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publication. Citation information: DOI 10.1109/LAWP.2014.2378174, IEEE Antennas and Wireless Propagation Letters, 2014.
IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, APRIL 2014 (REVIEWED ON SEPTEMBER 23, 2014)
1
FDTD Formulation for Graphene Modelling Based on Piecewise Linear
Recursive Convolution and Thin Material Sheets Techniques
Rodrigo M. S. de Oliveira∗ , Nilton R. N. M. Rodrigues† and Victor Dmitriev‡
Federal University of Pará (UFPA) - Institute of Technology (ITEC)
Belém, Pará, PO Box 8619, Zip Code 66073-900, Brazil.
E-mails: ∗ [email protected], † [email protected], ‡ [email protected]
Abstract—A finite-difference time-domain formulation based
on piecewise linear recursive convolution method and on thin
material sheets technique is developed for modelling terahertz
graphene antennas and some other photonic components. The
graphene sheets are modelled by specific recursive equations
obtained for tangential electric field components allowing one
to apply easily voltage or current sources between the sheets.
The effective conductivity of graphene sheets in Yee’s threedimensional lattice is calculated and used in simulations. A bowtie-like geometry is investigated, aiming at resonance tuning. The
developed numerical formulation is validated by comparison of
results with data published in literature.
Keywords—Graphene, Nanoantennas, FDTD method, piecewise
linear recursive convolution, thin material sheets, terahertz devices.
I.
I NTRODUCTION
RAPHENE consists of carbon atoms arranged in a
two-dimensional honeycomb hexagonal lattice forming
a sheet with the thickness of a single atom [1]. Owing to
its potential applications this material recently has attracted
tremendous research interest in many fields of technology [2]
including nanophotonics and in particular in terahertz antennas
[3]–[6]. The main attractive characteristics of graphene are a
possibility of dynamic spectral reconfigurability (by modifying
chemical potential) [5], relatively low losses in terahertz band
(in comparison with metals) [7] and a possibility to miniaturize
antennas (up to ∼ 6% of the wavelength in free space) due
to plasmonic effects [3]. We consider a small-signal regime
where the influence of electromagnetic field of surface plasmon
polariton (SPP) modes on graphene chemical potential is
negligible.
This way, graphene conductivity is modelled usually by
the Kubo formalism which can include both intraband and
interband components [8]. One of the challenges in the numerical calculations of such structures is different scales of
dimensions of the real devices. The length and width of the
devices can be 3-4 orders higher than their thickness [3]–[5],
[9], [10]. In several recently published works on the graphene
antennas and related topics, calculations were performed by
some commercial simulators [3]–[5]. Such calculations require
a considerable amount of computational resources because of
high discretization levels which are necessary to represent
graphene sheets. Besides, those simulators are usually based on
frequency domain techniques where one simulation is required
for each frequency of interest. In contrast, with time domain
G
techniques the impulsive response of a given structure can be
obtained with a single simulation run [11].
Recently, Transmission-Line Modeling (TLM) [12] and
analytical time-domain models [13] have been applied for
modelling graphene devices. A widely known, powerful and
yet relatively simple method in computational electrodynamics
is the Finite-Difference Time-Domain (FDTD) method [11].
Recently, FDTD has been used for modelling graphene-based
devices [9] [10]. In [9], a graphene conductivity formulation
based on multiple complex conjugate pole-residue pairs is presented. In [10], a precise surface boundary condition technique
is proposed for modelling graphene sheets laying at magnetic
field planes of Yee’s cell (exploiting the TM plasmonic mode
~ components can make
in graphene). However, the use of H
it difficult to establish controlled voltage or current between
two (or more) graphene sheets due to the staggered nature of
Yee’s cell [11].
In this work, we develop a FDTD formulation based on
piecewise linear recursive convolution (PLRC) [14] and on
thin material sheets [15] techniques for including graphene on
planes of Yee’s cells with electric field components tangential
to the sheets. This is possible because graphene plasmonic
mode is also characterized by tangential current densities [4],
[8]. The proposed formulation has two main advantages: 1)
it is mathematically very simple and precise, and 2) correct
numerical solutions are obtained for graphene sheets placed
~ field planes in Yee’s cells, simplifying FDTD
on transverse E
modelling photonic devices fed by voltage or current sources.
II.
