1
Supplementary Information:
Tunable Large Resonant Absorption in a Mid
-
I
nfrared
Graphene Salisbury Screen
Min Seok Jang
1,
†
, Victor W. Brar
2,3, †
, Michelle C. Sherrott
2
, Josue J. Lopez
2
, Laura
B
. Kim
2
,
Seyoon Kim
2
, Mansoo Choi
1,4
, and Harry A. Atwater
2,3
†
These authors contributed equally.
1) Global Frontier Center for Multiscale Energy Systems, Seoul National University, Seoul 151
-
747, Republic of Korea
2) Thomas J. Watson Laboratory of Applied Physics, California Institute of Technology,
Pasadena, CA 911
25
3) Kavli Nanoscience Institute, California Institute of Technology, Pasadena, CA91125
4) Division of WCU Multiscale Mechanical Design, School of Mechanical and Aerospace
Engineering, Seoul National University, Seoul 151
-
742
2
I. Device Fabrication
SiN
x
membranes were obtained commercially from Norcada, part #NX10500F. Electron beam
lithography at 100keV is used to pattern nanoresonator arrays in PMMA spun coated onto the
devices, and the pattern is transferred to the graphene via an oxygen plasma
etch. Our resonators
have widths varying from 20
–
60nm, with 9:1 aspect ratios and a pitch of
2
-
2.5 times
the width.
The resonators are spanned perpendicularly by graphene crossbars of a width equal to the
nanoresonator width. This aids conductivity a
cross the patterned arrays despite occasional
cracks and domain boundaries in the CVD
graphene sheet.
I
I
. Electromagnetic Simulations
We solve Maxwell’s equation by using finite element method.
G
raphene
is modeled
as a thin
layer of
the thickness
휏
and impose
the relative permittivity
휖
퐺
=
1
+
푖휎
/
(
휖
0
휔휏
)
. In actual
calculation,
휏
is chosen to be 0.1 nm which shows good convergence with respect to the
휏
→
0
limit
as seen in Figure S1
. The complex optical conductivity of graphene
휎
(
휔
)
is
evaluated
within local random phase approximation.
[
1
]
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
)
.
Here, the temperature
푇
is set as 300K. The intraband scattering rate
Γ
takes into account
scattering by impurities
Γ
imp
and by optical phonons
Γ
oph
.
By analyzing the absorption peak
width when the resonance energy is much lower than the graphene optical phonon energy
(~1600cm
-
1
), the impurity scattering rate can be
approximated
to be
Γ
imp
=
푒
푣
퐹
/
휇
√
푛휋
with the
mobility
휇
=
550cm
2
/Vs.
[
2
]
The rate of optical phonon scattering is estimated from theoretically
obtained self
-
energy
Σ
oph
(
휔
)
, as
Γ
oph
(
휔
)
=
2
Im
[
Σ
oph
(
휔
)
]
.
[
2
-
4
]
The frequency dependent
3
dielectric function
s
of Au and SiN
x
are taken from Palik
[
5
]
and Cataldo et al.
[
6
]
, respectively.
Finally, a constant factor, which accounts for experimental imperfections such as dead resonators
in the actual device, is multiplied to the simulated spectra.
Th
is
degradation factor is determined
to be 0.72 by
comparing simulation and measurement. Figure S
2
shows that the resulting
theoretical absorption spectra
reproduce
quite
well the experimental data.
II
I
. Determination of Carrier Density
The carrier density of
the
nanoresonators
was determined by
fitting
the peak frequencies of the
simulated absorption spectr
a
to the experimentally measured
absorption peaks of resonators
fabricated with different widths. The resulting carrier density values are comparable to those
calculated using a simple
parallel plate
capacitor model
with a 1
μ
m thick SiN
x
dielectric, as
shown in Figure S
3
a, yet there is some deviation
. We can attribute the
discrepancies
to a number
of possible effects. First, our SiN
x
membranes were obtained from a
commercial supplier
(Norcada) and t
heir stoichiometry and resulting DC dielectric constant
,
κ
, is not precisely known.
