Application of Synchronous Two-Port Resonators for ... - IEEE Xplore

ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 45, no. 4, july 1998

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Correspondence Application of Synchronous Two-Port Resonators for Measurement of SAW Parameters in Piezoelectric Crystals Waldemar Soluch, Senior Member, IEEE Abstract—The synchronous two-port resonator can be used for the determination of SAW parameters, including velocity, electromechanical coupling coefficient, and reflection coefficient of one strip. These parameters can be determined from a comparison between the measured and calculated transfer functions of the resonator. Using this technique, the SAW free surface velocity of the 45 XZ-cut of lithium tetraborate (Li2 B4 O7 ) crystal was found to be equal to 3436 m/s. The other measured parameters agree well with the values found in the literature.

I. Introduction he synchronous two-port resonator [1] (Fig. 1) consists of two reflectors and two interdigital transducers (IDTs). For the synchronous structure, it is assumed that the IDTs are integral parts of the reflectors. Furthermore, both the IDTs and the reflectors are identical, the IDTs are symmetrical, and the electrodes of the reflectors are short-circuited. Therefore, we have two regions with different SAW velocities: between the reflectors, and inside them. The surface between the reflectors can be free or metallized. The purpose of this work is to delineate how this type of resonator can be used for the determination of SAW parameters in piezoelectric crystals. By way of example, calculations and measurements will be presented for the 45◦ XZ-cut of lithium tetraborate (Li2 B4 O7 ) crystal [2].

T

II. Transfer Function The transfer function A12 of the two-port resonator can be written as [3]: A12 =

2 t[1 + r(S12 − S11 )]2 S13 2 2 )]2 , 2 2 (1 − rS11 ) − t [S11 + r(S12 − S11

(1)

where: t = exp(−j(βS d − π)), r = ΓTi exp(−j2βr l), βS = 2πf /vS , βr = 2πf /vr . Manuscript received July 29, 1997; accepted March 20, 1998. The author is with the Institute of Electronic Materials Technology, 01-919 Warsaw, Poland (e-mail: soluch [email protected]).

Fig. 1. SAW synchronous two-port resonator.

Here S11 , S12 , and S13 are the IDT scattering coefficients, Γ is the reflection coefficient of the reflectors, Ti is the loss coefficient, f is the frequency, and vS and vr are the SAW velocities in the areas between the reflectors and inside them, respectively. In the synchronous resonator, the IDTs are inside the reflectors and, therefore, d and l = p − p/4 (Fig. 1) are used in the expressions for t and r, respectively, instead of the distances between their centers. Also, an additional π radians phase shift is needed in the expression for t for correct phase shift calculation. When Γ = 0, we obtain: 2 2 |A012 | = tS13 /(1 − t2 S11 ).

(2)

which is equivalent to the transfer function of a delay line. The input admittance of the IDT may be calculated from the expressions given in [4], which take into account the influence of reflections. The scattering coefficients of the IDT and the reflection coefficient of the reflector may be calculated from the expressions given in [3]. To calculate the transfer function (1) as a function of frequency, it is advantageous to write a computer program. For precise measurements of the SAW parameters, the resonator should be properly designed. Usually, the SAW parameters are approximately known from calculations or from other measurements. The required dimensions and numbers of electrodes for the resonator can be chosen correctly by using the approximate SAW parameters in the computer program. The center frequency of the resonator should be chosen as low as possible in order to diminish the influence of errors in mask fabrication on the accuracy of the SAW parameter measurements. This frequency is limited by the maximum allowable dimensions of the resonator. To obtain reasonably high loaded Q for the resonance peaks in the main reflection band, the number of the electrodes in the IDTs and the aperture should be kept small for high coupling substrates. For low coupling substrates, these parameters should be made sufficiently large to obtain reasonable insertion loss. Equation (1) was applied using the published SAW parameters of the 45◦ XZ-cut lithium tetraborate (Li2 B4 O7 )