M ATHEMATICAL F ORMULATION
Graphene’s intraband dispersive behavior can be modelled
by the surface conductivity
σ∗
1
σ̃(ω) =
,
(1)
d jω + 2Γ
which is an approximation of Kubo’s formula [8]. In (1),
Γ = 1/(2τ0 ), d is the graphene sheet’s thickness [5] and
q2 k T
µc
−µc /kB T
σ ∗ = eπh̄B2
, where τ0 is the
kB T + 2 ln 1 + e
transport relaxation time, qe is the electron’s charge, kB is the
Boltzmann constant, T is temperature, h̄ is the reduced Plank’s
constant and µc is the chemical potential [8]. As demonstrated
in [11], [15], embedding a thin conductive sheet in FDTD
3D-lattice (in free space) produces an effective conductivity
σ̃eff = (d/∆)σ̃, in which ∆ is the edge of a cubic Yee cell. By
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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, APRIL 2014 (REVIEWED ON SEPTEMBER 23, 2014)
calculating the inverse Fourier transform of σ̃eff , one obtains
the effective conductivity
σ(t) = GC e−2Γt , t > 0,
becomes
(2)
∂Dy
+ Jy
∂t
in which GC = σ /∆. In Maxwell’s equations
(3)
~
~
~
~
with D(t)
= ǫE(t)
and B(t)
= µH(t),
for current J~ evaluation one can use the conductivity (2) since the convolution
~ =
J(t)
Z
t
~ − τ )σ(τ )dτ
E(t
(4)
0
is calculated for tangential current densities on graphene. For
the problem at hand, Ampère’s law in (3) becomes
ǫ
~
∂E
+
∂t
Z
t
~ − τ )σ(τ )dτ
E(t
0
~
= ∇ × H.
(5)
Taking into account the exponential behaviour of σ(t) in (2),
it is possible to evaluate the convolution in (5) by a recursive
procedure using the PLRC technique described in [14].
In order to simplify demonstration of the method, we shall
treat the problem for a single spatial dimension and the time.
We use Ey and Hz as field components for a wave propagating
along the x direction. Taking into account that τ = m∆t and
~ and H
~ are calculated respectively at n and n+ 1 FDTD
that E
2
∂D
discrete instants, we obtain for the term ∂ty + Jy in (3) and
(5) the following expression:
=ǫ
n+ 21
!
!
Jyn+1 + Jyn
Eyn+1 − Eyn
+
=
≈ǫ
∆t
2
!
Z
Eyn+1 − Eyn
1 (n+1)∆t (n+1)−m
+
Ey
σ(τ )dτ +
∆t
2 0
Z
1 n∆t n−m
Ey
σ(τ )dτ ),
(6)
+
2 0
∂Dy
+ Jy
∂t
where t = n∆t and the average current density
1
n+1
+ Jyn is used in (6) to provide the time synchro2 Jy
nization between Jy and Hz in (5).
The function Ey can be considered constant during each
individual FDTD time step. Therefore we can write the piecewise linear approximation
Jyn ≈
n−1
X
m=0
Jyn
Eyn−m
Z
Jyn+1 ,
(m+1)∆t
σ(τ )dτ.
(7)
m∆t
Applying (7) to
and
extracting the first term (m =
0) from Jyn+1 and grouping the remaining summations, (6)
+
≈ǫ
Eyn+1 − Eyn
∆t
!
+
Z ∆t
1
+ Eyn+1
σ(τ )dτ +
2
0
!
Z (m+2)∆t
Z (m+1)∆t
σ(τ )dτ +
σ(τ )dτ .
∗
~
~
∂D
~ ∂ B = −∇ × E
~
+ J~ = ∇ × H,
∂t
∂t
n+ 12
n−1
1 X n−m
E
2 m=0 y
(m+1)∆t
m∆t
(8)
To simplify notation in (8), we define the functions
Z (m+1)∆t
σm (i) =
σ(τ, i)dτ
(9)
m∆t
and
Sm (i) = σm (i) + σm+1 (i),
(10)
where x = i∆x. As a result, from Ampère’s law and (8)-(10),
we see that
1 n
n
~
+ ǫ(i)
∇×H
∆t Ey (i) − 2 Ψy (i)
y
n+1
,
(11)
Ey (i) =
ǫ(i)
1
∆t + 2 σ0 (i)
~ and Ψny (i) is
~
is the y-component of ∇ × H
where ∇ × H
y
the following convolution for the ith FDTD cell:
Ψny (i) =
n−1
X
Eyn−m (i)Sm (i).
