubes almost transparent up to 8 eV. The band gap of
BN nanotubes in DFT-LDA is 4 eV and is only weakly
dependent on radius and chirality [2,3]. Accordingly, the
absorption spectra for polarization parallel to the tube
axis are very similar for the (3,3), (5,5), and (6,0) tubes
which all display a strong absorption peak around 5 eV
and a second high peak around 14 eV. We expect these
structures to be stable also for tubes with larger diam-
eters. Only the fine structure of the spectra (e.g., the
absorption peak at 9 eV for the (3,3) tube) depends on
the details of the band-structure and varies for the dif-
ferent diameters and chiralities [20]. For the (6,0) tube,
the first high absorption peak is split and the dominant
peak shifted towards lower energy. This is a curvature ef-
fect which leads to a reduction of the band-gap for small
zigzag BN tubes.
In the next section we will see to which extent the
above findings for carbon and BN nanotubes can be un-
derstood by an analysis based on results for graphene
and BN sheets and graphite.
4 Optical Absorption Spectrum for graphite and
the graphene and BN sheets
Hexagonal graphite has the ABA Bernal stacking se-
quence of the graphene sheets. In the present work we
assumed the experimental lattice parameter ahex and
(c/a)hex ratio (2.46 ˚
A and 2.73, respectively [48]).
The calculated RPA optical absorption spectra [49],
with and without LFE, are shown in Fig. 3 for in-plane
light polarization (E⊥c) and in Fig. 4 (a) for polariza-
tion parallel to the caxis (Ekc). The in-plane spectrum
is dominated by a very intense peak structure at low fre-
quencies (up to 5 eV) and also another peak structure
of broader frequency range which sets in beyond 10 eV
and has a pronounced peak at 14 eV. The origin of these
peak structures is due to π→π⋆and σ→σ⋆interband
transitions, respectively, according to the earlier inter-
pretations by Bassani and Paravicini [50] who assumed
a two-dimensional approximation – no interaction be-
tween the graphene sheets – and the operative dipole
selection rules for this polarization. Our calculations of
the oscillator strength for specific transitions between
bands in the Brillouin zone (BZ) are in agreement with
their interpretation.
LFE are found to be nearly negligible for this polar-
ization. This is not surprising since for in-plane polar-
izations graphite is homogeneous in the long-wavelength
limit (q→0). The general aspects of the spectrum –
peak positions, their intensity and lineshape – are in
close agreement with the existing experimental results
[51] and the previous all-electron calculation of Ahuja et
al. [53] who neglected LFE.
The absorption spectrum of graphite for the light po-
larization parallel to the caxis is shown in Fig. 4 (a).
0 10 20 30
energy (eV)
0
5
10
15
20
optical absorption
with LFE
no LFE
π π∗graphite
in-plane polarization
σ σ∗
Fig. 3 Absorption spectrum of graphite for E⊥c. The star
symbol denotes the unoccupied electron states.
It is characterized by a weak intensity in the low fre-
quency range (0-5 eV) and important peaks in the fre-
quency range beyond 10 eV. For this polarization the
bandstructure does not play the exclusive direct role in
defining the absorption spectrum. Now LFE are very
important. When LFE are not considered the peak po-
sitions for this polarization are at 11 and 14 eV as in the
earlier DFT-RPA calculation [53]. However, when LFE
are included transitions are mixed and the absorption
spectrum is appreciably modified. The main effect of lo-
cal fields is to shift the oscillator strength at 10-15 eV to
higher frequencies. They decrease the intensity of the 11
eV peak and are responsible for the appearance of the
16 eV peak in the spectrum. The latter peak is seen in
experiments as a shoulder suggesting that the inclusion
of LFE is necessary.
The non-zero oscillator strength found below 5 eV is
attributed to the inter-layer interaction which is present
in the solid. It is also observed experimentally. The dipole
selection rules [50] for an isolated graphene sheet (i.e.
complete two-dimensionality) lead to vanishing matrix
elements and oscillator strength at this frequency range.
LFE do not have any influence on the lower part of the
spectrum (less than 10 eV).
The existing experimental evidence is not conclusive
for the dielectric function in the 11 eV frequency region.
The frequency-dependent Im[εM] obtained from electron
energy loss data [54, 55,56] displays a very sharp and in-
tense peak (Im[εM]=10) at 11 eV. On the other hand, on
the basis of optical measurements [52] the observed max-
imum at this frequency is of considerably smaller inten-
sity. The earlier interpretations [57] were based on semi-
empirical tight-binding bandstructure calculations in the
two-dimensional approximation (i.e. isolated graphene
layers) and they predicted peak structure between 13.5
and 16.5 eV.
It is therefore also important to understand the ef-
fect of the inter-layer interaction in the optical response
Optical properties of C and BN nanotubes 7
0
2
4
6
8
10
0
2
4
6
8
optical absorption
0 10 20 30
energy (eV)
0
2
4
6
8
2 (c/a)hex
3 (c/a)hex
graphite
no LFE
with LFE
E || c
(a)
(b)
(c)
Fig. 4 Absorption spectrum for Ekc, for graphite and the
graphene-sheet geometries with 2 (c/a)hex and 3 (c/a)hex.
and if this interaction is primarily responsible for the
occurrence of the intense peak at 11 eV. For this pur-
pose, we progressively increased the inter-layer spacing
(doublying and tripling the (c/a)hex ratio). This yields
stackings of graphene layers in the unit cell with much
weaker mutual interaction. The absorption spectra for
these graphene-sheet geometries are displayed in Fig. 4
(b,c). The first observation is that the oscillator strength
vanishes completely in the frequency region below 10 eV
in complete accordance with the predictions based on
the dipole-selection rules for an isolated graphene layer.
Therefore, the double inter-layer spacing leads to non-
interacting graphene layers as far as the RPA absorp-
tion spectrum (at this frequency range) is concerned.
When LFE are neglected, the peak structure in the 10-
15 eV range stays intact, now with a smaller intensity
due to the larger volume. With increasing interlayer sep-
aration, LFE become progressively more important. The
shift of oscillator strength induced by LFE is so big that
the absorption peaks at 10-15 eV are almost completely
suppressed. These findings demonstrate that both inter-
layer interaction and LFE influence considerably the in-
tensity of the absorption peak at 11 eV. Qualitatively,
the depolarization effects found in the tubes can hence
be explained by the local field effects observed in the
graphitic response perpendicular to the graphene sheets.
Likewise, the effect of intertube interaction in the spec-
tra for polarizations perpendicular to the tube axis is
also consistent with the increased propensity of LFE to
shift oscillator strength at higher frequencies when the
intersheet distance progressively increases (see Fig. 4).
We conclude this section with a comparative presen-
tation of the absorption spectra of the graphene and BN
sheets using the same inter-sheet distance for both cases,
shown in Fig. 5. Also we make comparisons with the
spectra of the corresponding tubes. The spectrum of the
graphene sheet for the in-plane polarization resembles
closely the in-plane polarization spectrum of graphite
(Fig. 3) (except for a scaling due to the change in the
volume), confirming once more, as in the case of the
tubes for the polarization parallel to the tube axis, that
the position of the absorption peaks and their lineshape
is only weakly influenced by the distance and inter-sheet
interaction in this case. The main difference between the
graphene and BN-sheet spectra for the in-plane polar-
ization is the complete absence of any feature below 4
eV in the BN spectrum. This is clearly related to the 4
eV LDA-band gap of BN. The out-of-plane polarization
spectra of the graphene and BN sheets are remarkably
similar. Most importantly, both display a transparency
up to 10 eV and both demonstrate a strong depolar-
ization effect with shift of oscillator strength to higher
energies [20].
