Fast Extraction of Coupling of Modes Parameters for Surface Acoustic Wave Devices Using Finite Element Method Based Simulation

Abstract

The finite element method (FEM) has been applied to extract the coupling-of-modes (COM) parameters of surface acoustic wave (SAW) devices for a long time. It always involves calculating the dispersive curves or harmonic admittance, which makes the extraction process and results sophisticated, time-consuming and inaccurate. Therefore, a simple method is proposed to extract all COM parameters of the SAW devices rapidly and accurately in this paper. It is based on the FEM and combines the stationary analysis with modal analysis. We have described in detail the basic principles and procedures of the proposed method, and made a comprehensive comparison between the proposed method and the other two existing methods. We have also examined the proposed method by extracting COM parameters of some common SAW substrate, and compared our extracted results with those reported in the other literatures. Results show that our proposed method holds higher accuracy and more efficiency (~s order) than the others (~h order). Moreover, our extracted COM parameters are in an excellent agreement with those reported in the other literatures.

1. Introduction

Surface acoustic wave (SAW) devices have expanded into diverse application fields such as filters, sensors and radio frequency identification (RFID) tags [1,2,3,4]. Accurate and efficient simulation methods are crucial for designing high-performance SAW devices. Several methods have been proposed for modeling and analyzing the characteristics of SAW devices [5,6,7,8], including the earlier δ-function models, equivalent circuit models and the later coupling-of-modes (COM) equations, finite element method/boundary element method (FEM/BEM), etc. However, the first two methods cannot take into account the interdigital transducer (IDT)’s internal reflections. Though the FEM/BEM model is considered to be the most accurate simulation method, it is time-consuming [7]. Therefore, the COM method is widely used for analyzing and designing SAW devices for the reason that it is extremely fast and holds enough accuracy. Regardless of the accuracy of the model itself, the accuracy of this method entirely depends on the accuracy of the COM parameters (SAW velocity v, reflectivity κ, transduction coefficient α, static capacitance C and propagation attenuation γ). Thus, obtaining accurate COM parameters is crucial for designing SAW devices. To obtain these key parameters, perturbation theory [9] and measurement harmonic admittance [10] have been proposed, but they are either costly or lack of accuracy. FEM-based methods FEM/spectral domain analysis (FEM/SDA) [11] and FEM/BEM [12] have also been employed in COM parameters extraction, but they are invalid for the multilayer structures. Recently, the pure FEMs, especially in combination with the existing commercial software such as ANSYS and COMSOL Multiphysics, have been attractively studied to extract these parameters [13,14,15]. They can conquer the problems mentioned above and several methods have been proposed, which can be divided into two branches: one is by calculating the dispersion characteristics [16,17]. The SAW velocity at different frequencies (dispersive curves) is calculated, which is realized by applying the Floquet boundary conditions, under short-circuit (SC) and open-circuit (OC) conditions. Then, SAW velocity v, reflectivity κ, transduction coefficient α and static capacitance C are obtained by finding the stopband edges and fitting the dispersive curves, respectively. Such methods are sophisticated and time-consuming for the reason that velocity at each frequency must be calculated. The other is by calculating the Harmonic Admittance (HA) [18]. The COM parameters are obtained from the HA in the same way as the former. The harmonic analysis is time-consuming because the computation time is proportional to the number of frequency points. Moreover, the accuracy totally depends on the step length of frequency, which makes the extracted parameters not exactly accuracy.

Thus, in this paper, we propose a simple and fast method, performed by COMSOL Multiphysics, to extract accurate COM parameters of SAW devices. It is based on the FEM and combines the stationary analysis with modal analysis. Firstly, we describe in detail the basic principles and procedures of the proposed method. Then, we make a comprehensive comparison between the proposed method and the other two existing methods which are both based on the harmonic analysis. Finally, we examine the proposed method by extracting COM parameters of some common SAW substrates (ST quartz, ST-25°X quartz and Y-51.25°Z LiTaO₃), and compare our extracted results with those reported in the other literatures.

2. Methods

2.1. Solutions of COM Equations

Figure 1 shows the schematic of COM model for the IDT, in which W is the aperture length, p is the pitch of the IDT, the wavelength (λ) is the twice of the pitch (λ = 2p). A drive voltage V connect over the bus bars of the IDT excites waves. In reverse, the waves propagating under the electrodes cause flow of the current I. The R(x) and S(x) represent the slowly varying amplitudes of the modes propagating in the positive and negative x-directions, respectively.

