The optical absorption properties of periodically patterned graphene plasmonic resonators are studied experimentally as the graphene sheet is placed near a metallic reflector. By varying the size and carrier density of the graphene, the parameters for achieving a surface impedance closely matched to free-space (Z_0 = 377Ω)  are determined and shown to result in 24.5% total optical absorption in the graphene sheet. Theoretical analysis shows that complete absorption is achievable with higher doping or lower loss. This geometry, known as a Salisbury screen, provides an efficient means of light coupling to the highly confined graphene plasmonic modes for future optoelectronic applications

Atwater, Harry A.

Brar, Victor W.

Choi, Mansoo

Jang, Min Seok

Kim, Laura

Kim, Seyoon

Lopez, Josue J.

Sherrott, Michelle C.

Caltech Authors - Main

1 
 
Supplementary Information: 
Tunable Large Resonant Absorption in a Mid-
Infrared Graphene Salisbury Screen 
Min Seok Jang1, †, Victor W. Brar2,3, †, Michelle C. Sherrott2, Josue J. Lopez2, Laura B. Kim2, 
Seyoon Kim2, Mansoo Choi1,4, and Harry A. Atwater2,3 
† 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 
 
 
 
 
 
 
 
 
2 
 
I. Device Fabrication 
SiNx 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 across the patterned arrays despite occasional 
cracks and domain boundaries in the CVD graphene sheet. 
 
II. Electromagnetic Simulations 
We solve Maxwell’s equation by using finite element method. Graphene is modeled as a thin 
layer of the thickness 𝜏 and impose the relative permittivity 𝜖𝐺 = 1 + 𝑖𝜎/(𝜖0𝜔𝜏). In actual 
calculation, 𝜏 is chosen to be 0.1 nm which shows good convergence with respect to the 𝜏 → 0 
limit as seen in Figure S1. The complex optical conductivity of graphene 𝜎(𝜔) is evaluated 
within local random phase approximation.[1] 
𝜎(𝜔) =
2𝑖𝑒2𝑇
𝜋ℏ(𝜔 + 𝑖Γ)
log [2 cosh (
𝐸𝐹
2𝑇
)] +
𝑒2
4ℏ
[𝐻 (
𝜔
2
) +
4𝑖𝜔
𝜋
∫ 𝑑𝜂
∞
0
𝐻(𝜂) − 𝐻 (
𝜔
2
)
𝜔2 − 4𝜂2
, 
where 
𝐻(𝜂) =
sinh(𝜂/𝑇)
cosh(𝐸𝐹/𝑇) + cosh(𝜂/𝑇)
. 
Here, the temperature 𝑇 is set as 300K. The intraband scattering rate Γ takes into account 
scattering by impurities Γimp and by optical phonons Γoph. 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 Γimp = 𝑒𝑣𝐹/𝜇√𝑛𝜋 with the 
mobility 𝜇 = 550cm2/Vs.[2] The rate of optical phonon scattering is estimated from theoretically 
obtained self-energy Σoph(𝜔), as Γoph(𝜔) = 2Im[Σoph(𝜔)].[2-4]  The frequency dependent 
3 
 
dielectric functions of Au and SiNx 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. This degradation factor is determined 
to be 0.72 by comparing simulation and measurement. Figure S2 shows that the resulting 
theoretical absorption spectra reproduce quite well the experimental data. 
 
III. Determination of Carrier Density 
The carrier density of the nanoresonators was determined by fitting the peak frequencies of the 
simulated absorption spectra 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 μm thick SiNx dielectric, as 
shown in Figure S3a, yet there is some deviation.  We can attribute the discrepancies to a number 
of possible effects.  First, our SiNx membranes were obtained from a commercial supplier 
(Norcada) and their 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 measurements were performed under 
FTIR purge gas (free of H2O and CO2), 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 graphene FET devices,[7-10] and we observe 
similar behavior in our devices, as shown in Figure S3b.   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 S3a we have included a theoretical estimate of those effects.[9]   In 
addition to atmospheric impurities, the SiNx 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 
presence of metallic impurities.[11]  Finally, we note that we have removed some of the 
graphene surface area in the process of fabricating 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 S3a 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 accumulate on the edges of the graphene nanoresonators, and thus alter their 
plasmonic resonances in more sophisticated ways.[12]  
 
IV. Peak Width Analysis 
In Figure S4a, 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 increasing 
resonance frequency and decreasing resonator width. The 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 electron scattering rate. However, in our sample, the interaction 
with SiNx substrate polar phonons results in a deviation of the plasmon scattering rate from the 
electron scattering rate. Therefore, we extract the intraband electron scattering rate (Γ) by fitting 
the FWHM of the simulated spectrum to the measured plasmon linewidth, as shown in figure 
S4b.  
We 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 𝜇, converted from the 
electron scattering rate via 𝜇 = 𝑒𝑣𝐹/Γ√𝑛𝜋 , is determined as 550cm
2/Vs with standard deviation 
50cm2/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 𝜏 and admittance 𝑌GR sitting atop a dielectric with thickness 𝑑 
and admittance 𝑌SiNx  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]  
𝑌 = 𝑌GR
𝑌′SiNx − 𝑖𝑌GR tan(𝑘1𝜏)
𝑌GR − 𝑖𝑌′SiNx tan(𝑘1𝜏)
, 
where  𝑌GR = √𝜖GR 𝜇GR⁄  and 𝑘1 = 𝜔√𝜖GR𝜇GR are the wave admittance and the wavevector 
inside the thin sheet, respectively. 𝑌′SiNx is the effective admittance of the dielectric as viewed 
from the position of the sheet, and is given by 𝑌′SiNx = 𝑌SiNx cot(𝑘2𝑑), where 𝑘2 is the 
wavevector inside the SiNx layer. For frequencies such that 𝑑 = 𝑚𝜆/4 and for 𝑘1𝜏 ≪ 1, then 
𝑌′SiNx ≪ 𝑌GR and tan(𝑘1𝜏) → 𝑘1𝜏, and the above equation reduces to Y = −iωετ. 
 
