Graphene-based nano-patch antenna for terahertz radiation
Ignacio Llatser
Nanonetworking Center in Catalunya (N3Cat), Universitat Politècnica de Catalunya, Jordi Girona 1-3, Campus Nord,
D6-008 08034 Barcelona, Spain
Institute of High-Frequency and Communication Technology, Faculty of Electrical, Information and Media Engineering,
University of Wuppertal, Rainer-Gruenter-Str. 21, D-42119 Wuppertal, Germany
Christian Kremers
Institute of High-Frequency and Communication Technology, Faculty of Electrical, Information and Media Engineering,
University of Wuppertal, Rainer-Gruenter-Str. 21, D-42119 Wuppertal, Germany
Albert Cabellos-Aparicio
Nanonetworking Center in Catalunya (N3Cat), Universitat Politècnica de Catalunya, Jordi Girona 1-3, Campus Nord,
D6-008 08034 Barcelona, Spain
Josep Miquel Jornet
Broadband Wireless Networking Laboratory, School of Electrical and Computer Engineering, Georgia Institute of Technology,
Atlanta, Georgia 30332, USA
Eduard Alarcón
Nanonetworking Center in Catalunya (N3Cat), Universitat Politècnica de Catalunya, Jordi Girona 1-3, Campus Nord,
D6-008 08034 Barcelona, Spain
Dmitry N. Chigrin
Institute of High-Frequency and Communication Technology, Faculty of Electrical, Information and Media Engineering,
University of Wuppertal, Rainer-Gruenter-Str. 21, D-42119 Wuppertal, Germany
Abstract
The scattering of terahertz radiation on a graphene-based nano-patch antenna is numerically analyzed. The
extinction cross section of the nano-antenna supported by silicon and silicon dioxide substrates of dierent
thickness are calculated. Scattering resonances in the terahertz band are identied as Fabry-Perot resonances
of surface plasmon polaritons supported by the graphene lm. A strong tunability of the antenna resonances
via electrostatic bias is numerically demonstrated, opening perspectives to design tunable graphene-based
nano-antennas. These antennas are envisaged to enable wireless communications at the nanoscale.
1. Introduction
Graphene has recently attracted intense atten-
tion of the research community due to its extraor-
dinary mechanical, electronic and optical proper-
ties [1]. Being a at monolayer of carbon atoms
tightly packed in a two-dimensional honeycomb lat-
tice, graphene allows to utilize novel physics in
a plethora of potential applications, ranging from
ultra-high-speed transistors [2] to transparent solar
cells [3], meta-materials [4] and graphene plasmon-
ics [5, 6, 7, 8, 9].
One particularly promising research eld is
that of graphene-enabled wireless communications.
Wireless communications among nanosystems, i.e.,
integrated systems with a size of a few microme-
ters, cannot be achieved by simply reducing the
Preprint submitted to Elsevier August 10, 2012
Figure 1: Sketch of the graphene-based nano-antenna under
consideration.
size of classical metallic antennas down to a few
micrometers. This approach presents several draw-
backs, such as the low mobility of electrons in nano-
scale metallic structures and especially the use of
very high resonant frequencies (up to infrared and
optical range), which result in a huge channel at-
tenuation and the diculty of implementing nano-
transceivers operating at such a high frequency. For
these reasons, using micrometer-size metallic anten-
nas to implement wireless communications among
nanosystems becomes unfeasible. However, owing
to its ability to support surface-plasmon polari-
tons (SPP) [10, 11], graphene is seen as the en-
abling technology for this emerging eld. Indeed,
a graphene-based nano-patch antenna with lateral
dimensions of just a few micrometers is predicted
to resonate in the terahertz band [12, 13], at a dra-
matically lower frequency and with a higher radia-
tion eciency with respect to their metallic coun-
terparts. In consequence, graphene-based nano-
antennas are envisaged to enable wireless commu-
nications among nanosystems [14].
In this work, we present a systematic numeri-
cal analysis of the terahertz radiation scattering
on a graphene-based nano-patch antenna (Figure
1). A graphene rectangular patch with length
L
and width
W
supported by a dielectric substrate of
thickness
D
is illuminated by a plane wave linearly
polarized along the patch length. The associated
scattering problem is then solved numerically, and
the extinction, absorption and scattering cross sec-
tions are analysed.