c 1998 IEEE 0885–3010/98$10.00

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crystal [2] to design a two-port resonator with the center frequency near 140 MHz. The following data were used: W = d = 2 mm, p = 12 µm, l = 9 µm, Nt = 31, and Nr = 300, where Nt and Nr are the number of electrodes in each of the IDTs and reflectors, respectively. It was assumed that the reflection coefficient per strip γ = −0.03 (for h/λ = 0.008, where h is the aluminum layer thickness and λ is the SAW wavelength [2], and γ stands for (∆V /V )gr as used in [3]), the electromechanical coupling coefficient k2 = 0.01 [2], Ti = 0.98 (as for quartz [3]), and the load resistance Rl = 50 Ω. Fig. 2 presents the results of the calculations. The delay line transfer function [Fig. 2(a)] is deformed by the reflections from the electrodes of the IDTs. Four resonance peaks are seen in the main reflection band of the reflector, and the transfer function of the resonator is asymmetrical. Below the main reflection band, the reflections from the reflectors add to the transfer function of the delay line, while above that, they subtract from it. This can be explained easily, because the output is a sum of two signals. An incident acoustic wave is first coupled directly by an IDT, and next, after reflection from a reflector, it is again coupled by the IDT. Fig. 2(a) thus means that the directly coupled and reflected signals are in phase in the first case (below the reflection band), and have opposite phases in the second case (above the reflection band). The source of these phase changes as a function of frequency are the variation of phase of the reflection coefficient. If we calculate the responses in a narrower frequency range [Fig. 2(b)], we see that at those frequencies for which |Γ| = 0, we have |A12 | = |A012 |. The very deep minimum of the resonator transfer function |A12 |, seen near the right-hand edge of the main reflection band, is the expected result of the sum when the two signals noted above have comparable amplitudes and opposite phases. This is a characteristic property of all synchronous resonators with γ < 0. When γ > 0 [Fig. 2(c)], the resonator transfer function changes its character and the deep minimum occurs at the lefthand edge of the main reflection band. Therefore, the sign of the reflection coefficient γ of one strip can be deduced easily.

III. Procedure for the Determination of SAW Parameters The calculated transfer function and selected characteristic frequencies are shown in Fig. 3. Here f1 and f2 are part of the series of frequencies at which the reflection coefficient |Γ| = 0. Three each of these frequencies both below and above the main reflection band of the reflector are marked by different arrows. These frequencies are located symmetrically around the center frequency fr of the reflector. At these frequencies, the insertion loss of the resonator is equal to that of the delay line [Fig. 2(b)] and depends on the electromechanical coupling coefficient k 2 of the piezoelectric substrate. The frequency fr depends on the period p (Fig. 1) and on the velocity vr in the re-

Fig. 2. Calculated responses (a) in a broad frequency range, (b) in a narrower frequency range, and (c) for γ > 0. The numbered curves are: reflection coefficient, transfer function |A012 | of a delay line, and transfer function |A12 | of the resonator.

soluch: two-part resonators and piezoelectric crystals

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the deep minimum at frequency f3 occurs at the righthand edge of the main reflection band, and for γ > it occurs at the left-hand edge. Next we change the value of γ in the computer program until the calculated and measured series of frequencies where |Γ| = 0 (f1 and f2 ) match. When the value of γ is correct, the differences between the frequencies of the resonance peaks (f5 -f4 , f6 -f5 , and f7 -f6 ), calculated and measured, should also be approximately equal. C. Velocity vS in the Area Between the Reflectors Using vr and γ as determined above, the velocity vS in the computer program is changed until the calculated and measured frequencies f4 , f5 , f6 , and f7 match. If the surface is free then vS = vf , and if it is metallized then vS = vm . D. Electromechanical Coupling Coefficient k 2 Fig. 3. Characteristic frequencies of the transfer function.

flector area. Frequency f3 is located near the right hand edge of the main reflection band [Fig. 2(b)]. The origin of this deep minimum in the transfer function was explained previously (Section II). The frequencies f1 , f2 , and f3 depend on the velocity vr and on the reflection coefficient per one strip γ. For larger γ, the bandwidth of the reflector is larger, and the frequencies are moved away from the center frequency. In this particular case there are four resonance peaks at frequencies f4 , f5 , f6 , and f7 . These frequencies depend on the velocities vS and vr in the area between the reflectors and inside them, respectively, and on the reflection coefficient γ. The differences between the frequencies are larger for larger γ. The loaded Q factors of the peaks depend on their positions in the reflection band, on the reflection coefficient γ, and on the loss coefficient Ti . By changing the SAW parameters used as the input data for the computer program, it is possible to match the calculated and measured transfer functions. After matching, the input parameters will be equal to the measured ones. The procedure for matching is as follows: A. Velocity vr in the Area of the Reflector This velocity can be determined from the known period p (Fig. 1) and from the measured center frequency fr of the reflector: vr = 2pfr . Because the frequencies f1 and f2 are located symmetrically around the frequency fr , then: fr = (f1 + f2 )/2. If the transfer function at the frequencies nearest to the main reflection band of the reflector is distorted, then the next pair of frequencies should be used to determine fr .

