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
A. B. Yakovlev et.al., IEEE Trans. Microwave Theory Tech., 57, Nov. 2009
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
A. B. Yakovlev et.al., IEEE Trans. Microwave Theory Tech., 57, Nov. 2009
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
A. B. Yakovlev et.al., IEEE Trans. Microwave Theory Tech., 57, Nov. 2009
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
A. B. Yakovlev et.al., IEEE Trans. Microwave Theory Tech., 57, Nov. 2009
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
solid lines: proper complex
dashed lines: improper complex
Wire-medium slab
Leaky Waves
A. B. Yakovlev et.al., IEEE Trans. Microwave Theory Tech., 57, Nov. 2009
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
solid lines: proper complex
dashed lines: improper complex
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
Mushroom HIS – Thin Metallic Patches
Proper complex and improper complex solutions
of mushroom HIS parameterized by metal thickness
solid lines: proper complex
dashed lines: improper complex
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
Mushroom HIS – Thin Metallic Patches
Phase constant
solid lines: proper complex
dashed lines: improper complex
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
Mushroom HIS – Thin Metallic Patches
Attenuation constant
solid lines: proper complex
dashed lines: improper complex
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)
Surface conductivity: 2 0.116 SD Surface conductivity: 2 0.058 SD
20
Mushroom HIS – Modal Interaction (1)
Modal Interchange
solid lines: proper complex
dashed lines: improper complex
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)
Surface conductivity: 2 0.06264 SD Surface conductivity: 2 0.06206 SD
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
Mushroom HIS – Modal Interaction (1)
Modal Interchange
solid lines: proper complex
dashed lines: improper complex
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)
Surface conductivity: 2 0.06264 SD Surface conductivity: 2 0.06206 SD
Attenuation constant
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
Mushroom HIS – Modal Interaction (1)
Pole dynamics
solid lines: proper complex
dashed lines: improper complex
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)
Surface conductivity: 2 0.06264 SD Surface conductivity: 2 0.06206 SD
23
Mushroom HIS – Modal Interaction (2)
Pole dynamics
solid lines: proper complex
dashed lines: improper complex
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)
Surface conductivity: 2 0.03944 SD Surface conductivity: 2 0.0377 SD
(0)nw
(1)nw
(2)/ 2n nw
• Type 0 branch points
- positive and negative solutions of dispersion equation
meet forming a second-order zero
• Type 1 branch points
- separate branches of complex-conjugate solutions of
dispersion equation (leaky-wave cutoff)
• Type 2 branch points
- 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
Complex Frequency-Plane Branch Points
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
Complex Frequency-Plane Branch Points
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
solid lines: proper complex
dashed lines: improper complex
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
solid lines: proper complex
dashed lines: improper complex
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
A. B. Yakovlev et.al., IEEE Trans. Microwave Theory Tech., 57, Nov. 2009
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
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