The remainder of this paper is organized as fol-
lows. In Section 2, the expression used to model
the electric conductivity of graphene is presented.
In Section 3, two dierent approaches to model a
graphene patch are described and compared. Sec-
Figure 2: Real and imaginary part of the total conductivity
(solid lines) and the intraband conductivity (dashed lines)
at room temperature (
T = 300 K
) and zero electrostatic bias
(
µ
c
= 0 eV
)
tion 4 explores the dependence of the graphene-
based nano-antenna cross sections as a function of
several pameters, such as the antenna dimensions or
the chemical potential. Finally, Section 6 concludes
the paper.
2. Graphene conductivity
Graphene-based nano-patch antennas are envis-
aged to have a length and width in the order of
a few micrometers [12, 13]. Since it has been
experimentally demonstrated that edge eects on
the graphene conductivity only appear in struc-
tures with lateral dimensions considerably smaller
than 100 nm [15], in this analysis we will disre-
gard the edge eects and will use the electrical
conductivity model developed for innite graphene
sheets [10, 11].
The surface conductivity of an innite graphene
lm can be calculated by means of the Kubo for-
malism [10, 11]. Within the random-phase approxi-
mation, the surface conductivity can be represented
in a local form with the Drude-like intraband con-
tribution
σ (ω) =
2e
2
π~
k
B
T
~
ln
2 cosh
µ
c
2k
B
T
i
ω + iτ
−1
,
(1)
and interband contribution given by
σ
i
(ω) =
e
2
4~
H
ω
2
+ i
4ω
π
ˆ
∞
0
dϵ
H(ϵ) − H (ω/2)
ω
2
− 4ϵ
2
.
Here
τ = 10
−13
s
is the relaxation time,
T
is the
temperature,
µ
c
is the chemical potential and
H (ϵ)
2
is dened as
H(ϵ) =
sinh (~ϵ/k
B
T )
cosh (µ
c
/k
B
T ) + cosh (~ϵ/k
B
T )
.
In the frequency region of interest (below 5 THz),
the intraband contribution (1) dominates [16]. This
can be seen in gure 2, where the frequency de-
pendence of the real and imaginary part of the in-
traband conductivity
σ
and the total conductivity
σ
total
= σ + σ
i
are compared at room temperature
(
T = 300 K
) and zero electrostatic bias (
µ
c
= 0 eV
).
In the following, we neglect the interband conduc-
tivity contribution and assume the conductivity of
the graphene patch to be purely intraband one.
3. Numerical methods
In order to nd the electromagnetic eld scat-
tered by a graphene structure, it is necessary to cou-
ple the phenomenological model of graphene con-
ductivity with Maxwell's equations. The major
challenge here is to model an innitesimally thin
graphene layer using a nite-size discretization of
space typical for numerical calculations.
There are two main methods that can be used to
model a graphene sheet. The rst technique con-
sists in approximating a graphene layer by an equiv-
alent thin slab with a small, but nite, width. The
propagation of the electromagnetic elds within the
slab is modeled by assigning to it a normalized ef-
fective conductivity [4, 7]
←→
σ =
σ/∆ 0 0
0 σ/∆ 0
0 0 0
(2)
where
∆
is the thickness of the equivalent slab and
the graphene sheet is located in the x-y plane. The
main drawback of this method is that a realistic
model of graphene will have a length
L
much larger
than its thickness
∆
, resulting in a very high aspect
ratio (
L/∆ ∼ 1000
). The numerical computation of
the electromagnetic elds in such a structure will
therefore require a very high mesh density, leading
to a high computational cost.
As an alternative to treat this problem with lower
computational costs, the graphene sheet can be
modeled as an equivalent impedance surface [13,
17]. The surface impedance
Z
s
connects the tan-
gential component of the electric eld on the surface
with the electric surface current,
E
τ
|
z=0
= Z
s
J
surf
.