 
VI. 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 𝜎eff, which can be evaluated from the far-field transmission and reflection 
coefficients. Consider a homogeneous thin film of conductivity 𝜎eff  placed on the interface (𝑧 =
0) between air (𝑧 < 0) and SiNx (𝑧 > 0) and a plane wave polarized along x direction is 
normally incident on the surface. The surface parallel electric field is continuous 𝐸𝑥
0+ = 𝐸𝑥
0− at 
the interface, while the magnetic fields are discontinuous due to the surface current, 𝐻𝑦
0+ −
𝐻𝑦
0− = 𝜎eff𝐸𝑥(𝑧 = 0) = 𝑌𝐸𝑥(𝑧 = 0). From these boundary conditions, the transmission (t) and 
reflection (r) coefficients satisfy the following equations, 
1 + 𝑟 = 𝑡, 
(1 − 𝑟) − 𝑛SiNx𝑡 = (
𝑌
𝑌0
) 𝑡, 
6 
 
where 𝑛SiNx is the refractive index of SiNx. The normalized surface admittance 𝑌/𝑌0 is then 
solely written in terms of transmission coefficient,  
𝑌
𝑌0
=
2
𝑡
− 1 − 𝑛SiNx . 
Because the fields of graphene plasmons are tightly confined near the surface with characteristic 
decay length similar to the width of the nanoresonators, we record the electric field of the 
transmitted wave at a position sufficiently far from the surface (𝑧0 = 1um) in order to exclude 
the evanescent field of graphene plasmons. The far field transmission coefficient is then obtained 
by accounting for the propagation factor 
𝑡 =
𝐸x(𝑧0) exp[−𝑖𝑛SiNx𝑘0𝑧0 ]
𝐸0(0)
, 
where 𝐸0 is the electric field of incident wave at the surface.  
Figure S5 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[𝑌] crosses zero, while Re[𝑌], which is directly proportional to 
the absorption cross section 𝜎Abs of individual resonator, has its maximum. As graphene 
becomes less lossy, the plasmon resonance gets sharper and stronger.  
  
 
 
 
 
 
 
7 
 
 
 
 
Figure S1: 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 calculations 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 1.42×1013cm-2.  
 
 
 
Figure S2: Comparison between experimental and theoretical absorption spectra. (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 with the carrier concentration of 
1.42×1013cm-2. The width of the resonators varies from 20 to 60nm.  
 
9 
 
 
Figure S3: 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 μm thick SiNx 
membrane with dielectric properties spanning those reported in literature.[15]  The blue line 
indicates a graphene/SiNx device that includes estimated doping effects due to atmospheric and 
substrate impurities that have been reported in SiO2.[7-10]  The black dotted line models a 
graphene/SiNx 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 resistance as applied gate bias 
swept up (dotted line) and down (solid line).  For (a, triangles), the CNP was assumed to occur at 
+80V, halfway between the two hysteric peaks. 
 
10 
 
 
Figure S4: Peak width and electron scattering rate. Frequency dependence of (a) the full 
width at half maximum (FWHM) linewidth of the absorption peaks, and (b) the fitted electron 
scattering rate. The resonator width ranges from 20 to 60 nm and the carrier density varies from 
0.66×1013 to 1.42×1013cm-2.  
11 
 
 
Figure S5: Surface admittance versus frequency. Frequency dependence of Re[𝑌/𝑌0] (solid) 
and Im[𝑌/𝑌0] (dashed) for 𝜇 = 550 (blue) and 4,000cm
2/Vs (red). An array of infinitely long 
graphene nanoribbons (40nm width, 40nm spacing) is assumed. The carrier density is set to 
1.42×1013cm-2.  
 
 
 
12 
 
 
Figure S6: Electric field distribution. Theoretical electric field profile of a 40nm graphene 
nanoresonator with the highest achieved carrier density (1.42×1013cm-2), obtained from an 
electromagnetic simulation assuming normal incidence. The quarter wavelength condition and 
plasmon resonance coincide at 1400cm-1 (left). At 2335cm-1, the optical thickness of SiNx is 
roughly half wavelength, resulting in vanishing electric field at the surface (middle). When the 
optical thickness of SiNx 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 S7: Peak 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 1.0−7.0×1013cm-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 SiNx 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 S8: Incidence 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 
angles (θ) and polarizations. (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 1.42×1013cm-2 and 550cm2/Vs, respectively.  
 
15 
 
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Tunable large resonant absorption in a midinfrared graphene Salisbury screen

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Tunable large resonant absorption in a midinfrared graphene Salisbury screen

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