The presence of the electrodes on the surface introduces a coupling between the modes. Assuming a linear coupling between the amplitudes due to reflectivity, and a linear transduction coefficient due to piezoelectricity, the final COM equation can be obtained [7]:

$$\left\{\begin{array}{l}\frac{\mathrm{d}R\left(x\right)}{dx}=-j\delta R\left(x\right)+j\kappa S\left(x\right)+j\alpha V,\\ \frac{\mathrm{d}S\left(x\right)}{dx}=j\delta S\left(x\right)-j{\kappa }^{*}S\left(x\right)-j{\alpha }^{*}V,\\ \frac{\mathrm{d}I\left(x\right)}{dx}=-2j{\alpha }^{*}R\left(x\right)-2j\alpha S\left(x\right)+j\omega CV,\end{array}\right.$$

(1)

where δ is the detuning parameter:

$$\delta =k-k_0=k_r-k_0-j\gamma =\frac{\omega }{\nu }-k_0-j\gamma ,$$

k is the complex wavenumber, k = k_r − jγ, k_r is the resonance wavenumber and k₀ is the center wavenumber, k₀ = 2π/λ = π/p. The COM parameters of the model are velocity v, reflectivity κ, transduction coefficient α, static capacitance per unit length C and propagation attenuation γ (superscript * denotes the complex conjugate). These parameters are determined from the simulation of an infinite periodic grating under different boundary conditions [19].

When considering no attenuation (γ = 0), the equation system (1) under SC conditions (i.e., V = 0) gives the relationships between stopband edges and velocity v, reflectivity κ:

$$\left\{\begin{array}{l}\nu =\lambda \frac{\left(f_{sc+}+f_{sc-}\right)}{2}\\ \left|\kappa \right|\lambda =2\pi \frac{f_{sc+}-f_{sc-}}{f_{sc+}+f_{sc-}}\end{array}\right.$$

(2)

Under OC conditions (i.e., I = 0), the equation system (1) gives the relationships between stopband edges and transduction coefficient α, directivity phase φ:

$$\left\{\begin{array}{l}\left|{\alpha }_n\right|=\sqrt{\omega C_n\lambda \pi \left(\frac{f_{oc+}+f_{oc-}}{f_{sc+}+f_{sc-}}-1\right)}\\ \cos \left(\phi \right)=\cos \left({\alpha }^2/\kappa \right)=\frac{{(f_{oc+}-f_{oc-})}^2-{(f_{sc+}-f_{sc-})}^2-{\left[\left(f_{oc+}+f_{oc-}\right)-\left(f_{sc+}+f_{sc-}\right)\right]}^2}{2\left(f_{sc+}+f_{sc-}\right)\left[\left(f_{oc+}+f_{oc-}\right)-\left(f_{sc+}+f_{sc-}\right)\right]}\end{array}\right.$$

(3)

where α_n is the normalized transduction coefficient, namely, α_n = αλ/√(W/λ). C_n is the normalized static capacitance, C_n = Cλ/W. The φ is the directivity phase, namely, φ = 2φ_α − φ_κ, where φ_α and φ_κ are the phases of α and κ. Equations (2) and (3) imply that all COM parameters, except the attenuation γ, can be obtained from the lower and upper edges of the SC and OC stopbands (f_SC−, f_SC+, f_OC−, f_OC+), if the C was previously extracted or given.

2.2. Extraction Method

Based on this, we propose a simple method, performed by COMSOL Multiphysics, to rapidly extract accurate COM parameters. It is divided into two steps: stationary analysis and modal analysis.

At first, stationary analysis is adopted to extract the C, which employs the FEM/PML model in this paper. Figure 2 shows a single wavelength primitive cell of the infinitely periodic IDTs on the piezoelectric substrate. A perfectly matched layer (PML) is placed at the bottom of the substrate to avoid reflections from the bottom [20]. An air layer is taken into account due to the charge on the IDT-air interfaces [21]. The solid mechanical field and electrostatic field of COMSOL are coupled. The detailed boundary conditions are listed in

Table 1. The boundary conditions of the stationary analysis.

