Accepted Manuscript Emergence of a nonlinear plasmon in the electronic response of doped graphene Antonio Politano, Gennaro Chiarello PII: S0008-6223(14)00062-1 DOI: http://dx.doi.org/10.1016/j.carbon.2014.01.026 Reference: CARBON 8704 To appear in: Carbon Received Date: 23 September 2013 Accepted Date: 17 January 2014 Please cite this article as: Politano, A., Chiarello, G., Emergence of a nonlinear plasmon in the electronic response of doped graphene, Carbon (2014), doi: http://dx.doi.org/10.1016/j.carbon.2014.01.026 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Transcript
Accepted Manuscript
Emergence of a nonlinear plasmon in the electronic response of doped graphene
Received Date: 23 September 2013Accepted Date: 17 January 2014
Please cite this article as: Politano, A., Chiarello, G., Emergence of a nonlinear plasmon in the electronic responseof doped graphene, Carbon (2014), doi: http://dx.doi.org/10.1016/j.carbon.2014.01.026
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customerswe are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, andreview of the resulting proof before it is published in its final form. Please note that during the production processerrors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Around the K point of the Brillouin zone, the π bands of freestanding graphene have a linear disper-
sion, which is the solution of the relativistic Dirac equation for massless particles [1], so as to form
the so-called Dirac cones. However, the chiral nature of charge carriers in the two equivalent sublat-
tices of graphene [2] induces deviations from a perfect cone. The cone becomes squeezed along the
K–K’ directions [1, 3] (trigonal warping).
Such anisotropy may be tailored for engineering novel technological applications [4, 5] in the field
of nano-devices.
While some theoretical attempts are existing [6, 7], the effects of trigonal warping on plasmon
modes remain completely unexplored by experimentalists.
On the other hand, the comprehension of collective electronic excitations of graphene plays a
key role for engineering many technological applications in electronics, photonics, nano-
medicine etc. [8]. In particular, acoustic plasmons in doped graphene are attracting much interest
in recent years from both experimentalists [9-11] and theoreticians [6, 12, 13] for both fundamental
interest and possible technological applications. In fact, the usual square-root-like dispersion
of plasmons in the two-dimensional electron gas (2DEG) makes a distortionless propa-
gation of nonmonochromatic signals inherently impossible, since the different frequencies compo-
nents propagate at different velocities. This problem can be overcome in the case of materials exhi-
biting a plasmon mode with a linear dispersion (acoustic plasmon) [14].
Acoustic plasmons have been predicted [15-18] and experimentally revealed in many other systems
[14, 19-21], whenever a 2DEG spatially coexist with the underlying 3D electron gas.
Among graphene systems, the epitaxial growth of monolayer graphene (MLG) on Pt(111) is par-
ticularly interesting [22-28] as a consequence of the weak graphene-Pt interaction [23], in contrast
with MLG grown on other transition-metal substrates [29-31]. In fact, the graphene-Pt distance
(3.30 Å) lies close to the c-axis spacing in graphite. The electronic structure of MLG on Pt(111) re-
sembles that of isolated graphene [22]. In particular, the linear dispersion of π bands, which gives
rise to many manifestations of massless Dirac fermions, is preserved.
Angle-resolved photoemission spectroscopy (ARPES) experiments [22] do no show any remarkable
hybridization of graphene π states with metal d states. They just represent a superposition of gra-
phene and metal-derived states, with minimal interaction between them. The MLG is hole-doped by
charge transfer from the Pt substrate (Fermi level below Dirac point) [24]. The Fermi energy of the
graphene layer shifts 0.30±0.15 eV below the Dirac-energy crossing point of the π bands. Thus,
MLG/Pt(111) represents an ideal playground for studying plasmon modes in doped free-standing
graphene.
Herein, we study the experimental plasmon spectrum of self-doped graphene. In the long wave-
length limit and for small energies, a gap appears in the dispersion relation of plasmon modes. Two
well-distinct plasmon modes can be resolved. Apart from the usual acoustic plasmon, also a nonli-
near plasmon is observed. Its existence is a consequence of the anisotropy of the π bands of gra-
phene[6]. While the acoustic plasmon represents the in-phase oscillation of the two type of elec-
trons in the anisotropic π bands, the low-energy nonlinear mode is an out-phase oscillation.
