of 7
PHYSICAL REVIEW APPLIED
10,
054053 (2018)
Modulated Resonant Transmission of Graphene Plasmons Across a
λ
/50
Plasmonic Waveguide Gap
Min Seok Jang,
1,
*
Seyoon Kim,
1,2
Victor W. Brar,
3
Sergey G. Menabde,
1
and Harry A. Atwater
2,4
1
School of Electric Engineering, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea
2
Thomas J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California
91125, USA
3
Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA
4
Kavli Nanoscience Institute, California Institute of Technology, Pasadena, California 91125, USA
(Received 7 June 2018; revised manuscript received 7 September 2018; published 26 November 2018)
We theoretically demonstrate the nontrivial transmission properties of a graphene-insulator-metal
waveguide segment of deeply subwavelength scale. We show that, at midinfrared frequencies, the
graphene-covered segment allows for the resonant transmission through the graphene-plasmon modes as
well as the nonresonant transmission through background modes, and that these two pathways can lead to
a strong Fano interference effect. The Fano interference enables a strong modulation of the overall optical
transmission with a very small change in graphene Fermi level. By engineering the waveguide junction, it
is possible that the two transmission pathways perfectly cancel each other out, resulting in a zero transmit-
tance. We theoretically demonstrate the transmission modulation from 0% to 25% at 7.5-
μ
m wavelength
by shifting the Fermi level of graphene by a mere 15 meV. In addition, the active region of the device
is more than 50 times shorter than the free-space wavelength. Thus, the reported phenomenon is of great
advantage to the development of on-chip plasmonic devices.
DOI:
10.1103/PhysRevApplied.10.054053
I. INTRODUCTION
Graphene has been recently proposed as a candidate
material for the electro-optic devices operating in mid-
infrared that can control the phase and intensity of light at
high data rates. The tunable light-graphene interaction can
be achieved by controlling the graphene interband tran-
sitions [
1
3
] as well as by tuning the properties of the
plasmon modes supported by the free carriers in a graphene
sheet [
4
6
]. The graphene plasmons are particularly inter-
esting for application purposes because the semimetallic
and two-dimensional nature of graphene allows for these
modes to be highly tunable and deeply subwavelength,
with the plasmon wavelength shown to be around 100
times shorter than the free-space wavelength [
7
9
]. Fur-
thermore, while the interband absorption efficiency of
graphene is limited to a quantum of optical conductance
(2.3%), the strong oscillator strength of the plasmonic
modes allows for a higher dynamic range of control as well
as more efficient light modulation within a smaller active
area. These unconventional properties promise a creation
of graphene-based active plasmonic devices that have
high-modulation depth and can be integrated on a chip at
length scales approaching those of electronic transistors.
*
jang.minseok@kaist.ac.kr
Thus far, the plasmonically driven graphene nanores-
onators have shown tunable absorption from terahertz to
mid-infrared frequencies [
10
16
], and it has been exper-
imentally demonstrated that the modulation depth can
be significantly improved toward perfect modulation effi-
ciencies when they are combined with noble metal plas-
monic structures [
15
,
16
]. The graphene-plasmonic devices
designed for the modulation of the free-space light have
been mostly based on patterning a large number of quasi-
identical resonators on a graphene sheet that supports plas-
monic resonances that are collectively tuned by controlling
the carrier density of the graphene sheet. Therefore, these
devices have a large footprint (typically approximately
50
×
50
μ
m
2
) and do not offer an opportunity to study
single plasmonic cavity physics. In addition, the graphene
Fermi level has to be tuned by a few hundred meV, which
requires a large gate bias in order to completely shift
the plasmon resonance into and out of the desired fre-
quency range. Moreover, the
Q
factors of those devices
are typically quite low (approximately 10) due to the edge
roughness of the graphene ribbons caused by the lithogra-
phy process [
12
,
14
17
]. Thus, while graphene ribbons of
moderate mobility have been predicted to drive the total
absorption with narrow resonance peaks, the extra loss
introduced at the ribbon edges prevents such a performance
from being experimentally observed.
2331-7019/18/10(5)/054053(7)
054053-1
© 2018 American Physical Society
MIN SEOK JANG
et al
.