Boundary	Mechanical Boundary Conditions	Electrical Boundary Conditions
ΓT (top of air)	Free	Zero charge
ΓA (air-solid interfaces)	Free	Continuity
Γ+, Γ− (±electrode-solid interfaces)	Free	VΓ+ − VΓ− = ΔV
ΓB (bottom of substrate)	Fixed	Zero charge
ΓL, ΓR (left and right boundaries)	Periodic continuity boundary condition
ΓF, ΓE (front and back boundaries)	Periodic continuity boundary condition

. The stationary analysis is performed with a voltage drop ΔV loaded between the left and right electrode. When the analysis is done, the normalized static capacitance C_n can be deduced from the electrostatic field energy W_e, namely:

$$C_n=\frac{2W_e}{{\left(\Delta V\right)}^{2}W}.$$

(4)

Thus, the static capacitance per unite C can be extracted.

Then, modal analysis is adopted to extract the remaining COM parameters. That is, model analysis is performed to solve the eigenvalues under SC and OC conditions. These eigenvalues correspond to f_SC−, f_SC+, f_OC−, f_OC+, which are the key values to determine the remaining COM parameters. The SC and OC conditions are realized by setting zero voltage and charge, respectively, at the electrode-solid interfaces (Γ₊, Γ₋). The detailed boundary conditions for the modal analysis are given in

Table 2. The boundary conditions for the three different methods.

Boundary	Mechanical Boundary Conditions	Electrical Boundary Conditions
ΓT	Free	Zero charge
ΓA	Free	Continuity
ΓB	Fixed	Ground
Γ+, Γ−	Free	1: V = 0 (SC); $\int D_i·n_{i}dS=0$ (OC) 2 and 3: VΓ+ = 1V; VΓ− = 0V
ΓL, ΓR	1 and 2: Periodic continuity boundary condition 3: Floquet boundary condition: kx = (1 + Δ)k0
ΓF, ΓE	Periodic continuity boundary condition

Notice: 1, 2, 3 represent methods 1, 2 and 3.

. Finally, meshing the primitive cell. Mesh division determines the solution mode of the model and directly affects the calculation time and accuracy. In this work, the boundary surface is firstly meshed, and then the whole model is meshed along the thickness direction.

3. Comprehensive Comparison of Different Extraction Methods

In this section, we make a comprehensive comparison between the proposed method and the other two methods, which are both based on the harmonic analysis. The analysis calculates the harmonic admittance, of which resonance and anti-resonance frequencies (f_r, f_a) correspond to edges of the stopbands. Thus, finding the resonance and anti-resonance frequencies can determine the COM parameters. However, in some particular cases, the two required modes cannot be simultaneously excited, but it can be conquered by adding a tiny directivity [22]. To achieve this, there are two methods. One is adding a tiny directivity naturally, that is, slightly changing the Euler angle to make the substrate unidirectional. The other is adding a tiny directivity structurally, that is, the wavelength (λ) slightly differs from the period (2p). It can be realized by the Floquet boundary conditions with a tiny detuning: kx = (1 + Δ)k₀, where the Δ is a tiny proportion. For the sake of convenience, in the following text, we use method 1, 2 and 3 to denote our proposed method and the other two methods, respectively. The detailed settings of boundary conditions are listed in Table 2 for the three different methods, where D_i is the electrical displacement and n_i is the normal vector of S.

We have extracted all COM parameters of the ST quartz substrate with the three different methods. We considered the same configuration as presented in [23], where the pitch of IDT was p = 3.932 µm, the aluminum thin film thickness was 160 nm, and the finger width to pitch ratio was m/p= 0.5. The material constants were taken from ref. [24]. The sparse direct solver PARDISO was employed in the FEM calculation. The calculation was run on a 20 cores 2.2 GHz Xeon computer with 128 GB of RAM. The computation time of the method 1 was 30 s, while that of the others were both 1 h 3 min.

After the calculation, three groups of COM parameters were obtained. Then, we calculated the relative admittances (Y11) by these COM parameters and COMSOL, respectively. As depicted in Figure 3, the Y11 calculated by the first group of COM parameters, extracted by the method 1, is in the closest correspondence to the COMSOL calculating Y11. The slight difference is caused by the PML, which introduces additional attenuation in the calculation [25]. Thus, it could be concluded that our proposed extraction method holds excellent characteristics of higher accuracy and more efficiency simultaneously.

