S u p p l e m e n t a
l
M
a t e r i a l
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, CA 91125,
United States
3
Department of Physics, University of Wisconsin
-
Madison, Madison, WI
53706, United States
4
Kavli Nanoscience Institute, California Institute of Technology, Pasadena, CA 91125, United States
†
These authors contributed equally to this work
*
jang.minseok@kaist.ac.kr,
**
haa@caltech.edu
1.
Electromagnetic Simulations
The elect
romagnetic response of the structure
was numerically simulated using COMSOL Multiphysics
–
a commercial full
-
wave simulation software based on finite element method (FEM). The port boundary
condition was applied
to the entrance portion of the input MIM waveguide in order to launch the TM
0
mode and to make it transparent to the reflected TM
0
mode. The same boundary condition was applied
to the exit portion of the output MIM waveguide without the mode excitation. T
he half space over the
device was enclosed by perfectly matched layers (PMLs) to mimic the open boundaries for the unbound
waves radiat
ing
from the waveguide gap. We adopted the triangular mesh,
the size of
which
was chosen
to
range from 1 to 10 nm in the
MIM waveguide region, from 0.2 to 5 nm in the waveguide gap region,
and from 5 to 200 nm in the air region to ensure that the mesh size
was
sufficiently smaller than the
spatial variation of the electromagnetic fields. This was confirmed by checking the si
mulation conversion,
so that further mesh refinement had only a negligible effect on the resulting electromagnetic fields profile.
2. Optical Conductivity of Graphene
In our
FEM
simulations, graphene is modeled as a thin layer
with a thickness
of
훿
= 0.3 nm
and
possessing the relative permittivity
휖
퐺
=
1
+
푖휎
/
(
휖
0
휔훿
)
. The complex optical conductivity of graphene
휎
(
휔
)
is evaluated within the local random phase approximation [
23
]:
휎
(
휔
)
=
2
푖
푒
2
푇
휋
ℏ
(
휔
+
푖
Γ
)
log
[
2
cosh
(
퐸
퐹
2
푇
)
]
+
푒
2
4
ℏ
[
퐻
(
휔
2
)
+
4
푖휔
휋
∫
푑휂
∞
0
퐻
(
휂
)
−
퐻
(
휔
2
)
휔
2
−
4
휂
2
]
,
where
퐻
(
휂
)
=
sinh
(
휂
/
푇
)
cosh
(
퐸
퐹
/
푇
)
+
cosh
(
휂
/
푇
)
.
The temperature
푇
is set as 300 K, and the intraband scattering rate is
Γ
=
푒
푣
퐹
/
휇
√
푛휋
, where
휇
is the
carrier mobility of graphene [
20 in the main text
]. Here we ignore the scattering by th
e optical phonons
in graphene because our target frequency
휔
=
0.165eV is lower than the bottom of the graphene optical
phonon band
휔
oph
~
0.2eV [
34
]
,
as explained in the Section 3 below
. The dielectric functions of Au
and SiO
2
are adopted from Palik [
19
].
3. Selection of the Operation Frequency Window
The operating frequency range of our devices
is
fundamentally
limited by two kinds of phonons: the
polar phonons in the SiO
2
substrate (having energy
of ~
0.133 eV [
19
]) and the intrinsic optical phonons
in graphene (having energy
of ~
0.2 eV [
20
]). Figure S1 shows the real and
the
imaginary permittivity of
SiO
2
as reported by Palik [
19
]. One finds that, although the main phonon peak is located around 0.13 e
V,
the real part of the permittivity is still negative up to 0.155 eV. Therefore, in order to guarantee
a
low
plasmonic loss, the operation frequency should be above 0.16 eV. Regarding the graphene optical
phonons, due to the thermal broadening of 0.026 eV
at room temperature, the interaction between the
graphene plasmons and optical phonons starts to manifest itself at around 0.175 eV. Therefore, the actual
frequency window of low
-
loss operation is between 0.16 eV (
λ
0
≈ 7.75 μm) and 0.175 eV (
λ
0
≈ 7.1 μm).
If the operational wavelength approach
es
either of the phonon modes, the device performance may
deteriorate due to the phonon
-
induced damping. Therefore, we selected the operation frequency near the
middle of the phonon
-
free energy band at 0.165 eV (
λ
0
≈
7.5 μm).
Fig. S1. Permittivity of SiO2 measured by Palik [
19
].
4
.
Condition for Graphene Plasmon Resonance
The
graphene Fermi level
E
res
corresponds to a plasmonic resonance at which the denominator of the
expression for
t
G
,
1
−
푟
GM
2
exp
(
2
푖
푛
퐺
푘
0
퐿
)
, is
minimized
, providing the maximum transmission
.
Then,
t
he condition for the
lowest order resonance is
given by
휙
GM
푟
(
퐸
res
)
+
Re
{
푛
퐺
(
퐸
푟푒푠
)
}
푘
0
퐿
=
2
휋
,
where
n
G
is the effective index and
휙
GM
푟
(
퐸
res
)
is the refl
ection phase of graphene plasmon.
For
퐸
퐹
≫
푘
퐵
푇
and
휔
≫
훾
, where
훾
is the scattering rate of electrons in graphene, the wavevector of
the
propagating
graphene plasmon mode can be analytically expressed as [
20
]
:
푛
퐺
푘
0
=
휋
ℏ
2
휔
2
휖
0
(
1
+
휖
SiO2
)
푒
2
퐸
퐹
.
However, t
he reflection phase,
휙
GM
푟
(
퐸
res
)
, is difficult
to express analytically because the calculation
involves boundary conditions with non
-
trivial geometry
,
and thus we calculate
휙
GM
푟
(
퐸
res
)
numerically
in this work.