Observation of Dirac plasmons in a topological insulator.
ABSTRACT Plasmons are quantized collective oscillations of electrons and have been observed in metals and doped semiconductors. The plasmons of ordinary, massive electrons have been the basic ingredients of research in plasmonics and in optical metamaterials for a long time. However, plasmons of massless Dirac electrons have only recently been observed in graphene, a purely two-dimensional electron system. Their properties are promising for novel tunable plasmonic metamaterials in the terahertz and mid-infrared frequency range. Dirac fermions also occur in the two-dimensional electron gas that forms at the surface of topological insulators as a result of the strong spin-orbit interaction existing in the insulating bulk phase. One may therefore look for their collective excitations using infrared spectroscopy. Here we report the first experimental evidence of plasmonic excitations in a topological insulator (Bi2Se3). The material was prepared in thin micro-ribbon arrays of different widths W and periods 2W to select suitable values of the plasmon wavevector k. The linewidth of the plasmon was found to remain nearly constant at temperatures between 6 K and 300 K, as expected when exciting topological carriers. Moreover, by changing W and measuring the plasmon frequency in the terahertz range versus k we show, without using any fitting parameter, that the dispersion curve agrees quantitatively with that predicted for Dirac plasmons.
- SourceAvailable from: Jian Zhen Ou[Show abstract] [Hide abstract]
ABSTRACT: Tunable plasmon resonances in suspended 2D molybdenum oxide flakes are demonstrated. The 2D configuration generates a large depolarization factor and the presence of ultra-doping produces visible-light plasmon resonances. The ultra-doping process is conducted by reducing the semiconducting 2D MoO3 flakes using simulated solar irradiation. The generated plasmon resonances can be controlled by the doping levels and the flakes’ lateral dimensions, as well as by exposure to a model protein.Science of Advanced Materials 03/2014; · 2.51 Impact Factor
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ABSTRACT: The recent developments in the emerging field of plasmonics in graphene and other Dirac systems are reviewed and a comprehensive introduction to the basics models and techniques is given. We discuss intrinsic plasmon excitations of single and bilayer graphene via hydrodynamic equations and the random phase approximation, but also comment on double and multilayer structures. The retardation limit and Dirac systems with large spin-orbit coupling will be covered, too.10/2013;
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ABSTRACT: The development of metamaterials, data processing circuits and sensors for the visible and ultraviolet parts of the spectrum is hampered by the lack of low-loss media supporting plasmonic excitations. This has driven the intense search for plasmonic materials beyond noble metals. Here we show that the semiconductor Bi1.5Sb0.5Te1.8Se1.2, also known as a topological insulator, is also a good plasmonic material in the blue-ultraviolet range, in addition to the already-investigated terahertz frequency range. Metamaterials fabricated from Bi1.5Sb0.5Te1.8Se1.2 show plasmonic resonances from 350 to 550 nm, while surface gratings exhibit cathodoluminescent peaks from 230 to 1,050 nm. The observed plasmonic response is attributed to the combination of bulk charge carriers from interband transitions and surface charge carriers of the topological insulator. The importance of our result is in the identification of new mechanisms of negative permittivity in semiconductors where visible range plasmonics can be directly integrated with electronics.Nature Communications 01/2014; 5:5139. · 10.74 Impact Factor