4. Extraction Results

In order to examine the proposed method, we have extracted the COM parameters of some common SAW substrate. They were ST quartz, ST-25°X quartz and Y-51.25°Z LiTaO₃. In this section, the reference plane of coupled-mode coefficients is located at the point for Arg(α) = 0°.

4.1. ST Quartz

We considered infinitely long IDTs on the ST quartz substrate, where the pitch of IDT was p = 3.932 µm, the aluminum thin film thickness was h = 160 nm, the finger width to pitch ratio was m/p= 0.5. The material constants were taken from ref. [24].

Table 3. COM parameters for ST quartz substrate.

COM Parameter	Measured Value [23]	Calculated Value [26]	Our Results
v (m/s)	3146	3152	3143
κp (= κλ)	−0.015	−0.021	−0.023
Arg(κ) [deg]	—	—	0
αn (f0)($\sqrt{1/\Omega \lambda }$)	2.71 × 10−4	2.84 × 10−4	2.85 × 10−4
Cst (pF)	1.95	1.92	1.953

shows our extracted COM parameters, measured [23] and calculated values [26]. It is apparent that our results of v, α_n, C_st, agree well with the measured ones, where the C_st is defined as in [23]. On the other hand, the κ_p is larger than the measured value, but close to the calculated value in ref. [26].

4.2. ST-25°X Quartz

We considered infinitely long IDTs on the ST-25°X quartz substrate, where p = 10 µm, h = 0.2 µm and m/p= 0.5. The material constants were also taken from ref. [24].

Table 4. COM parameters for ST-25°X quartz substrate.

COM Parameter	Measured Value [27]	Calculated Value [27]	Our Results
v (m/s)	3239.7	3242.1	3240.7
κp (= κλ)	0.00584	0.0049	0.00575
Arg(κ) [deg]	95.68	103.17	97.96
$\alpha \lambda /\sqrt{\omega C\lambda }$	0.0214	0.0205	0.0224
Cn (pF/m)	50.77	49.27	49.11

shows our extracted COM parameters, measured and calculated values [27]. Excepted for normalized transduction coefficient $\alpha \lambda /\sqrt{\omega C\lambda }$ and static capacitance C_n, our extracted results agree better with the measured ones [27], than the calculated ones.

4.3. Y-51.25°Z LiTaO₃

We considered infinitely long IDTs on the Y-51.25°Z LiTaO₃ substrate, where p = 8 µm, h = 0.2 µm and m/p= 0.5. The material constants were taken from ref. [28]. Table 4 shows our extracted COM parameters, measured and calculated values [27]. Clearly, all of our extracted parameters agree better with the measured ones [27] than the calculated ones.

From Table 3, Table 4 and

Table 5. COM parameters for Y-51.25°Z LiTaO₃ substrate.

COM Parameter	Measured Value [27]	Calculated Value [27]	Our Results
v (m/s)	3199.8	3206.7	3200.23
κp (= κλ)	0.0098	0.0106	0.0106
Arg(κ) [deg]	101.6	113.91	100.4
$\alpha \lambda /\sqrt{\omega C\lambda }$	0.0455	0.0407	0.0459
Cn (pF/m)	455	446.6	447.68

, it is confirmed that the difference between our extracted parameters and the measured ones [23] is slightly large for ST quartz compared with the ST-25°X quartz and the Y-51.25°Z LiTaO₃ substrates. This seems to be due to the fact that the measured parameters are extracted by fitting the input admittance characteristics of a one-port SAW resonator with 25-electrode pairs and 100-wavelength aperture width [23]. Therefore, the finite IDT period, the finite aperture width, and the residual parasitics (such as resistance of busbar, which are not considered in our theoretical model) influence the measured values.

5. Conclusions

In this paper, we propose a simple and fast method to extract accurate COM parameters for designing SAW devices. It is based on the FEM and combines the stationary analysis with modal analysis. We have made a comprehensive comparison between the proposed method and the other two existing methods. Results show that the proposed method holds higher accuracy and more efficiency than the other FEM-based methods. We have also examined the proposed method by extracting COM parameters of some common SAW substrates (ST quartz, ST-25°X quartz and Y-51.25°Z LiTaO₃), and compared our extracted results with those reported in the other studies. Results show that the extracted COM parameters by the proposed method are in an excellent agreement with the reported values in the other literatures, which implies that the method can be utilized in the practical design of SAW devices.