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LAYERED GRAPHENE FOR ENHANCED
TRANSMISSION AT LOW TERAHERTZ
Alexander B. Yakovlev
Department of Electrical Engineering
University of Mississippi
1ST
WORKSHOP ON METAMATERIALS AND PHOTONIC
CRYSTALS, MEPHOC’2013
ARMENIA, COLOMBIA, SEPT. 30-OCT. 4, 2013
Introduction and Motivation
Graphene-Dielectric Stack
Dual Capacitive/Inductive Nature of Graphene Patches
Results and Discussions
Conclusions
2
OUTLINE
Enhanced Transmission at low-THz
3
Extremely thin conducting layers are almost opaque.
However, multilayer metal-dielectric PBG-like structures become
transparent within certain frequency bands in the optical regime.
I. R. Hooper et al., Opt. Express, 16, 17249 (2008)
M. R. Gadsdon et al., J. Opt. Soc. Am. B, 26, 734 (2009)
Induced transparency in the optical regime
BACKGROUND AND MOTIVATION (1)
4
Can a similar effect be observed at
microwaves??
The metal films are substituted by perforated metal layers
Microwave transmissivity of a metamaterial-dielectric stack
Butler et al., Appl. Phys. Lett., 95, 174101 (2009)
Yakovlev et al., 3rd Int. Congress on Advan. Electromag. Materi. in Microwa. and Optic.,(2009)
BACKGROUND AND MOTIVATION (2)
Scalora et al., Jour. App. Physics , 83, 2377–2383 (1998).
Feng et al., Phys. Rev. B. , 82, 085117(2005)
5 6 7 8 9 10 11 12 13 14 15 160
0.2
0.4
0.6
0.8
11
Frequency (GHz)
|S21|2
|S21|2 FEM model
|S21|2 Experimental
|S21|2 Analytical
A
B C D
BACKGROUND AND MOTIVATION (3)
Characteristic Band
What is the nature of these
resonances ??
can we tune them
can we predict the band
The number of transmission peaks is equal to the number of
layers (resonators)5
Butler et al., Appl. Phys. Lett. , 95, 174101 (2009)
Kaipa et al., Opt. Express, 18, 174101 (2010)
6
GRAPHENE-DIELECTRIC STACK
7
SURFACE CONDUCTIVITY OF GRAPHENE
220
2
2 2
0
1
, , ,
4
d d
c
d d
f fd
jje jT
f fd
j
Surface conductivity of graphene [Kubo formula]
, which at low-terahertz frequencies behaves as a low-loss
inductive surface.
1sZ
G. W. Hanson, J. Appl. Phys., 103, 064302 (2008)
1ln2
2
2
intra
Tk
B
cB BceTkj
Tkej
j
jje
c
c
2
2ln
4
2
inter
Intraband contributions Interband contributions
-e : charge of electron, T : temperature, : energy
: angular frequency, : reduced Planck’s constant
: chemical potential, : phenomenological scattering rate
2h
c
In the far-infrared regime, the contribution due to the interband electron
transition is negligible
,B ck T
8
Solid lines: approximate closed-form expressions (intraband + interband)
Dashed lines: numerical integration [Kubo formula]
SURFACE CONDUCTIVITY OF GRAPHENE
2 5
min
1/ 1.32 meV, 0.5 ps, 300 K
/ 2 6.085 10 S
T
e h
0.2 eVc 0.5 eVc
1 3 5 7 9 11 13 150
50
100
150
Frequency [THz]
Co
nductiv
ity
Re (/min
)
Im (/min
)
9Single sheet of graphene is highly reflective at low-THz frequencies
Behaves similar to an Inductive grid (metallic meshes) at microwaves.