Experiments were carried out in a ultra-high vacuum (UHV) chamber operating at a base pressure of
5·10-9 Pa. The sample was a single crystal of Pt(111). The substrate was cleaned by repeated cycles of
ion sputtering and annealing at 1300 K. Surface cleanliness and order were checked using Auger elec-
tron spectroscopy (AES) and low-energy electron diffraction (LEED) measurements, respectively.
Graphene was obtained by dosing ethylene onto the clean Pt(111) substrate held at 1150 K. The
high temperature of the sample during depositions favours the increase of the size of monolayer
graphene islands [32] and, moreover, allows maintaining the substrate clean so as to avoid any con-
taminant-induced effect on graphene growth. Monolayer graphene has been reached upon an expo-
sure of 3·10-8 mbar for ten minutes (24 L. 1 L=1.33·10-6 mbar·s).
)sinE
E1(sin
mE2q S
p
lossi
pθ−−θ=
�
In these conditions, Raman spectroscopy unambiguously indicates the formation of a single layer of
graphene, as we reported elsewhere [33].
The characterization of the MLG has been carried out by identifying phonon modes which are fin-
gerprints of graphene formation [27, 34-36].
HREELS experiments were performed by using an electron energy loss spectrometer (Delta 0.5,
SPECS). The energy resolution of the spectrometer was degraded to 5 meV so as to increase the
signal-to-noise ratio of loss peaks. Dispersion of the loss peaks, i.e., Eloss(q||), was measured by
moving the analyzer while keeping the sample and the monochromator in a fixed position. To
measure the dispersion relation, values for the incident energy Ep and incident angle θi were chosen
were chosen so as to obtain the highest signal-to-noise ratio. The primary beam energies used for
the dispersion, Ep=7-12 eV, provided, in fact, the best compromise among surface sensitivity, the
highest cross-section for mode excitation and q|| resolution [37].
As
the parallel momentum transfer, q|| depends on Ep, Eloss, θi and θs according to:
where Eloss is the energy loss and θs is the electron scattering angle [38].
Accordingly, the integration window in reciprocal space [39] is
where α is the angular acceptance of the apparatus (±0.5° in our case). To obtain the energies of loss
peaks, a polynomial background was subtracted from each spectrum. The resulting spectra were fit-
ted by a Gaussian line shape (not shown herein).
All measurements were made at room temperature.
Figure 1 shows the evolution of HREELS spectra as a function of the primary electron beam en-
ergy Ep for a fixed scattering geometry. Loss spectra show a remarkable dependence on Ep. In par-
ticular, for primary energies below 6 eV, only a peak around 0.5 eV is observed. At Ep=8 eV two
well-distinct plasmon modes may be spectroscopically resolved. The second plasmon mode around
1.65 eV increases its spectral weight so as to be predominant in the spectrum recorded for Ep=16
eV. Together with these peaks, also optical phonons of graphene at 0.1 and 0.2 eV are clearly ob-
served [34, 35].
This striking dependence on the primary energy explains why previous experimental studies
were not able to resolve the two plasmon modes [10, 11].
Figure 1 Dependence of the HREELS spectrum of MLG/Pt(111) on the primary electron beam
energy. Spectra have been recorded in the same scattering geometry (incidence angle and
scattering angle , respectively, measured with respect to the sample normal). In the top
axis, the energy scale is reported in THz.
The HREELS spectra for Ep=7 eV, recorded as a function of the scattering angle, are reported in
Figure 2. The two plasmon modes develop with the scattering angle. The higher-energy mode
emerges in off-specular geometries, for values of lower than 50° with respect to the sample
normal. This mode is characterized by a linear dispersion (acoustic plasmon). By contrast, the low-
energy mode has a nonlinear dispersion relation (Figure 3). Its intensity reaches a maximum and
then decreases for higher momenta.
The existence of a trigonal warping in graphene could imply the existence of two plasmon
modes: one plasmon mode in which these two types of electrons oscillate in phase and another one
of lower frequency in which the two types of electrons oscillate out-of-phase. The former mode is
characterized by a linear dispersion (acoustic plasmon), while the latter is the nonlinear mode.
Figure 2 HREELS spectra for MLG/Pt(111), recorded as a function of the scattering angle. The
primary energy is 7 eV, while the incidence angle is fixed at 55° with respect to the sample normal.