(12)
m=0
At this point, one should notice that the calculation of (11)
is computationally prohibitive because of the present form of
(12). Therefore, the aim here is to obtain a recursive FDTD
equation for Ψny in (12). Substituting (2) in (9), one obtains
Z (m+1)∆t
σm (i) =
GC e−2Γt dτ
(13)
m∆t
and after integration
GC −2Γm∆t
(14)
e
− e−2Γ(m+1)∆t .
2Γ
i
From (14), one can notice that
GC −2Γ(m+1)∆t
e
− e−2Γ(m+2)∆t =
σm+1 (i) =
2Γ
i
= e−2Γ∆t σm (i).
(15)
σm (i) =
Additionally, from (10) and (15), one has the recursive relation
Sm+1 (i) = e−2Γ∆t Sm (i).
(16)
As a consequence, expressions (12) and (16) lead to the
following simple recursive FDTD updating equation for (12):
Ψny (i) = Eyn (i)S0 (i) + e−2Γ∆t Ψyn−1 (i),
(17)
with Ψ0y (i) = 0. Once (17) is calculated, Ey can be updated
−2Γ∆t
with (11). Observe
GC /(2Γ) and
that σ0 = 1 − e
−4Γ∆t
S0 = 1 − e
GC /(2Γ) are obtained from (14) and (10).
DE OLIVEIRA et al.: A FDTD FORMULATION FOR GRAPHENE SHEETS BASED ON PLRC TECHNIQUE
3
Fig. 3. Transient signals: (a) total currents flowing into electrode B, (b) gap
voltages.
Fig. 1. Graphene antenna [3] modelled for validation of the proposed method.
Current Source
Graphene
sheet
.
Graphene
z
y
x
P.E.C.
Electrode B
Ey
(i,j,k+1)
Sheet
.
Current
Source
Ex
P.E.C.
Electrode A
.
(i,j,k)
.
Graphene
sheet x
Ex
(i+1,j,k)
Ey
(a)
(b)
y
Fig. 2. Graphene modelled in FDTD space: (a) a part of sheet in 3D space
and (b) two graphene sheets, the P.E.C. electrodes A and B and current source.
Ex and Ey are electric field components on graphene sheets.
III.
R ESULTS
A. Validation of the proposed numerical formulation
In order to validate the FDTD methodology developed in
this work, two graphene-based antennas proposed in [3] have
been modelled with our FDTD solver. Both antennas consist
of two coplanar graphene sheets placed on a semi-infinite glass
substrate with ǫr = 3.8 and σ = 0 as illustrated by Fig.1. The
sheets are separated by a gap Sg = 3µm for both antennas.
Antenna 1 is described by the parameters L = 17µm, W =
10µm and µc = 0.13eV and antenna 2 is characterized by
L = 23µm, W = 20µm and µc = 0.25eV.
Fig.2(a) shows a part of a graphene sheet in 3D FDTD
~
lattice. The E-field
components Ex and Ey on the graphene
sheets (shown in Figs.2 (a) and (b)) were calculated by using
the proposed technique. These fields produce the current (4).
Other field components were calculated by their corresponding
original FDTD equations. In order to avoid numerical rounding
problems when evaluating (11) and (17), σ0 , S0 , Gc and the
coefficient e−2Γ∆t were calculated by using the open source
Multiple Precision library GMP/MPFR from GNU.
As long as the results in [3] were obtained by using the commercial frequency-domain FEM solver HFSS, we conceived
the excitation structure of Fig.2(b) which consists of current
source placed on the antenna’s plane between two perfect
conducting (PEC) electrodes A and B. The metallic electrodes
were included in our FDTD model exclusively because they
are used in HFSS excitation port. The width of the electrodes is
500 nm. Soft current source [11] was implemented by forcing
current densities Jx at the antenna gap to follow a monocycle
pulse. The most part of energy of the pulse is contained in the
spectral range 0.2-1.8 THz. The antennas’ impedances were
˜
calculated by using the Fourier
P transforms Ṽ (f ) and I(f ) of
the transient voltage V (t) =
Ex ∆x produced between the
electrodes and of the total transient current I(t) flowing into
electrode B (Fig.2(b)). The obtained voltages V (t) and total
currents I(t) for both antennas are shown by Fig.3.