This allows for a range of possible values for
κ
, which can lead to significant differences
in the
induced carrier density in a graphene device. Second, our measurement
s
were performed under
FTIR purge gas (free of H
2
O and CO
2
), but atmospheric impurities were likely still present
during the measurements
. Those types of impurities have previously been shown to induce
hysteresis
effects in the conductance curves of graphen
e FET devices,
[
7
-
10
]
and we observe
similar behavior in our devices, as shown in Figure S
3
b. Because the concentration of those
impurities can depend on the applied gate bias, they can also
alter
the carrier density vs
. gate bias
curves, and in Figure S
3
a we have included a theoretical est
imate of those effects.
[
9
]
In
addition to atmospheric i
mpurities, the SiN
x
surface itself can contain charge traps that fill or
empty with the applied gate bias. Such charge traps could induce anomalous behavior in the
conductance curves of the graphene FET devices, similar to what has been observed in the
pr
esence of metallic impurities.
[
11
]
Finally, we
note that we
have removed some of the
graphene surface area in the process of fabricati
ng the nanoresonators. This
difference in total
available surface area should alter the carrier density dependence assumed by the capacitor
model, such that more charge is likely packed into a smaller area. In Figure S
3
a we have
4
provided a simple estimate of this effect based on the
assumption that an equal amount of
induced carriers are distributed equally across the smaller available surface area, leading to larger
carrier densities. However, theoretical predictions have shown that the extra carrier density
should preferably accum
ulate on the edges of the graphene nanoresonators, and thus alter their
plasmonic resonances in more sophisticated ways.
[
12
]
I
V
. Peak Width Analysis
In Figure S
4
a
, we plot the full with at half maximum (FWHM) of the absorption peaks of
graphene nanoresonator arrays with various sizes and doping levels.
The linewidth, which can be
interpreted as the plasmon scattering rate, almost monotonically increases with increa
sing
resonance frequency and decreasing resonator
width
.
T
he lifetime of plasmon is estimated as
10
-
50 fs
from inverse linewidth
.
When the substrate medium is lossless and dispersionless, the scattering rate of graphene
plasmon is simply equal to the elect
ron scattering rate. However, in our
sample
, the interaction
with
SiN
x
substrate
polar
phonon
s
results in a deviation of
the plasmon scattering rate from the
electron scattering rate.
Therefore, we
extract the intraband electron scattering rate (
Γ
)
b
y
fitting
the
FWHM of the simulated spectrum to the measured
plasmon
linewidth, as shown in figure
S
4
b.
W
e
found
that
there is no noticeable difference in
the electron scattering rates
among
nanoresoantors wider than 40nm. Because those nanoresonators
oscillate
at frequencies much
lower than graphene optical phonon (~1600cm
-
1
),
the dominant damping mechanism in this
regime is scattering
from
impurities.
[
2
,
3
]
The
average
carrier mobility
휇
, converted from the
electron scattering rate via
휇
=
푒푣
퐹
/
Γ
√
푛휋
, is
determined
as 550cm
2
/Vs
with stan
dard deviation
50cm
2
/Vs.
On the other hand, at frequencies higher than 1600cm
-
1
,
the electron scattering rate of
20nm nanoresonators dramatically increases as the carrier density increases (and thus the
plasmon
frequency increases), possibly due to coupling with graphene optical phonons.
5
V. Derivation of Surface Admittance
of a Thin Layer
Consider a
thin layer of thickness
휏
and admittance
푌
GR
sitting
atop a dielectric with thickness
푑
and admittance
푌
SiN
x
deposited on a reflecting mirror
as diagramed in the inset of Figure 1a
. For
normally incident light, the effective
surface
admittance of the stack is given
by
[
13
,
14
]
푌
=
푌
GR
푌
′
SiN
x
−
푖
푌
GR
tan
(
푘
1
휏
)
푌
GR
−
푖
푌
′
SiN
x
tan
(
푘
1
휏
)
,
w
here
푌
GR
=
√
휖
GR
휇
GR
⁄
and
푘
1
=
휔
√
휖
GR
휇
GR
are the wave admittance and
the
wavevector
insi
de the thin sheet, respectively.
