Piezoelectric thin films have been widely used in

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1 ieee transactions on utrasonics, ferroeectrics, and frequency contro, vo. 50, no. 3, march Resonant Spectrum Method to Characterize Piezoeectric Fims in Composite Resonators Yuxing Zhang, Zuoqing Wang, and J. David N. Cheeke, Senior Member, IEEE Abstract In this paper, we present a direct method to characterize a piezoeectric fim that is sandwiched with two eectrodes and deposited on a substrate to form a fourayer thickness extension mode composite resonator aso known as over-moded resonator. Based on the parae and series resonant frequency spectra of a composite resonator, the eectromechanica couping factor, the density and the eastic constant of the piezoeectric fim can be evauated directy. Experimenta resuts on sampes consisting of ZnO fims on fused quartz substrates with different thickness are presented. They show good agreement with theoretica prediction. The mechanica effect of the eectrode on the method is investigated, and numerica simuation shows that the effect of the eectrodes can be propery corrected by the modified formuae presented in this paper. The effect of mechanica oss in piezoeectric fim and in substrate on this method aso has been investigated. It is proven that the method is insensitive to the osses. I. Introduction Piezoeectric thin fims have been widey used in high frequency buk acoustic wave BAW and surface acoustic wave SAW devices, such as fiters, resonators, actuators, and sensors. For sef-supported, singe piezoeectric fims, various measurement techniques have been deveoped to evauate the acoustic properties of these fims. However, for some piezoeectric ceramic fims e.g., soge PZT fims on meta substrate [1] and for very high frequency acoustic wave devices e.g., zinc oxide fims on fused quartz [], [3] and AN fims on siicon [4], the fims are not sef-supported and the fim properties depend on the substrates. Therefore, the methods recommended by the IEEE standard [5] are not avaiabe. In the case of thin fims deposited on very thick substrates, which are used as high frequency transducers, Bahr and Court [6] deveoped a method to determine the couping coefficient from measuring the transducer input admittance. By puse measurements, an infinite ong deay ine is imitated, and thus the propagation characteristics of the acoustic wave in the substrate does not affect the measured vaues of the input admittance. The eectromechanica couping coefficient kt was determined by data-fitting Manuscript received January 10, 00; accepted October 1, 00. This work was supported by the Natura Sciences and Engineering Research Counci of Canada NSERC. Y. Zhang is with the Wireess Technoogy Lab, Norte Networks, Ottawa, Ontario KH 8E9, Canada e-mai: zhangy@ nortenetworks.com. Z. Wang is with the TXC Corporation, Tao-Yuan County, Taiwan, R. O. China. J. D. N. Cheeke is with the Physics Department, Concordia University, Montrea, Quebec H3H 1M8, Canada. the input admittance curve versus frequency. Meitzer and Sittig [7] improved the method by taking the effects of the eectrode and the interconnection ayers into account. Because the vaue of kt is evauated from the magnitude of the input admittance, which depends on a the eectrica factors, a compicated measurement system and very accurate caibration of the system are necessary. In the case of a piezoeectric fim deposited on a thin substrate to form a thickness extension mode composite over-moded resonator, Hickerne [3] and Naik et a. [4] introduced methods to extract the kt vaue by fitting the eectric input impedance and admittance data with the equivaent circuit anaysis resuts on mutimode resonance of a composite resonator. We previousy reported a direct method to characterize a piezoeectric fim coated on an isotropic substrate to form a two-ayer composite resonator [8], [9]. By knowing the resonant spectra of a composite resonator, three parameters of the piezoeectric fim i.e., the eectromechanica couping coefficient, the eastic constant, and the density coud be determined. The vaidity of this method was demonstrated with simuations when eectrodes were ignored. For the high frequency devices, however, the eectrode effect cannot necessariy be ignored. In this paper, the resonant spectrum method is extended to the case in which the eectrodes are taken into account. A set of expicit formuae that forms the foundation of the method are first presented, foowed by the experimenta resuts on high frequency ZnO/fused quartz composite resonators. The vaidation and the accuracy of the method are proven by numerica simuation in Section IV. The impact of the mechanica effect of eectrodes is discussed and the improvement of the method for the eectrode effect is demonstrated in Section V. Because the mechanica osses of the materias pay an important roe on the eectric impedance of high-frequency resonators, from which the resonant frequencies are determined, the effect of the mechanica osses on the resonant spectrum method is investigated by numerica simuation in Section VI. In a the simuations, the experimentay measured data for the ZnO thin fim obtained from Section III are taken as the standard parameters; other parameters are taken from iterature. A detaied derivation of the reated formuae is presented in the Appendix. II. The Resonant Spectrum Method The resonant spectrum method is based on two groups of approximate formuae, which are derived from the eec /$10.00 c 003 IEEE

2 3 ieee transactions on utrasonics, ferroeectrics, and frequency contro, vo. 50, no. 3, march 003 Fig. 1. A typica four-ayer composite resonator, with the definitions of the materia parameters and dimensions. tric input impedance expression of a four-ayer composite resonator. A composite resonator has mutipe resonances, determined mainy by the acoustic properties of the materias and the thickness of the four ayers. Three parameters i.e., ongitudina veocity V, density ρ, and eectromechanica couping coefficient kt of the piezoeectric fim can be determined directy from the parae and series resonant spectra of a composite resonator. The detaied derivation of the formuae wi be given in the Appendix. In this section ony the resuts are presented. The eectric input impedance of a four-ayer thickness extension mode composite resonator, as shown in Fig. 1, is: Z in = 1 jωc 0 [ 1 k t γ ] z 1 + z sinγ + j1 cos γ, 1 z 1 + z cosγ + j1 + z 1 z sinγ where C 0 = ε S 33S/ is the static capacitance of the resonator, S is the area of the eectrodes, ε S 33 and are the permittivity and thickness of the piezoeectric ayer; kt is the eectromechanica couping coefficient of the piezoeectric ayer; γ = ω/v is the phase deay of the ongitudina acoustic wave with veocity V in the piezoeectric ayer, ω is the anguar frequency; z 1 and z are the acoustic oading impedances on both sides of the piezoeectric ayer normaized to Z 0 = SρV, the acoustic impedance of the piezoeectric ayer, ρ is the density of the piezoeectric ayer. Fig. shows the eectric input impedance of a resonator consisting of a PZT fim deposited on a stainess stee pate. The parameters used are isted as sampe III in Tabe I. In the cacuation, an arbitrary sma imaginary part 0.1% is added to the veocities for both the fim and the substrate to avoid singuarities. It is shown in Fig. that the impedance response has a goba hyperboic decrease, which is determined by the static capacitance C 0 ; and, in the frequency region corresponding to the fundamenta mode of the piezoeectric fim, there is a series of resonant peaks, each peak corresponds to a resonant mode of the composite resonator. The static capacitance C 0 is ony reated with the hyperboic decrease of the impedance response, not with the resonant frequencies. Contrariy, kt shoud be reated ony with the resonant frequencies as it is in the singe piezoeectric pate Fig.. The simuated eectric impedance of a PZT/stainess stee composite resonator as a function of frequency. The curve ceary shows the hyperboic decrease due to the static capacitance and the peaks due to mutipe resonances in the substrate. case. In the derivation of the method, we foowed a simiar procedure as in the singe piezoeectric pate case to derive the parae and series resonance frequency equations from 1. From the resonant frequency equations, we derived the formuae that are the principes of the proposed resonant spectrum method. By definition in the IEEE standard [5], the parae resonant frequency corresponds to the maximum resistance, which is the rea part of Z in. It can be derived from 1 by setting the denominator to zero and is given by: z 1 + z cosγ + j1 + z 1 z sinγ =0. The series resonant frequency, which corresponds to the maximum of the conductance, can be derived from 1 by setting the numerator to zero and is given by: z 1 + z cosγ + j1 + z 1 z sinγ k t γ z 1 + z sinγ + j1 cos γ =0. 3 It wi be seen in the Appendix that z 1 and z are purey imaginary functions if the materia parameters are rea, so and 3 are not compex equations. For a mutimode composite resonator, by defining the spacing of the parae resonant frequencies SPRF f p m =f p m +1 f p m, 4 we derived two approximate formuae that reate the ongitudina veocity V and the density ρ of the piezoeectric ayer with two characteristic vaues of the SPRF, f N and f T, which can be obtained from the SPRF distribution versus the resonant mode order m. The norma regions are the areas in which γ is cose to an integer mutipication of π. At the center of the first norma region i.e., γ π, the SPRF is given by: f N = f 0 1+ ρ 1 e1 e1 + ρ e e +, 5 ρ sb sb

3 zhang et a.: characterization of piezoeectric fim in composite resonator 33 TABLE I The Parameters of Composite Resonators Used in Simuation. Piezoeectric fim Substrate Eectrodes ρ V k t ρ sb V sb sb ρ e V e e Sampe Materia kg/m 3 m/s % µm kg/m 3 m/s µm z sb kg/m 3 m/s µm R I ZnO/fused quartz II ZnO/fused quartz III PZT/stainess stee where f 0 = V sb / sb is the SPRF of the bare substrate pate so it is a constant; ρ sb, ρ e1,andρ e are the densities of the substrate, the top and the midde eectrodes, respectivey; sb, e1,and e are the thicknesses of the substrate, the top and the midde eectrodes, respectivey. For the case in which the acoustic impedance of the piezoeectric ayer is greater than that of the substrate referred as soft substrate e.g., ZnO fim on quartz substrate, f N corresponds to the first minimum of the SPRF. In the case in which the acoustic impedance of the piezoeectric ayer is ess than that of the substrate, referred to as hard substrate e.g., PZT on stainess stee substrate, f N corresponds to the first maximum of the SPRF. The transition regions are the areas in which γ is cose to a haf integer mutipication of π. At the center of the first transition region i.e., γ π/, the SPRF is given by: f T = f 0 1+ ρ sbvsb ρv + ρ sbvsb e sb ρ e V + ρ sbρ e1 Vsb sb ρ V e 1 e1, sb 6 where V, V sb, V e1,andv e are the veocities of the piezoeectric ayer, substrate, and top and midde eectrodes, respectivey. When the acoustic impedance of the piezoeectric ayer is greater than that of the substrate, f T corresponds to the first maximum of the SPRF. In the other case f T corresponds to the first minimum of the SPRF. It is shown in 5 and 6 that the density ρ and the ongitudina veocity V of the piezoeectric ayer can be evauated by obtaining the two characteristic vaues, f N and f T, from the SPRF distribution and by knowing the thickness of the piezoeectric fim and other parameters of the eectrodes and the substrate. The eastic constant c D 33 then is given by the formua c D 33 = ρv. Of course, these two formuae can be used to evauate other parameters if ρ and/or V are known. In anaogy to the formua for the couping coefficient of a singe-ayer piezoeectric resonator when the couping coefficient is sma [5]: k t,singe π 4 f s 1 f s, 7 f p f p we introduce an effective couping coefficient k eff m, which indicates the eectromechanica couping intensity of a specific resonant mode of a composite resonator: keff π f s m m = 1 f sm, 8 4 f p m f p m where f p m, f s m arethem-order parae and series resonant frequencies, respectivey. From the series resonant frequency determination 3, we derived the reationship between kt and k eff m as foows. At the first norma region kt is given by: kt = 1 + m N z sb 1+ ρe1e1+ρee+ρ sb sb k eff m N +1, 9 where z sb = Z sb /Z 0 is the normaized acoustic