Tunable
Large Resonant Absorption in a Mid-IR
Graphene Salisbury Screen
Min Seok Jang
1,
†
, Victor W. Brar
2,3, †
, Michelle
C.
Sherrott
2
, Josue J. Lopez
2
, Laura K. 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-
744, Republic of Korea
2) Thomas J. Watson Laboratory of Applied Physics, California Institute of Technology,
Pasadena, CA 91125
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
Enhancing the interaction strength between graphene and light is an important objective
for those seeking to make graphene a relevant material for future optoelectronic
applications. Plasmonic modes in graphene offer an additional pathway of directing
optical energy into the graphene sheet, while at the s
ame time displaying dramatically
small optical confinement factors that make them an interesting means of coupling light to
atomic or molecular emitters. Here we show that graphene plasmonic nanoresonators can
1
be placed a quarter wavelength from a reflect
ing surface and electronically tuned to mimic
a surface with an impedance closely matched to freespace (
Z
0
= 377 Ω)
. This geometry –
known in early radar applications as a Salisbury screen – allows for an order of magnitude
(from 2.3 to 24.5%) increase of the optical absorption in the graphene and provides an
efficient means of coupling to the highly confined graphene plasmonic modes.
The ability to interact strongly with light is important for a material to be useful in optics
-based
applications. Monolayer graphene exhibit
s a number of interesting
optical phenomena
including
a novel photo-
thermoelectric effect,
4,5
strong non-
linear behavior
,
6,7
and the potential for ultra
-
fast photodetection.
8
However
, t he absolute magnitude of these effects
is limited by the amount
of light absorbed by the graphene sheet, which is typically 2.3% at infrared and optical
frequencies
9,10
- a small value that reflects the single atom thickness of graphene. T
o increase
total overall graphene
-light interaction, a number of novel light scattering and absorption
geometries have recently been developed. These include coupling
graphene to
resonant metal
structures
11
-16
or optical cavities
where the electromagnetic fields are enhan
ced
17
-19
, or draping
graphene over optical waveguides
to
effectively increase the overall optical p
ath length along the
graphene.
20
,21
While those methods rely on enhancing
interband absorption
processes
, graphene
can als
o be patterned and
doped so as to excite plasmonic modes that
display strong resonant
absorption in the terahertz to mid
-infrared regime
.
22
-26
The plasmonic modes are highly sensitive
to their environment, and they have been shown to display large absorption
when
embedd
ed in
liquid salts
22 ,27
or by sandwiching dopants between several graphene layers.
26
However
without
blocking the transmission of light, it is not
possible to achieve unity absorption in th
ese
previously demonstrated geometries
22
,27
26
. Moreover, i
n order to
access nonlinear or high
2
frequency modulation as well as
the high confinement factors characteristic of graphene
plasmons, device geometries with open access to the graphene surface that operate with field
effect gating
at low doping are desirable.
Plasmonically active metallic and semiconductor structures can achieve near
-perfect
absorption of radiation at specified frequencies using a resonant interference absorption
method.
28
-32
The electromagnetic design of these structures
derives
in part
from the original
Sali
sbury screen design, but with the original resistive sheet replaced by an array of resonant
metal structures used
to achieve a low surface
impedance at optical frequencies
. The high
optical interaction strength of these structures has made them useful in such applications as
chemical sensing,
29
,33
and it was
recently proposed that similar devices c
ould be possible using
graphene to achieve near
per
fect
absorption from
THz to
Mid
-IR.
34
,35
Such a device would offer
an efficient manner of coupling micron
-scale freespace light into nanoscale plasmonic modes,
and would further allow for electronic control of that in
-coupling process. In this Letter, we
construct a device based on that principle, using tunable
graphene nanoresonators placed a fixed
distance away from a metallic reflector
to drive a dramatic increase in optical absorption into the
graphene
.
A schematic of our device is shown in Figure 1
a. A graphene sheet grown using
chemical vapor deposition on copper foil is placed on a 1
μ
m thick
low stre
ss silicon nitride
(SiN
x
) membrane with 200nm of Au deposited on the opposite side
that is used as both a
reflector and a backgate electrode.
