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High-Impedance Surfaces with g pGraphene Patches as Absorbing
Structures at MicrowavesStructures at Microwaves
A B Yakovlev G W Hanson and A MafiA. B. Yakovlev, G. W. Hanson, and A. Mafi
Third International Congress on Advanced Third International Congress on Advanced Electromagnetic Materials in Microwaves and Optics
London, United Kingdom30 August 4 September 200930 August – 4 September, 2009
IntroductionIntroduction
Homogenization models for the analysis of high-impedance surfaces with g pgraphene (two-dimensional semi-metal) patches with and without vias
Dynamic model for HIS with graphenepatches (no vias)
grid impedance of graphene patches- grid impedance of graphene patches- circuit theory model
Non-local model for mushroom-type HIS with graphene patches- Additional Boundary Condition (ABC)
22
- spatial dispersion of wire-medium slab
GrapheneGraphene
•• GrapheneGraphene is a mono-atomic layer of graphite•• A single-wall carbon nanotube is a rolled-up sheet of graphenegrapheneof graphenegraphene
• Although graphene has been long studied to explain the properties of carbon systems, it was long thought p p y , g gthat graphene itself did not exist
2004 – graphene found!2004 graphene found!
33
GrapheneGraphene
GrapheneGraphene is moderately easy to make, and is visible in an optical microscope when residing on oxidized Si with
44
a opt ca c oscope e es d g o o d ed S ta certain Si02 thickness due to a weak interference effect
Graphene Graphene –– New Generation of TransistorsNew Generation of Transistors
Graphene can be gated, and has long spin-coherence length and high mobility at room temperaturelength and high mobility at room temperature
μ greater than 15,000 cm2/Vs havebeen measured and 200 000 cm2/Vs
55
been measured, and 200,000 cm /Vsare predicted to be possible
Surface Conductivity of GrapheneSurface Conductivity of Graphene
• No magnetic bias field• Spatial dispersion – not important at microwaves
1ln2
2/
2
2Tk
B
cB BceTkj
Tkej
2 BTkj
-e: charge of an electronk B lt ’ t tkB: Boltzmann’s constantμc: chemical potential (electrostatic bias)Γ : electron scattering rate (1012 Hz)g ( )T : temperature (300 K)
66
PlanePlane--Wave Incidence Wave Incidence Analytical Modeling of Graphene HIS StructuresAnalytical Modeling of Graphene HIS Structures
• Dynamic solution of 2D stripgrid scattering problem
• Averaged impedance boundary• Averaged impedance boundarycondition
• Approximate Babinet principle
Transmission-line networkGraphene patches
g ds
g d
Z ZZ
Z Z
HISHISZdηo Zg
g d
Zs
77
Grid Impedance of Graphene Patches and StripsGrid Impedance of Graphene Patches and Stripsy 1
p
py
220
2
1( ) 2 11 sin
2
effTEg
eff
pZ jp g k
k
g
gx
ff
( ) 2effTM
gpZ j
p g
ln csc
2effk p g
p
pw gpw
E
220
21 sin( ) 2
effTMg
eff
kpZ jp w k
E( ) 2
effTEg
pZ jp w
ln csc
2effk p w
p
p g
E
220
2
1( ) 2
1 sin
effTEg
ff
pZ jp g k
k
88
effk
( ) 2effTM
gpZ j
p g
Surface Impedance of Grounded SlabSurface Impedance of Grounded Slab
Dielectric Impedance
zd
TMd hkj,Z
2
20 sin1tani
TM
hkjZ zdTEd tan
sin,
20
rr
2sin
TEr sin
2sin00 rk zdvertical component of thewave vector of the refracted wave
y
z
x
reh
99
Graphene HIS at Oblique IncidenceGraphene HIS at Oblique Incidence
D = 2 mm, g = 0.2 mm, h = 1 mm 10.2r
TM polarization
-10
-5
0
R fl ti ti
-25
-20
-15
|R|,
dB
0
c = 0.5185 eV
Reflection properties are sensitive to the angle of incidence
-40
-35
-30 = 0
= 30
= 45
= 60
10108 9 10 11 12 13 14 15 16 17