PHYS. REV. APPLIED
10,
054053 (2018)
In this work, we report on a device geometry that uti-
lizes a single graphene-plasmonic cavity to actively tune
the transmission through a metal-insulator-metal (
M-I-M
)
waveguide. In this geometry, the transmission modula-
tion is achieved via Fano interference effect, whereby
the transmission through resonant graphene plasmons can
destructively or constructively interfere with the trans-
mission through nonresonant background modes. This
mechanism provides multiple advantages over previously
proposed structures. First, the proposed geometry utilizes
the Fano interference property of being highly sensitive
to the plasmonic resonance frequency of the graphene
cavity, and thus a small variation of the graphene Fermi
level (approximately 0.01 eV) creates a large change in
the transmission intensity compared to the conventional
graphene-based modulators solely utilizing the plasmon
resonance or the interband transition in graphene. Sec-
ond, we demonstrate an efficient light modulation with
a single graphene cavity instead of using a large array
of resonators [
10
16
] or an elongated waveguide struc-
ture [
1
3
] employed to enhance the optical response. This
compact structure ensures a deeply subwavelength-device
footprint, leading to an ultrafast and energy-efficient opera-
tion as well as providing means to study the single plasmon
behavior in graphene such as nonlinear effects [
18
]and
single-emitter coupling [
4
]. Finally, the graphene plasmons
in our device are laterally confined by the graphene-metal
interface rather than the physical edges of graphene, thus
the fabrication of our device does not require a litho-
graphic graphene patterning. Consequently, we expect that
our device will avoid performance-limiting issues caused
by patterned graphene, including the edge roughness and
formation of the edge states [
12
,
14
17
], and therefore,
could exhibit sharper plasmonic resonance compared to the
previously demonstrated devices relying on the physically
patterned graphene.
II. RESULTS AND DISCUSSION
A. Transmission through a composite plasmonic
waveguide
A schematic of the device is shown in Fig.
1
.Two
identical
M-I-M
waveguides, consisting of a SiO
2
slab
core and gold claddings, are separated by a small gap.
In the gap region, the top gold layer is replaced with a
sheet of graphene to form a short section of a graphene-
insulator-metal (
G-I-M
) waveguide. The graphene layer is
electrically grounded and its carrier density can be actively
tuned by applying a gate voltage to the bottom gold layer.
The thickness of the core
d
and the cladding
h
, and the size
of the gap
L
are chosen as
d
=
h
=
100 nm and
L
=
140 nm.
The operating photon energy of our device is chosen to be
ω
=
0.165 eV, corresponding to the free-space wavelength
λ
0
of 7.5
μ
m. This frequency is selected in order to sup-
press absorption losses due to the vibration modes in SiO
2
FIG. 1. Two identical
M-I-M
waveguides are separated by a
narrow gap covered by a sheet of graphene. The thicknesses of
the SiO
2
(
d
) and the gold
h
layers are both 100 nm, and the width
of the gap
L
is 140 nm.
(approximately 0.133 eV) [
19
] and the optical phonons in
graphene (
0.2 eV) [
20
], as well as to avoid plasmon-
phonon coupling effects that are known to occur near the
substrate phonon energies [
21
]. The input and output
M-
I-M
waveguides support only the fundamental transverse
magnetic mode (TM
0
) because their thickness is far below
the diffraction limit [
22
]. On the contrary, in the subwave-
length gap region, there exist graphene plasmons, weakly
bound surface plasmon polaritons (SPPs) on the bottom
gold surface, and a continuum of unbound eigenmodes due
to the semi-infinite free space above the graphene sheet.
In order to study the transmission properties of this
composite waveguide structure, we begin by numerically
investigating the coupling characteristics between
M-I-M
TM
0
and graphene plasmon modes across just a single
M-I-M
G-I-M
junction [Fig.
2(a)
]. In this calculation,
we obtain the steady-state solution of Maxwell’s equa-
tions from full-wave simulations using the finite element
method, and then decompose the total fields into the eigen-
modes [
21
]. The optical properties of gold and SiO
2
are
taken from Palik [
19
]. The graphene is considered as a
thin film having a thickness of
δ
=
0.3 nm and a dielec-
tric function of

G
=
1
+
i
σ
/
ωδ
. The optical conductivity
of graphene
σ
(
ω
) is calculated within the local random-
phase approximation [
23
], assuming a carrier mobility
μ
of 10 000 cm
2
V
–1
s
–1
. The effective mode indices of
M-I-M
TM
0
(
n
M
I
M
) and graphene plasmons (
n
G
) are calculated
to be
n
M
I
M
=
1.11
+
0.033
i
and
n
G
=
27.1
+
0.27
i
at
Fermi energy
|
E
F
|=
0.4 eV. Figure
2(a)
shows the result-
ing steady-state electric-field profile when a TM
0
mode
is continuously excited from the
M-I-M
waveguide and
propagates to the
M-I-M
G-I-M
interface at
|
E
F
|=
0.4 eV.