For modelling graphene antennas by the FDTD method, we
have used cubic Yee cells with ∆ = 500nm (spatial step) and
set a 96 × 90 × 71 grid. Antenna 1 was also simulated with
∆ = 250nm and ∆ = 125nm. The computational domain was
truncated by using the CPML formulation [11] and numerical
stability was observed when ∆t was set to 99% of Courant’s
limit [11].
Fig.4 shows a comparison among impedance curves obtained with the PLRC-FDTD formulation proposed in this
paper and the results of [3] for the same antennas. It can
be observed that for ∆ = 500nm the results agree well
in terms of resonance frequencies (where imaginary part of
impedance is zero). Small deviations are observed for the peaks
of real part of the impedance for antenna 1 (about 4%) and
antenna 2 (approximately 1.5%). Fig.4 also shows impedance
curves obtained for antenna 1 with higher discretization levels
(∆ = 250nm and ∆ = 125nm). It is observed that 1) all
resonance frequencies are still close to those given in [3]
and very small shifts are observed, as expected, and 2) peaks
of real part of impedance converges to the values given by
[3]. This means that the proposed formulation can model the
graphene thickness d independently of ∆, as described in [15],
since ∆ > d/2 and ∆ ≤ λmin /10. Modifications in results
are simply due to improved precision in numerical evaluation
of derivatives in Maxwell’s equations and the corresponding
reduction of the numerical dispersion [11].
Finally, Fig.5 shows the spatial current distribution for
antenna 1 at the resonance frequency of 1.02 THz, obtained
via discrete Fourier transforms. This result also fully agrees
with the current diagram in [3].
B. Tuning resonance frequency
In this section, we use our technique for analysis of one
example of graphene antennas shown in Fig.1. It consists
of gradual increasing of the width of the graphene sheets,
such as illustrated by Fig.6(a). Thus, we deal with a bow-tie
antenna. For the antenna in Fig.6, the width from antenna 1 is
progressively incremented starting from the metallic electrodes
with 10µm to 16µm at the end. The length of the original
device (antenna 1) was preserved. This geometric modification
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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, APRIL 2014 (REVIEWED ON SEPTEMBER 23, 2014)
D = 500 nm
Antenna 1 – FDTD:
(RE) - (D = 0.25 mm)
(IM) - (D = 0.25 mm)
(RE) - (D = 0.125 mm)
(IM) - (D = 0.125 mm)
1.02 1.17 1.35
1.53
Resonances
electric field components are present. This allows one to easily
excite the antennas by voltage or current sources coplanarly
placed to the sheets. Two rectangular antennas with graphene
sheets placed on a dielectric substrate were modelled and full
agreement with results published in literature was observed. A
bow-tie geometry was analyzed and it was shown that in such
antenna some tuning of the device’s resonances is possible.
Finally, notice that the presented formulation can be used not
only for analysis and design of graphene antennas and but also
for modelling other photonic devices.
R EFERENCES
[1]
Fig. 4. Validation of the proposed method: impedances of antennas 1 and 2.
[2]
[3]
[4]
[5]
Fig. 5. Current distribution on graphene sheets for antenna 1, f = 1.02THz.
[6]
[7]
[8]
[9]
Fig. 6. Bow-tie graphene antenna: (a) geometry and current distribution,
f = 0.90THz, (b) impedance Z.
[10]
of the usual rectangular antenna can be used for controlling
the resonance frequency. Fig.6(b) shows a comparison of
impedance curves for antenna 1 and the bow-tie antenna. It
can be seen that the resonances were displaced from 1.35 THz
to 1.237 THz and from 1.02 THz to 0.90 THz. This effect can
be understood by inspecting Figs. 5 and 6(a), where we see
that the bow-tie geometry forces the most part of the current to
flow over longer trajectories in comparison with the rectangular
geometry.
[11]
[12]
[13]
[14]
IV. F INAL R EMARKS
In this paper, we presented a PLRC-FDTD formulation
for including the effective conductivity of graphene sheets in
FDTD lattices. With the formulation, it is possible to model
graphene sheets at planes in Yee’s grid on which tangential
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