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
1National Nano Device Laboratories, Tainan, Taiwan
2Department of Computer and Communication, SHU TE University, Kaoshsiung, Taiwan
3Institute 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 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 started on designing a periodic configuration as similar as a quantum-well, called as photonic bandgap (PBG). These PBG materials can be applicable to a wide range of frequencies and recently are designed 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 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 microstrip circuit on silicon substrate. First, we discuss the plane wave expansion method to be applied to 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 then the EM simulator using finite element full wave simulation is also performed to compare the theoretical calculation.
II. Theoretic Analysis
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 field vector is parallel to the 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 composed of periodic arrays of air rods inserted into silicon substrate. The 2D photonic lattice consists of cylindrical rods arranged parallel to one another in a hexagon lattice structure below the 50 Ωmicrostrip line with lattice constant 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 the four Maxwell equations determining the propagating phenomenon of electromagnetic wave. [9]
(1)
(2)
(3)
(4)
In the uniform non-magnetic (μ=μ0)dielectric with lossless and charge-free (σ=0 andρ=0), the above four equations can rewritten as:
(5)
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:
, where a is cell distance (or called as lattice constant) and l is integer.
Since the Fouries transform has the property as , in which g(q) is the plane wave coefficient of wave vector of q, so . Another, the electric and magnetic fields distribution in the photonic crystal, can be expressed as the Bloch theorem: [10]
, where is also a periodic function. Therefore, the magnetic fields and effective dielectric constant distribution are written as:
(6)
(7)
(8)
(9)
where G and G” are all designed reciprocal lattice vector.
Finally, the eigenfunction of the TE Mode (H-Polarization) and TM Mode (E-Polarization) are expressed as:
TE-Mode: (10)
TM-Mode:
(11)
If different reciprocal G’ values are replaced into the Maxwell equations, then the following equations can be obtained:
(12)
Where A is the coefficient matrix of the equation group. The above theoretical descriptions are then written in to the Matlab program to calculate the bandgap effect of the PBG structure. The designed parameters are the cell distance and the filling factor on the different substrate. The filling factor is also the function of the PBG type, so the HC lattices are used to create the bandgap effect in this paper.
Fig. 2 shows the pattern of the HC lattice. We can find the filling factor () and cell distant in the 2-D PBG pattern, which the values of a and radius r are chosen to yield a filling factor = 0.6. The proposed structure has been fabricated on the 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 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.
III. Results and Discussions
Theoretical results of photonic bandgap structure using the plane wave expansion method are shown in Fig. 3, in which a sizable bandgap is created. The specific values of the Brillouin zone analyzed for the HC lattices are shown 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) 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. 5. It is found that increasing number of 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.
IV. Conclusions
The PBG structure for microstrip circuit on silicon substrate is verified to have bandgap effect at the C band of microwave region is investigated. The 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 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.
References
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Fig.1 Three-dimensional view of PBG structure
Fig. 2 Pattern of the honeycomb-circle (HC) lattice.
(a) TE-mode
Fig. 3(b) TM- mode
Fig. 3 The photonic band structure.
Fig. 4 S-parameters of EM simulation for the PBG microstrip line on silicon substrate.
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.