M INTERACTIONS IN MUSHROOM T METAMATERIALS WITH …yakovlev/...MODAL INTERACTIONS IN MUSHROOM- TYPE METAMATERIALS WITH THIN METAL/GRAPHENE PATCHES A. B. Yakovlev and G. W. Hanson SESSION:

MODAL INTERACTIONS IN MUSHROOM-

TYPE METAMATERIALS WITH THIN

METAL/GRAPHENE PATCHES

A. B. Yakovlev and G. W. Hanson

SESSION: CARBON AND QUANTUM METAMATERIALS 2

THURSDAY, OCTOBER 13, 2011

Metamaterials 2011: The Fifth International Congress on

Advanced Electromagnetic Materials

in Microwaves and Optics

Barcelona, SPAIN

October 10-15, 2011

2

Introduction

Nonlocal Homogenization Model for Mushroom

Surfaces with Thin Metal/Graphene Patches Generalized Additional Boundary Condition

Natural Waves in the Transition of Mushroom Surface

to Wire-Medium Slab Modal Interaction Phenomena

Complex Frequency-Plane Branch Points

Conclusion

Outline

x

z

a

a

g

2r0

Ground plane (PEC) z

a k

r

g

L

Lk

La

r0

1

y

E

H

y = 0

3

0)(

0

Ly

ydy

ydI

000

Ly

y

xz

y

rdy

dHk

dy

dE

In terms of “macroscopic” field components:

y = L

In mushroom-type HIS structures with short vias, the presence of patches significantly

reduces the spatial dispersion in the wire-medium slab resulting in the uniform current

along the vias

Introduction – Mushroom-Type HIS

Additional boundary condition for the “microscopic” current at the via-ground plane

connection ( ) and via-patch connection ( ): Ly 0y

A. Demetriadou et.al., J. Phys.: Condens. Matter, 20, 295222, 2008

O. Luukkonen et.al., IEEE Trans. Microwave Theory Tech., 57, Nov. 2009

A. B. Yakovlev et.al., IEEE Trans. Microwave Theory Tech., 57, Nov. 2009

M. Silveirinha et al., New J. Phys., 10, 2008


The results of natural modes [surface waves and leaky waves] based on non-local

(SD+ABC) and local (ENG) homogenization models are in excellent agreement

Nonlocal and Local Homogenization Models PEC patches

TMz Period: 2 mm

Gap: 0.2 mm

Radius of vias: 0.05 mm

Slab thickness: 1 mm

Dielectric permittivity: 10.2

4

SD+ABC ENG

5

8 9 10 11 12 13 14 15 16 17 18-30

-25

-20

-15

-10

-5

0

Frequency (GHz)

|R|, d

B

ABC - PEC

HFSS

PEC

x

z

a

a

g

2r0

L y

z

x

Metal patches

Metal Thickness: 1nm

y

E

H

k

Mushroom HIS – Thin Metal Patch

A. B. Yakovlev et.al., IEEE Trans. Microwave Theory Tech., 59, Mar. 2011

7

3 5.8 10 (S/m)D

TM polarization: 30˚

6

Generalized Additional Boundary Condition (GABC)

Thin Metal/Graphene Patch

z

y

y

E

H

PEC

rL

t

cJ

n̂

Leontovitch Boundary Condition:

c

metalr

t JE

,

0

Good conductor

c

D

t JE

2

1

Thin metal tDD 32

1ln2

Γ22

2intra

2

Tk

B

cBD

BceTkj

Tkej

Surface Conductivity of Graphene

0

3, 1

Dmetalr j

G. W. Hanson, J. Appl. Phys., 103, 2008

-e: charge of an electron

kB: Boltzmann’s constant

μc: chemical potential (electrostatic bias)

Γ : electron scattering rate (1012 Hz)

T : temperature (300 K)

ћ : modified Planck’s constant

S. A. Tretyakov, Analytical Modeling in Applied Electromagnetics,

Artech House: Norwood, MA, 2003

7

j

Jcss

Principle of Conservation of Surface Charge

0)()(

0

2

Lyc

c

r

D yJdy

ydJ

j

Generalized Additional Boundary Condition (GABC) at the via-thin

metal/graphene connection, : Ly

dy

ydJ

jc

s

)(1

Enrs

ˆ0

Surface charge density for thin metal/graphene

at the connection point, : Ly

c

D

rs J

2

0

Thin wires

z

y

y

E

H

PEC

rL

t

cJ

n̂

Generalized Additional Boundary Condition (GABC)

Thin Metal/Graphene Patch

A. B. Yakovlev et.al., IEEE Trans. Microwave Theory Tech., 59, Mar. 2011

2

2 21

p

xx

r xk

/

r r

c

8

Nonlocal Model – SD + ABC

• Wire medium slab as uniaxial continuous material characterized by

tensor effective permittivity

• Spatial dispersion

0 ˆˆ ˆˆ ˆˆeff r xxxx yy zz 2

2

0

2 /

ln 0.52752

p

a

a

r

is the plasma wavenumber p

M. Silveirinha et al., IEEE Trans. Antennas Propagat., 56, Feb. 2008

Mushroom Array with Thin Metal/Graphene Patches

TMz

| | | |z z g y yx L x L x L x LE E Z H H

9

SD + ABC Model

• Magnetic field of TMz natural modes:

0

cos cosh

z

z

x jk z

y

jk z

y TEM r TM TM

H Ae e

H B x B x e

air region

wire medium slab

• Two-sided impedance boundary condition at :

In terms of field components:

In terms of field components:

x L

2

0

( )( ) 0cD c x L

r

dJ xJ x

j dx

2 0 00

0yxD

r z r x z y x Lr

dHdEk E k H

j dx dx

0

( )0c

x

dJ x

dx

0 00

yxr z x

dHdEk

dx dx

• Generalized additional boundary condition at the via-thin metal/graphene patch

connection, : x L

•Additional boundary condition at the via-ground plane connection, : 0x

a

g

g

a

z

y

22

eff

D

g jga

aZ

a

gakeff

2cscln

10

S. A. Tretyakov, Analytical Modeling in Applied Electromagnetics, Artech House: Norwood, MA, 2003

O. Luukkonen et al., IEEE Trans. Microwave Theory Tech., 57, Nov. 2009

Grid Impedance of Thin Metal/Graphene Patches

eff

eff

0 effeff kk 0

2

1 reff

TMz

00 0

1( , ) coth cot 0z TM r

g

H k K L L jZ k

2 2 2

TM p z rk

11

• TMz natural modes

2 2

0 zk

Dispersion Equation

where

2

2 21

pTM

xx

z pk

In the limiting case (transparent patches) it turns to wire-medium slab:

M. Silveirinha et al., IEEE Trans. Antennas Propagat., 56, Feb. 2008

In the limiting case (PEC patches) it turns to mushroom HIS:

2 2

0 0

2 2

0 0

11 tanh 1 1 tan

11 coth cot

D TM D rTM rTM

xx r r

r D TM TM D rTM rTM

r xx r r r

L Lj j

K

L Lj j

02 D

D2


12

Mushroom Surface with Metallic Patches

tDD 32

7

3 5.8 10 (S/m)D

TMz

Period: 2 mm

Gap: 0.2 mm

Radius of vias: 0.05 mm

Slab thickness: 1 mm

Dielectric permittivity: 10.2

TMz natural modes

0 3, 2 / Dt

13

Mushroom HIS – PEC Patches

Transition from proper to improper complex leaky waves:

from backward to forward radiation


14

Transition of Mushroom HIS to WM Slab

Phase constant

solid lines: proper complex

dashed lines: improper complex

Wire-medium slab

Leaky Waves

16 18 20 22 24 26 28 30-2

-1

0

1

2

3

4

5

6

Frequency (GHz)

Re

al(

k z/k

0)

a = 1 mm

a = 2 mm

a = 3 mm


Physical leaky waves of the mushroom HIS structure in the transition to

the wire-medium slab continue to physical leaky waves of wire-medium slab

7

3 5.8 10 (S/m)D tDD 32

10 15 20 25 30-3

-2

-1

0

1

2

3

Frequency (GHz)

Re

(kz/

k0)

2d

= 0.0058 S

2d

= 5.810-4

S

2d

= 0.0029 S

2d

= 5.810-6

S

2d = 0.058 S

2d

= 5.8 S

15

Transition of Mushroom HIS to WM Slab

Attenuation constant



Wire-medium slab

Leaky Waves


Physical leaky waves of the mushroom HIS structure in the transition to

the wire-medium slab continue to physical leaky waves of wire-medium slab

16 18 20 22 24 26 28 30-6

-4

-2

0

2

4

6

Frequency (GHz)

Ima

g(k

z/k

0)

a = 1 mm

a = 2 mm

a = 3 mm

7

3 5.8 10 (S/m)D tDD 32

10 15 20 25 300

1

2

3

4

5

6

Frequency (GHz)

Im(k

z/k

0)

2d

= 0.058 S

2d

= 5.8 S

2d

= 0.0058 S

2d

= 0.0029 S

2d

= 5.810-4

S

2d

= 5.810-6

S

16

Mushroom HIS – Thin Metallic Patches

Surface conductivity:

Proper complex and improper complex solutions

of mushroom HIS parameterized by metal thickness



8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3

Frequency (GHz)

Re

(kz/

k0)

8 10 12 14 16 18 20-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Frequency (GHz)

Im(k

z/k

0)

2 5.8 SD

17


Proper complex and improper complex solutions

of mushroom HIS parameterized by metal thickness



8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3

Frequency (GHz)

Re

(kz/

k0)

8 10 12 14 16 18 20-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Frequency (GHz)

Im(k

z/k

0)

Surface conductivity: 2 0.58 SD

18


Phase constant



8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3

Frequency (GHz)

Re

(kz/

k0)

8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3

Frequency (GHz)

Re

(kz/

k0)