We compare now the calculated sheet-absorption spec-
tra for in-plane polarization with the corresponding tube-
absorption spectra for polarization parallel to the tube
axis (Fig. 1 for carbon and Fig. 2 for BN). In the carbon
case, the strong absorption feature of the sheet between
0 and 4 eV maps onto one or several absorption peaks of
the tubes in this energy range. The details of this map-
ping are, however, sensitively dependent on the diameter
and the chirality of the tubes, since dipole selection rules
play a strong role in these highly symmetric systems. In
particular, for tubes with larger diameter than the ones
calculated in this article, the distance E11 of the first
van Hove singularities strongly depends on whether the
tube is metallic or semiconducting. Therefore, calcula-
tions on graphite or graphene alone will not be sufficient
to predict absorption spectra of small-diameter carbon
nanotubes. For BN, in contrast, the first high absorption
peak at 5.5 eV maps directly onto a corresponding peak
in the tubes. Also the high energy absorption feature
around 14 eV is very similar for the BN sheet and the
BN tubes. Due to the large band gap of BN, the com-
parison between sheet and tube spectra is much more
favourable for BN than for carbon tubes.
On the other hand, in the case of carbon, the ex-
trapolation of results from graphite and graphene to the
tubes is much more straightforward for the electron en-
ergy loss spectra as discussed in the following section.
5 EELS spectra in the limit of small q: tubes
versus the layered structures
Having seen the strong interaction effects that occur in
optical spectra for a polarization perpendicular to the
planes or tube axis, it is interesting to make an excur-
sion to a different type of spectroscopy, namely electron
8 A.G. Marinopoulos, L. Wirtz, et al
0
5
10 C-sheet
0
5
10
optical absorption
510 15
energy (eV)
0
5
10 ⊥
||
aver.
0
4
8BN-sheet
0
4
8
510 15
energy (eV)
0
4
8⊥
||
aver.
Fig. 5 Calculated absorption spectra for the graphene and hexagonal BN sheets [58].
energy loss (EELS) [59]. Although also in this case one
can measure, like in the absorption experiment, a mo-
mentum transfer close to zero, there is a substantial dif-
ference in the definition of the response function: in the
absorption measurement, one detects the response to the
total macroscopic field, whereas in EELS the response to
the external field is reported. Therefore, as pointed out
above, absorption is linked to the imaginary part of the
macroscopic dielectric function, but EELS is linked to
the imaginary part of the inverse of the latter. Math-
ematically this translates into the fact that the crucial
response function for EELS is governed by equation (2),
whereas in the case of absorption one can use slightly dif-
ferent quantity: the macroscopic dielectric function can
in fact be rewritten as
εM(ω) = 1 −limq→0VC(q, G = 0)¯χG=G′=0(q;ω) (5)
where VC(G= 0) is the long-range component of the
Coulomb potential, and it has a 1/q2divergence for van-
ishing qand the quantity ¯χobeys an equation similar to
(2): ¯χ=χ0+χ0(¯
VC+fxc) ¯χ, but setting the G= 0 com-
ponent of VCto zero ( ¯
VC) [30]. Therefore, this seemingly
tiny difference is responsible for the difference between,
e.g., the position of the main absorption peaks and that
of the valence plasmons in solids, and one can also ex-
pect that it will be crucial when interaction effects on
the spectra are discussed. In particular, EELS spectra
should show, due to the presence of this long-range term,
stronger interaction effects than absorption spectra.
This is in fact the case, as we will illustrate in the
following for the total π+σplasmon in graphite. This
plasmon represents the collective excitation mode of all
the valence electrons in graphite. Before discussing the
results we stress here that the tubes cannot be consid-
ered as completely isolated objects in this calculation
of the loss function with the 5.5 ˚
A intertube distance.
We refer to the tubes in the latter geometry as distant.
Fig. 6 shows the RPA loss function, −Im(1/εM), for the
(3,3) tube, in the range 15-35 eV and for a vanishing
momentum transfer qparallel to the tube axis. For this
orientation LFE are negligible. A strong shift of the π+σ
plasmon from 22 to 28 eV due to intertube interactions in
the solid can be seen [60]. The magnitude of this shift re-
veals a strong dependence of the plasmon position upon
the intertube distance (hence the average valence elec-
tron density) essentially following the plasmon-frequency
dependence in the case of the homogeneous electron gas
[61].
