Transmitted and Reflected Graphene Surface
Waves Due to Substrate Discontinuities
Stamatios A. Amanatiadis1 , Alexandros I. Dimitriadis2 Theodoros T. Zygiridis3 , and Nikolaos V. Kantartzis1
1 Dept.
of Electrical & Computer Engineering, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
of Condensed Matter Physics, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
3 Dept. of Informatics & Telecommunications Engineering, University of Western Macedonia, 50100 Kozani, Greece
2 Institute
Abstract—The transmission and reflection coefficients of
graphene surface waves, owing to the discontinuities of its
substrate, are systematically studied and numerically computed
in this paper. In essence, the propagation properties of these
graphene-supported waves are theoretically extracted, revealing
their strong dependence on the substrate material. The latter
leads to the disruption of a propagating surface wave to a
reflected and a transmitted one, because of any potential discontinuity. Finally, the coefficients and the propagation angles
of these waves are accurately evaluated by means of a robust
finite-difference time-domain algorithm that treats graphene as
a surface boundary condition.
Index Terms—effective index, FDTD methods, graphene, reflection, surface wave, transmission.
I. I NTRODUCTION
Lately, graphene, the truly two-dimensional carbon allotrope
of atoms bonded in a honeycomb pattern, has been acknowledged by the research community as a major scientific
breakthrough with an abundance of intriguing applications [1].
Principally, its unique electromagnetic properties at a wide
frequency range have rendered graphene-based devices as very
competitive counterparts of various traditional apparatuses. In
particular, its capability to support the propagation of highly
confined transverse magnetic (TM) surface plasmon polariton
(SPP) waves, at the far-infrared spectrum, can really enhance
the design of compact-sized planar arrangements.
Basically, the propagation properties of graphene TM SPP
waves depend on the material’s surface conductivity and the
relative permittivity of its surrounding media [2]–[5]. Hence,
the presence of a discontinuity at the substrate material,
considering a constant conductivity for graphene and the
upper medium, can disrupt the propagating incidence wave,
producing a reflected and a transmitted one. It is the purpose of the present work to comprehensively investigate and
computationally extract the coefficients of these waves, in
terms of the incidence one, and to study the propagation
angle of the transmitted wave when the incident surface wave
advances to the discontinuity at oblique angles. All numerical
results are extracted through an appropriately modified finitedifference time-domain (FDTD) method, which can efficiently
treat graphene as a surface boundary condition [6].
II. T HEORETICAL A SPECTS
Graphene is viewed as a truly two-dimensional material
and is sufficiently characterized by its surface conductivity
σ(ω, µc , Γ, T ); where ω is the radian frequency, µc the chemical potential controlled either by chemical doping or by an
applied gate voltage, Γ a phenomenological scattering rate
assumed to be independent of energy, and T the temperature.
The prior conductivity is evaluated by the compact expression
resulting from the Kubo formula, considering only the dominant (at the examined frequencies) intraband contributions,
σintra (ω, µc , Γ, T )
[
]
µc
e2 kB T
+ 2 ln(e−µc /kB T + 1) ,
= −j 2
π~ (ω − j2Γ) kB T
(1)
with −e the electron charge and ~, kB the reduced Planck and
Boltzmann constant, respectively.
It is well-known that graphene is capable of supporting
strongly confined TM SPP waves on its surface, mainly beyond
the far-infrared regime, whose propagation properties depend
on surface conductivity. In the general case, that an infinite
graphene layer is placed between two dielectrics, of relative
permittivity εr1 and εr2 , the complex propagation constant of
the surface wave, kρ , is obtained through [3]
√
εr1
εr2
+√
= jση0 ,
2
(kρ /k0 ) − εr1
(kρ /k0 )2 − εr2
(2)
where k0 and η0 are the vacuum wavenumber and waveimpedance, correspondingly, while σ is graphene’s surface
conductivity (1). In this context, the effective index neff ,
defined as the ratio of the SPP wavenumber to the vacuum
one, can now be easily estimated.
III. E XTRACTION OF SPP WAVE C OEFFICIENTS
A. Substrate Effect on Graphene Surface Waves
Initially, the influence of the substrate’s relative permittivity
on the propagation properties of a graphene surface wave,
assuming the vacuum as its superstrate, is examined. Specifically, the effective index is extracted through (2) for a set of
typical graphene parameters, i.e. µc = 0.2 eV, Γ = 0.5 meV at
room’s temperature T = 300 K. In this framework, the effective index, depicted in Fig. 1 at 2 THz, increases linearly with
the relative permittivity, resulting in stronger confined SPP
waves of decreased wavelength. Therefore, the variation of
the substrate is indeed able to alter the propagation properties,
despite the constant surface conductivity of graphene.