Observation of Dirac plasmons in a topological insulator
P. Di Pietro,1M. Ortolani,2O. Limaj,3A. Di Gaspare,4V. Giliberti,2F.
Giorgianni,3M. Brahlek,5N. Bansal,5N. Koirala,5S. Oh,5P. Calvani,1and S. Lupi6
1CNR-SPIN and Dipartimento di Fisica, Universit` a di Roma ”La Sapienza”, Piazzale A. Moro 2, I-00185 Roma, Italy
2CNR-IFN and Dipartimento di Fisica, Universit` a di Roma ”La Sapienza”, Piazzale A. Moro 2, I-00185 Roma, Italy
3Dipartimento di Fisica, Universit` a di Roma ”La Sapienza” and INFN, Piazzale A. Moro 2, I-00185 Roma, Italy
4CNR-IFN, Via Cineto Romano, 00000 Roma
5Department of Physics and Astronomy Rutgers,
The State University of New Jersey 136 Frelinghuysen Road Piscataway, NJ 08854-8019 USA
6CNR-IOM and Dipartimento di Fisica, Universit` a di Roma ”La Sapienza”, Piazzale A. Moro 2, I-00185 Roma, Italy
PACS numbers: 71.30.+h, 78.30.-j, 62.50.+p
Plasmons are the quantized collective oscilla-
tions of electrons in metals and doped semi-
conductors.The plasmons of ordinary, mas-
sive electrons are since a long time basic ingre-
dients of research in plasmonics and in optical
metamaterials . Plasmons of massless Dirac
electrons were instead recently observed in a
purely two-dimensional electron system (2DEG)
like graphene , and their properties are promis-
ing for new tunable plasmonic metamaterials in
the terahertz and the mid-infrared frequency
range . Dirac quasi-particles are known to ex-
ist also in the two-dimensional electron gas which
forms at the surface of topological insulators due
to a strong spin-orbit interaction . Therefore,
one may look for their collective excitations by
using infrared spectroscopy. Here we first report
evidence of plasmonic excitations in a topological
insulator (Bi2Se3), that was engineered in thin
micro-ribbon arrays of different width W and pe-
riod 2W to select suitable values of the plasmon
wavevector k. Their lineshape was found to be
extremely robust vs. temperature between 6 and
300 K, as one may expect for the excitations of
topological carriers. Moreover, by changing W
and measuring in the terahertz range the plas-
monic frequency νP vs. k we could show, without
using any fitting parameter, that the dispersion
curve is in quantitative agreement with that pre-
dicted for Dirac plasmons.
A topological insulator (TI) is a quantum electronic
material with an insulating gap in the bulk, of spin-orbit
origin, and gapless surface states at the interface with the
vacuum or another dielectric. The latter states are metal-
lic and associated with massless Dirac quasi-particles
[4–6]. The transport properties of these states, which
exhibit chirality, are protected from back-scattering by
the time-reversal symmetry and cannot be destroyed or
gapped by scattering processes which do not involve mag-
netic impurities. Since their discovery TI’s raised great
interest, not only for their outstanding physical prop-
erties, like axionic electromagnetic response [7, 8], and
exotic superconductivity [9, 10], but also for the poten-
tial applications in quantum computing [11, 12], tera-
hertz (THz) detectors  and spintronic devices .
Like for other compounds, some of these foreseen appli-
cations may benefit from the exploitation of the 2DEG
collective excitations, namely from plasmonics. Among
the 2DEGs, the TI surface states present the advantage
that they spontaneously provide a two-dimensional Dirac
system from the bulk material, without physically imple-
menting an atomic monolayer like in graphene. More-
over, thanks to the momentum-spin lockage, TI plasmons
may potentially preserve the coherence of the electronic
states up to room temperature. This would be a major
step forward in quantum mechanics applications.
In the 2DEG at the TI surface, collective excitations
(plasmons) like those recently detected in graphene 
are indeed expected to exist.
be directly excited by electromagnetic radiation because
their dispersion law is such as to prevent the conserva-
tion of momentum in the photon absorption process. In
other 2D systems, the necessary extra-momentum was
provided through a patterning of the surface with a sub-
wavelength grating [2, 15]. Here we have applied this
methodology to thin films of Bi2Se3.
terned in form of micro-ribbon arrays of different widths
W and periods 2W (Fig. 1-a,c). We could thus detect, in
the THz range, the optical absorption from collective os-
cillations of the electrons confined within the TI ribbons,
namely, the plasmons of the topological insulator. Their
2D character was confirmed by varying W and measur-
ing the corresponding peak frequency νP, as the plasmon
wavevector is related to W by k ? π/W [16, 17]. We
thus found the expected two-dimensional dispersion law
νP∝√k. Finally, we could assign the plasmon to mass-
less Dirac electrons by observing its robustness against
temperature changes from 6 to 300 K and by a compari-
son with theoretical calculations. Indeed, the experimen-
tal dispersion law was exactly reproduced by that of a 2D
Dirac plasmon [19, 20] without using any fitting parame-
ter, but just using the experimental values for the Fermi
However, they cannot
They were pat-
arXiv:1307.5974v1 [cond-mat.mes-hall] 23 Jul 2013
velocity vF, the 2D charge density nD, and the dielectric
constant ?, measured in Bi2Se3films grown in the same
experimental conditions [21, 22].