The incident
M-I-M
TM
0
mode, which is launched from
the left side, splits into a backward-propagating
M-I-
M
TM
0
mode (reflection), forward-propagating graphene
plasmons (transmission), and the background modes com-
posed of unbound radiation and weakly bound gold SPPs.
We can characterize the mode-coupling relationship
between the
M-I-M
TM
0
and the graphene-plasmon
054053-2
MODULATED RESONANT TRANSMISSION . . .
PHYS. REV. APPLIED
10,
054053 (2018)
0
0.3
-0.3
0
-1.0
0
.
2
0
.
1
0
.
2
-
(a)
0
-1
1
Au
Au
SiO
2
Graphene
(b)
(c)
(d)
(e)
0.3
0.4
0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.3
0.4
0.5
0
0.0
0.2
0.4
0.6
0.8
1.0
0
Reflection
Reflection
Reflection
Reflection
Transmission
Transmission
Transmission
Transmission
FIG. 2. (a) The electric-field distribution at the
M-I-M
G-I-
M
junction at
E
F
=
0.4 eV. (b),(c) The amplitude and phase
of the reflection (
r
MG
, blue dashed) and transmission (
t
MG
,red
solid) coefficients for incoming
M-I-M
TM
0
mode as a func-
tion of graphene Fermi level. (d),(e) Same as (b),(c), but for the
opposite geometry with the graphene plasmons propagating from
the
G-I-M
segment into the
M-I-M
waveguide. The geometric
parameters are chosen as
d
=
h
=
100 nm.
modes in the forward-propagating direction in terms
of the complex transmission
t
MG
=|
t
MG
|
exp
(
i
φ
t
MG
)
and
reflection
r
MG
=|
r
MG
|
exp
(
i
φ
r
MG
)
coefficients as shown in
Figs.
2(b)
and
2(c)
. Likewise, by launching the wave in
the opposite direction, we can also calculate the
t
GM
=
|
t
GM
|
exp
(
i
φ
t
GM
)
and
r
GM
=|
r
GM
|
exp
(
i
φ
r
GM
)
coefficients
for incoming graphene plasmons hitting the
M-I-M
waveg-
uide [Figs.
2(d)
and
2(e)
]. The amplitude of each coeffi-
cient is defined as
|
t
|
2
=
P
t
/
P
i
and
|
r
|
2
=
P
r
/
P
i
, where
P
i
,
P
t
,and
P
r
are the time-averaged power flows carried by the
incident, transmitted, and reflected modes, respectively.
The phase term is determined such that
φ
t
,
r
=
arg[
E
t
,
r
x
/
E
i
x
],
where the complex electric-field components
E
i
x
,
E
t
x
,and
E
r
x
are evaluated at the junction. By comparing Figs.
2(b)
and
2(d)
, a clear correspondence can be observed between
the transmission coupling coefficients
|
t
MG
|
2
and
|
t
GM
|
2
from the
M-I-M
TM
0
to the graphene plasmon and from
the graphene plasmon to the
M-I-M
TM
0
, respectively. As
the graphene Fermi level (
E
F
) is increased, the electromag-
netic fields of the graphene-plasmon mode are less tightly
confined to the graphene surface [
20
], and thus match bet-
ter with the field profile of the TM
0
mode. As a result,
their coupling efficiency rises from 4% to 9% as the Fermi
level is varied from
E
F
=
0.3 eV to 0.5 eV. We also point
out that
|
r
MG
|
2
+|
t
MG
|
2
is only around 0.8, which indi-
cates that almost 20% of the incident power carried by
the
M-I-M
TM
0
mode is transferred to the background
modes, which include the continuum of unbound radiation
and the weakly bound gold surface plasmons. In contrast,
the graphene plasmons do not significantly scatter into
the background modes (
|
r
GM
|
2
+|
t
GM
|
2
1). Finally, we
note that the vertical air-metal interface at the waveguide
junction imposes a
π
phase shift upon reflection of the
graphene plasmon
r
GM
π)
, but little phase change on
the reflecting
M-I-M
TM
0
mode
r
MG
0
)
, as shown in
Figs.
2(c)
and
2(e)
.
B. Fano interference between multiple modes in
the gap
When two
M-I-M
waveguides are slightly separated
from each other with a graphene-covered gap in between,
a complex mode-coupling dynamics occurs between the
guided
M-I-M
TM
0
mode and the gap modes. This effect
produces a nontrivial Fermi-level dependence on the trans-
mission from input to output ports. As shown in Fig.