Electric field (mV/m)
10
! er2 = 2
0
-5
-10
40
Fig. 1.
The effective index of graphene SPP waves versus the relative
permittivity of its substrate, with vacuum as the superstrate and µc = 0.2 eV,
Γ = 0.5 meV at 2 THz.
! er2 = 10
5
60
80
100
120
Distance (µm)
140
160
Fig. 4. Electric field distribution at a line parallel to the SPP wave propagation
for different substrate discontinuities. The source is located at 250 µm.
1
RSW
e2
ISW
ISW
e1
0.8
TSW
|T|, |R|
RSW
TSW
e2
0.6
0.4
0.2
Fig. 2. Graphical depiction of the analysis setup. Graphene is modelled as
a surface boundary and the source is located at its center.
0
1
Transmission
Reflection
6
11
16
Relative permittivity of substrate
21
Fig. 5. Transmission and reflection coefficients versus the relative permittivity
of the substrate for the considered setup.
B. Numerical Findings
The transmitted and reflected surface waves, due to the
substrate’s discontinuity, are acquired via the setup of Fig. 2,
where the substrate material is altered from ε1 = ε0 to ε2 at
a specific point. The numerical analysis is conducted in terms
of a properly tailored FDTD algorithm that can reliably treat
graphene as a surface impedance [6] at the xz-plane, while the
500 µm×200 µm×500 µm computational domain is divided
er2 = 2
er2 = 10
Fig. 3. Electric field distribution at a plane perpendicular to the surface of
graphene for different substrate discontinuities, indicated by the dashed line.
into 140 × 56 × 140 cells of ∆x = ∆y = ∆z = 3.57 µm.
Moreover, the time-step is selected as ∆t = 6.837 fs and the
infinite space is terminated by a 16−cell perfectly matched
layer (PML), appropriately adjusted to efficiently annihilate
the propagating surface waves on graphene. Hence, the transmission coefficient is straightforwardly calculated through the
wave after the discontinuity, whereas the reflection coefficient
is evaluated prior to the substrate alteration, by subtracting
the influence of the incident wave. It is noteworthy to stress
that both coefficients are computed as close as possible to
the discontinuity in order to avoid the effect of the different
propagation lengths, that can degrade the results.
The distribution of the normal to graphene electric field
component, given in Fig. 3, reveals that for an increased
substrate relative permittivity the amplitude of the transmitted
wave is significantly decreased. Similar outcomes are observed
in Fig. 4, where the same electric field component is plotted
along the propagation of the incident wave. Moreover, having
performed several numerical simulations to explore the effect
of the relative permittivity on the transmitted and reflecting
waves, Fig. 5 illustrates the computed reflection and transmission coefficients. It can be observed that as the discontinuity
becomes steeper, surface waves are mainly reflected.
In the previous analysis we considered that the propagation of the incident surface wave is normal to the substrate
discontinuity, while the case of the oblique incidence is now
Transmission angle qt (o)
60
qi =30o
qi =45
40
o
o
qi =60
20
0
1
6
11
16
Relative permittivity of substrate
21
Transmission angle qt (o)
Fig. 6. Angle of the transmitted wave for different values of the incidence
angle for the considered setup.
80
er2 = 2
60
er2 = 5
studied and the propagation angle of the resulting waves is
extracted. Initially, the angle of the reflected surface wave
θr is observed independent to the substrate’s permittivity and
equal to the incident one θi . On the other hand, the angle of
the transmitted surface wave θt is decreased as the relative
permittivity is increased for any incident angle, as depicted in
Fig. 6. Additionally, the dependence on the incident angle is
illustrated in Fig. 7, where it is observed that the transmitted
angle is always less that the incident one for dielectrics with a
relative permittivity larger than air’s. Finally, the distribution
of the normal-to-graphene electric field component is depicted
in Fig. 8 to visualize the propagation of surface plasmon
polariton waves. The excitation source is a dipole at the center
of graphene, with the incident, reflected, and transmitted waves
easily detected at the appropriate regions.
IV. C ONCLUSION
er2 = 10
40
20
0
0
20
40
Incidence angle qi (o)
60
80
Fig. 7. Angle of the transmitted wave for different relative permittivity values
of the substrate for the considered setup.
Fig. 8. Surface plasmon polariton wave propagating onto graphene with
substrate discontinuity εr2 = 5.
The transmission and reflection coefficients of graphene SPP
waves due to its substrate discontinuity have been thoroughly
analyzed in this paper. To this aim, the normal incidence
scenario has been studied, unveiling that a steep difference
between the relative permittivity of the adjacent materials leads
to a strong reflected wave rather than a transmitted one.
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