Six thin films of Bi2Se3were grown by Molecular Beam
Epitaxy on 0.5 mm thick sapphire substrates (Al2O3)
[21, 22]. Three out of them had a thickness d = 120 quin-
tuple layers (QL), where 1 QL ? 1 nm, the other three d
= 60 QL. Transport characterization through resistivity
and Hall measurements show that both Dirac electrons
generated by topology and bulk massive electrons due
to band-bending effect [21, 22] participate to the surface
conduction. One film was kept as grown for sake of com-
parison, the other five were patterned by electron-beam
lithography and reactive-ion etching in form of parallel
ribbons of widths W =2, 2.5, 4, 8, 20 µm, and periods
L = 2W. Therefore, the filling factor was 0.5 for all
patterned samples (Fig.1-a, c).
The transmittance T(ν) of the six films was measured
in the THz range by a Fourier-transform interferometer
from 6 to 300 K. The corresponding extinction coeffi-
cients E(ν) = 1 − T(ν) are reported in Fig 1-b for the
as-grown sample and for the patterned films (d, e), once
normalized by the respective peak values, both at 6 K
(blue lines) and 300 K (red lines). The as-grown film in
Fig. 1-b exhibits two peaks at the frequencies already re-
ported for Bi2Se3single crystals, namely, the α phonon
mode at 1.85 THz (61 cm−1) and the barely discernible β
phonon mode at 4.0 THz (132 cm−1)  which broaden
at 300 K. Phonon lines are superimposed to a Drude ab-
sorption, which was mainly attributed to Dirac surface
The extinction coefficients of the patterned samples
are reported in Fig 1-d for the radiation field paral-
lel to the ribbons. In this case the absorption, and
its T-dependence, are very similar to those of the non-
patterned film.This comparison also shows that the
patterning procedure did not affect at all the physical
properties of the samples (see also Table S1 of the Sup-
plementary Information). Moreover, in Fig. 1-d one may
remark that the phonon frequencies neither appreciably
change with W, nor with d.
Figure 1-e shows instead the extinction coefficient for
the radiation polarized perpendicularly to the ribbons.
As this direction is that of the reciprocal-lattice vec-
tors needed for the energy-momentum conservation in
the plasmonic absorption, it is the one where the plas-
mon can be observed. As one can see, either at 6 K and
at 300 K the α phonon is replaced by a double absorp-
tion, where both peak frequencies strongly depend on
W. We assign these features to the α phonon and to the
plasmon of Bi2Se3, mutually interacting via a Fano in-
terference. This produces a renormalization of both the
phonon and the plasmon frequency, with a hardening of
the mode at higher frequency and a softening of that at
lower frequency, independently of their nature. A similar
effect is reported in the literature for doped graphene on
SiO2. A Fano effect on the weak β phonon line is also
observed in the top panel of Fig. 1-e, as it becomes closer
to the plasmon resonance, through an inflection point in
E(ν) at the bare phonon frequency (magenta line).
In order to extract from the data in Fig. 1-e the bare
plasmon (νP) and phonon (νph) frequency we have fit the
experimental data to the following Equation, obtained by
Giannini et al. :
E(ν?) =(ν?+ q(ν?))2
The expressions for the renormalized frequency ν?which
depends on νphand for the (Fano) parameter q, that is
the ratio between the probability amplitude of exciting
a discrete state (phonon) and of exciting a continuum
or quasi-continuum state (plasmon), are reported in the
Methods section. ΓP is the plasmon linewidth and g is
the coupling factor of the radiation with the plasmon.
Equation 1 takes into account the frequency separation
between the plasmon and the phonon excitations which
here changes with W.
The above model satisfactorily reproduces the experi-
mental data as shown in Fig. 2 by the black line. The
green and red lines describe instead the bare phonon
and plasmon contributions, respectively, reconstructed in
terms of a Lorentzian shape. As one can see, the bare
phonon line does not change with W and its frequency is
the same as for the parallel polarization in Fig. 1-d, while
the plasmon softens and narrows as W increases. This ef-
fect is better shown in the inset of Fig. 2 (bottom panel)
where the plasmon linewidth ΓP is shown vs. W at 6 K.