3(a)
,
the overall transmittance of an incident TM
0
wave in the
M-I-M
G-I-M
M-I-M
geometry now displays a number
of sharp peaks and dips at particular
E
F
values. Most
notably, the overall transmittance shows a sharp peak
at
|
E
F
|=
E
max
=
0.38 eV, but suddenly drops down and
almost vanishes at
|
E
F
|=
E
min
=
0.395 eV. The overall
absorption and radiation loss in the active region of the
device are also calculated and plotted in Fig
3(b)
.The
absorption loss is calculated by integrating the ohmic
power dissipation in graphene, and the radiation loss
is obtained by integrating the Poynting vector of the
(c)
(d)
(e)
0
1
Au
Au
SiO
2
0.1
0.2
0.3
0.4
0.5
0
5
10
15
20
25
0.1
0.2
0.3
0.4
0.5
0
10
20
30
40
50
0
20
40
60
80
100
0
10
20
30
40
50
(a)
(b)
c
d
e
d
e
c
FIG. 3. (a),(b) The transmittance (red solid), reflectance (blue
dashed), absorption (purple solid), and radiation loss (orange
dashed) of the plasmonic modulator. The geometrical parame-
ters are chosen as
d
=
h
=
100 nm and
L
=
140 nm. (c–e) The
amplitude of the electric field
|
E
x
|
at (c)
E
F
=
0.385 eV (the fun-
damental resonance mode), (d) at 0.3 eV (off resonance), and (e)
at 0.22 eV (second-order resonance mode), all plotted in the same
scale.
054053-3
MIN SEOK JANG
et al
.
PHYS. REV. APPLIED
10,
054053 (2018)
radiated waves directed to the upper-half space. Unlike
the asymmetric line shape of transmission, the absorp-
tion exhibits a symmetric Lorentzian peak centered at
|
E
F
|=
E
res
=
0.385 eV. Recognizing the fact that the major
absorption occurs in the graphene layer, we attribute the
absorption peak at
|
E
F
|=
E
res
to the Fabry-Perot res-
onances of graphene plasmons in the waveguide gap.
By plotting the electric-field distribution of the device
at 7.5
μ
mfor
E
F
values corresponding to the absorp-
tion peaks [Figs.
3(c)
and
3(e)
], we find an intense, sin-
gle node E field distribution at
|
E
F
|=
0.385 eV, while
for
|
E
F
|=
0.22 eV, a double-node structure is revealed,
indicating first- and second-order plasmonic resonances,
respectively. In contrast, when the system is off resonance
at
|
E
F
|=
0.3 eV, the field concentration in the gap region
is negligible, as exemplified in Fig.
3(d)
.
While the absorption of the
M-I-M
G-I-M
M-I-M
junction is associated with the Fabry-Perot resonance in
the
G-I-M
cavity, the line shape of the overall transmission
displayed in Fig.
3(a)
requires an understanding of the mul-
tiple possible transmission pathways. The first transmis-
sion channel is through the resonant graphene-plasmonic
mode, and its transmission coefficient,
t
G
, can be analyti-
cally obtained from the Fabry-Perot interferometer model
in terms of the transmission and reflection coefficients at
the
M-I-M
G-I-M
junction,
t
G
=
t
MG
t
GM
exp
(
in
G
k
0
L
)
1
r
2
GM
exp
(
2
in
G
k
0
L
)
,
where
k
0
is the free-space wavenumber. As seen in
Fig.
4(a)
, the amplitude of
t
G
exhibits a conventional
symmetric-resonance curve centered at
|
E
F
|=
0.385 eV
that agrees well with the absorption maximum obtained
from the full-wave simulations. The second transmis-
sion channel is through the background modes of the
junction—both free-space modes and conventional metal-
surface plasmons. The transmission coefficient through
these modes,
t
B
, is calculated numerically, and approxi-
mated as a Femi-level independent constant
t
0
B
as shown
in Fig.
4(a)
.
Unlike the graphene-plasmon transmission, the back-
ground transmission is of a nonresonant nature, and slowly
varies as a function of
E
F
in order to maintain eigen-
modes’ orthogonality in the waveguide as the variation in
the graphene-plasmon mode alters the spatial profiles of
the background modes. At the same time, the overall trans-
mission across the
G-I-M
segment is highly dependent
on the phase difference between the modes of two trans-
mission channels. When the Fermi energy of graphene is
lower than
E
res
, the
t
B
-and
t
G
-associated modes interfere
constructively as they are roughly in phase, as shown in
Fig.