Nanoresonators with widths ranging from 20-
60nm are then
patterned over 70×70
�
m
2
areas into
the graphene using 100keV electron beam lithography
(see
Methods)
. An atomic force microscope (AFM) image of the resulting graphene nanoresonators is
shown in the inset of Fig. 1b. T
he device was placed under a Fourier transform infrared (FTIR)
3
microsc
ope operating in reflection mode, with the incoming light polarized perpendicular to the
resonators. The carrier density of the graphene sheet was varied
in situ
by applying a voltage
across the SiN
x
between the gold and the graphene, and the resulting c
hanges in resistance were
continuously monitored using source and drain electrodes connected to the graphene sheet
(Fig
1b)
. The carrier density of the graphene nanoresonators was determined from experimentally
measured resonant peak frequencies
(see Section I & II in Supplementary Information).
The total absorption in the device
–
which includes absorption in the SiN
x
and the
graphene resonators
- is determined
from the difference in the reflected light from the
nanoresonator arrays and an adjacent gold mirror
. F
or undoped and highly doped 40nm
nanoresonators
, the total absorption is shown in Figure 2a, revealing large absorption at
frequencies below 1200cm
-1
, as well as an absorption peak that varies strongly with doping at
1400cm
-1
and a p
eak near 3500cm
-1
that varies weakly with doping.
In order to distill absorption
features
in the graphene from the environment
( i.e., SiN
x
and Au back reflector
), we plot the
difference in absorption between the undoped and doped nanoresonators
, a s shown in Figure 2b
for 40nm nanoresonators. This normalization removes
the low frequency feature below 1200cm
-
1
,which is due
to the broad optical phonon absorption in the SiN
x
and is independent of graphene
doping
. T
he
absorption feature
at 1400cm
-1
, however, shows a dramatic dependence on the
graphene sheet carrier density, with absorption into the graphene nanoresonators varying from
near 0% to 24.5% as the carrier density is raised to 1.42 × 10
13
/cm
2
. Because the absorption
increases with carrier density, we associate
it with
resonant absorption in the confined plasmons
of the nanoresonators.
22
-25 ,36
In
Figure 2b we also
see that absorption at 3500cm
-1
exhibits an
opposite
trend
relative to
the lower energy peak, with graphene-
related
absorption decreas
ing
with higher carrier densit
y. This higher energy feature is due to interband
graphene
absorption,
4
where
electronic transitions are Pauli blocked by state filling at higher carrier densities.
37
For
spectra taken from the bare, gate
-tunable graphene surface, this effect leads to
~8
% absorption,
roughly twice the intensity observed from patterned areas. Finally, in Figure 2c, we investigated
the graphene nanoresonator ab
sorption a
s the
resonator
width is varied from 20 to 60nm
at f
ixed
carrier density
. Th
is figure shows that the lower energy, plasmonic absorption peak has
a strong
frequency and intensity dependence on resonator width, with the maximum absorption occurring
in the 40nm ribbons
.
The carrier
density
dependent plasmonic dispersion of this system
is shown in Figure 3a
.
The observed
resonance frequency
varies from 1150-
1800cm
-1
, monotonically increas
ing
with
larger
carrier densit
ies
and smaller
resonator widths
. T
he plasmon energy
asymptotically
approaches
~ 1050cm
-1
due to a polar phonon in the SiN
x
that strongly reduces the dielectric
function of the substrate at that energy
.
38
This
coupling between the substrate polar phonon and
the graphene plasmon has also been previously observed in back-
gated SiO
2
devices
.
23
,25 ,39
In
Figure 3b we plot the
intensity
of the plas
monic absorption as a function of frequency at varying
carrier densities
, revealing that for all carrier densities, the maximum in absorption always
occurs at 1400cm
-1
. ,
The experimental behavior observed in Figures 2 and 3 has som
e similarities with
graphene plasmonic resonators patterned on back
-gated SiO
2
devices
, however the
re are some
significant differences. Most notably, the absolute absorption observed in this device is one
order of magnitude greater that what has previously been observed. Furthermore
the maximum
absorption in this device always occurs near 1400cm
-1
, in contrast to previous graphene
plasmonic devices where lower frequency resonances showed greater intensity due to fewer loss
pathways and better
k
-vector matching between the graphene plasmons and freespace light
.