ΓP can be assumed to be the sum of the following inde-
pendent contributions: i) the Drude linewidth obtained
by fitting the spectra of Fig. 1-d (see Table S1 in SI); ii)
the Landau damping rate, through both the creation of
hole-electron pairs, and a phonon-assisted process ;
iii) radiative decay into photons; iv) finite size effects
. Contribution i) is independent of W and provides
the background marked by the black dotted line in the
inset. The other effects are expected to increase with
increasing plasmon frequency (decreasing W) and there-
fore should be responsible for the behavior of ΓP shown
in the inset.
The bare plasmon frequencies obtained from the Fano
fits are plotted vs. the wavevector k in Fig. 3 and vs.
W−1/2in its inset. The additional point (green diamond)
refers to a seventh sample with W =1.8 µm, and period
L =4 µm (filling factor 0.45 or L = 2.2W), whose raw
data are reported in Fig. S1 of the SI. Incidentally, the
agreement with the dispersion curve of the green dia-
mond with L = 2.2W suggests that small changes in the
filling factor do not affect the plasmon absorption whose
characteristic frequency is mainly driven by W . As
shown in the inset, νP ∝ W−1/2, as expected for a 2D
plasmonic excitation. In the main panel of the Figure,
the same points are plotted vs. k, based on the relation
k ? π/W. Here, the approximation accounts both for
a possible depletion in the electron density at the rib-
bon edges as observed in graphene  and for scattering
from the edges . As both phenomena affect only a
range of tens of nanometers, smaller than our W by two
orders of magnitude, the corresponding correction can
be neglected. The effective value of k may be also influ-
enced by the excitation of edge modes [16, 17] as those
predicted in graphene, but those results cannot be easily
transferred to the present material where, for example,
the electron mobility is lower by three orders of magni-
tude than that considered in those calculations. In any
case, by assuming k = π/W we find that our points are
in good agreement with the dispersion law νP∝√k pre-
dicted by the Equations discussed below and reported by
the black dashed line in Fig. 3.
Once established that the observed plasmon is that
of the 2D electron gas at the TI surface, one won-
ders whether it should be ascribed mainly to the Dirac
fermions or to the massive electrons. As we discuss in
the following, both qualitative and quantitative argu-
ments support the former assignment. The qualitative
argument starts from the observation that the plasmon
absorption linewidths in Fig. 1-e are very similar at 6
K and 300 K (see also the Fano fit parameters in Ta-
bles S2 and S3 of the SI). This shows that the plasmon
absorption is remarkably robust against a variation in
temperature by a factor of 50. This would be hard to ex-
plain if a major contribution to the absorption were due
to collective excitations of conventional electrons. Due
to the momentum-spin lockage, the Dirac quasiparticles
in TI’s are instead virtually unaffected by the scatter-
ing mechanisms - except for those with magnetic impu-
rities (here absent) - and therefore by their temperature
dependence. Indeed, a weak broadening of TI’s states
with temperature has been observed in Bi2Se3by Angle-
Resolved Photoemission Spectroscopy , consistently
with the present results, even if the electron-phonon in-
teraction is reported to be strong . This suggests that
the TIs Dirac electrons are protected not only from im-
purity scattering but also from scattering by phonons.
One may therefore speculate that the weak temperature
dependence of the plasmon linewidth is related to the
topological protection of the electrons. The infrared ab-
sorption of ”massive” plasmons in 2D electron gases was
reported only below rather low temperatures  because
the electron-phonon scattering is their main decay chan-
nel [29, 30]. Those temperatures, moreover, decrease as
the leading longitudinal optical phonon frequency ?ωLO
decreases: they are close to that of liquid nitrogen in
GaN heterostructures with ?ωLO= 92 meV , to that
of liquid helium in GaAs heterostructures with ?ωLO=
36 meV . As in Bi2Se3?ωLO∼ 19 meV , therein
we should expect the linewidth of 2D massive plasmons
being so large that their absorption is confused in the
background, even at the lowest T (6 K). A robustness vs.
temperature of the plasmonic excitations was found also
in graphene , where Dirac-plasmon absorption peaks
were observed at 300 K. Therein, however, one has a
much higher ?ωLO= 137 meV.