Reflectivity and Transmissivity for normal incidence
FREE-STANDING GRAPHENE
1/ 1.32 meV
0.5 ps
300 K
1 eVc
T
10
TWO-SIDED GRAPHENE STRUCTURE
Thickness (h): 10 μm
Permittivity: 10.2
K 300 T ps, 0.5
meV 23.11
Transmission resonance appears at low frequencies
Graphene sheets effectively increase the electrical length
FP-type resonance of dielectric slab loaded with graphene sheets
μc = 0.5 eV
11
POWER TRANSMISSION SPECTRA
The number of transmission peaks is equal to the number of
dielectric slabs within the characteristic frequency band
Thickness (h): 10 μm
Permittivity: 10.2
K 300 T ps, 0.5
meV 23.11
h h
h h
1.0 eVc
I
II
12
POWER TRANSMISSION SPECTRA
Enhanced transmission at low-THz
Fabry-Perot resonances of the individual open/coupled cavities
Thickness (h): 10 μm
Permittivity: 10.2
K 300 T ps, 0.5
meV 23.11
4 layer graphene structure
- 4 dielectric slabs
- 5 graphene sheets
h h
h h
A
BC
D
13
ELECTRIC FIELD DISTRIBUTIONS
1.0 eVc
14
ELECTRIC FIELD DISTRIBUTIONS-ANIMATION PLOTS
0.005
0.078
0.370
0.443
0.881
1.173
Mode D
0.002
0.053
0.258
0.411
0.615
0.717
Mode B
0 20 40 60 80 100 120 140 160 1800
2
4
6
8
d (degrees)F
requency (
TH
z)
15
PB -I
SB
PB -II
PB -I
PB -II
BRILLOUIN DIAGRAMS – PASSBANDS AND STOPBANDS
Thickness (h): 10 μm
Permittivity: 10.21.0 eVc
StopBand
Multi-layer graphene-dielectric stack
SB: StopBand
PB: PassBand
0 20 40 60 80 100 120 140 160 1803.5
4
4.5
5
5.5
6
6.5
d (degrees)F
requency (
TH
z)
16
Thickness (h): 150 μm
Permittivity: 2.21.0 eVc
3.5 4 4.5 5 5.5 6 6.50
0.2
0.4
0.6
0.8
1
Frequency [THz]
|T|2
PB -I
PB -II
PB -III
PB -IV
PB -I PB -II PB -III PB -IV
SB SB SB SB
GRAPHENE THICK SLABS BRILLOUIN DIAGRAMS
StopBand
StopBand
StopBand
StopBand
Four-layer graphene-dielectric stack
SB: StopBand
PB: PassBand
Exhibits a series of bandpass regions separated by bandgaps
A thick dielectric slab is sometimes needed for mechanical handling
17
0 5 10 15 200
0.2
0.4
0.6
0.8
1
Frequency [THz]
|T|2
N = 4
N = 10
N = 20
N = 40
0 5 10 15 200
0.2
0.4
0.6
0.8
1
Frequency [THz]
|T|2
N = 4
N = 10
N = 20
N = 40
μc = 0.5 eV μc = 1 eV
1r
z
x
y
mh 10
Number of peaks correspond to number
of layers (N)
With increase in ‘N’, all peaks lie in a
characteristic frequency band
Acts as a Wideband Bandpass filter
BROADBAND PLANAR FILTERS
18
h: 30 μm,
Period (D) = 20 μm,
Strip width (w) = 2 μm,
t = 0.4 μm,
Dielectric permittivity: 1
Graphene-air stack mimics the behavior of Fishnet-air stack at THz
Five-layer graphene/meshgrid stacks separated by free-space
0 2 4 6 8 10 12 14 16 18 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency [THz]
|T|2
Meshgrid
Graphene, c = 1eV
BROADBAND PLANAR FILTERS
low-terahertz frequenciesinductive surface reactance
capacitive surface reactance
low-frequency passband followed by
alternating stop and passbands as
frequency increases
inductive surface reactance
low-frequency stopband, followed
by alternating pass and stopbands as
frequency increases
Metallic mesh-grids
microwave/low-terahertz
frequencies
microwave/low-terahertz
frequencieslow-terahertz frequencies
Graphene sheets
low-loss inductive surface reactance
low-frequency stopband, followed
by alternating pass and stopbands as
frequency increases
low-terahertz frequencies
Combining the properties of the above
three structures in a single configuration
low-loss inductive and capacitive surface
reactance
Combined filtering properties of the above
three configurations
Advantage of the Graphene Metasurface
Metallic patches
LAYERED GRAPHENE PATCHES
19
20
For interior patches
For patches at top and
bottom interfaces
SURFACE IMPEDANCE OF GRAPHENE PATCHES
21
D = 10 μm; g = 1 μm; εr = 4;
μc = 0.5 eV
• For the graphene patches, the surface impedance changes from capacitive to inductive as frequency
increases
At low frequencies the behavior of graphene patches is similar to that of the metallic patches
(capacitive)
At high frequencies the behavior of graphene patches becomes similar to that of a graphene sheet
(inductive)
D = 10 μm; g = 1 μm; εr = 4;
μc = 1eV
SURFACE IMPEDANCE OF GRAPHENE PATCHES
22
D = 10 μm; g = 1 μm; εr = 4; h = 10 μm
Five-Layer Stack of Graphene Patches
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
Frequency [THz]
|T|2
Graphene Patch, c = 0.5 eV, Analytical
Graphene Patch, c = 0.5 eV, HFSS
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
Frequency [THz]
|T|2
Graphene Patch, c = 1 eV, Analytical
Graphene Patch, c = 1 eV, HFSS
The analytical results are in good agreement
with the full-wave simulations
μc = 0.5 eV
μc = 1 eV
TRANSMISSIVITY
23
D = 10 μm; g = 1 μm; εr = 4; h = 10 μm
Fig. 1
Fig. 2
Fig. 3Fig. 4
μc = 0.5 eV
μc = 1 eV
TRANSMISSIVITY
24
D = 10 μm; g = 1 μm; εr = 4; h = 10 μmAnalytical Results
• One can clearly notice the low passband (starting from zero frequency), followed by
a deep stopband, and then a second passband
• Also, it can be noticed that with an increase in the number of layers, the number of
transmission peaks which corresponds to the number of coupled layers increases,
still maintaining the same characteristic frequency bands
TRANSMISSIVITY VERSUS NUMBER OF LAYERS
25
Analytical Results The Brillouin diagrams perfectly predict
the passband and stopband regions of the
corresponding finite-layer structures
BLOCH-WAVE ANALYSIS
D = 10 μm; g = 1 μm;
εr = 4; h = 10 μm;
μc = 0.5 eV
• For mode B, the field value is low in the middle graphene
patch and is concentrated more near the remaining
graphene patches, which is consistent with the electric-field
distributions shown in the inset
For mode A, the field
value is zero near the
middle graphene patch,
which is consistent with
the electric-field
distribution shown in the
inset
26
ELECTRIC FIELD DISTRIBUTIONS
27
D = 100 nm; g = 10 nm; εr = 1; h = 166 nm Four-layer thin-metal patch-dielectric stack
The results are calculated using a fit based on measured data [1] for the permittivity,
utilizing an augmented Drude model [2]
1. P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972)
2. A. Vial and T. Laroche, J. Phys. D: Appl. Phys., 40, 7152 (2007)
SURFACE IMPEDANCE AND TRANSMISSIVITY
THIN-METAL PATCHES
28
CONCLUSIONS
Graphene patches have a dual (capacitive and inductive) nature at low-
terahertz frequencies
We mimic the enhanced transmission at optical frequencies with a
metal-dielectric stack and in the microwave regime with stacked-
metascreens, at low-THz using stacked-graphene
The range of frequencies where the peaks are expected for a finite
graphene-dielectric stacked structure can be analytically and
accurately estimated from the Bloch analysis
At low-frequencies, it behaves similar to that of the multilayer
stack of metallic patches (capacitive, wherein the geometric patch
capacitance dominates over the weak metal inductance), and at higher
frequencies similar to that of a multilayer stack of contiguous graphene
sheets (inductive due to a large kinetic inductance of the material),
thus combining both the properties in a single configuration