푌
′
SiN
x
is
the effective admittance of the dielectric as viewed
from the position of the sheet
, and is given by
푌
′
SiN
x
=
푌
SiN
x
cot
(
푘
2
푑
)
, where
푘
2
is the
wavevector inside the SiN
x
layer
. For frequencies such that
푑
=
푚휆
/
4
and for
푘
1
휏
≪
1
, then
푌
′
SiN
x
≪
푌
GR
and
tan
(
푘
1
휏
)
→
푘
1
휏
, and
the above equation
reduces to
Y
=
−
iωετ
.
V
I
. Calculation of Surface Admittance of Graphene Nanoresonator Arrays
The surface admittance
Y
= −
iωετ
of a graphene nanoresonator array is equivalent to its effective
sheet conductivity
휎
eff
,
which
can be evaluated from the far
-
field transmission and reflection
coefficients.
Consider a homogeneous thin film of conductivity
휎
eff
place
d
on the interface
(
푧
=
0
)
between air
(
푧
<
0
)
and SiN
x
(
푧
>
0
)
and
a
plane wave
polarized along
x
direction is
normally incident on the surface.
The surface parallel electric field is continuous
퐸
푥
0
+
=
퐸
푥
0
−
at
the interface, while the magnetic fields are discontinuous due to the surface current,
퐻
푦
0
+
−
퐻
푦
0
−
=
휎
eff
퐸
푥
(
푧
=
0
)
=
푌
퐸
푥
(
푧
=
0
)
.
From these boundary conditi
ons,
the transmission (
t
) and
reflection (
r
)
coefficients satisfy the following
equations,
1
+
푟
=
푡
,
(
1
−
푟
)
−
푛
SiN
x
푡
=
(
푌
푌
0
)
푡
,
6
where
푛
SiN
x
is the refractive index of SiN
x
.
The normalized surface admittance
푌
/
푌
0
is then
solely written in terms of transmission coefficient
,
푌
푌
0
=
2
푡
−
1
−
푛
Si
N
x
.
Because the fields of graphene plasmons are tightly confined near the surface with characteristic
decay length similar to the width of the
nano
resonators
,
w
e record the
electric field of the
transmitted wave at
a position sufficiently far from the surface (
푧
0
=
1
um
)
in order to exclude
the evanescent field of graphene plasmons.
The far field transmission coefficient is then obtained
by accounting for the propagation factor
푡
=
퐸
x
(
푧
0
)
exp
[
−
푖
푛
Si
N
x
푘
0
푧
0
]
퐸
0
(
0
)
,
where
퐸
0
is the
electric field of incident wave at the surface.
Figure S
5
plots
the resulting complex surface admittance of a graphene nanoribbon array
as a
function of frequency
.
As in
figure 4, both ribbon width and the spacing between the ribbons are
set as 40nm.
On resonance,
Im
[
푌
]
crosses zero, while
Re
[
푌
]
, which
is directly proportional to
the absorption cross section
휎
Abs
of individual resonator,
has its maximum
.
As graphene
be
comes less lossy, the
plasmon
resonance
gets
sharper and stronger.
7
Figure S
1
:
Convergence of electromagnetic simulations with graphene thickness
.
Finite
element method (FEM) calculations for plasmon
peak frequency (top), line
-
width (middle), and
maximum absorption
difference
(bottom)
of
graphene nanoribbons
with varying width
under
8
normal incidence
. Four different thicknesses of graphene (
τ
= 0.05, 0.1, 0.34, and 1.0 nm) are
investigated. The calcula
tions for
τ
= 0.1 nm and 0.05 nm are almost
identical
, assuring that the
simulations converge to
τ
→0
limit. For all cases,
the carrier concentration
is
1.42×10
13
cm
-
2
.
Figure S
2
:
Comparison between experimental and theoretical absorption spectr
a
.
(a)
Experimental and (b) theoretical change in absorption with respect to the absorption at the charge
neutral point (CNP) in 40nm wide graphene nanoresonators at various doping levels. (c)
Experimental and (d) theoretical absorption difference spectra wit
h the carrier concentration of
1.42×10
13
cm
-
2
. The width of the resonators varies from 20 to 60nm.