The quantitative argument for the assignment of the
plasmon peaks in Fig. 2 to Dirac quasiparticles proceeds
as follows. In TI thin films, the 2D free-electron layers at
the TI-substrate and at the TI-vacuum interfaces inter-
act through an effective Coulomb potential [19, 20]. This
mechanism, if one neglects interlayer tunneling, leads to
the appearance of two longitudinal collective excitations,
i.e.an optical plasmon (with νP ∝
acoustic plasmon (with νP ∝ k) [19, 20]. The acous-
tic mode has been estimated  to be degenerate with
the continuum, i. e., to be strongly Landau-damped 
and, then, unobservable. Therefore, the THz spectrum
of TI’s should be characterized by a single optical mode
to which, in principle, would participate both massive
and Dirac fermions. In the long wavelength limit k → 0,
the dispersion laws of Dirac and massive plasmons can
be written, respectively [19, 20],
√k), and an
where the spin (gs) and the valley (gv) degeneracies
are both equal to 1 .
films grown in the same conditions as the present ones,
nM = 7.5 ± 3.5 × 1012cm−2, nD= 3 ± 1 × 1013cm−2,
vF = 6 ± 1 × 107cm/s , and the effective mass of
the parabolic bulk band is m∗= 0.15±0.01 me[21, 22].
Neither Eq. 2 nor Eq. 3 contain the film thickness or
the bulk dielectric function. They instead depend on the
average ?=(?1+?2)/2 between the dielectric function of a
vacuum (?1= 1) and that of the substrate (?2∼ 10). The
theoretical Dirac and massive plasmon dispersions were
calculated by Eqs. 2 and 3 at the selected wavevectors
k = π/W by using for nD, vF(Dirac), nMand m∗(mas-
sive) their central experimental values reported above.
The results are compared with the experimental data re-
ported by dots in Fig. 3. Therein, the dashed black curve
is the dispersion predicted for a Dirac plasmon in Bi2Se3,
the dotted blue line that of massive particles. The much
better agreement with data of the Dirac plasmon disper-
sion with respect to the ”massive” one, obtained without
using free parameters, strongly supports the above qual-
itative argument, namely that the plasmonic excitations
Moreover, in Bi2Se3 thin
observed in this experiment must be ascribed to Dirac
In conclusion, we have reported the first observation
of Dirac plasmon resonances in thin films of Bi2Se3. We
have shown, based on both qualitative arguments and
the comparison with theoretical calculations, that those
plasmons should be assigned to Dirac quasi-particles at
the metallic surface of the topological insulator.
strong Fano-like interference between the Dirac plasmons
and the phonons of Bi2Se3, here observed, opens promis-
ing perspectives. Indeed, similar interference effects ob-
served in conventional plasmonic systems between bright
(dipole) modes and dark (quadrupole) modes  were
used to increase the quality factor of resonances. Simi-
lar opportunities can be exploited in TI’s, for example,
to implement novel sensors in the terahertz range. Fi-
nally, the plasmon tunability, here engineered through
microribbon arrays of different width W and period 2W,
can be generalized to produce more complex plasmonic
designs in these intriguing topological materials.
T = 6 K
W = 2 µm
νP (k) (THz)
FIG 1: Extinction coefficients of the micro ribbon arrays of Bi2Se3 topological insulators in the
terahertz range. a. Scanning Electron Microscope (SEM) image of the W=2.5 µm patterned film. b. Extinction
coefficient of the as-grown, unpatterned film, at 6 K (blue lines) and 300 K (red lines). c. Optical-microscope
images of the five patterned films with different widths W and periods 2W; the red arrows indicate the direction
of radiation polarization, either perpendicular or parallel to the ribbons.
the images. d. Extinction coefficient at 6 K (blue lines) and 300 K (red lines) of the five patterned films, with
the radiation field parallel to the ribbons. e. Extinction coefficient of the five patterned films, with the radiation
polarized perpendicularly to the ribbons, at 6 K (blue lines) and 300 K (red lines). All data are normalized by their
respective peak values.
The film thickness is reported under
FIG 2: Search for the bare plasmon frequencies. Normalized extinction coefficient of the five patterned
films, with the radiation polarized perpendicularly to the ribbons, at 6 K (circles) and fits to Eq. 1 (black lines).
The bare plasmon and phonon contributions, extracted through the fits, are reported by the red and the green line,
respectively. The inset in the bottom panel displays the plasmon linewidth ΓP vs. the ribbon width W at 6 K. The
dotted line is the Drude contribution extracted from data with polarization parallel to the ribbons, the dashed line
is a guide to the eye. Both in the top and bottom panel the bump at 2.6 THz is instrumental and due to a bad
compensation of the Mylar beamsplitter absorption.