23
,25
5
These new
absorption feature
s can be understood by considering the role of the gold reflector.
At 1400cm
-1
the optical path length of the SiN
x
is
λ
/4
n
and the gold reflector creates a standing
wave between the incident and reflected light that maximizes the electric field on the SiN
x
surface. As a consequence, when the graphene nanoresonators are tuned to absorb at 1400cm
-1
,
a double resonance condition is met, and the dissipation of the incoming radiation is greatly
enhanced.
In order to illustrate the ro
le of the interference effect, t
he frequency dependence of
the electric
field
intensity on the bare nitride surface
is plotted as a dashed
curve
in Figure 2c.
As can be seen in this figure, the intensity of the plasmonic absorption displays a frequency
dependence that is similar to the calculated field intensity.
Full wave finite element electromagnetic simulations
are performed
i n order to better
understand the performance of our device and the underlying mechanisms driving the large
observed absorption.
23
The conductivity of the graphene
sheet is modeled using the local random
phase approximation
40
with the intraband scattering rate Γ including both 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 is
approximated to be Γ
imp
=
ev
F
/
μ
√
푛푛푛푛
, with a carrier mobility of
550cm
2
/Vs.
36
The rate of optical phonon scattering is estimate from the theoretically obtained
self
-
energy Σ
oph
(
ω
), as Γ
oph
(
ω
)=2Im[Σ
oph
(
ω
)].
25 ,36 ,41
In order to match
the calculations to our
experimentally determined spectra, we multiply the theoretical spectrum by a constant factor of
0.72. This factor accounts for experimental imperfections in the device such as electronically
isolated resonators caused by cracks in the graphene sheet, or resonators that contain a graphene
grain boundary. Our resulting theoretical curves for the frequency and intensity dependence of
the resonant absorption are shown in Figs. 3a and 3b, respectively. As seen in Figure 3b, the
6
theory
and the measurement show
similar features -
a maximum plasmonic absorption
consistently occurs around 1400cm
-1
for a given charge density regardless of the resonator width.
The field profiles from our calculations are shown in Fig. 3c, revealing a strong plasmonic
response in the graphene nanoresonators for the
λ/4
n
condition where the electric field is
maximized on the surface and the resonators match the correct resonance conditions.
A more complete understanding of the large resonant absorption observed in this
graphene
Salisbury screen comes from viewing the effect in terms of
impedance matching
,
where the graphene meta
surface is modified in such a way that it mimics a load whose
admittance is
close to the free space wave admittance
Y
0
=
�
휖휖
0
/
휇휇
0
, and thus a
llows for all
incident light to be absorbed in the graphene she
et.
1,3
This description is diagramed in the inset
of Figure 1a. To understand this model, we can consider the effective admittance of a thin layer
of thickness
휏휏
and admittance
푌푌
GR
=
�
휖휖
GR
휇휇
GR
⁄
sitting atop a dielec
tric with thickness
푑푑
and
admittance
푌푌
SiN
x
deposited on a reflecting mirror. For frequencies such that
d
=
mλ
/4 and for
휏휏 ≪
1
, the total effective admittance of the stack is given by
푌푌
=
−푖푖푖푖휖휖
GR
휏휏
(see Section IV in
Supplementary Information). For normally incident light, the amount of absorption is given by
A
= 1
−
|(
Y
0
−
Y
)/(
Y
0
+
Y
)|
2
when the layer is located a quarter wavelength away from the back
reflector.
3
T
hus, the absorption approaches unity
as the relative admittance
Y
/
Y
0
approaches 1.
Typically, t
he admittance of
an unpatterned graphene sheet
is quite low, and equivalent to
its sheet
conductivity
σ
. Thus for unpatterned graphene,
Y
=
σ ≈ e
2
/4
ħ
=
παY
0
≈
0.023
Y
0
when the
photon energy is sufficiently higher
than the Pauli
-blocked interband transition energies
, where
α
is the fine structure constant
. As a result, the absorption by a pristine graphene
monolayer
in the
7
Salisbury screen configuration can be
calculated as
A
≈ 8.8% ≈ 4
πα
which is consistent
with our
experimental observation
s of the higher energy feature at 3500cm
-1
shown in Figure 2b.
With optical resonators patterned into the
graphene layer, however, t
he surface electric
admittance can be dramatically increased
. When
the resonators are sparsely spaced so that they
barely
interact with each other,
one can obtain the effective permittivi
ty of the resonator array by
simply multiplying the
spatial
density of the resonators
by
the polarizability of an individual
resonator
a
(
ω
). The admittance is then
Y
= -
i
ωa
(
ω
)/
S
, where
S
is the area of the unit cell
. On
resonance, there is a dramatic increase in
Im[
a
] , while
Re[
a
] crosses zero.
35
Recognizing that the
absorption cross
-section of a dipole is
σ
Abs
= (
ω
/c)Im[
a
/ε
0
], the surface admittance is given by
Y
=
(
σ
Abs
/
S
)
Y
0
on resonance. This is physically intuitive because it says that complete absorption
occurs when the absorption cross section of the resonator
array
is large enough to cover the
entire surface.
As the resonators become closer to each other, the resonance frequency redshi
fts
due to inter
-resonator coupling, yet the condition for perfect
absorption remains valid.
35
For our
device at its highest doping level
,
σ
Abs
/
S
is estimated to be 0.13
Y
0
, which is
much
higher than
πα
,
and this allows for the large absorption we observe in our graphene nanoresonators shown in
Figure 2.
Increasing carrier density leads to better coupling between the incoming light and the
graphene plasmons
, resulting in a stronger plasmon resonance. Therefore, at a given resonance
frequency, higher doping enhances the absorption performance
as seen in Figure 3b
and S6.
Finally, we point out that
the resonant absorption can be further increased if the resistive
damping in the
graphene is reduced.
In Figure 4a, we plot the calculated carrier mobility
dependence of
the surface admittance for a
n array of graphene nanoribbons
on a 1
μ
m SiN
x
/Au
layer
. The highest achieved carrier density 1.42×10
13
/cm
2
is assumed, and the width of the
ribbons
is chosen to
be
4 0nm in order to match the plasmon resonance with the quarter
8
wavelength condition of the SiN
x
layer (~1
400
cm
-1
). Because the resonator
absorption cross
-
section increases as
the
graphene becomes less lossy
, the resonant surface admittance increases
with increasing mobility and crosses the free space admittance
Y
0
at a carrier
mobility of
μ
≈
4,000cm
2
/Vs. As
Y
exceeds
Y
0
, the maximum absorption starts decreasing.
However,
i t should be
noted that
in this
high mobility
regime
, perfect
absorption can still be achieved by shifting the
quarter wavelength condition from the plasmon resonance frequency
via changing the SiN
x
thickness
in order to decrease the coupling between the free wave and the graphene plasmon. To
illustrate
this, Figure 4b shows the simulated peak absorption in the same resonator array as
a
function of both the mobility and the thickness of the nitride layer.
Indeed, for
Y
>
Y
0
the perfect
absorption occurs at two di
fferent thickness values: one
thinner and another
thicker than 1
μ
m.
This
deviation becomes larger as the
graphene mobility increases, and for mobilities reaching
10,000cm
2
/Vs the device will show total absorption for nitride layers with thicknesses of 700nm
or 1.3
μm
.
In summary, we have experimentally demonstrated that graphene plasmonic resonators
placed a quarter wavelength away from a back reflector
can absorb
almost 25%
of incoming M
id
infrared light
- more than 10 times higher than the case of unpatterend graphene
without a
reflector (~2.3%). The frequency and the amount of absorption can be largely tuned by
controlling the plasmon resonance of the nanoresonators via electrostatic gating or varying the
resonator size.
This strong optical response allows for graphene to be considered relevant as a
serious material to be used in optoelectronic devices. Furthermore, our modeling predicts that
modestly increasing the graphene mobility or decreasing the resonator line roughness can lead to
100% absorption, a tangible and important goal.
