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

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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