Two Dimensional Photonic Bandgap Structure for
Microstrip Circuits on Silicon Substrate
Min-Hung Weng1, Shich-Chuan Wu1, Tsung Hui Huang2, Han-Ding Hsueh1, Ru Yung
Yang3, and Mau-Phon Houng3
1
National Nano Device Laboratories, Tainan, Taiwan
2
Department of Computer and Communication, SHU TE University, Kaoshsiung,
Taiwan
3
Institute of Microelectronics, Department of Electrical Engineering,
National Cheng Kung University, Tainan, Taiwan
Abstract-In this paper, the photonic bandgap (PBG) structure of the henoycomb-circle
(HC) type is verified to have bandgap effect for microstrip circuit on silicon substrate
at microwave region. The electromagnetic wave of the TE-mode and TM-mode are
obtained to understand the bandgap effect, according to the filling factor and the cell
distance of the PBG structure by using the Plane Wave Expansion method. And then
the full wave electromagnetic simulation tool is also used to simulate and verify the
bandgap effect of the proposed PBG structure. Therefore, the designed circuit will be
suitable to applicable to suppress the spurious response in the MMIC.
Key-words: photonic bandgap, PBG, microstrip circuit, silicon substrate
I. Introduction
Recently, photonic crystals made of
artificial materials in one, two or three
dimensional periodic dielectrics have
been proved to have interesting
characteristics, not yet available with the
started on designing a
configuration
as
similar
quantum-well, called as
bandgap (PBG). These PBG
can be applicable to a wide
frequencies and recently are
periodic
as
a
photonic
materials
range of
designed
ordinary materials. These new periodic
structures act on electromagnetic waves
in a similar way as natural crystals act
on electron waves, and provide a wide
rejection band or forbidden frequency
bands in some frequency range. [1-3]
The proceeded researches or scientific
literatures about microwave circuits are
for active or passive circuits, broadband
absorbers and antennas substrates in
microwave and millimeter wave
domains.[4-9] A novel configuration of
stacked uniplanar compact photonic
bandgap (UC-PBG) plane has been
published in[4]. Practically, the PBG
configurations
for
conventional
microstrip
circuits
usually
have
a
field
vector
is
parallel
to
the
difficulty in evaluating relationships
between their physical dimensions and
bandgap range. Another, the PBG
microstrip circuits are fabricated on the
commercial microwave substrate, which
are not suitable for integrating with the
MMIC.
In this paper, the objective is to
present a rigorous analysis method of
predicting the bandgap effect on PBG
symmetrical axis of the rods to obtain
bandgap effect. Connected structures are
excited by an H polarization in which
the magnetic field vector is parallel to
symmetrical axis of the rods to obtain
bandgap effect. We consider only
disconnected structures in this paper, but
all presented results can be applied for
connected structures. Fig.1 presents the
first studied PBG structure which is
microstrip circuit on silicon substrate.
First, we discuss the plane wave
expansion method to be applied to
composed of periodic arrays of air rods
inserted into silicon substrate. The 2D
photonic lattice consists of cylindrical
analyze the PBG simple microstrip
structure described in Fig.1. By
calculating the electromagnetic wave of
the TE-mode and TM-mode, the
bandgap effect can be understood
according to the filling factor and the
cell distance of the PBG structure, and
rods arranged parallel to one another in a
hexagon lattice structure below the 50
Ωmicrostrip line with lattice constant
then the EM simulator using finite
element full wave simulation is also
performed to compare the theoretical
calculation.
the four Maxwell equations determining
the propagating phenomenon of
electromagnetic wave. [9]

  H (r , t )  0
(1)


   (r ) E (r , t )  0
(2)
II. Theoretic Analysis
1 


  E (r , t ) 
H (r , t )  0
c t
There are two types of structures
developed in precedent papers[4-8]: one
is composed of parallel air holes in
electric material and called connected
structure and another order one is
composed of parallel dielectric rods in
air and called disconnected structure.
Excitation is the main difference
between
these
two
structures.
Disconnected structures are excited by
an E polarization in which the electric
a=10mm and rod radius d=2.67mm, also
called heneycomb-circle (HC) type.
To analyze the bandgap effect of the
proposed structure, the beginning from
(3)
1  


  H (r , t )   (r ) E (r , t )  0 (4)
c
t
In the uniform non-magnetic (μ=
μ 0)dielectric with lossless and
charge-free (σ=0 andρ=0), the above
four equations can rewritten as:

(
 H

)
2
c2

H
(5)
r (r )
Three popular numerical method to
calculate the PBG bandgap are (1)
Plane-wave expansion method, (2)
Transfer matrix method and (3) Finite
Difference Time Domain (FDTD). In
this paper, the plane-wave expansion
method is used. In the phonitic crystal
with periodic dielectric, the electric and
magnetic fields are periodic function
having the non-shifting displacement.
Therefore, the function of the dielectric
constant can be written as:



 ( r  l a )   ( r ) , where a is cell
distance (or called as lattice constant)
and l is integer.
Since the Fouries transform has the
property as


f ( r )   dqg ( q )e
 
i qr
, in
which g(q) is the plane wave coefficient
of
wave
vector
of
q,
so



 
 

TE-Mode:

 
( ) 2 D( k , G ) 
c



(10)

 ( k  G ' )  ( k  G )
1


 


 
(G  G ' )  D ( k , G ' )

G'
TM-Mode:

 
( ) 2 B( k , G ) 
c



(11)

 ( k  G' )  ( k  G)
1
(G  G ' )  B ( k , G ' )

G'
If different reciprocal G’ values are
replaced into the Maxwell equations,
then the following equations can be
obtained:

 
f ( r  R)   dqg ( q )ei q  r ei q  R  f ( r )   dqg ( q )ei q  r
2
(12)
AX

X
. Another, the electric and magnetic
c2
fields distribution in the photonic crystal,
can be expressed as the Bloch theorem:
Where A is the coefficient matrix of
[10]
the equation group. The above
 


i( k  r )
,
where
theoretical descriptions are then written
H k ( r )  u k ( r )e



in to the Matlab program to calculate the
uk ( r  l a )  uk ( r ) is also a periodic
bandgap effect of the PBG structure.
function. Therefore, the magnetic fields
The designed parameters are the cell
and effective dielectric constant
distribution are written as:
distance and the filling factor on the
  

different substrate. The filling factor is
(6)
H k ( r )   H  e i ( k  G ) r
G

also the function of the PBG type, so the
G
  



HC lattices are used to create the
E z (r )   A(k , G )  e i ( k G )r (7)
G
bandgap effect in this paper.
  
 

H z (r )   D(k , G )  ei ( k G )r (8)
Fig. 2 shows the pattern of the HC
G



1
lattice. We can find the filling factor
(9)
 1 (G" )  e iG"r
 

 ( r ) G"
( Filling  Factor  A1 / A2 ) and cell distant
where G and G” are all
in the 2-D PBG pattern, which the
designed reciprocal lattice vector.
values of a and radius r are chosen to
Finally, the eigenfunction of the
yield a filling factor = 0.6. The proposed
TE Mode (H-Polarization) and TM
structure has been fabricated on the
Mode (E-Polarization) are expressed as:
silicon substrate with 0.65 mm thick and
dielectric constant εr of 11.74. The
characteristic equation is scanned along
the edge of the Brillouin zoon, which
represents the irreducible values of the
5. It is found that increasing number of
propagation vectors in the lattice. All
other propagation vectors are comprised
of these vectors. For each of the
propagation vector on the Brillouin zone,
the eigenvalue corresponding to a
frequency of propagation is determined.
rods n has a real effect on bandgap depth,
namely, the insertion loss decreases
from –3 dB to –15 dB. It may be
because the boundary conditions
imposed by the microstrip dimensions
satisfy the field distributions of the
propagation excited mode.
III. Results and Discussions
Theoretical results of photonic
bandgap structure using the plane wave
expansion method are shown in Fig. 3,
IV. Conclusions
The PBG structure for microstrip
in which a sizable bandgap is created.
The specific values of the Brillouin zone
analyzed for the HC lattices are shown
circuit on silicon substrate is verified to
have bandgap effect at the C band of
microwave region is investigated. The
along the x axis of Fig. 3. From the
theoretical calculation, the bandgap of
the TE-mode and TM-mode are obtained
and therefore the bandgap width (a/λ)
filling factor and the cell distance are
key components to control the bandgap
width and bandgap depth. By using the
plane wave expansion method, the
electromagnetic wave of the TE-mode
and TM-mode are obtained to
understand the bandgap effect. And then
of the HC pattern with a filling factor =
0.6 on the silicon substrate with 0.65
mm is 0.42-0.45, namely the bandgap
frequency is located from 3.67 to
3.94GHz.
Fig. 4 shows S-parameters of EM
simulation for the PBG microstrip line
on silicon substrate, by suing the HFSS
simulation tool. The simulated results of
the reflection (S11) and transmission (S21)
coefficients of a 50Ω microstrip line on
HC PBG structure have the bandstop
frequency response at 2.4 to 3.5GHz,
which are larger than those of the
theoretical calculation results. Further
investigating the effect of the PBG, the
S-parameters of EM simulation as
functions of number of rods n along
propagation axis for the PBG microstrip
line on silicon substrate are shown in Fig.
the electromagnetic simulation tool is
also used to simulate and verify the
bandgap effect of the henoycomb-circle
(HC) type for microstrip circuit on the
silicon
substrate.
Although
the
theoretical calculation results and the
EM simulation results are some different,
there is still enough evidence to obtain
the bandgap effect at C band. Therefore,
the designed circuit will be suitable to
applicable to suppress the spurious
response in the MMIC.
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Fig.1 Three-dimensional view of PBG
structure
Fig. 3(b) TM- mode
Fig. 3 The photonic band structure.
Fig. 2 Pattern of the honeycomb-circle
(HC) lattice.
Fig. 4 S-parameters of EM simulation
for the PBG microstrip line on silicon
substrate.
(a) TE-mode
Fig. 5 S-parameters of EM simulation as
a function of number of rods along
propagation axis for the PBG microstrip
line on silicon substrate.