-50
-45
Frequency, GHz
Tunable Graphene HISTunable Graphene HIS
D = 2 mm, g = 0.2 mm, h = 1 mm 10.2r r
TM polarization
Reflection minima obtained at different incident angles gby adjusting the chemical potential
Solid lines – analytical modelDashed lines – FEM results(C l M l i h i
1111
(Comsol Multiphysics, http://www.comsol.com)
Tunable Graphene HISTunable Graphene HIS
D = 2 mm, g = 0.2 mm, h = 1 mm 10.2r r
TE polarization
Reflection minima at different incident angles are obtained in a narrow frequency range by adjusting the chemical potentialj g
Solid lines – analytical model
1212
HIS with PEC Patches and Lossy Dielectric SlabHIS with PEC Patches and Lossy Dielectric Slab
D = 2 mm, g = 0.2 mm, h = 1 mm
TM polarization
-10
-5
0
Reflection minima are sensitiveto the angle of incidence
-25
-20
-15
R|,
dB = 0
= 10
r = 10.2(1 - j*0.26)
Solid lines – analytical model
-40
-35
-30
|R = 10
= 20
= 30
= 40
= 50
60
13139 10 11 12 13 14 15
-50
-45
Frequency, GHz
= 60
Mushroom Array with Graphene PatchesMushroom Array with Graphene Patches
NonNon Local ModelLocal Model SD + ABCSD + ABCyH
0r ag a
xa
g
NonNon--Local Model Local Model –– SD + ABCSD + ABC
aE
ak eff
g aa
g
z
g
g
L
Ground planezr
z
02rGraphene patches• Wire medium slab as anisotropic material characterized by effective permittivity• Spatial dispersion
i th l b
2
1 p
0 ˆ ˆ ˆ ˆ ˆˆeff r yyxx yy zz
22 2 /
l 0 2p
aa
is the plasma wavenumberp
2 21 pyy
r yk
/
r r
c
14140
ln 0.52752
ar
Silveirinha et al., IEEE Trans. Antennas Propagat., 56, Feb. 2008
SD + ABC ModelSD + ABC ModelTM-polarized incident plane wave excites TEM and TM modes in the wireTM-polarized incident plane wave excites TEM and TM modes in the wire medium slab
0 0
cos cosh
z
z
y y jk zx
jk zx TEM r TM TM
H e e e
H B y B y e
air region
wire medium slab
| | | |z z g x xy L y L y L y LE E Z H H
• Two-sided impedance boundary condition at y=L:
• Additional boundary condition at the via-graphene patch connection at y=L:
( ) ( ) 0y L
dI y I yj dy
0 00
0y xr z r y z x y L
r
dE dHk E k Hj dy dy
In terms of field components:0
yrj dy
• Additional boundary condition at the via-ground plane connection at y=0:
0
( ) 0y
dI ydy
1515
dy
In terms of field components:0 0
0y xr z y
dE dHkdy dy
Reflection CoefficientReflection Coefficient
0
0 0
0
1coth cot
1th t
TM rg
L L K jZ k
L L K j
• Reflection coefficient
0
0 0
coth cotTM rg
L L K jZ k
where 1 1 tanh 1 1 tanrL L
0 0
1 tanh 1 1 tan
1 1 coth cot
rTM rTM
yy r r
r TM TM rTM rTM
L Lj j
KL L
j j
2 2 2TM p z rk 2 2
0 zk
0 0r yy r r rj j 2
2 21 pTMyy
z pk
In the limiting case 0 (transparent patches) it turns to wire-medium slab: Silveirinha et al., IEEE Trans. Antennas Propagat., 56, Feb. 2008
I th li iti
1616
In the limiting case (PEC patches) it turns to mushroom HIS: Luukkonen et al., IEEE Trans. Microwave Theory Tech., 2009 (to appear)
Yakovlev et al., IEEE Trans. Microwave Theory Tech., 2009 (to appear)
Mushroom Array with Graphene PatchesMushroom Array with Graphene Patches
a
yH
0r ag a
xa
gaE
ak eff
g aa
gg
Gz
Lr
z
Ground plane
02rGraphene patches
Period: 2 mmGap: 0.2 mmRadius of vias: 0.05 mm
1717
Substrate thickness: 1 mmDielectric permittivity: 10.2
Mushroom HIS with Graphene PatchesMushroom HIS with Graphene Patches
C i ith h HIS ith t iC i ith h HIS ith t iComparison with graphene HIS without viasComparison with graphene HIS without viasTM polarization
Mushroom HIS with HIS with grapheneMushroom HIS with graphene patches
HIS with graphene patches (no vias)