4(b)
. The phase of the graphene-plasmon transmission
is flipped by
π
across the resonance while the phase of the
background transmission experiences little change, leading
Analytic
Simulation
0.3
0.35
0.4
0.45
0.5
0.0
0.1
0.2
0.3
0.4
0.5
0.0
0.1
0.2
0.3
0.3
0.35
0.4
0.45
0.5
-0.8
-0.6
-0.4
0
-1.0
-0.8
-0.6
-0.4
-0.2
-0.2
-1.0
(c)
(d)
(a)
(b)
0.4
Analytic
Simulation
FIG. 4.
(a) The amplitude of the transmission through the
graphene-plasmon resonance (red solid), and through the back-
ground modes (blue dashed) vs
E
F
. (b) The phase difference
between
t
G
and
t
0
B
,
φ
G
φ
0
B
=
arg
(
t
G
)
arg
(
t
0
B
)
. (c),(d) The
dependence of the amplitude and phase of the total transmis-
sion coefficient,
t
total
=
t
G
+
t
B
; the approximated analytic model
(red) is in good accordance with the full simulation result (blue
dashed) by capturing the distinctive asymmetric line shape.
to a destructive interference between two transmission pro-
cesses for
E
F
>
E
res
. As a consequence, the maximum and
minimum in overall transmission occur at either side of the
plasmon resonance peak, producing the asymmetric-line
shape as seen in Figs.
3(a)
and
3(b)
.
In order to clearly illustrate the mechanism of this res-
onant interference, we provide an approximated analytic
model in which we assume the background transmission
as a Fermi energy-independent constant
t
0
B
. The value
of
t
0
B
is estimated as the average of total transmission
coefficients at
E
F
=
0.3 and 0.5 eV, where the system
is far off resonance, and thus the contribution from the
plasmon transmission is minimal. The overall transmis-
sion coefficient is then obtained by simply adding both
graphene-plasmon and background contributions,
t
G
+
t
0
B
.
As shown in Fig.
4(b)
, the phase difference between two
transmission channels varies from around
0.2
π
(con-
structive) to
1.2
π
(destructive) as
E
F
sweeps across
E
res
. Figures
4(c)
and
4(d)
show that the resulting ana-
lytic model has a strong correspondence with the full-wave
simulation result, including the distinctive sharp-switching
behavior. This type of resonant interference, which was
first explained by Fano in the context of inelastic electron-
scattering processes [
24
], generally occurs when a dis-
crete state is embedded in a continuum [
25
,
26
]. In our
structure, the discrete state is represented by the resonant
graphene plasmons, while the continuum is represented by
the background modes including the unbound radiations.
C. Modulation of the resonant transmission
The sharp-modulation behavior described above allows
for large changes in transmission with very small changes
054053-4
MODULATED RESONANT TRANSMISSION . . .
PHYS. REV. APPLIED
10,
054053 (2018)
in graphene Fermi level (
E
F
=
0.015 eV), and therefore,
requires a small variation of gate voltage for the switch-
ing operation. Estimated carrier-concentration difference
between the maximum and minimum transmission states is
n
=
(
E
2
min
E
2
max
)/π

2
v
2
F
8.5
×
10
11
cm
2
.Thespe-
cific capacitance of the device is
C
G
=
34.5 nF/cm
2
,
leading us to the conclusion that the gate voltage
V
G
one needs to apply between the bottom metal layer and
graphene in order to switch the transmission from its mini-
mum to maximum is only
V
G
=
e
n
/
C
G
3.96 V. More-
over, the fundamental graphene-cavity mode involved in
the resonant transmission has an extremely small mode
volume. By conservatively assuming a diffraction-limited
width [
27
]
W
λ
0
, where
λ
0
is the free-space wave-
length, the active volume of the device is approximated
as
V
(
d
+
2
h
)
LW
10
3
λ
3
0
. This extreme miniaturiza-
tion stems from the highly confined nature of the graphene
plasmons. The small active area required for the resonant
transmission along with superior electrical transport prop-
erties of graphene could result in an exceedingly short
RC
time constant, which, in principle, enables a fast switching
speed.
The modulation intensity and the resonance condition of
the device can be altered by engineering the geometry of
the system. Most notably, the gap size
L
between the two
M-I-M
waveguides determines the Fermi energy at which
the graphene plasmons are in resonance (
E
res
) at a given
frequency [
21
]. From the dispersion relation of graphene
plasmons [
10
,
28
]
ω
E
1
/
2
F
k
1
/
2
G
, where
k
G
=
n
G
k
0
is the
wavenumber of the graphene plasmon, we deduce that an
increasing gap size
L
lowers the resonance frequency at a
fixed Fermi energy, but raises
E
res
at a fixed frequency. At
the same time, the devices with wider gaps show higher
modulation intensity as presented in Figs.
5(a)
and
5(b)
,
because higher
E
res
induces stronger plasmon resonance as
the coupling of the
M-I-M
TM
0
to the graphene plasmons
becomes more efficient (Fig.
2
). The thickness of the metal
cladding
h
and the dielectric core
d
of the
M-I-M
waveg-
uides do not significantly alter the resonance Fermi energy,
but affect the transmission characteristics by modulating
the relative phase and intensities of the graphene-plasmon
transmission and the background transmission. Thickening
the metal cladding
h
of the
M-I-M
waveguides monoton-
ically increases the maximum overall transmission
T
max
by suppressing the radiative loss and enforcing the back-
ground transmission. While a thicker dielectric core
d
is
also favorable for higher background transmission, the
thicker core broadens the
M-I-M
-guided mode, resulting in
worse spatial-mode matching with the graphene plasmons,
deteriorating
t
G
. The dependence of
T
max
on
d
is, therefore,
nonmonotonic, as summarized in Fig.
5(c)
.
We emphasize that the overall transmission can be
entirely suppressed by inducing total destructive inter-
ference between the graphene-plasmon and background
transmissions,
t
G
=−
t
B
. This condition for the zero
22
0.3
0.35
0.4
0.45
0.5
0
10
20
30
120
140
160
180
0.3
0.35
0.4
0.45
0.5
120 nm
170 nm
80
100
120
140
160
160
140
120
100
80
80
100
120
140
160
(%)
34
30
26
(dB)
≥60
40
20
0
(c)
(d)
(a)(b)
FIG. 5. (a) The transmittance
T
vs
E
F
for the gap size
L
from
120 nm (red) up to 170 nm (blue) with a 10-nm step; here, the
transmittances are obtained from full-wave simulations using the
finite element method. (b) Dependence of
E
res
on
L
; the analyt-
ically obtained
E
res
(red dashed) perfectly agrees with the FEM
simulation results (red circles). (c) The maximum transmittance
T
max
, and (d) the ratio of the maximum and minimum transmit-
tance
T
max
/
T
min
plotted as a function of
d
and
h
.The
T
max
/
T
min
values over the 60 dB correspond to the total destructive inter-
ference between the resonant (graphene) and nonresonant (back-
ground) transmission. In (c) and (d),
L
is set to 140 nm as in the
cases of Fig.
3
.
transmission can be achieved by carefully adjusting the
geometrical parameters of the system. The maximum
amplitude of the graphene plasmon transmission at res-
onance should be greater than the background transmis-
sion (
|
t
G
(
E
res
)
|
>
|
t
B
|
), and their phase difference
φ
G
φ
B
should also be controlled by tuning the mode-coupling
characteristics at the waveguide junctions. Figure
5(d)
plots the simulated ratio of the maximum and minimum
transmission
T
max
/
T
min
as a function of the core thick-
ness
d
and the cladding thickness
h
. Indeed, the on/off
ratio diverges for a certain set of parameter values as indi-
cated in Fig.
5(d)
, showing that the complete suppression
of transmission is achievable.
D. Modulation efficiency dependence on the graphene
quality
As a final remark, we discuss how the overall transmis-
sion depends on the carrier mobility of graphene. Since
the propagation loss of graphene plasmons is inversely
proportional to its carrier mobility [
20
], we calculate the
transmittance as varying the carrier mobility of graphene
μ
from 10
3
to 10
5
cm
2
V
1
s
1
depending on the fabrica-
tion method [
29
31
] and the substrate material [
32
,
33
].
Figure
6(a)
shows that
T
max
can be as high as 50% for
μ
=
5
×
10
4
cm
2
V
1
s
1
and decreases down to 11% at
μ
=
1000 cm
2
V
–1
s
–1
. In Fig.
6(b)
, the minimum nearly
054053-5
MIN SEOK JANG
et al
.
PHYS. REV. APPLIED
10,
054053 (2018)
0.3
0.35
0.4
0.45
0.5
0
10
20
30
40
50
0246810
0
5
10
15
20
50,000
20,000
10,000
5,000
2,000
(a)(b)
FIG. 6. (a) The transmittance
T
vs
E
F
for different carrier
mobility
μ
(as denoted). (b) Dependence of
T
max
(blue squares)
and
T
min
(red circles) on
μ
.
vanishes for
μ
5000 cm
2
V
–1
s
–1
, which corresponds to
the condition of
|
t
G
(
E
res
)
|≥|
t
B
|
.The
T
max
/
T
min
ratio is pre-
dicted to be 2.5 times lower for a low-quality graphene of
μ
=
1000 cm
2
V
1
s
1
.