Surface conductivity: 2 0.116 SD Surface conductivity: 2 0.058 SD

19





8 10 12 14 16 18 20-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Frequency (GHz)

Im(k

z/k

0)

8 10 12 14 16 18 20-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Frequency (GHz)

Im(k

z/k

0)


20

Mushroom HIS – Modal Interaction (1)

Modal Interchange



8 9 10 11 12 13 14 15 16 170

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Frequency (GHz)

Re

(kz/

k0)


Phase constant

8 9 10 11 12 13 14 15 16 170

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Frequency (GHz)

Re

(kz/

k0)

21


Modal Interchange



8 9 10 11 12 13 14 15 16 17-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Frequency (GHz)

Im(k

z/k

0)



8 9 10 11 12 13 14 15 16 17-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Frequency (GHz)

Im(k

z/k

0)

22


Pole dynamics



0.4 0.6 0.8 1 1.2 1.4-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Re(kz/k

0)

Im(k

z/k

0)

0.4 0.6 0.8 1 1.2 1.4-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Re(kz/k

0)

Im(k

z/k

0)

Modal interaction (modal exchange) occurs with varying metal thickness

(surface conductivity)


23


Pole dynamics



Modal interaction (modal exchange) occurs with varying metal thickness

(surface conductivity)

0.4 0.6 0.8 1 1.2 1.4-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Re(kz/k

0)

Im(k

z/k

0)

0.4 0.6 0.8 1 1.2 1.4-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Re(kz/k

0)

Im(k

z/k

0)


(0)nw

(1)nw

(2)/ 2n nw

• Type 0 branch points

- positive and negative solutions of dispersion equation

meet forming a second-order zero


- separate branches of complex-conjugate solutions of

dispersion equation (leaky-wave cutoff)


- connect n and n+2 different modes within a given class (TE or TM)

, , 0z z

z zk kH k H kw w w

, , 0z

z zkH k H kw w

Complex frequency-plane branch points:

Hanson and Yakovlev, Radio Sci., 34, Nov.-Dec. 1999

Hanson et al., IEEE Trans. Antennas Propag., 51, Feb. 2003

Yakovlev and Hanson, IEEE Trans. Antennas Propag., 51,

Apr. 2003

A complete rotation about in the complex frequency plane results in the smooth interchange of the n and n+2 TM (TE) modes of the metamaterial structure

(2)/ 2n nw


24

25

Evolution of Branch Points

The branch points control the modal interaction and the position of the branch points

(below, above, and on the real frequency axis) corresponds to different modal behavior


13.135 13.14 13.145 13.15 13.155 13.16 13.165 13.17 13.175-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Re{f} (GHz)

Im{f

} (G

Hz)

2d

= 0.0754 S

2d

= 0.06438 S

2d

= 0.06257336554 S

2d

= 0.0609 S

2d

= 0.0522 S

26

Mushroom HIS – Modal Interaction

Modal Exchange and Modal Degeneracy



Modal interaction (modal exchange) and modal degeneracy occur due to presence of

branch points in the complex frequency plane

13 13.05 13.1 13.15 13.2 13.25 13.30.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

0.92

Frequency (GHz)

Re(k

z/k

0)

2d

= 0.06438 S

2d

= 0.06257336554 S

2d

= 0.06438 S

2d

= 0.0609 S

13 13.05 13.1 13.15 13.2 13.25 13.3-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

Frequency (GHz)

Im(k

z/k

0)

2d

= 0.06257336554 S

2d = 0.06438 S

2d

= 0.0609 S

2d

= 0.0609 S

27

Perturbed Wire-Medium Slab

Wire-medium slab



Phase constant

0 5 10 15 20 25 30 350

0.5

1

1.5

2

2.5

3

3.5

4

Frequency (GHz)

Re

al(k z

/k0)

TEM

SD+ABC

improper real

HFSS.....

proper real


Proper complex solution of mushroom HIS with thin metallic patches in the

limiting case (transparent patches) continues to a proper real (bound) mode

of the wire-medium slab

7

3 5.8 10 (S/m)D tDD 32

15 17.5 20 22.5 25 27.5 300

0.5

1

1.5

2

2.5

3

3.5

4

Frequency (GHz)

Re

(kz/

k0)

2d = 5.810

-4 S

2d

= 5.810-6

S

28

Conclusions

A Nonlocal (SD+ABC) homogenization model with generalized ABC is proposed for the analysis of TM natural waves of mushroom structures with thin metal/graphene patches

It is observed that for some values of surface conductivity (metal thickness) modal interactions (modal exchange) occur, which can be explained by the evolution of complex frequency-plane branch-point singularities across the real frequency axis

Modal propagation of leaky waves and surface waves is studied in the

transition of mushroom-type surface to bed-of-nails-type wire media

Documents

M INTERACTIONS IN MUSHROOM T METAMATERIALS WITH …yakovlev/...MODAL INTERACTIONS IN MUSHROOM- TYPE METAMATERIALS WITH THIN METAL/GRAPHENE PATCHES A. B. Yakovlev and G. W. Hanson SESSION: