Inverse Determination of Coupling of Modes Parameters of Surface Acoustic Wave Resonators

Inverse Determination of Coupling of Modes Parameters of Surface Acoustic Wave Resonators

T.-T. WU
, S.-M. WANG, Y.-Y. CHEN, T.-Y. WU1, P.-Z. CHANG, L.-S. HUANG, C.-L. WANG1, C.-W. WUand C.-K. LEE Institute of Applied Mechanics, National Taiwan University, Taipei, Taiwan 106, R.O.C.

1_{TXC Corporation, Tao-Yuan, Taiwan, R.O.C.}

(Received May 1, 2002; accepted for publication August 5, 2002)

Extraction of coupling of modes (COM) parameters from the manufacturing process of a surface acoustic wave device (SAW) is important in practice. In this study, we employed an inverse algorithm to recover the COM parameters from the measured admittance of a one-port SAW resonator. For completeness, the COM equations, the P-matrix representation and the related COM parameters have been summarized. The admittance of a one-port resonator was derived based on the P-matrix representation. One-port and two-port SAW resonator filters were fabricated and the related responses were measured. The simplex method was then used to determine the COM parameters inversely. The recovered COM parameters of the one-port SAW resonator have been demonstrated to be applicable to simulate not only the frequency response of the one-port SAW resonator but also that of the two-port one. [DOI: 10.1143/JJAP.41.6610]

KEYWORDS: surface acoustic wave, coupling of modes, IDT, inverse, resonator filter

1. Introduction

The development of mobile phone systems has aroused much interest in the investigations of surface acoustic wave (SAW) devices. In the SAW device design, the coupling of modes (COM) model has been used successfully in modeling surface acoustic wave (SAW) devices for many years.1–3) In the COM analysis, parameters of the COM differential equations are obtained either by measurements or by borrowing from some related theoretical modeling.4–6) COM parameters computed from the theoretical model have provided excellent guidelines for the design of SAW filters, nevertheless, experimental values of the COM parameters are still very important in practice to realize high precision. Although there is some literature on the analysis of various types of SAW filters, literature on the measurements of COM parameters are limited. In the past, Hartmann and Hartmann7)used a five-transducer test structure for measur-ing COM parameters of Rayleigh-type SAW. Later, Hartmann and Plessky8) applied the same test structure to measure the leaky Rayleigh-type SAW devices. In their studies, they conducted a massive number of experiments and reported the effects of IDT finger width to pitch ratio and electrode thickness on the COM parameters.

In this paper, we present a complete procedure to recover the COM parameters from the measured admittance of a one-port SAW resonator. For completeness, the COM equations, the P-matrix representation and the related COM parameters have been summarized first. The admit-tance of the one-port resonator was derived based on the P-matrix representation. One-port and two-port SAW resonator filters were fabricated and the related responses were measured. Then, the simplex method was used to determine the COM parameters inversely. The recovered COM parameters of the one-port SAW resonator have also been used to predict the frequency response of the two-port SAW resonator fabricated under a similar manufacturing process.

2. Coupling of Modes Model for a Uniform SAW

Transducer

Coupling of modes equations have been derived and utilized for analyzing SAW transducers with constant or

arbitrary reflectivity weighting.5,6) In the formulations, the effects of propagation loss, electrode reflections, electrical transduction, acoustic reception, thin film loss and the distributed finger capacitance have been included. In this paper, we adopted the COM formulation presented by Abbott.4) For a uniform transducer, the spatial dependence of the COM parameters disappeared, and the COM equations governing SAW mode amplitudes Rðx; !Þ, Sðx; !Þ propagating in the x directions (Fig. 1) can be arranged in a concise form as4)

dRðxÞ dx ¼ jkERðxÞ þ jKRe j2k0x_{SðxÞ þ j} RV0ejk0x ð1Þ dSðxÞ dx ¼ þjkESðxÞ  jKSe þj2k0x_{RðxÞ  j} SV0eþjk0x ð2Þ dIðxÞ dx ¼ þj2SRðxÞe þjk0x_þj2 RSðxÞejk0x j 3!CF= T 3 þ j!RFCF
 V0; ð3Þ

where T is the wavelength of transduction, V0 is the

voltage across the IDT, k0 ¼ 2

T is the transducer’s

synchronous wave-number and

kE ¼ þ ! R  2 2_!C FR2F T 9 þ ð!RFCFÞ2


( )

( )

(

)

(

)

)

(

(

)

(

)

Fig. 1. Coordinates of the IDT.
_{Correspondence author. E-mail address: wutt@spring.iam.ntu.edu.tw}

Part 1, No. 11A, November 2002

#2002 The Japan Society of Applied Physics

j "  þ 6 2_R F T 9 þ ð!RFCFÞ2
# ð4Þ R¼ 3eþjT 3 þ j!RFCF ð5Þ S¼ 3ejT 3 þ j!RFCF ð6Þ KR¼ þKeþjBþ 2j2RF Tej2T 3 þ j!CFRF ð7Þ KS¼ þKejBþ 2j2RF Teþj2T 3 þ j!CFRF : ð8Þ

In the equations, B¼=2 and T¼ are the phase offsets

of the grating and the potential, respectively. The origin of the coordinate system is located at a distance T=4 away

from the center of the first IDT finger (Fig. 1). The COM parameters are

R: Rayleigh wave velocity of the substrate,

: transduction coefficient,

RF: thin film resistance in one transduction period,

CF: interdigital capacitance in one transduction period,

K: reflection parameter,

: propagation loss per unit length.

To facilitate the cascading of uniform transducer ele-ments, solutions of the COM equations can be presented in the P matrix form. In the P matrix representation, the acoustic ports are treated as scattering ports and the electric port as the admittance port as

Sð0Þ RðLÞ I 2 6 4 3 7 5 ¼ P11 P12 P13 P21 P22 P23 P31 P32 P33 2 6 4 3 7 5 RIð0Þ SIðLÞ V0 2 6 4 3 7 5; ð9Þ where P11 ¼ þjKSsinðDLÞ D cosðDLÞ þ j sinðDLÞ ð10Þ P12 ¼ D D cosðDLÞ þ j sinðDLÞe jk0L _ð11Þ P13 ¼ þjL sinðDL=2Þ DL=2
  SD cosðDL=2Þ þ jðKSRþSÞsinðDL=2Þ D cosðDLÞ þ j sinðDLÞ   ð12Þ P22 ¼ þjKRsinðDLÞ D cosðDLÞ þ j sinðDLÞe j2k0L _ð13Þ P23 ¼ þjL sinðDL=2Þ DL=2
  RD cosðDL=2Þ þ jðKRSþRÞsinðDL=2Þ D cosðDLÞ þ j sinðDLÞ   ejk0L ð14Þ P33 ¼ j2 KS2RþKR2Sþ2SR D3
  DL D sinðDLÞ þ jð1  cosðDLÞÞ D cosðDLÞ þ j sinðDLÞ   2 ðKS 2 RþKR2SÞ þ2KRKSRS D3
  1  cosðDLÞ D cosðDLÞ þ j sinðDLÞ
 þj 3!CFL= T 3 þ j!RFCF
 ð15Þ and  ¼ kEk0, D ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2_K RKS p .

3. One-Port SAW Resonator

In this section, we express the admittance of a one-port SAW resonator in terms of the components of a P matrix. Figure 2 shows schematics of the one-port SAW resonator, which consists of an interdigital transducer for both voltage input and output, and two reflected gratings on both sides. D2 and D4 are the distances between the IDT and the left

grating and right grating, respectively. As an external voltage input to the resonator, the IDT converts electrical signals into surface acoustic waves, which then propagate away from the IDT along the x directions. As the waves hit the metal gratings, they are reflected back to the IDT and form a resonant cavity. To enhance the reflection, the metal gratings are usually in short circuit. In that case, the reflection coefficient  of the metal gratings can be obtained from the IDT’s P matrix equation [eq. (9)] by setting V0¼0,

RFðxÞ ¼ 0 and CFðxÞ ¼ 0. It is easily seen that for wave

incidents from the left side of a grating, the reflection coefficient is P11, and for wave incidents from the right side

of a grating, the reflection coefficient is P22.

Utilizing the grating reflection coefficients, section A of the one-port resonator shown in Fig. 3 can be simplified into the form, which consists of a 2  2 P-matrix as9)

bA₁ iA₂ " # ¼ P A 11 PA12 PA₂₁ PA 22 " # aA₁ uA₂ " # ; ð16Þ where A¼P22e2jkD2 and /4 D2 /4 D4 = /2 W

Substrate IDT Grating

Fig. 2. Schematics of the one-port SAW resonator.

2 D D₄ A 1 A a 1 A b 2 A i 2 A u

P

PA₁₁ ¼P22þ A_P 12P21 1  A_P 11 ð17Þ PA₁₂ ¼P23þ AP13P21 1  A_P 11 ð18Þ PA₂₁ ¼P32þ AP12P31 1  A_P 11 ð19Þ PA₂₂ ¼P33þ A_P 13P31 1  A_P 11 : ð20Þ

In addition, using the reflection from the grating on the right side of section A, we have aA

1 ¼ B_bA 1, where  B_¼ P11e2jkD4. If we substitute aA1 ¼ B_bA

1 into eq. (16), the

relationship between the current iA

2 and the voltage u A 2 can be

written as

iA₂ ¼YuA₂; ð21Þ

where Y is called the admittance of the one-port SAW resonator and is equal to

Y ¼ PA₂₂þ  B_PA 12P A 21 1  B_PA 11 : ð22Þ

The real part and the imaginary part of Y are the conductance and susceptance of the resonator, respectively.

4. Theoretical Calculation of the COM Parameters The frequency response of a SAW resonator can be calculated by using the COM-based P matrix provided that the COM parameters are available. The COM parameters are: surface wave velocity vR, reflection parameter K,

propagation loss , transduction parameter , thin film finger capacitance CFand finger resistance RF. In the following, the

formulae or values we used in this study are given.

4.1 Surface wave velocity vR

The perturbation of surface wave velocity by a thin metal film grating was calculated using the free software FEMSDA developed by Hashimoto and Yamaguchi.10)

4.2 Transduction parameter 

The transduction coefficient , which is responsible for the excitation efficiency of the IDT, can be derived as11)

ð!Þ ¼QFðkÞ T ffiffiffiffiffiffiffiffiffiffiffiffiffiffi !WS 2 r ð23Þ

with the elemental charge density QF defined as

QFðkÞ ¼ "Sð1Þ 2 sinðsÞ Psðcos Þ Pmðcos Þ for m kp 2m þ 1; ð24Þ

where W is the aperture of the IDT,  ¼ a=p and a=p is the electrode width to grating period ratio, "Sð1Þis the surface

effective permittivity as a function of the slowness, m is an integer, s ¼_ð2Þkp m, Psðcos Þ is a Legendre function

and Pmðcos Þ is a Legendre polynomial. The coefficient S

is defined as11) 1 s ¼ k0 d"SðkÞ dk   k0 : ð25Þ 4.3 Reflection parameter K

The reflection parameter K represents the reflectivity of the thin film finger in the IDT or grating. It arises from two causes: the electrical loading and the mechanical loading, and can be expressed by12,13)

K ¼ Rk k2 e 2
 þRm h
 sin a p
  ₁ p; ð26Þ

where Rkand Rmdenote the electrical effect and mechanical

effect, respectively. h is the metal film thickness, is the wavelength of the surface wave and k2_e is the electromecha-nical coupling coefficient. The functions Rk and Rm are

Rk¼   2 cosðÞ þ Ps½cosðÞ Ps1½cosðÞ   ð27Þ Rm¼  k2_e "Sð1Þ " U1 ’
2 ð1þ02fÞ þ U2 ’
2 ð2þ02fÞ þ U3 ’
2 0v2_f # ; ð28Þ

where U1U2U3 are the displacements of the surface wave, ’

the surface electrical potential, 0, 0, 0are the density and Lame constants of the thin film electrode, and

1 ¼

40_{ð þ }0_Þ

0_þ20 ; 2¼

0_{: ð29Þ}

The electromechanical coefficient k2

ecan be obtained from

14) k2_e¼2s"ð1Þs : ð30Þ

4.4 Thin film finger resistance RF

For a single electrode type IDT, the thin film finger resistance can be determined by11)

RF¼2W=3aNp; ð31Þ

where  is the sheet resistance of a metal film, W is the aperture, a is the finger width and Np the number of IDT

pairs. For an aluminum thin film with thickness h (m) between 0.05 m to 0 .3 m, the sheet resistance can be approximated as

  0:04=h ð /squareÞ: ð32Þ

4.5 Thin film finger capacitance CF

For ST-X quartz with an aluminum finger on top and the finger width to pitch ratio a=p ¼ 0:5, the finger capacitance can be approximated as.15)

C0¼0:503 ðpF=cmÞ:

4.6 Propagation loss 

For ST-X quartz, the propagation loss can be approxi-mated as11)

  0:47f þ 2:62f2 ðdB=sÞ ð33Þ

where the operating frequency f is in GHz.

On employing eq. (22) and the COM parameters described in this section, the admittance of a one-port resonator can be calculated. Shown in Fig. 4 is the result of simulated conductance and susceptance of a one-port

synchronous SAW resonator. The substrate of the resonator is ST-quartz, the period of the fingers is p ¼ 3:932 m, the aperture is W ¼ 100
2p, the number of IDT pairs is Nt¼25, and the number of gratings is Ng¼300. Parametric

simulation results revealed that the position of the stopband is predominantly controlled by the surface wave velocity R,

and the width of the stopband is controlled by the reflection parameter K. The amplitudes of the conductance and susceptance are influenced by the transduction parameter  and the propagation loss . The thin film resistance RFand

capacitance CFare related to the DC shift of the conductance

and susceptance curves.

5. Experimental

A one-port synchronous SAW resonator similar to the schematics shown in Fig. 2 was fabricated. The substrate of the resonator was ST-quartz, the period of the fingers was p ¼ 3:932 m, the aperture was W ¼ 100
2p, the number of IDT pairs was Nt¼50, and the number of gratings was

Ng¼300. The aluminum thin film thickness was h ¼

160 nm, and the finger width to pitch ratio was m=p ¼ 0:5. The solid lines shown in Figs. 5 and 6 are the measured conductance and susceptance of the resonator, respectively. The measurements were conducted through the utilization of a wafer probe. For later comparison, a standard two-port resonator filter was also fabricated in the same wafer. The material of the substrate and the number of IDT pairs and gratings are all the same as those of the one-port SAW resonator. The distance between the grating and the IDT was 3/8 wavelength and the delay of the two IDTs was 15 wavelength. The solid line shown in Fig. 7 denotes the measured insertion loss of the two-port resonator filter. On comparison with the calculated insertion loss (dotted line), a downshift of the center frequency was found.

6. Inversion of the COM Parameters

In previous sections, we have shown that the conductance and susceptance of a one-port SAW resonator can be calculated using the P-Matrix formulation. We have also demonstrated that the six COM parameters affect the characteristics of the one-port resonator’s admittance in Fig. 4. Simulated conductance and susceptance of a one-port synchronous

SAW resonator.

Fig. 5. Measured conductance of the one-port synchronous SAW resonator.

Fig. 6. Measured susceptance of the one-port synchronous SAW reso-nator.

Fig. 7. Measured (solid line) and simulated (dotted line) insertion loss of the two-port resonator filter.

various ways. For example, the surface wave velocity R

affects the center frequency of the stopband and the reflection parameter K affects the bandwidth of the stopband. On having the forward simulation model and the measured admittance, the COM parameters can be deter-mined inversely through the utilization of an inverse algorithm.

In this study, we adopted the downhill simplex meth-od16,17)as the inverse optimization algorithm. We note that this method is totally different from the simplex method in linear programming, the down hill method can be used in the nonlinear optimization problems.18) An error function e, which is proportional to the absolute value of the complex admittance, was defined as

e ¼X

N

i¼1



½CmðiÞ  CgðiÞ2þ ½SmðiÞ  SgðiÞ2



ð34Þ

where Cm, Sm are the measured conductance and

suscep-tance and Cg, Sg are the guessed conductance and

susceptance. i represents the discrete frequency and N is the number of data points utilized in the inversion process. In the inversion process using the simplex method, if there are n unknowns to be recovered, then, n þ 1 sets of unknown vectors have to be guessed initially. To speed up the inversion efficiency and avoid the local minimum, initial guesses of the COM parameters are important. Furthermore, imposed constraints on the COM parameters are also important for obtaining a correct result.

In the following, we explain the guidelines for making the initial guesses and the necessary constraints. We used the center frequency of the stopband in the measured admittance curve to estimate the surface wave velocity and serve as the initial guess. The upper and lower margin frequencies of the stopband were used to estimate the constraints of the surface wave velocity. As for the reflection parameter, we use the information carried by the bandwidth of the stopband that the larger the reflection parameter, the bigger the width of the stopband.

