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New J. Phy s. 24 (2022) 083016 https://doi.org/10.1088/1367-2630/ac85d5
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PAPER
Discussion of temperature-dependent epsilon-near-zero effect
in graphene
Yevhenii M Morozov1,2,∗, Anatoliy S Lapchuk2and Ivan V Gorbov2
1Biosensor Technologies, AIT-Austrian Institute of Technology, 3430 Tulln, Austria
2Department of Optical Engineering, Institute for Information Recording of NAS of Ukraine, 03113 Kyiv, Ukraine
∗Author to whom any correspondence should be addressed.
E-mail: Yevhenii.Morozov@ait.ac.at
Keywords: graphene, isotropic model, epsilon-near-zero effect, temperature dependence, theoretical analysis
Supplementary material for this article is available online
Abstract
In the present paper, we discuss the temperature-dependent epsilon-near-zero (ENZ) effect in
graphene arising in the framework of its isotropic model. The effect was theoretically investigated
in detail using a simplified model design of the slot line containing a graphene layer in which all
other effects are eliminated allowing us to focus solely on the ENZ effect. With the reduction of
graphene effective temperature, the ENZ effect in the near-IR wavelength range was found to
become pronounced even for structures and metasurfaces for which it has been considered
neglectable and has not been previously observed at room temperatures. This
temperature-dependent behaviour was interpreted analytically within the approximation in which
the real part of the graphene dielectric constant is considered vanishingly small in comparison
with the imaginary part (this condition is always satisfied at the ENZ point in graphene).
Furthermore, the results presented in the paper may be potentially helpful in the construction of
an experiment designed to finally prove or disregard the applicability of the isotropic model of
graphene.
1. Introduction
Graphene-supported devices and metadevices have been intensively investigated to be used in the fields
ranging from electronic–photonic integrated circuits and all-optical switchers/modulators to biosensors
[1–9]. In addition, the epsilon-near-zero (ENZ) effect arising in graphene being considered as a 3D
isotropic material [10] has been also considered in transverse magnetic (TM)-modes-supporting devices to
further increase their performance [11–16]. Besides, the optical properties of graphene were also
theoretically studied considering it as an isotropic material [17,18]. At the same time, despite controversial
results and still ongoing dispute if the ENZ effect can exist in graphene in reality [10,19–22], the
controversial nature of the isotropic graphene model has never been considered as an issue for devices
designed to be used without TM-modes as fundamental/guided modes (i.e., devices with guided modes
which electric field components lie in the plane of graphene layer)—this is due to the fact that under this
condition ENZ effect in graphene is considered as neglectable and has been never observed in such kind of
devices in the near-IR wavelength range at room temperatures [3,23]. In other words, the isotropic
graphene model has been considered applicable without any concerns for such kind of devices. However,
willing to reveal the real nature of the temperature-dependent modulation effect shown in [23], in the
present work we highlight that with the reduction of graphene effective temperature, the ENZ effect in the
near-IR wavelength range becomes essential even for such kind of structures and can greatly affect their
performance. This temperature-dependent behaviour was interpreted analytically within the approximation
when the real part of the graphene dielectric constant is considered vanishingly small compared to the
imaginary part (this condition is always satisfied at the ENZ point in graphene).
© 2022 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft
New J. Phy s. 24 (2022) 083016 YMMorozovet al
Figure 1. 3D view of the original design of the modulator. Reproduced from [23]. © 2022 The Author(s). Published by IOP
Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft.CC BY 4.0.
2. Details of the theoretical analysis
2.1. Model of the considered problem
The original design of the graphene-based modulator considered in [3,23]isshowninfigure1.Aplane
wave with the electric field polarization along x-ory-axis(i.e.,lyingintheplaneofthegraphenelayer)
propagating in the negative direction along z-axis excites the structure. In [23]itwasshownthatunder
specific values of the graphene effective temperature Tand chemical potential μc, the propagating wave
acquires an increased absorption which allowed to increase the modulation depth of the device up to three
times. However, the real nature of the modulation was not revealed, and (in accordance with the obtained
results) the effect was attributed to an additional resonance excitation in the graphene layer (e.g., a
plasmonic one). The observed effect was almost impossible to relate at that moment to the ENZ effect in
graphene due to the geometrical and electromagnetic configuration of the problem.
Using the original design (shown in figure 1), it is extremely difficult to interpret results obtained in [23]
duetothefactthatitcontainstwoslot lines (orthogonal to each other) that form a cross-shaped structure,
and this cross-shaped structure is excited by the plane wave simultaneously all over the structure’s surface.