Finally, t
hese results clearly demonstrate that
the extremely small mode volumes of graphene plasmonic modes
can be made accessible to free
9
space probes despite the large d
iscrepancies in w
avelength that suppress such coupling. Because
the large light confinement of graphene plasmons allow them to couple efficiently to nearby
dipole emitters, this technique could indirectly allow for a robust interaction between molecular
scale emitters and
free
space light.
Methods
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 t
he nanoresonator width. This aids conductivity across the
patterned arrays despite occasional cracks and domain boundaries in the CVD graphene sheet.
Figure Captions
Figure 1. Schematic of experimental device.
(a) 70×70μm
2
graphene nanoresonator array
is
patterned on 1μm thick silicon nitride (SiN
x
) membrane via electron beam lithography. On the
opposite side, 200nm of gold layer is deposited that serves as both a mirror and a backgate
electrode. A gate bias was applied across the SiN
x
layer in order to modulate the carrier
concentration in graphene. The reflection spectrum was taken using a Fourier Spectrum Infrared
(FTIR) Spectrometer attached to an infrared microscope with a 15X objective. The incident light
10
was polarized perpendicul
ar to the resonators
. The inset schematically illustrates the device with
the optical waves at the resonance condition. (b)
DC resistance of graphene sheet as a function
of the gate voltage.
The inset is an atomic force microscope image
of 40
nm nanores
onators.
Figure 2.
Gate
-induced modulation of absorption in graphene nanoresonator arrays.
(a)
The total absorption in the device for undoped (red dashed) and highly hole doped (blue solid)
40nm nanoresonators. Absorption peaks
at 1400cm
-1
and a peak at 3500cm
-1
are strongly
modulated by varying the doping level, indicating these features are originated from graphene.
On the other hand, absorption below 1200cm
-1
is solely due to optical phonon loss in SiN
x
layer.
(b) The change in absorptio
n with respect to the absorption at the charge neutral point (CNP) in
40nm wide graphene nanoresonators at various doping levels. The solid black curve represents
the abs
orption difference spectrum of
bare (unpatterned) graphene. (c) Width dependence of t
he
absorption difference with the carrier concentration of 1.42×10
13
cm
-2
. The width of the
resonators varies from 20 to 60nm. The dashed curve shows the theoretical intensity of the
surface parallel electric field at SiN
x
surface when graphene is absent. N
umerical aperture of the
15X objective (0.58) is considered.
Figure 3. Dispersion and peak absorption of plasmon resonances in graphene
nanoresonator arrays.
(a) Peak frequency as a function of resonator width. Solid curves and the
symbols plot the theor
etical and measured peak frequencies respectively. The width of the
nanoresonators are determined by AFM measurements. (b) Frequency dependence of the
11
maximum absorption difference with varying doping level (symbols). The solid curves indicate
the theoreti
cal values obtained from finite element electromagnetic simulations multiplied by a
constant factor 0.72 which takes into account the fabricational imperfections such as dead
resonators. (c) 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
incidence. The quarter wavelength condition and plasmon resonance coincide at 1400cm
-1
.
Figure 4.
Carrier mobility dependence
. (a) Dependence of normalized surface admittance
Y
/
Y
0
of 40nm graphene nanoribbon array on resonance (red) and the maximum absorption (right) on
the carrier mobility
μ
(intraband scattering rate Γ=
ev
F
/
μ
√
푛푛푛푛
). The thickness of the SiN
x
layer
and the pitch are assumed to be 1um and 40nm, respectively. The admittance is monotonically
increases as the mobility increases, but is not directly proportional to it due to the loss in SiN
x
substrate. 100% absorption occurs at
μ
≈ 4,000cm
2
/Vs. (b)
Maximum absorption in the device as
a function of the SiN
x
thickness and the mobility. Impedance matching condition (
Y
=
Y
0
) is
indicated as the grey dashed line. The red dotted curve indicates the condition for perfect
absorption.