This shows that the tubes respond as homogeneous
and highly polarizable objects for parallel qin the long-
wavelength limit (q→0). A direct consequence would
then be that the atomic arrangement, orientation of bonds
and helicity may play a secondary role in the response in
this frequency and qrange. Therefore, the result for the
π+σplasmon (shown in Fig. 6) would be representa-
tive of either of the three tubes since all of them – being
of nearly the same diameter – possess the same aver-
age electron density. This is indeed the case as it can be
seen in the inset of Fig. 6 where an almost indistinguish-
able π+σplasmon was also obtained for the (5,0) tube.
Hence, in contrast to an optical absorption experiment,
small-q loss measurements of the π+σplasmon cannot
determine tubular helicities for a given tube diameter.
Optical properties of C and BN nanotubes 9
The governing factor for the π+σplasmon must
be traced to the in-layer graphitic response as it can
be seen in Fig. 6 where the loss function for graphite
and graphene is also shown (for in-layer qorientation)
at comparable (to the tubes) average electron densities
(to within 10 %) [35] (dashed curves in Fig. 6). This
shows that the loss function of the tubes for parallel qin
this frequency range is governed by the average-density-
dependent part of the in-layer graphitic response. Similar
plasmon shifts, therefore, can also be expected in other
carbon systems with graphene-based structural blocks
e.g. multiwalled tubes. The present results outline the
significance of the π+σplasmon as a key measurable
spectroscopic quantity which could gauge the intertube
distances and interactions in real samples of carbon nan-
otubes.
10 15 20 25 30 35
loss energy (eV)
0
1
2
3
loss function
10 20 30
0
1
2π + σ plasmon
solid
distant
Fig. 6 Calculated RPA loss function for the carbon (3,3)
distant tubes (continuous thin line) and solid (continuous
thick line) of tubes. qis vanishing with an orientation parallel
to the tube axis. The dashed curves denote the results for
graphite (thick) and graphene (thin) for small in-layer q[35].
The loss function for the (5,0) tube is shown in the inset as
a dotted curve. LFE are negligible for this qorientation.
In order to gain a more complete understanding of
the in-layer graphitic response at small q’s and how the
latter is influenced by the inter-layer interaction we de-
termined both the loss and dielectric function for various
graphene-like geometries i.e. varying the interlayer spac-
ing or, equivalently, the (c/a)hex ratio. In these calcula-
tions we also looked at the lower-frequency πplasmon.
Bearing in mind the discussion in the previous para-
graph, these calculations could then serve as benchmarks
for predicting the position of the π+σplasmon in nan-
otubes as a function of the intertube distances for low
q’s.
The loss and dielectric function for small in-plane
q(0.22 ˚
A−1) is shown in Fig. 7 for graphite and the
graphene geometries with multiple (c/a)hex ratios. It can
be seen that the peak positions of both plasmons have
shifted to lower frequencies when the inter-layer sepa-
0
1
2
3
loss function
0
5
10
0 10 20 30 40
energy (eV)
0
5
10
15
2 (c/a)hex
3 (c/a)hex
5 (c/a)hex
graphite
Imε
Μ
small in-plane q
Reε
Μ
π
π + σ
π
σ
π∗
σ∗
Fig. 7 Loss and dielectric function for small in-plane q (=
0.22 ˚
A−1) for graphite and the graphene-like geometries with
multiple (c/a)hex ratios. Only the results without LFE are
presented since LFE are negligible in this q range.
ration is increased. This effect is very pronounced for
the π+σplasmon position. These results reaffirm that
the total (π+σ) plasmon is extremely sensitive to the
inter-layer interaction for small in-plane q’s. This can
be explained as follows: with increasing (c/a)hex, i.e.
interlayer spacing, the system becomes an assembly of
nearly isolated graphene sheets. This leads to a decrease
of screening (Re[εM]→1; see Fig. 7) with the effect
that −Im[1/εM]→Im[εM], namely a coincidence of
the loss and absorption functions. Since for the in-plane
polarization the positions of the absorption peaks do not
change with increasing (c/a)hex (see Im[εM] in Fig. 7),
then the loss function undergoes important changes, in
particular the π+σplasmon. The latter is displaced at a
much faster rate to lower frequencies towards the 14 eV
peak of the absorption spectrum whereas the πplasmon
peak is rather insensitive to (c/a)hex since it is already
located very close to the 0-5 eV peak structure of the
π→π⋆transitions in Im[εM].