At the same time, due to the symmetry of the considered geometry, one of the slot lines (which is extended
along the plane wave polarization) cannot qualitatively affect the observed results. Therefore, in the present
work, we replaced the original design with a model one (this model with notations and dimensions is
shown in figure 2). The model design is a single slot line rotated by π/2 (in respect to the slot line of the
original design) in order to be extended along the excitation mode propagation. A short section (1000 nm
long) of the slot line with strips made of the perfect electric conductor was added at the beginning of the
line to be able to launch the fundamental quasi transverse electro-magnetic (TEM) mode at the Ag-strips
line beginning (see figure 3). This model design allows one to excite the quasi-TEM mode of the slot line
from one side and analyse its propagation along the line while changing the graphene physical parameters.
Importantly that all other electromagnetic configuration features (such as excitation wave polarization
relative to the graphene layer) remain intact in the model design.
Numerical simulations were performed in the frequency domain (finite element method). The following
parameters were used for the design model: the width wof the slot line is 90 nm; thickness of the Ag layer is
100 nm; thickness of the SiO2buffer dielectric layer is 10 nm; thickness of the graphene layer Δis 0.34 nm.
Worth to note that all slot line parameters (such as width w=90 nm and thickness of the Ag layer of
100 nm) were kept the same as for the original design. An adaptive mesh convergence analysis was carried
out before the main investigation (see figure S1 (https://stacks.iop.org/NJP/24/083016/mmedia)in
supporting information).
2.2. Parameters of the graphene layer
Graphene, a monolayer of carbon atoms, is considered to have a finite thickness Δof 0.34 nm. Its effective
dielectric constant εgcan be represented by a diagonal tensor with in-plane components εxx and εyy and a
component εzz (which is normal to the graphene surface in our configuration). In-plane components εxx
and εyy can be calculated as [10]
εxx =εyy =ε∞ +iσ
ωε0Δ,(1)
2
New J. Phy s. 24 (2022) 083016 YMMorozovet al
Figure 2. 3D (a) front and (b) back views and (c) 2D front-view schematic of the model design with notations and dimensions.
Figure 3. Magnitudes of the (a) absolute value, (b) x-component, and (c) z-component of the electric field of the fundamental
quasi-TEM-mode. The scale bar is 100 nm.
where ε0and ε∞ are vacuum and background dialectic constants, respectively, ωis radian frequency, and
σis graphene in-plane conductivity. In the similar way, surface-normal component εzz can be calculated
via the graphene surface-normal conductivity σ⊥as
εzz =ε⊥∞ +iσ⊥
ωε0Δ,(2)
where ε⊥∞ is background dialectic constant. In many cases, it can be assumed that ε∞ =ε⊥∞ . Besides, in
the framework of graphene isotropic model, σ=σ⊥and, therefore, from equations (1)and(2):
εxx =εyy =εzz. In this case, given the graphene (isotropic) surface conductivity σg[24,25], its effective
dielectric constant εgcan be deduced as
εg=1+iσg
ωε0Δ.(3)
Carrier density nin graphene can be tuned by varying the chemical potential μcwhich, in turn, can be
actively tuned by varying the applied voltage Vg[3,23,26]. Therefore, real and imaginary parts of the
graphene dielectric constant εgcan be tuned by changing its chemical potential μc(see figure 4). As it is
seen from figure 4, the permittivity of graphene reaches the ENZ point (εg=−0.019 +i0.411) at
μc=0.5 eV and f=193 THz (λ=1.553 μm), however this effect was not observed in ‘non-TM-mode’
devices at room temperatures [3,23] (this issue has been described above). Electron relaxation time in
graphene, τ, was set to 0.1 ps for all values of the temperature T.
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New J. Phy s. 24 (2022) 083016 YMMorozovet al
Figure 4. Real and imaginary parts of the dielectric constant εgof the graphene layer under consideration as a function of the
chemical potential μc. Parameters: f=193 THz (λ=1.553 μm), T=293.15 K.
Figure 5. Distribution of the x-component of the electric field along the center of the slot line (x=0, z=0) for the cases of (a)
T=293.15 K, f=193.000 THz and (b) T=135.25 K, f=191.573 THz, respectively.
