Effective Fermi Level Modulation of Coupled Graphene-
Metal Surface Plasmon Polaritons at Sub-THz Frequency
Range
Hameda Alkorre 1*, Gennady Shkerdin
2, Johan Stiens
1, Youssef Trabelsi
3, Roger Vounckx
1
1Department of Electronics and Informatics (ETRO), Laboratory for micro- and photon electronics (LAMI),Vrije
Universiteit Brussel (VUB), Pleinlaan 2, B-1050 - Brussels – Belgium. 2Institute of Radio Engineering and Electronics RAS-Russian.
3Faculty of Science of Monastir, Department of physics, 5019-Monastir-Tunisia.
*Phone: +32 2 629 10 24; Fax: +32 2 629 2883 ; E-mail: [email protected]
Abstract- In this paper we investigate how one can
take additional advantage of coupled
metal-graphene plasmons. The dispersion relation
for coupled metal-graphene plasmons is presented
here for a proposed multilayer structure.
Graphene and couple metal graphene modes
supported by this structure are shown below, these
modes can depend very strongly on Fermi energy
level values in the graphene layer at the sub-THz
frequency range, which eventually can be used in a
wide range of applications.
Index Terms- coupled metal graphene, multilayer
structure, sub-THz, plasmon.
I. INTRODUCTION
Discovered only a few years ago, a single layer
of carbon atoms with honeycomb structure and
its derivative have attracted intense attention in
the fields of chemical, physical and biological
sciences. These emerging nanostructures exhibit
unique electrical, optical, thermal, and
mechanical properties, due to its unique conical
and symmetric band structure. The atomically-
thin sheets could be potentially assembled by the
existing thin-film techniques. However, to date
there are only a few studies of graphene – based
devices in the sub-THz frequency range.
Graphene behaves as an essentially 2D electronic
system. In the absence of doping, conduction and
valence bands meet at a point – the so – called
Dirac point-which is also the position of the
Fermi energy. The band structure, calculated in
the tight binding approximation is shown in
Fig.1, for low energies the dispersion around the
Dirac point can be expressed as:
(1)
Where the Fermi velocity is ,
and n =1 for conduction band, and n = −1 for
the valence band [1]. Recent publications show
that the conductivity of graphene to be Drude-
like in the THz regime, indicating the dominance
of intraband conductivity over interband at low
frequencies [2]. In recent years, an enormous
interest has been surrounding the field of
plasmonics, because of the variety of
tremendously exciting and novel phenomena it
could enable.
The surface plasmon polariton is the
electromagnetic excitation which propagates
along the interface between conductor and
dielectric. The wave is evanescently confined in
the perpendicular direction.
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Fig.1. The band structure of a representative three-
dimensional solid (left) is parabolic, and The energy
bands of two- dimensional graphene (right) are
smooth-sided cones, which meet at the Dirac point,
figure adapted from [1].
This effect occurs at different frequencies than
the bulk plasma oscillations [3].
The theory of graphene plasma was developed by
several researchers immediately after the
discovery of graphene. On the other hand, only a
few experimental results of graphene plasmons
are available in the literature. This theory was
adapted from that of metals, as there are many
similarities, but also some differences. For
instance in metals, parameters such as the
conductivity, charge density, wave number and
confinement are fixed, whereas they can be tuned
in graphene by applied electrical field or optical
stimulation or chemical doping. Also, in
graphene – the plasmon mass depends on the
electron concentrations, because of the linear
band structure. Furthermore, the material
parameters of graphene like excitation
wavelengths of interest range from the
microwave to the mid – infrared regimes. Lately
this drew lots of interest for nanophotonics [4]. In
contrast, plasmons in metal structures are mainly
in the near-infrared and visible regions because
of the optical properties of the noble metals used
[5].
Therefore, the fact that plasmons in graphene
could have low losses for certain frequencies
since the losses can be controlled by varying the
electron concentrations [6] and flexibility [7]
make them potentially interesting for
nanophotonic applications.
To study the coupling between electromagnetic
waves and the metal-graphene system, Maxwell’s
equations were solved in our previous study to
achieve an expression for the dispersion relation
of the multilayer structure [8].
The coupling of graphene plasmons with surface
metal plasmons in a structure containing a
graphene layer and metal substrate separated by
air gap was studied in [9]: the solution of the
coupled plasmon mode shows a linear dispersion
behavior in a specific parameter range.
This work has been divided into Three parts. In
the first part we describe our model of our three-
layer waveguide with graphene using Maxwell’s
equations. Next we discuss the dependence of the
dispersion relation on the electron concentration
in the graphene layer at the sub-THz frequency
range. In the last section we draw some
conclusions.
II. THE MODEL OF COUPLED GRAPHENE-
METAL SURFACE PLASMONS STRUCTURE
In our work, we focused on the alteration of the
following structure: Air/graphene/buffer
layer/metal Fig.2.
Fig.2. A multi-layer structure under consideration,
which consists of Air/graphene/buffer
layer(SiO2)/metal (gold).