III. CONCLUSION
In summary, we propose a graphene-based mid-infrared
plasmonic waveguide modulator, which exhibits a sharp
resonant transmission coming from the Fano interference
between the plasmon resonance in graphene and the back-
ground transmission. In the proposed device, the resonant
interference between the graphene-plasmon transmission
and the background transmission can entirely suppress the
overall transmission, resulting in a very high-modulation
efficiency, which requires a very small change in the
graphene Fermi level (on the order of 10 meV) for switch-
ing. Moreover, the active volume of the device is about
a thousand times smaller than the diffraction limit, mak-
ing it a promising building block for deep-subwavelength
ultrafast optical-integrated devices.
ACKNOWLWDGMENTS
The authors acknowledge support from the National
Research Foundation of Korea (NRF) (Grant No.
2017R1E1A1A01074323, M.S.J.) and KAIST Global
Center for Open Research with Enterprise (GCORE)
(Grant No. N11180017, S.G.M.) funded by the Ministry
of Science and ICT, and Basic Science Research Program
through NRF funded by the Ministry of Education (Grant
No. 2017R1D1A1B03034762, M.S.J). M.S.J., V.W.B., and
H.A.A. acknowledge support from the Air Force Office of
Scientific Research through Grant No. FA9550-16-1-0019.
V.W.B. thanks the Wisconsin Alumni Research Foundation
for support.
M.S.J., and S.K. contributed equally to this work
[1] M.Liu,X.Yin,E.Ulin-Avila,B.Geng,T.Zentgraf,L.
Ju, F. Wang, and X. Zhang, A graphene-based broadband
optical modulator,
Nature
474
, 64 (2011).
[2] C. T. Phare, Y.-H. D. Lee, J. Cardenas, and M. Lipson,
Graphene electro-optic modulator with 30 GHz bandwidth,
Nat. Photonics
9
, 511 (2015).
[3] V. Sorianello, M. Midrio, G. Contestabile, I. Asselberghs, J.
Van Campenhout, C. Huyghebaert, I. Goykhman, A. K. Ott,
A. C. Ferrari, and M. Romagnoli, Graphene–silicon phase
modulators with gigahertz bandwidth,
Nat. Photonics
12
,
40 (2018).
[4] F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo,
Graphene plasmonics: A platform for strong light–matter
interactions,
Nano Lett.
11
, 3370 (2011).
[5] A. N. Grigorenko, M. Polini, and K. S. Novoselov,
Graphene plasmonics,
Nat. Photonics
6
, 749 (2012).
[6] F. J. García de Abajo, Graphene plasmonics: Challenges
and opportunities,
ACS Photonics
1
, 135 (2014).
[7] J. Chen, M. Badioli, P. Alonso-González, S. Thongrat-
tanasiri, F. Huth, J. Osmond, M. Spasenovi
́
c, A. Centeno,
A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J.
García de Abajo, R. Hillenbrand, and F. H. L. Koppens,
Optical nano-imaging of gate-tunable graphene plasmons,
Nature
487
, 77 (2012).
[8] Z.Fei,A.S.Rodin,G.O.Andreev,W.Bao,A.S.McLeod,
M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G.
Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F.
Keilmann, and D. N. Basov, Gate-tuning of graphene plas-
mons revealed by infrared nano-imaging,
Nature
487
,82
(2012).
[9] P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno,
A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens,
A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand,
Controlling graphene plasmons with resonant metal anten-
nas and spatial conductivity patterns,
Science
344
, 1369
(2014).
[10] L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H.
A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang,
Graphene plasmonics for tunable terahertz metamaterials,
Nat. Nanotechnol.
6
, 630 (2011).
[11] S. Thongrattanasiri, F. H. L. Koppens, and F. J. García
de Abajo, Complete Optical Absorption in Periodically
Patterned Graphene,
Phys. Rev. Lett.
108
, 047401 (2012).
[12] H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F.
Guinea, P. Avouris, and F. Xia, Damping pathways of
mid-infrared plasmons in graphene nanostructures,
Nat.
Photonics
7
, 394 (2013).
[13] V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A.
Atwater, Highly confined tunable mid-infrared plasmonics
in graphene nanoresonators,
Nano Lett.
13
, 2541 (2013).
[14] M. S. Jang, V. W. Brar, M. C. Sherrott, J. J. Lopez, L. Kim,
S. Kim, M. Choi, and H. A. Atwater, Tunable large reso-
nant absorption in a midinfrared graphene Salisbury screen,
Phys.Rev.B
90
, 165409 (2014).