Acknowledgement
s
We
gratefully acknowledge support from the Air Force Office of Scientific Research Quantum
Metaphotonic MURI program under grant FA9550-
12-
1- 0488 and use of facilities of the DOE
“Light
-Material Interactions in Energy Conversion” Energy Frontier Research Cen
ter (DE
-
12
SC0001293). M. S. Jang and M. Choi acknowledge support from the Global Frontier R&D
Program on Center for Multiscale Energy System
s funded by the National Research Foundation
under the Ministry of Science, ITC & Future Planning, Korea (
2011
-0031561, 2011-
0031577).
M.S. Jang acknowledge
s a post
-doctoral fellowship from the POSCO TJ Park Foundation. V.W.
Brar gratefully acknowledges a post
-doctoral fellowship from the Kavli Nanoscience Institute.
M.C. Sherrott gratefully acknowledges graduate fello
wship support from the Resnick
Sustainability Institute at Caltech
. S
. Kim and M.S. Jang acknowledges support from a Samsung
Fellowship.
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Figure 1. Schematic of experimental device.
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Figure 2.
Gate
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Figure 4.
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Supplementary Information:
Tunable Large Resonant Absorption in a Mid-IR
Graphene Salisbury Screen
Min Seok Jang
1,
†
, Victor W. Brar
2,3, †
, Michelle C. Sherrott
2
, Josue J. Lopez
2
, Laura K. 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 91125
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
1
I. Electromagnetic Simulat
ions
We solve Maxwell’s equation by using finite element method. G
raphene is modeled
as a thin
layer of the thickness
휏휏
and impose the relative permittivity
휖휖
퐺퐺
=
1 +
푖푖푖푖
/(
휖휖
0
휔휔휏휏
)
. In actual
calculation,
휏휏
is chosen to be 0.1 nm which shows good convergence with respect to the
휏휏
→
0
limit. The complex optical conductivity of graphene
푖푖
(
휔휔
)
is evaluated
within local random
phase approximation.
1
푖푖
(
휔휔
)
=
2
푖푖푒푒
2
푇푇
휋휋ℏ
(
휔휔
+
푖푖Γ
)
log
�
2 cosh
�
퐸퐸
퐹퐹
2
푇푇
��
+
푒푒
2
4
ℏ
[
퐻퐻 �
휔휔
2
�
+
4
푖푖휔휔
휋휋
� 푑푑푑푑
∞
0
퐻퐻
(
푑푑
)
−퐻퐻�
휔휔
2
�
휔휔
2
−
4
푑푑
2
,
where
퐻퐻
(
푑푑
)
=
sinh
(
푑푑
/
푇푇
)
cosh
(
퐸퐸
퐹퐹
/
푇푇
)
+
cosh
(
푑푑
/
푇푇
)
.
Here, the temperature
푇푇
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
=
푒푒푣푣
퐹퐹
/
휇휇
√
푛푛휋휋
with the
mobility
휇휇
=
550cm
2
/Vs.
2
The rat
e of optical phonon scattering is estimated from theoretically
obtained self
-energy
Σ
oph
(
휔휔
)
, as
Γ
oph
(
휔휔
)
=
2Im
�Σ
oph
(
휔휔
)
�
.
2-4
The frequency de
pendent
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 S1 shows that the resulting
theoretical absorption spectra
reproduce quite
well the experimental data.
2
II. 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 usin
g a simple parallel plate
capacitor model
with a 1
μ
m thick SiN
x
dielectric, as
shown in Figure S2a, 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 their stoichiometry and resulting DC dielectric constant,
�
, is not precisely known.
This allows for a range of possible values for
�
, 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 i
n the conductance curves of graphene FET devices,
7-10
and we observe similar
behavior in our devices, as shown in Figure S2b. 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 S2a we have included a theoretical est
imate of those effects.
9
In addition to
atmospheric impurities, the SiN
x
surface itself can contain charge traps that fill or empty with the
applied gate bias. Such charge traps coul
d induce anomalous behavior in the conductance curves
of the graphene FET devices, similar to what has been observed in the presence 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 S2a we have provided a simple estimate of this
effect based on the assumption that an equal amount of induced carriers are distributed equally
across the smaller avail
able surface area, leading to larger carrier densities. However, theoretical
predictions have shown that the extra carrier density should preferably accumulate on the edges
of the graphene nanoresonators, and thus alter their plasmonic resonances in more sophisticated
ways.
12
3