Parametric simulation results on the effect of finger resistance and capacitance on the admittance of a one-port resonator have shown that they only caused the DC shift of the admittance curves. The position and bandwidth of the stopband are not influenced by their changes. The resistance of a one-port resonator consists of the finger resistance, the resistance of the bus bar electrode and the contact resistance. These effects cannot be decoupled easily from a single admittance measurement. In this paper, instead of recovering the finger resistance RF, we recovered the total resistance RP

of the resonator. This was done by setting RF¼0 in the

COM model first, and then, added a total resistance RP in

series with the impedance of the resonator19) as shown in Fig. 8.

As for the initial guess of the transduction parameter , the propagation loss  and the thin film capacitance CF,

results showed that the values calculated by the aforemen-tioned COM model could serve as good initial guess. Numerical experience has shown that as long as those initial guesses are in the same order of the theoretical calculated one, a stable inversion can be achieved.

In the inversion process, the COM parameters have been

redefined as 0_{¼K
, the reflectivity of electrode system}

per wavelength, 0_{¼
, the attenuation per wavelength,}

0_{¼
, the electromechanical coupling constant, and}

Cst ¼ CF
Nt, the static capacitance of the transducer. The

initial guesses and the recovered COM parameters are listed in Table I. The dotted lines in Figs. 5 and 6 are the conductance and susceptance curves calculated based on the inversely determined COM parameters. We note that the measured and the inversely calculated results were in perfect agreement. Except for the surface wave velocity, we note that the initial guesses of the COM parameters can be about 30% different from the true values.

In the following, we show that the COM parameters determined inversely from the one-port resonator can be utilized in the design of a two-port SAW resonator as well. The dotted line in Fig. 7 denotes the calculated insertion loss of the two-port SAW resonator filter. The COM parameters were calculated based on the formulae (or values) given in §4. Although the basic characteristics of the two-port SAW resonator filter has been predicted by the COM simulation, a frequency shift exists between the measured and simulated insertion loss. The dotted line in Fig. 9 denotes the simulated insertion loss based on the COM parameters inversely determined from the one-port resonator. In this case, we note that the measured and the simulated resonant peaks of the insertion loss match quite well.

7. Conclusion

Extraction of the COM parameters from the SAW manufacturing process is important in practice. In this

Rp

Za = Ra + j

(

)

Ra

( )

jXt

( )

jXa

( )

Fig. 8. Simplified equivalent circuit of the one-port resonator.

Table I. Guessed and recovered COM parameters from the one-port SAW resonator admittance. Initial guess 0(m/s) 0(%) 0ð1= ffiffiffiffi p Þ Cst (pF) 0_{ðneper= Þ R} Pð Þ 3146.1 1:5 2:98  104 _{1.8878 2:96  10}4 ₁₆ 3147.2 1:95 2:58  104 _{1.4565 3:34  10}4 ₁₅ 3146.4 1:05 2:62  104 _{2.1785 4:24  10}4 ₁₇ 3146.3 1:385 3:54  104 _{2.2388 3:07  10}4 ₂₀ 3146.5 1:67 1:93  104 _{2.038 5:57  10}4 _25.5 3146 1:165 2:68  104 _{1.3475 2:10  10}4 ₂₁ 3145.9 1:47 2:29  104 _{2.5691 4:05  10}4 _12.5 Inversed result v0(m/s) 0(%) 0ð1= ffiffiffiffi p Þ Cst (pF) 0_{ðneper= Þ R} Pð Þ 3146.3621 1:4973 2:705  104 _{1.9474 3:823  10}4 _21.6397

paper, we have presented an inverse method for recovering the COM parameters from the admittance measurement. To serve as the forward simulation of the admittance of the one-port resonator, the COM equations, the P-matrix representa-tion and the related COM parameters have been summarized briefly. One-port and two-port SAW resonator filters were fabricated through the standard lithography process and the related responses were measured. The simplex method has been used to determine the COM parameters inversely and the results are in agreement with the standard values. The recovered COM parameters have also been used to calculate the frequency response of the two-port SAW resonator. Results have shown that the calculated response well matches the measured one.

Acknowledgment

The authors acknowledge the financial support for this research provided by National Science Council of Taiwan (NSC90-2212-E-002-156) and TXC corporation.

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