3. Results and discussion
Figures 5(a) and (b) show distribution of the x-component of the electric field along the center of the slot
line (x=0, z=0—see figures 2and 3) for the cases of T=293.15 K, f=193.000 THz and T=135.25 K,
f=191.573 THz, respectively. Frequency fof 191.573 THz for T=135.25 K was chosen as it corresponds
to the point of anomalous absorption observed in [23] and it was found that permittivity of graphene
reaches the ENZ point (εg=0.015 +i0.115) at μc=0.489 eV for these values of fand T(see figure S2).
Electric field distribution shown in figures 5(a) and (b) are typical standing waves formed by forward and
backward (reflected from the line terminal) waves. In addition, in figures 5(a) and (b) dashed lines
connecting third and fifth antinodes and second and fourth nodes (relatively to the line beginning) are
shown. Due to the fact that conditions at the slot line terminal are the same for all values of μc(and,
therefore, return loss values are the same in all cases), the slope of these lines can be used to characterize the
linear losses in the slot line for different values of the graphene chemical potential μc. For more clearance,
insets for figure 5are presented in figure S4. From figure S4 it is clear that at ENZ points (μc=0.500 and
0.489 eV for figures S4(a) and (b), respectively) linear loss increases sharply, which leads to an increased
absorption of the quasi-TEM mode of the slot line. It is also clear from figure 5that at T=135.25 K linear
loss increases much greater than that at T=293.15 K.
Increased linear loss observed in figure S4 cannot be attributed to a plasmonic resonance excitation in
graphene because under such conditions real part of the graphene dielectric constant (see figures 4and S2)
4
New J. Phy s. 24 (2022) 083016 YMMorozovet al
Figure 6. (a) Ex-and(b)Ez-field distributions for different frequencies in plane of y=2287 nm in the case of T=135.25 K,
μc=0.489 eV. The scale bar is 100 nm; the color bar is the same as in figure 3.
do not satisfy the plasmonic dispersion relation. However, due to the fact that this increased linear loss
occurs at ENZ points of graphene, it can be related to the ENZ effect in graphene despite the fact that
z-component of the quasi-TEM mode electric field (which is orthogonal to the graphene layer) is negligible
all over the slot line cross-section except for the edges of the metallic strips (see figure 3(c)). To test this
assumption, we examined electric field distributions in the plane orthogonal to the quasi-TEM mode
propagation and coming across fourth of the considered in figure 5antinodes. Ex-andEz-field distributions
for different frequencies in plane of y=2287 nm in the case of T=135.25 K, μc=0.489 eV are depicted in
figures 6(a) and (b), respectively. In figure 6, the very left pictures are overall views at f=180 THz and
right pictures are magnified views of the corner marked by white circle in the left pictures for different
frequencies.
From figure 6(a) it is clear that magnitude of the Ex-component of the quasi-TEM mode is slightly
lowered at f=191.573 THz. At the same time, at f=191.573 THz, Ez-component starts to concentrate
almost completely in the graphene layer (which is a property of the ENZ effect). This effect can be
understood from the following consideration: the field amplitude in the one optical medium (ε
a) relatively
to another one (ε
b) can be increased (close to their interface or in the volume in the case of a thin layer) if
the ratio of |ε
a/ε
b|is large. This is attributable to the discontinuity of the normal components of the electric
field at the interface: Dna =Dnb ⇒Enaε
a=Enbε
b⇒Enb =|(ε
a/ε
b)|Ena. If the medium bis lossy (ε
b>0),
then the field undergoes propagation loss which is proportional to Enb ε
b(for the exact expression, see
equation (4)below).
To make results presented in figure 6more clear, distributions of the electric field Ez-component along
the white horizontal dashed line (see left picture in figure 6(b)) for different frequencies are presented in
figures S5(a) (figure 7(a) shows the inset of figure S5(a)). On the contrary, there is no ENZ effect at
T=135.25 K and μc=0.100 eV (see figures S5(b) and 7(b)).
It is important to note here that the same behaviour shown in figures 6and 7in the case of
T=135.25 K, μc=0.489 eV was observed in the case of T=293.15 K, μc=0.500 eV. However, the
question is arising here— why ohmic loss is so drastically different in both cases (see figure 8where ohmic
5
New J. Phy s. 24 (2022) 083016 YMMorozovet al
Figure 7. Ez-field distributions for different frequencies along the line in the case of (a) T=135.25 K, μc=0.489 eV and (b) T
=135.25 K, μc=0.100 eV.
Figure 8. Ohmic losses in graphene for (a) T=293.15 K, μc=0.500 eV and (b) T=135.25 K, μc=0.489 eV.