At present, existing measurements [62,63] of the loss
spectra of samples of single-wall carbon nanotubes have
given a π+σplasmon in the frequency range 21–24 eV
for small momentum transfer q. Our predicted frequency
of the π+σplasmon for the case of the distant tubes
is within this frequency range (Fig.6). However, a direct
comparison of the present results for the loss function of
(3,3) and (5,0) tubes with the measured loss data is not
straightforward. The difficulty stems from two factors,
tube diameter and alignment, which have a competing
10 A.G. Marinopoulos, L. Wirtz, et al
effect on the π+σplasmon position: a) Bulk samples
of single-wall tube material possess a mean diameter of
14 ˚
A [63], which is considerably larger to the range of
4˚
A studied in the present work. Assuming a common
intertube spacing, then larger diameters would give rise
to a displacement of the π+σplasmon towards lower
frequencies since the diametrically-opposed wall parts of
the same tube are facing each other at larger distances,
i.e. a situation resembling a stacking of graphene sheets
with larger (c/a)hex ratios (see Fig. 7). For instance, the
5 (c/a)hex ratio corresponds to an intersheet separation
of 16.8 ˚
A and the corresponding π+σplasmon peak is
at 16 eV. b) The alignment of the tubes in the samples is
not perfect; therefore it is to be expected that also out-
of-plane excitations of graphitic origin will contribute to
the response. These excitations [35] should tend to pro-
duce a more diffuse shape for the loss spectrum, heavily
dampening the π+σplasmon and shifting the observed
peak towards higher frequencies.
The type of response described just above can be
clearly seen in Fig. 8 which shows the loss function of
the distant (3,3) tubes for qorientation perpendicular
to the tube axis. For this orientation in-plane as well
as out-of-plane graphene excitations contribute to the
tube response. The latter cause the diffuse shape of the
loss function (see inset of Fig. 8). LFE are now very
important and the peak position of the π+σplasmon
is at 28 eV.
0 10 20 30 40 50
energy (eV)
0
1
2
loss function
0 10 20 30 40 50
0
1
2graphene 2 (c/a)hex
distant (3,3) tubes
q perpendicular
to tube axis
Fig. 8 Calculated RPA loss function for the carbon (3,3)
distant tubes for small q(0.19 ˚
A−1) of orientation perpen-
dicular to the tube axis. In the inset the loss function of
graphene [35] for 2 (c/a)hex and for a similar qof orienta-
tion perpendicular to the sheet is displayed. Continuous and
dashed curves denote results obtained with and without LFE,
respectively.
For completeness, we show in Fig. 9 also the calcu-
lated EELS spectra for small BN nanotubes with mo-
mentum transfer q→0 along the tube axis. As in the
case of carbon tubes, two main features are clearly pro-
nounced: the π-plasmon at 6-7 eV and the high energy
collective oscillation (π+σplasmon) around 20 eV. The
exact position of the peaks depends on the radius and
chirality of the tubes and on the intertube distance. In
order to compare with experimental EELS-spectra on
multi-wall BN tubes [64], an extrapolation to larger-
diameter tubes is needed and the inter-wall interactions
have to be taken into account.
0510 15 20 25
energy (eV)
0
0.5
1
1.5
Ioss function
BN(3,3)
BN(5,5)
BN(6,0)
π
π+σ
Fig. 9 Loss spectra for three different BN nanotubes with
momentum transfer q→0 along the direction of the tube
axis. The spectra are calculated with an inter-tube distance
of 7.4 ˚
A. Since the spectra with and without LFE are almost
indistinguishable, only the calculation without LFE is shown.
6 Discussion and Conclusions
In this closing section we should like first to comment
on the validity of RPA and TDLDA for the description
of the optical spectra of the tubes. It is well known that
RPA and TDLDA often give excellent results for loss
spectra [30]. However, in this work we have seen that
also very good absorption spectra were obtained for the
carbon tubes, even at the RPA level, despite the fact
that both RPA and TDLDA are known to fail badly in
the description of absorption spectra of many bulk ma-
terials (silicon and argon being two representative cases)
[30]. Our explanation in this regard is the following: for
the light polarization parallel to the tube axis, the en-
suing screening is significant and, therefore, XC effects
are damped. For the perpendicular polarization and for
the larger inter-wall distance (Dt= 5.5 ˚
A) the tubes be-
have essentially like isolated systems, where strong can-
cellations are known to occur between self-energy cor-
rections and the electron-hole interaction, i.e. between
XC effects (see Ref.[30]). The experimental precision is
then not high enough (also due to the almost vanished
absorption intensity in the relevant frequency range) to
discern to which extent XC effects should be better de-
scribed by approximations beyond the TDLDA. In view
of these considerations, we can be confident regarding
the quality of the calculations. On the other hand, the
Optical properties of C and BN nanotubes 11
results for the strongly interacting tubes (smaller inter-
tube distance) should be regarded as qualitative since
the system becomes then more similar to a solid where
the cancellations may be more incomplete, and since no
direct comparison to experiment is possible, at present,
in this case.
Still, excitonic effects are not included in the calcula-
tions. They could play a role in both carbon and BN nan-
otubes, leading to redistribution of oscillator strength
and/or appearence of new peaks in the bandgap (bound
excitons). They might be the reason for the anoma-
lous E11/E22 ratio [65] measured recently for carbon
tubes [15]. We expect a stronger deviation of measured
optical spectra from theoretical ones in the case of BN
nanotubes. For this large band gap material, two effects
will most likely play an important role: Quasi-particle
corrections will widen the band gap as in the case of
hexagonal BN, where a GW calculation has demonstrated
a band-gap increase from 4 eV to 5.5 eV [3]. Excitonic
effects, in contrast, will lead to isolated states in the
band gap or to an overall reduction of the band-gap.
To which extent quasi-particle corrections and excitonic
effects cancel each other is presently not clear. Work
along these lines is in progress [66]. For the role of quasi-
particle corrections and excitonic effects in carbon tubes,
we refer the reader to Ref. [46].
In conclusion, the present ab initio calculations of
the optical absorption of small-diameter carbon and BN
nanotubes give good agreement with the available ex-
perimental data. The inclusion of local field effects in
the response to a perturbation with perpendicular po-
larization is necessary for a proper description of the
depolarization effect leading to a suppression of the low-
energy absorption peaks for both types of tubes. This
suppression can also explain recent findings in near-field
Raman microscopy of single-wall carbon nanotubes [67].
The proper analysis of the polarization dependence of
the absorption cross section is very important in order to
describe the surface enhanced Raman scattering exper-
iments in isolated carbon nanotubes [18, 68]. In carbon
tubes the position of the first absorption peak strongly
varies with the tube indices while in BN tubes the first
peak is determined by the band gap of BN and is there-
fore mostly independent of (n, m). For the BN tubes
some of the fine-structure which distinguishes tubes of
different chirality is only visible in the UV region which
gives rise to the hope that this energy regime will be
probed in the future.
The intertube interaction was also found to be very
important. For the carbon tubes this interaction is the
governing factor which determines the position of the
higher-frequency π+σplasmon. This plasmon, hence,
may prove to be a very useful spectroscopic quantity
probing intertube interactions and distances in real sam-
ples. Finally, the corresponding results for the layered
constituents — graphene and BN sheets — revealed that
some aspects of the tubular dielectric response can be ex-
plained even at a quantitative level from the in-layer and
interlayer response of the sheets.
7 Acknowledgments
This work was supported by the European Community
Research Training Networks NANOPHASE (HPRN-CT-
2000-00167) and COMELCAN (HPRN-CT-2000-00128),
by Spanish MCyT(MAT2001-0946) and University of
the Basque Country (9/UPV 00206.215-13639/2001). The
computer time was granted by IDRIS (Project No. 544),
DIPC and CEPBA (Barcelona). The authors also grate-
fully acknowledge fruitful discussions with Thomas Pich-
ler and Nathalie Vast.
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