The dispersion relation of the plasma frequency of the acoustic plasmonASPω recorded in our ex-
periments (Figure 3) is well described by a linear relationship between the energy the momentum:
qAASP =ω� ||
where A=(7.2+0.5) eV·Å
The group velocity of the sheet plasmon in MLG/Pt(111) (≈106 m/s) is similar to the Fermi velocity
in graphene. The linear behavior of its dispersion implies that both phase and group velocities of the
collective excitation are the same, so signals can be transmitted undistorted along the surface.
Instead, the dispersion curve of the nonlinear plasmon is well fitted by a power law:
n
nonlinear qB ||=ω�
with B=(0.66+0.02) eV·Ån and n= (0.12+0.01)
Figure 3 Plasmon dispersion in MLG/Pt(111). The size of the markers reflects the intensity of the
plasmon modes, normalized to that of the elastic peak. Markers are coloured following a colour
scale reported in the right. The dotted line represents an interpolation of the nonlinear mode for
small momenta, for which the intensity is vanishing.
The nonlinear plasmon has been firstly predicted by Gao et al. [6] for doped free-standing graphene.
It should be noticed that by using an oversimplified isotropic graphene band structure in the near-
ness of the K point, it cannot be observed in theoretical works [40].
Its nonlinear dispersion, together with its high spectral weight and its lower energy makes it a natu-
ral candidate for applications in graphene-based nonlinear plasmonics [41, 42].
It should be reminded that in undoped graphene, both acoustic plasmon and the nonlinear mode dis-
appear [6, 43]. In the case of MLG/Pt(111), the p-type doping of the graphene sheet occurring in
this interface is responsible of the existence of the above-mentioned two plasmon modes.
We also note that when the acoustic plasmon coexists in the HREELS spectrum with the nonlinear
mode, its intensity in the range 0.6-1.0 eV is particularly weak (as highlighted by the size of mark-
ers in Figure 3) with respect to the case of impinging energies above 12 eV, for which only the
acoustic plasmon is recorded. Thus, the coexistence of the two modes induces a gap in the disper-
sion relation when considering the intensity of the plasmon modes.
Finally, in order to investigate whether the nonlinear mode depends on the graphene/metal inter-
face, we have also studied its dispersion for MLG/Ru(0001) (Figure 4). We find that two plasmon
modes exist also in this case, even if the dispersion of the nonlinear mode is slightly different for
the peculiar properties of this periodically rippled graphene/metal interface (see Refs. [29, 44, 45]
for more details and Figure S2 of the Supplementary Information for a comparison). Note that at 0.2
Å-1 the energy of the nonlinear plasmon is 1.2 eV in MLG/Ru(0001), while it is about 0.5 eV in
MLG/Pt(111).
We suggest that the nonlinear plasmon in MLG/Ru(0001) is ascribed to hills, for which the Dirac
cone survives (see calculations in the Supplementary Information).
Figure 4 Behavior of the nonlinear mode as a function of the parallel momentum transfer q|| for
MLG/Ru(0001). The intensity of the plasmon modes, normalized to that of the elastic peak, is re-
flected by the size of the markers, which are coloured following a colour scale reported in the right.
The solid line represents the best fit. In the inset, a selected loss spectrum for MLG/Ru(0001) has
been reported. Two distinct loss features are present.
In conclusion, we have reported direct evidence of the existence of two well-distinct plasmon
modes in the electronic response of doped graphene. The mode at higher energies is the usual
acoustic plasmon, while a nonlinear mode is observed at lower energies. The emergence of a
nonlinear mode is a direct consequence of the trigonal warping in graphene, that is the anisotropy of
π bands of graphene in the nearness of the K point. The acoustic plasmon represents an in-phase os-
cillation of electrons in π bands, while the nonlinear mode is an out-of-phase oscillation of π elec-
trons. These findings are particularly relevant in the field of graphene-based plasmonics, which has
a significant potential for technological applications (see for example Refs. [46-50]).
Acknowledments
We thank dr. Silkin for many helpful discussions. AP is also grateful to dr. Silkin’s institution
(DIPC) for hospitality during various stages of this research.
Furthermore, we also thank dr. Polini.
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