-5
0
-5
0
-20
-15
-10
= 0 5185 eV-20
-15
-10
0 5185 V
-35
-30
-25
|R|,
dB
= 0
= 30
= 45
= 60
c = 0.5185 eV
-35
-30
-25
|R|,
dB
= 0
= 30
= 45
= 50
c = 0.5185 eV
8 9 10 11 12 13 14 15 16 17-50
-45
-40
Frequency, GHz
8 9 10 11 12 13 14 15 16 17-50
-45
-40
Frequency (GHz)
= 55
= 60
1818
Mushroom HIS with graphene patches results in stable resonance frequencies (in the vicinity of the plasma frequency) for different angles of incidence
q y,
Tunable Mushroom HIS with Graphene PatchesTunable Mushroom HIS with Graphene Patches
TM polarization
-20
-10
0
-20
-10
0 = 50, c
= 0.6335 eV
= 55, c = 0575 eV
= 60, c = 0.467 eV
50
-40
-30
|R|,
dB
= 0, c = 0.5185 eV
50
-40
-30
|R|,
dB-70
-60
-50 = 30, c = 0.661 eV
= 45, c = 0.6875 eV
-70
-60
-50
In mushroom HIS with graphene patches the reflection minima can
10 10.5 11 11.5 12 12.5 13 13.5 14-80
Frequency (GHz)
10 10.5 11 11.5 12 12.5 13 13.5 14
-80
Frequency (GHz)
1919
In mushroom HIS with graphene patches the reflection minima can be obtained at different incident angles by adjusting the chemical potential
Mushroom HIS with PEC Patches and Lossy Dielectric SlabMushroom HIS with PEC Patches and Lossy Dielectric Slab
C i ith HIS ith t iC i ith HIS ith t iComparison with HIS without viasComparison with HIS without vias
TM polarization
-5
0
-5
0
Mushroom HIS Patch array HIS (no vias)
-20
-15
-10
B 0
r = 10.2(1 - j*0.26)-20
-15
-10
B
r = 10.2(1 - j*0.26)
-35
-30
-25
|R|,
dB = 0
= 10
= 20
= 30
= 40
= 50
-35
-30
-25
|R|,
dB = 0
= 10
= 20
= 30
= 40
9 10 11 12 13 14 15-50
-45
-40
Frequency, GHz
= 50
= 60
9 10 11 12 13 14 15-50
-45
-40
Frequency, GHz
= 50
= 60
2020
Mushroom HIS results in better absorption, however, in both cases the reflection coefficient is sensitive to the angle of incidence
Microscopic Current Along the ViasMicroscopic Current Along the Vias
1 1.02
8 GHz 14 GHz
0.98
0.99
0.96
0.98
1
0.96
0.97
I(y)/I
(0)
c =0.5 eV
[ = 2.9410-2-j7.3910-4 S]
c =0.25 eV
[ = 1.4710-2-j3.710-4 S]0.9
0.92
0.94
I(y)/I
(0)
c =0.25 eV
[ 1 47 10-2 j6 46 10-4 S]
c =0.5 eV
[ = 2.9410-2-j1.310-4 S]
0.93
0.94
0.95 c =0.1 eV
[ = 5.9510-3-j1.510-4 S]
0.82
0.84
0.86
0.88 [ = 1.4710 2-j6.4610 4 S]
c =0.1 eV
[ = 5.9410-3-j2.6110-4 S]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1y/L
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1y/L
For large chemical potential [electrostatic bias field] the microscopic current along the vias is close to uniform and the
2121
microscopic current along the vias is close to uniform, and the spatial dispersion effects in wire medium are significantly reduced
ConclusionsConclusions
Dynamic model for HIS with graphene patches and non-local(SD+ABC) model for mushroom HIS with graphene patchesare proposed for the analysis of absorption properties atmicrowavesmicrowaves
The reflection minima in HIS structures with graphenepatches (with and without vias) can be obtained at differentp ( )incident angles by adjusting the chemical potential(electrostatic bias field)
For large values of chemical potential the microscopic For large values of chemical potential the microscopiccurrent along the vias is close to uniform, and the spatialdispersion effects in wire-medium are significantly reduced
2222
AcknowledgementAcknowledgement
The authors are thankful to Mário G. SilveirinhaThe authors are thankful to Mário G. Silveirinhafor helpful discussions regarding the Additionalfor helpful discussions regarding the Additionalg gg gBoundary ConditionBoundary Condition
2323