Taking into account that the current induced in the
Figure 3: Extinction cross section per unit width of a 10
µ
m
wide graphene patch with length
L = 5 µ
m. Results of the
surface impedance model (dashed black line) and the equiva-
lent slab model (solid lines ) are shown at room temperature
(
T = 300 K
) and zero electrostatic bias (
µ
c
= 0 eV
). The
thicknesses of the equivalent graphene slab
∆
are 500 nm,
200 nm and 5 nm, from left to right.
graphene layer is purely supercial and it is related
to the tangential component of the electric eld via
the surface conductivity
σ
as
J
surf
= σ E
τ
|
z=0
, one
can dene the boundary conditions at the graphene
interface
ˆ
n ×
H|
z=+0
− H|
z=−0
= J
surf
=
1
Z
s
E
τ
|
z=0
,
(3)
where
Z
s
= 1/σ
is the equivalent surface impedance
of the graphene. The boundary conditions (3) fully
determine the electromagnetic problem and can be
solved numerically using a computational scheme of
choice.
Figure 3 contains a comparison between the
equivalent slab model and the surface impedance
model. In all numerical calculations the method of
moments with surface equivalence principle [18] has
been employed. The solid lines show the extinction
cross section (see Sec.4 for details) of the antenna as
a function of frequency when the antenna is mod-
eled as a thin slab, with an eective conductivity
as dened in (2), for dierent antenna thicknesses:
500 nm, 200 nm and 5 nm, from left to right. The
dashed line corresponds to the surface impedance
model. One can see that the equivalent slab model
converges to the surface impedance model as the
equivalent slab thickness is reduced, while simul-
taneously requiring denser mesh (higher computa-
tional costs) for smaller equivalent thickness. In
what follows the surface impedance model is used
to numerically characterize the graphene patch due
3
to its accuracy and eciency.
4. Scattering properties
In order to study the performance of a graphene-
based nano-patch antenna, it is interesting to in-
vestigate the scattering, absorption and extinction
cross sections of a graphene patch. The scattering
(absorption) cross section is dened as a ratio of the
scattered (dissipated) power to the incident power,
namely
σ
scat
=
¸
S
d
2
r S
s
· n
|S
inc
|
,
(4)
and
σ
abs
=
¸
S
d
2
r S · n
|S
inc
|
.
(5)
Here the surface integration is performed over a
surface enclosing the graphene antenna.
n
is the
surface normal.
S
,
S
s
and
S
inc
are the Poynting
vectors of the total, scattered and incident elds,
respectively. The extinction cross section is given
by the sum of the scattering and the absorption
cross sections
σ
ext
= σ
scat
+ σ
abs
.
(6)
The calculated scattering and absorption cross
sections of a graphene patch with length
L = 1 µm
and width
W = 100 µm
normalized to the patch ge-
ometrical area are shown in gure 4. The antenna
is supported by an innite silicon substrate with
dielectric constant
ε = 11.9
. Room temperature
(
T = 300 K
) and zero electrostatic bias (
µ
c
= 0 eV
)
are assumed. The interaction of the terahertz ra-
diation with the antenna is dominated by the ab-
sorption, with the scattering being three orders of
magnitude weaker due to the large wavelength mis-
match between the electromagnetic excitation in
the graphene layer and in the far-eld. The to-
tal extinction cross section is equal to a few per-
cents of the graphene area and demonstrates a clear
resonant character. The obtained absorption cross
sections are consistent with experimental results re-
ported for graphene micro-ribbon arrays [19].
In order to understand the resonant behavior, we
consider a simple Fabry-Perot (FP) model for the
graphene patch. An innite graphene layer placed
on the air-dielectric interface supports transverse-
magnetic (TM) SPP waves with a dispersion rela-
Figure 4: The absorption (black line) and scattering (dashed
red line) cross sections of a graphene antenna on an innite
silicon substrate, obtained using numerical simulations, as
compared with the absorption (blue line with dots) and scat-
tering (dashed green line with dots) cross sections obtained
using the Fabry-Perot model.
tion given by [5]
1
β
2
−
ω
2
c
2
+
ε
β
2
− ε
ω
2
c
2
= −i
σ (ω)
ωε
0
,
(7)
where
β
and
ω
are the wave-vector and frequency
of the SPP wave and
ε
is the dielectric constant
of the substrate. While the air-dielectric interface
does not support SPP waves, a termination of the
graphene lm acts as a mirror and a FP type res-
onator can be realized when the following condition
is satised
L
eff
= L + 2δL = m
λ
2
= m
π
β
.