[15] S. Kim, M. S. Jang, V. W. Brar, Y. Tolstova, K. W. Mauser,
and H. A. Atwater, Electronically tunable extraordinary
optical transmission in graphene plasmonic ribbons cou-
pled to subwavelength metallic slit arrays,
Nat. Commun.
7
, 12323 (2016).
[16] S. Kim, M. S. Jang, V. W. Brar, K. W. Mauser, L. Kim, and
H. A. Atwater, Electronically tunable perfect absorption in
graphene,
Nano Lett.
18
, 971 (2018).
[17] Z. Fei, M. D. Goldflam, J.-S. Wu, S. Dai, M. Wagner, A.
S. McLeod, M. K. Liu, K. W. Post, S. Zhu, G. C. A. M.
054053-6
MODULATED RESONANT TRANSMISSION . . .
PHYS. REV. APPLIED
10,
054053 (2018)
Janssen, M. M. Fogler, and D. N. Basov, Edge and surface
plasmons in graphene nanoribbons,
Nano Lett.
15
, 8271
(2015).
[18] M. Gullans, D. E. Chang, F. H. L. Koppens, F. J. García de
Abajo, and M. D. Lukin, Single-Photon Nonlinear Optics
with Graphene Plasmons,
Phys.Rev.Lett.
111
, 247401
(2013).
[19] E. D. Palik,
Handbook of Optical Constants of Solids
(Academic Press, New York, 1998).
[20] M. Jablan, H. Buljan, and M. Soljacic, Plasmonics in
graphene at infrared frequencies,
Phys.Rev.B
80
, 245435
(2009).
[21] See Supplemental Material at
http://link.aps.org/supple
mental/10.1103/PhysRevApplied.10.054053
for details on
the electromagnetic simulation methods, optical conductiv-
ity of graphene, and discussion of the operation frequency
window and graphene plasmon resonance.
[22] M. Born and E. Wolf,
Principles of Optics: Electromag-
netic Theory of Propagation, Interference and Diffrac-
tion of Light
(Cambridge University Press, Cambridge,
1999).
[23] L. A. Falkovsky and A. A. Varlamov, Space-time disper-
sion of graphene conductivity,
Eur. Phys. J. B
56
, 281
(2007).
[24] U. Fano, Effects of configuration interaction on intensities
and phase shifts,
Phys. Rev.
124
, 1866 (1961).
[25] B. Luk’yanchuk, M. I. Zheludev, S. A. Maier, N. J. Halas,
P. Nordlander, H. Giessen, and C. T. Chong, The Fano
resonance in plasmonic nanostructures and metamaterials,
Nat. Mater.
9
, 707 (2010).
[26] E. Miroshnichenko, S. Flach, and Y. S. Kivshar, Fano res-
onances in nanoscale structures,
Rev. Mod. Phys.
82
, 2257
(2010).
[27] S. A. Maier, Effective mode volume of nanoscale plasmon
cavities,
Opt. Quant. Electron.
38
, 257 (2006).
[28] E. H. Hwang and S. D. Sarma, Dielectric function, screen-
ing, and plasmons in two-dimensional graphene,
Phys. Rev.
B
75
, 205418 (2007).
[29] S. Park and R. S. Ruoff, Chemical methods for the produc-
tion of graphenes,
Nat. Nanotechnol.
5
, 217 (2010).
[30] J. Kim, H. Park, J. B. Hannon, S. W. Bedell, K. Fogel,
D. K. Sadana, and C. Dimitrakopoulos, Layer-resolved
graphene transfer via engineered strain layers,
Science
342
,
833 (2013).
[31] J. H. Lee, E. K. Lee, W. J. Joo, Y. Jang, B. S. Kim, J. Y.
Lim, S. H. Choi, S. J. Ahn, J. R. Ahn, M. H. Park, C. W.
Yang, B. L. Choi, S. W. Hwang, and D. Whang, Wafer-scale
growth of single-crystal monolayer graphene on reusable
hydrogen-terminated germanium,
Science
344
, 286 (2014).
[32]K.I.Bolotin,K.J.Sikes,J.Hone,H.L.Stormer,and
P. Kim, Temperature-Dependent Transport in Suspended
Graphene,
Phys. Rev. Lett.
101
, 096802 (2008).
[33] C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S.
Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shep-
ard, and J. Hone, Boron nitride substrates for high-quality
graphene electronics,
Nat. Nanotechnol.
5
, 722 (2010).
054053-7