Table 1. Real (ε) and imaginary (ε) parts of the graphene εg.
T=293.15 K, μc=0.500 eV T=135.25 K, μc=0.489 eV
ε−0.019 0.015
ε 0.411 0.115
losses in graphene for both cases are shown) which leads to the observed fact that ENZ effect almost does
not appear at room temperatures and was not observed for the considered modulator geometries [3,23]
while it exists in principle under such conditions. Especially, taking into account that in the case of
T=293.15 K, μc=0.500 eV imaginary part of the graphene εgis about 4 times higher than that for
T=135.25 K, μc=0.489 eV (see figure S3 and table 1).
While it appears senseless under the ‘normal’ operation, at the ENZ effect a condition of |ε|ε is
occurred and, when this condition is satisfied, decreasing of the imaginary part can lead to the ohmic loss
increasing. To understand this, consider a capacitor filled with vacuum and having a thin lossy layer inside
it, to which a varied voltage is applied. Ohmic losses in the lossy layer can be calculated as:
P=Re1
2
Vl
→
E
→
j∗dV=ωε0
2
Vl
ε →
E
→
E∗dV,(4)
where
Eis the electric field in the layer,
j∗is the complex conjugate current density and the integration is
carried out over the entire volume of the layer Vl. Consider the electric field in the layer: in approximation
6
New J. Phy s. 24 (2022) 083016 YMMorozovet al
of a thin layer, electric fields above and below the layer are the same: E0=Vg/h,whereVgis the applied
voltage and his a distance between the capacitor plates. For the electric field component orthogonal to the
layer at the interface between air and the layer one can write therefore:
Dn0=Dnl =E0=Eεl,(5)
where εlis a complex dielectric constant of the layer. Therefore, E=E0/εland substituting this into the
equation (4), one obtains:
P=Re1
2
Vl
→
E
→
j∗dV=ωε0
2
Vl
ε
→
E0
→
E∗
0
(ε)2+(ε)2dV.(6)
In the approximation of |ε|ε (which always takes place at the ENZ point), one can rewrite
equation (6) as follows:
P=Re 1
2
Vl
→
E
→
j∗dV=ωε0
2
Vl
→
E0
→
E∗
0
ε dV.(7)
From equation (7) is it thus clear that decreasing of the imaginary part ε of the lossy layer can lead to
the ohmic loss increasing inside it.
This analysis can account for the observed ohmic loss increasing (see figure 8) at the decreased graphene
effective temperature T—because at the decreased Treal and imaginary part curves come closer to each
other at the ENZ point (see figure S3) which, in turn, leads to a significant imaginary part decreasing
(see table 1) at that point and, therefore, to the ohmic loss increasing (according to equation (7)).
4. Conclusion
ENZ effect may arise in graphene if its effective dielectric constant is treated as an isotropic one, i.e.,
εxx =εyy =εzz (called the graphene isotropic model). Despite still ongoing dispute if it is possible for the
ENZ effect to exist in graphene in reality [10,19–22], the controversial nature of the graphene isotropic
model has never been considered as an issue for devices designed to be used without TM-modes as
fundamental/guided modes. Therefore, the graphene isotropic model has been considered as applicable
without any concerns for such kind of devices. In the present paper, we however showed that the ENZ effect
in the near-IR wavelength range can arise in graphene even in these structures with reducing the graphene
effective temperature. The observed temperature-dependent behaviour was interpreted analytically within
the approximation when the real part of the graphene dielectric constant is considered vanishingly small in
comparison with the imaginary part (this condition is always satisfied at the ENZ point in graphene). The
obtained results show the conditions when ENZ in graphene is more pronounced and, therefore, they may
be potentially helpful in the construction of an experiment designed to finally prove or disregard the
applicability of the isotropic model of graphene. This study can thus significantly contribute to the field of
investigation of the fundamental properties of graphene.
Acknowledgments
This work was supported by the National Academy of Sciences of Ukraine (Grant 0119U001105). YMM
acknowledges support from the Austrian Science Fund (FWF) through the Lise Meitner Programme (Grant
M 2925). ASL and IVG acknowledge support from the National Research Foundation of Ukraine (Grant
2020.02/0090). Open Access is funded by the Austrian Science Fund (FWF).
Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.
ORCID iDs
Yevhenii M Morozov https://orcid.org/0000-0001-9689-8641
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New J. Phy s. 24 (2022) 083016 YMMorozovet al
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