FIG 3: Experimental and theoretical dispersion of plasmons in Bi2Se3. Inset: linear dependence of νP on
W−1/2, where W = π/k is the ribbon width. Main panel: experimental values at 6 K (blue full circles) compared
with the plasmon dispersion for Dirac (dashed black line) and massive electrons (dotted blue line) calculated with
no fitting parameters by Eq. 2 and Eq. 3, respectively. The green diamond refers to a sample with W =1.8 µm and
period L = 2.2W, see text.
The high quality Bi2Se3 thin-films were prepared by
MBE using the standard two-step growth method devel-
oped at Rutgers University [21, 22]. The 10 × 10 mm2
Al2O3substrates were first cleaned by heating to 750oC
in an oxygen environment to remove organic surface con-
tamination. The substrates were then cooled to 110oC,
where an initial 3 QL of Bi2Se3was deposited. This was
followed by heating to 220oC, where the remainder of the
film was deposited to attain the target thickness. The Se
and Bi flux ratio was kept to be approximately Se/Bi
10/1, to minimize Se vacancies.
cooled, they were removed from the vacuum chamber,
and vacuum-sealed in plastic bags within two minutes,
and shipped to the University of Rome.
Bi2Se3 ribbons were fabricated by electron-beam
lithography (EBL) and subsequent Reactive Ion Etching
(RIE). The Bi2Se3 film was spin-coated with a double
layer of electron-sensitive resist polymer PMMA (Poly-
(methyl methacrylate)) up to a total thickness of 1.4 mi-
crons. The ribbon pattern with different W was then
written in the resist by EBL. In order to obtain a litho-
graphic pattern with re-alignment precision below 10 nm
over a sample area suitable for Terahertz spectroscopy of
10x10 mm2, we used an electron beam writer equipped
with a XY interferometric stage (Vistec EBPG 5000).
The patterned resist served as mask for the removal of
Bi2Se3by RIE at low microwave power of 45 W to pre-
vent heating of the resist mask. Sulfur hexafluoride (SF6)
was used as the active reagent.
etched at a rate of 20 nm/min, which was verified by
Atomic Force Microscopy (AFM) after soaking the sam-
ple in acetone to remove PMMA. The in-plane edge qual-
ity after the RIE process, as inspected by AFM, closely
follows that of the resist polymer mask, i.e. edge rough-
ness smaller than 20 nm. The vertical profile of the edge
forms an angle of about 45 degrees with the substrate
plane, because our RIE process has no preferred etching
The absorption spectra in the Terahertz range were
obtained by using a Bruker IFS-66V Michelson interfer-
ometer and a liquid-He cooled bolometer. The Bi2Se3
film and a co-planar Al2O3bare substrate were mounted
on the cold finger of a He-flow cryostat and kept at a
pressure of about 10−6mbar. The radiation was polar-
ized either along, or perpendicular to, the ribbons by a
THz polarizer having a degree of polarization > 99.5 %.
The extinction coefficient reported in Fig. 2 was obtained
from the film transmittance T, defined as the ratio be-
Once the films were
The Bi2Se3 film was
tween the intensity transmitted by the thin film and that
transmitted by the bare substrate.
The Fano fits were obtained by replacing in Eq. 1 
ν − νph
Γph(ν)/2−ν − νP
the plasmon-coupled phonon linewidth
and the Fano factor
Γph(ν)/2+ν − νP
Therein, w and g are the coupling factors of the radiation
with the phonon and the plasmon, respectively, v mea-
sures the phonon-plasmon Fano interaction, and ΓP is
the width of the plasmon line, assumed to be Lorentzian.
We thank M. Polini for fruithful discussions about
Dirac and massive plasmonic dispersions.
M. B., N. B., N. K. and S. O. fabricated and character-
ized Bi2Se3films. M.O. and A. D.G. and V.G. performed
EBL lithography and etching. P.D.P, F. G., O.L. and M.
O. carried out the terahertz experiments and data anal-
ysis. P.C., M. O., and S. L. were responsible for the
planning and the management of the project with inputs
from all the co-authors, especially from P.D.P., F. G. S.
O. and O.L. All authors extensively discussed the results
and the manuscript that was written by P.C., M.O. and
The authors declare no competing financial interests.
Correspondence and requests for materials should be ad-
dressed to S.L. (email@example.com)
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