The value of k2 in the computer program is changed until the calculated and measured insertion losses at the frequences f1 and f2 match. This coefficient also can be determined from the measured velocities vf and vm because k2 = 2(vf − vm )/vf . However, in this case we must take into account the influence of the finite thickness of the metal layer. E. Loss Coefficient Ti The loaded Q factors of the resonance peaks are defined as Qn = fn /Bn , where Bn is the 3 dB bandwidth of the nth peak (in our case n = 4, 5, 6, and 7). The loss coefficient Ti in the computer program is changed until the calculated and measured Qn factors match.

IV. Measured SAW Parameters of the 45◦ XZ-Cut of Li2 B4 O7 Crystal A synchronous two-port resonator was fabricated using the design parameters and dimensions as discussed in Section II. A negative photomask was made with electron beam photolithography. Aluminum electrodes were deposited on the Li2 B4 O7 crystal substrate by the lift-off method. The electrode thickness was h = 0.2 µm. The measured response in a 50 Ω system (HP Network Analyzer 8752A) is shown in Fig. 4. By matching the measured and calculated responses, the following results were obtained: vr = 3426 m/s, vf = 3436 m/s, γ = −0.03 (h/λ = 0.0083), k2 = 0.01, and Ti = 0.98.

V. Discussion B. Reflection Coefficient γ Per One Strip of the Reflector As shown previously, the sign of γ can be deduced from the type of asymmetry of the transfer function. For γ < 0,

The shape of the measured response (Fig. 4) is in good agreement with the calculated shape (Fig. 3). Some discrepancies exist for the frequencies f1 and f2 nearest to

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ment between the calculated and measured vf is much better for the set of constants presented in [6] than for those sets of constants published by other authors [7], [8].

VI. Conclusions The synchronous two-port resonator easily can be used for measurements of the major SAW parameters. The free surface SAW velocity for the 45◦ XZ-cut lithium tetraborate crystal has been measured using this technique. The measured value of 3436 m/s is in good agreement with the calculated value. The measured reflection coefficient per one strip and electromechanical coupling coefficient are in good agreement with both published and calculated values.

References

Fig. 4. Measured response. TABLE I Calculated vf and k2 for the 45◦ XZ-cut Li2 B4 O7 Crystal Using Different Sets of Material Constants. Reference

[6]

[7]

[8]

vf [m/s] k2

3433 0.0092

3407 0.0095

3465 0.0080

the reflection band, thus the next set of frequencies in the series were used for matching the measured and calculated responses. The free surface SAW velocity (3436 m/s) for the 45◦ XZ-cut lithium tetraborate crystal has not been measured previously. There is a good agreement between γ and k2 as measured in this work and as previously published [2]. As listed in Table I, both vf and k2 have been calculated for different sets of material constants [5]. The agree-

[1] P. S. Cross, W. R. Shreve, and T. S. Tan, “Synchronous IDT SAW resonators with Q above 10,000,” Proc. IEEE Ultrason. Symp., 1979, pp. 824–829. [2] S. Matsumura, T. Omi, N. Yamaji, and Y. Ebata, “A 45 ◦ X cut Li2 B4 O7 single crystal substrate for SAW resonators,” Proc. IEEE Ultrason. Symp., 1987, pp. 247–252. [3] L. A. Coldren and R. L. Rosenberg, “Scattering matrix approach to SAW resonators,” Proc. IEEE Ultrason. Symp., 1976, pp. 266–271. [4] Y. Koyamada, S. Yoshokawa, and F. Ishihara, “One-port SAW resonators using long IDTs and their application in narrow band filters,” Rev. Electr. Commun. Lab. (Japan), vol. 27, pp. 445– 458, May-June 1979. [5] M. Lysakowska and W. Soluch, “Calculation of surface acoustic wave parameters in piezoelectric crystals,” Elektronika, no. 10, pp. 14–16, 1997 (in Polish). [6] J. A. Kosinski, Y. Lu, and A. Ballato, “Pure-mode measurements of Li2 B4 O7 material properties,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 41, pp. 473–477, July 1994. [7] N. M. Shorrocks, R. M. Whatmore, F. W. Ainger, and I. M. Young, “Lithium tetraborate—a new temperature compensated piezoelectric substrate material for surface acoustic wave devices,” Proc. IEEE Ultrason. Symp., 1981, pp. 337–340. [8] M. Adachi, T. Shiosaki, H. Kobayashi, O. Onishi, and A. Kawabata, “Temperature compensated piezoelectric lithium tetraborate crystal for high frequency surface acoustic wave and bulk wave device applications,” Proc. IEEE Ultrason. Symp., 1985, pp. 228–232.