In our study we have taken Au (gold) as metal,
and we selected SiO2 (silicon dioxide) as
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substrate because it has a minimum absorption,
since the imaginary part of its refractive index is
quite small in our interested frequency range.
Here, we consider TM modes in the geometry
depicted in Fig.2. It is assumed that modes
propagate along the Z-axis where electric field
and magnetic field, , and Ex=0,
Hy= Hz=0 [3].
The dispersion equation is given as follows:
3 2 2 3 1 2 2 1 (2)
Where for j=1, 2, 3. Where
are the dielectric permittivity of air,
SiO2, gold, respectively,
is the
electromagnetic wavevector in vacuum, c is the
velocity of light, and is the electromagnetic
wave angular frequency,
.
Where σ is the graphene surface conductivity. In
general the graphene conductivity is defined as:
(3)
Where intra(), and inter
() are the
contributions of intraband and interband
transitions respectively [10].See Appendix for
more details.
As we mentioned above in this frequency range
the graphene conductivity is dominated by
intraband contribution which is given by:
(4)
Where e is the electron charge, , kB is
Boltzmann constant and t is temperature in
Kelvin. At room temperature and low Fermi
energy level (4) leads to the semi-classical model
[11], [12].
(5)
In this work we consider the latter equation, for
the conductivity calculation of graphene.
Where EF is the Fermi energy level of graphene
and is electron momentum relaxation time,
electron surface concentration ns is connected to
Fermi energy by:
(6)
And
(7)
Where is the mobility; we use in our simulation [5], [13].
The Fermi energy level value can easily connect
into an electron concentration value by means of
(6). The relation between graphene electron
concentration and Fermi energy plotted in Fig.3.
Fig.3. The electron concentration on the graphene
versus Fermi energy level.
From the general four-layer dispersion relation,
various three-layer and two-layer dispersion
relations can be described. The solutions of these
less complicated structures give insights for the
understanding of the dispersion relation of the
full four-layer structure [14].
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From (2) the dispersion of a three-layer structure
containing air, graphene, and an unlimited buffer
layer is given by:
(8)
This equation is described by the 4th power
equation for the q2 value and its general solutions
are too cumbersome to write them down.
However, the solution is considerably simplified
for the case when the solution of
the dispersion relation for graphene plasmon-
polaritons [9] can be written down as follows:
(8-a)
Metal plasmons in a two-layer structure –
containing metal and unlimited buffer layer for
are described as:
(9)
The solution of this equation leads to the well-
known dispersion relation for surface metal
plasmon (see, for example, [15]).
Metal plasmons in a two-layer structure –
containing metal and air for are described
as:
(10)
The solution of this equation also given in[15].
This structure exhibits different modes; these are
classified with respect to their magnetic field
profile inside the structure as antisymmetric and
symmetric modes. The antisymmetric mode is
called short range surface plasmons polariton,
and features greater confinement and higher
propagation losses, while the symmetric mode is
called long range SPP, since it exhibits lower
confinement and greater propagation distance
and this mode is our interests for possible
applications.
III. NUMERICAL RESULTS AND DISCUSSIONS
Numerical calculations were performed for the
structure shown in Fig.2. Parameters for the
calculations were taken from the Ref.s [12], [16],
and [17].
The solution of the dispersion relation of
decoupled metal plasmon for a 2-layer structure:
SiO2 /gold, and graphene polariton for a 3-layer
structure: air /graphene / SiO2 at Fermi energy
EF=0.5 eV versus polariton frequency are plotted
in Fig.4.
The figure shows that the difference of
wavevectors of decoupled metal and graphene
polaritons decreases for sub-THz frequency
range. Therefore the stronger coupling effects in
this range of frequencies are expected.
Fig.5(a), and Fig.5(b) show the solution of the
dispersion relation for the 3–layer structure
air/graphene/SiO2 (8) versus frequency. It is clear
that, these graphene modes are modulated by the
Fermi level values of the graphene layer, but
these modes are short range, since the imaginary
part of the wavevector is big.
Fig.4. The solution of the dispersion relation of
decoupled metal and graphene plasmon for Fermi
Energy EF=0.5 eV; exact and approximate dispersion
dependences for graphene plasmon, and, metal
plasmon dispersion for wide buffer layer thickness.
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Fig.5(a). The solution of the dispersion relation of
air/graphene/SiO2 structure at different Fermi energy
levels EF, real part.
Fig.5(b). The solution of the dispersion relation of
air/graphene/SiO2 structure at different Fermi energy
levels EF, imaginary part.
On the other hand, decoupled – graphene
metal plasmon modes that are shown in
(Figs. 6(a),(b) & 7(a),(b)) are long range
modes, the variation of the imaginary part of
the wavevector for different Fermi energy
level values is extremely sensitive to the
variation of Fermi energy levels in the
frequency range below 2.5 THz, and we
noted a remarkable oscillation with change
EF.
Nevertheless, the variation of the real part of
wavevector is only remarked for high Fermi
energy levels: at exactly EF=0.64 eV in this
simulation, we find a considerable variation
of about 500 cm-1
for 1 THz.