(8)
Here
λ
is the SPP wavelength,
m
is an integer,
L
eff
is the eective resonator length and
δL
is a mea-
sure of the eld penetration outside the graphene
patch. Solving the dispersion relation (7) with the
FP condition (8) for a given eective resonator
length
L
eff
results in a set of
m
complex frequen-
cies
ω
m
(resonator modes). The coupling of the
incident radiation with those modes leads to reso-
nances in the extinction spectra. Taking into ac-
count that the modes of the resonator are orthog-
onal, one can model them as a set of independent
driven harmonic oscillators with angular frequen-
cies
ω
0m
=
(Reω
m
)
2
+ (Imω
m
)
2
and damping
rates
γ
m
=
−
2Im
ω
m
. Introducing the dipole po-
larizabilities of the oscillators
α
m
as
µ
m
=
f
m
ω
2
0m
− ω
2
− iγ
m
ω
E
inc
= α
m
E
inc
,
(9)
4
with
f
m
being the oscillator strength, the scatter-
ing and absorption cross sections of the graphene
antenna can be calculated as a sum of the normal-
ized scattered and dissipated power of individual
oscillators [20], namely
σ
sca
=
ω
4
6πε
2
0
c
4
N
m=1
|α
m
|
2
(10)
and
σ
abs
=
ω
ε
0
c
N
m=1
Imα
m
.
(11)
In gure 4, the absorption and scattering cross
sections obtained with the FP model are compared
with the results of a direct numerical simulation.
The eective length of the FP resonator is set
to
L
eff
= 1.36 µm
, where the penetration length
δL = 0.18 µm
has been estimated based on the nu-
merical simulations. With the oscillator strength
f
m
used as a t parameter, a reasonable agreement
with numerical results can be achieved even if only
the rst FP mode (
f
1
= 0.073
) is taken into ac-
count.
While the operation of the graphene patch an-
tenna is based on the SPP resonances, it is impor-
tant to analyze the SPP properties. In gure 5
the SPP wavelength is shown as a function of the
frequency. One can see that in the terahertz fre-
quency range the wavelength of SPP is of the order
of a few micrometers, matching the expected size
of envisaged graphene-based nano-antennas. In the
same time the propagation length, shown in Fig. 5
in units of the corresponding SPP wavelength, is
relative small staying below one SPP wavelength.
This result is consistent with the SPP behavior ob-
served at infrared frequencies [21]. However, since a
resonant graphene patch antenna will have dimen-
sions of approximately half the SPP wavelength,
the sub-wavelength propagation ohmic losses in the
graphene sheet are not expected to hamper the per-
formance of graphene antennas considerably.
5. Scattering tuning
The spectral position of the SPP resonance can
be adjusted by an appropriate choice of the res-
onator length and width. The dependence of the
resonant frequency on the resonator length, calcu-
lated using equivalent surface impedance method,
is shown in gure 6 for dierent resonator widths.
Figure 5: Logarithmic plot of the plasmon wavelength (black
line, left vertical axis) and the ratio of the propagation length
to the plasmon wavelength (red line, right vertical axis) in a
free-standing graphene layer as a function of frequency. The
frequency ranges from 0.1 to 5 THz.
Figure 6: Dependence of the rst resonant frequency of the
graphene antenna on its length,
L
, for dierent widths,
W =
100 µm
,
W = 5 µm
,
W = 0.5 µm
and
W = 0.2 µm
.
The results correspond to a graphene patch on in-
nite silicon substrate at room temperature and
zero electrostatic bias. A wide frequency range in
the terahertz band can be covered by choosing an-
tenna dimensions. For a given length, antennas
with smaller width possess resonance at lower fre-
quencies. This eect might be attributed to the
higher connement of surface plasmons in a narrow
graphene patch, which in turn leads to higher eec-
tive permittivity and lower resonance frequency. In
the same time, the resonance shifts towards higher
frequencies for shorter antennas, in full agreement
with the resonance condition (8).
The dielectric constant and thickness of the sub-
strate inuence both the spectral position and mag-
nitude of the resonance. In gure 7 (top), the
extinction cross section of the graphene antenna
with length
L = 1 µm
and width
W = 0.5 µm
5
