Effective Fermi Level Modulation of Coupled Graphene ... · is the Fermi energy level of graphene and is electron momentum relaxation time, electron surface concentration . n. s

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.

INTERNATIONAL JOURNAL OF MICROWAVE AND OPTICAL TECHNOLOGY,

VOL.9, NO.6, NOVEMBER 2014

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mailto:[email protected]

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.


four-layer structure at different Fermi energy levels EF

For d2 (SiO2 thickness)=50 µm, real part.



For d2 (SiO2 thickness)=50 µm, imaginary part.



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for d2 (SiO2 thickness)=140 µm, real part.



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.

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Effective Fermi Level Modulation of Coupled Graphene ... · is the Fermi energy level of graphene and is electron momentum relaxation time, electron surface concentration . n. s