Fig.6(a). The solution of the dispersion relation of
four-layer structure at different Fermi energy levels EF
For d2 (SiO2 thickness)=50 µm, real part.
Fig.6(b). The solution of the dispersion relation of
four-layer structure at different Fermi energy levels EF
For d2 (SiO2 thickness)=50 µm, imaginary part.
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Fig.7(a). The solution of the dispersion relation of
four-layer structure at different Fermi energy levels EF
for d2 (SiO2 thickness)=140 µm, real part.
Fig.7(b). The solution of the dispersion relation of
four-layer structure at different Fermi energy levels EF
for d2 (SiO2 thickness)=140 µm, imaginary part.
VI. CONCLUSION
Various optimal structures with high
tuneability were presented for the sub-THz
frequency range. The dispersion relation and
the solution for these structures were
achieved. It was shown that a graphene sheet
can have an important impact on the structure
under consideration characteristics, so the
coupled graphene-metal plasmon modes
depend very strongly on the changing of
Fermi Energy level in the graphene layer,
and this gives advantages to use in a wide
range of applications in sub-THz frequency
range for example as sensors and modulators.
APPENDIX
The electromagnetic wave field’s tangential
components of the structure are described by the
equations below:
For air layer ( )
;
(A-1)
For buffer layer ( )
Where
(A-2)
And
(A-3)
For metal layer (
,
(A-4)
The boundary conditions for electromagnetic
wave field components are:
0 (A-5)
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The Solution of these boundary condition
equations leads to the dispersion (2) in the main
text.
ACKNOWLEDGMENT
The authors acknowledge Vrije Universiteit Brussel (VUB) through the SRP-project M3D2, the European Science Foundation (ESF, NEWFOCUS), and the Libyan Ministry of Higher Education.
REFERENCES
[1] G. Filippo, S. Sushant, and R. Vito, Electronic
Properties of Graphene Probed at the Nanoscale,
Physics and Applications of Graphene –
Experiments, Dr. Sergey Mikhailov (Ed.), ISBN:
978-953-307-217-3, 2011.
[2] C. J. Docherty, M. B. Johnston, “Terahertz
properties of graphene”, J. Infrared Milli,
Terahertz Waves 33: 797–815, Jun 2012.
[3] A. S. Maier, Plasmonics: Fundamentals and
Applications, Springer, 2007.
[4] M. Jablan, H. Buljan, and M. Soljačić,
“Plasmonics in graphene at infrared frequencies”,
Phys. Rev. B 80, 245435, December 2009.
[5] P. David and X. Fengnian, “ Graphene versus
metal plasmons”, Nature Photonic:7, 420, April
2013.
[6] K. V. Sreekanth, and Y. Ting, “Long range
surface plasmons in a symmetric graphene system
with anisotropic dielectrics”, J. Opt. 15, February
2013.
[7] H. Alkorre, G. Shkerdin, C. De Tandt, R.Vounckx
and J. Stiens, “Tunable coupled graphene-metal
plasmons in multi-layer structures at GHz and
THz frequencies” , ImagineNano, conference,
Bilbao: Spain, April 2013.
[8] L. W. Bing, W. Zhu, J. Xu Hong, H. N. Zhen , G.
D. Zheng, and T. Jun, “Flexible transformation
plasmonics using graphene”, Cui-Optics Express,
Vol. 21, Issue 9, pp. 10475-10482, May 2013.
[9] J. Norman H. Morgenstern, “Coupling of
graphene and surface plasmons, Phys. Rev. B, 80,
193401, 2009.
[10] L. A. Falkovsky, “Optical properties of graphene
and IV-VI semiconductors”, Phys. Usp. 51, 887-
897, March 2008.
[11] S. A. Mikhailov, “Non-linear graphene optics for
terahertz Applications”, Microelectron. J. 40, 712,
2009.
[12] G.W. Hanson, “Quasi-transverse electromagnetic
modes supported by a graphene parallelplate
waveguide”, J. Appl. Phys. 104, 084314, October
2008.
[13] R. Kitamura, L. Pilon, and M. Jonasz, “Optical
constants of silica glass from extreme ultraviolet
to far infrared at near room temperature”, Applied
Optics, Vol. 46(33), pp. 8118-8133, November
2007.
[14] J. Stiens, H. Alkorre, G. Shkerdin and R.
Vouncks, “The lower and sub-THz frequency
range: the optimal spectrum for coupled graphene-
metal like plasmon polaritons”, Optical and
Quantum Electronics, issue special edition, 2014.
[15] L. Landau, E. Lifshitz, Electrodynamics of
Pergamon and continuous media , London:Press,
1960.
[16] T. J. Parker, K. A. Maslin, and G. Mirjalili,
Handbook of Optical Constants of Solids, edited
by E. D. Palik, Academic Press, San Diego: USA,
1st edition, 1998.
[17] P. B. Johnson, and R. W. Christy, “Optical
constants of the noble metals”, Phys. Rev. B,
6(12):4370–4379, December 1972.
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