9
Figure S
3
:
Carrier
density
and resistance
versus back gate voltage.
(
a
) Theoretical (lines)
carrier density dependence on gate bias
for a graphene
FET device on
a 1
μ
m thick SiN
x
membrane
with dielectric properties spanning those reported in literature.
[
15
]
The blue line
indicates a graphene/SiN
x
device that includes estimated doping effects due to atmospheric and
substrate impurities that have been reported in SiO
2
.
[
7
-
10
]
The black dotted line models a
graphene/SiN
x
device that contains a graphene surface patterned such that 45
% of the sheet has
been removed
and the surface charge is concentrated into a smaller area. The triangles indicate
the calculated carrier densities of our device
determined by fitting the simulated peak position to
the experimental
results.
(b) Hysteresis effects in graphene/SiN resis
tance as applied gate bias
swept up (dotted line) and down (solid line). For (a, triangles), t
he CNP was assumed to occur at
+80V, halfway between the two hysteric peaks.
10
Figure S
4
:
Peak width and electron scattering rate.
Frequency dependence of
(a) t
he
full
width at half maximum (FWHM) linewidth of the absorption peaks
, and
(b) the
fitted
electron
scattering rate.
The r
esonator width ranges from 20 to 60 nm and the carrier density varies from
0.66×10
13
to 1.42×10
13
cm
-
2
.
11
Figure S
5
:
Surface
admittance versus frequency.
Frequency dependence of
Re
[
푌
/
푌
0
]
(solid)
and
Im
[
푌
/
푌
0
]
(dashed) for
휇
=
550 (blue) and 4,000cm
2
/Vs (red).
An array of infinitely long
graphene nanoribbons (40nm width, 40nm spacing) is assumed. T
he carrier density
is set to
1.42×10
13
cm
-
2
.
12
Figure S
6
:
Electric field distribution.
Theoretical electric field profile of a 40nm graphene
nanoresonator with the highest achieved carrier density (1.42×10
13
cm
-
2
), obtained from an
electromagnetic simulation assuming normal incidenc
e. The quarter wavelength condition and
plasmon resonance coincide at 1400cm
-
1
(left). At 2335cm
-
1
, the optical thickness of SiN
x
is
roughly half wavelength, resulting in vanishing electric field at the surface (middle). When the
optical thickness of SiN
x
becomes three quarters of the wavelength, the surface electric field is
maximized again, but the higher order plasmon resonance at this frequency is very weak (right).
13
Figure S
7
:
P
eak absorption
versus
carrier density
and resonator width
.
Maximum
theoretical
absorption in a graphene nanoribbon array as a function of
the carrier density
and the resonator
width.
The carrier density 1.0
−
7.0
×10
13
cm
-
2
,
which
is equivalent
to the
Fermi energy
of
0.37
−
0.98eV
.
Increasing carrier density leads to better coupling between the incoming light and
the graphene plasmons, resulting in stronger plasmon resonance. Therefore
,
higher doping
tends
to
enhance the absorption performance
.
The spacing between ribbons is equal to
the ribbon width
and the SiN
x
thickness is set to 1um
. The carrier mobility is assumed to be 550cm
2
/Vs, and the
interaction with graphene optical phonon is considered
.
[
2
-
4
]
14
Figure S
8
:
I
ncidence angle dependence
.
(a)
Theoretical change in absorption with respect to the
absorption at the charge neutral point in 40nm wide graphene
nanoribbons
for
various incidence
angle
s (
θ
)
and polarization
s
. (b) Dependence of theoretical maximum absorption difference on
the incidence angle for
s
(blue) and
p
(red) polarized illumination. The average maximum
absorption (
black) does not
vary
much
for
θ
≤
35
°
, which corresponds to the numerical aperture
of the
objective used in the experiment (N.A
.
= 0.58).
The carrier
density and the
mobility
are
assumed to be
1.42
×10
13
cm
-
2
and
550cm
2
/Vs
, respectively.
15
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