impedance of the substrate, Z sb = Sρ sb V sb,m N + 1 is the resonant mode order at the center of the first norma region. At the first transition region, kt is given by: kt = [m T +1/z sb +1] 1+ ρv / ρ sb V ρ ev sb / + ρv / sb e /e + ρe1e1 keff m T +1, Γ 10 where m T +1 is the resonant mode order at the center of the first transition region. Γ is a correction factor, which is caused by the difference between the m T + 1 order series resonant frequency and the center of the first transition region. Γ is given by: Γ=1 ρv 1+ πf s π V πfs ρ sb Vsb m T +1/π + πf s e. 11 ρ sb V sb ρ e V e where f s is the series resonant frequency at the center of the first transition region. By knowing the thicknesses, the densities, and the veocities of four ayers and by evauating the keff m vaue from the experiment/simuation data, kt can be obtained from 9 or 10. Usuay, the maximum vaue of keff m shoud be used in the evauation for the best accuracy. When Z 0 >Z sb,themaximumkeff m isocatedinthe first norma region and when Z 0 < Z sb,themaximum keff m is ocated in the first transition region. By introducing the ratio of the haf waveength resonant frequency of the top resonator f c to the haf waveength resonant frequency of the substrate f sb : V sb + 1 ρ / e e ρsb R = f c f sb = V sb + ρe1 e1 / ρ + 1 ρ e e / ρ, 1

4 34 ieee transactions on utrasonics, ferroeectrics, and frequency contro, vo. 50, no. 3, march 003 we can get approximate expressions of the mode order at the first norma region and the first transient region: m N = the nearest integer of R, 13 R 1 m T = the nearest integer of. 14 When R is cose to an integer or R is so arge that the fractiona part can be ignored, 9 can be simpified to: kt = + ρ sb sb + ρ e1 e1 + ρ e e + ρe1 e1 + 1 ρ keffm N +1. e e 15 When R is cose to an odd integer, or say R 1/ is cose to an integer, or R is very arge, 10 can be simpified to: k t = 1+ ρe1e1 + ρv ρ sb V sb sb + ρ V ρ sb V +1 sb + ρ e1 e1 + 1 ρ e e 1 ρee keff m T Γ These simpified equations aso have a cear physica interpretation. Eq. 15 shows when the eectrodes can be ignored, kt of the piezoeectric fim is the effective couping factor at the center of the first norma region mutipied by the mass ratio of the whoe resonator to the piezoeectric fim. III. Measurement of ZnO/SiO Composite Resonators In order to demonstrate the feasibiity of the resonant spectrum method, some composite resonator sampes have been measured. The resonators are composed of ZnO fims deposited on fused quartz substrates [10]. The thickness of the ZnO fims is about 5 µm and the thicknesses of the quartz substrates are about 6 mi Sampe I and 9 mi Sampe II. There is a circuar ground eectrode of auminum approximatey 0.4-µm thick underying the ZnO fim, and there are four sma circuar eectrodes of 0.4 µm auminum on the top of ZnO fim, forming 4 composite resonators Fig. 3. The refection coefficients s 11 of the resonators were measured with an HP8753D network anayzer Agient, Pao Ato, CA from 100 MHz to 800 MHz. Because the maximum number of measurement points of an HP8753D is ony 1601 and we need an utrahigh resoution resonant spectrum, the whoe frequency span of 700 MHz was divided into tens of narrow spans, 10 MHz each for 6-mi thickness sampes and 8 MHz each for 9-mi thickness sampes. The network anayzer was caibrated ony once for 700 MHz span, and an interpoated caibration was used for each measurement. A LabWindow program Nationa Instruments Corp., Austin, TX running in a computer was used to contro the network anayzer for setting the centra frequency and span, running the measurement, and downoading the measured data to the computer. Thus, high-resoution frequency responses for the whoe span were acquired. Fig. 3. ZnO/SiO composite resonators used in the experiment. The bottom eectrode is common to a four resonators. After further interpoating the measured data near each resonant peak and then converting s 11 into impedance Z in and admittance Y in, the parae and series resonate frequencies of these sampes were cacuated. Distributions of the SPRF and the keff m of the sampes are shown in Figs. 4 and 5, from which the two characteristic vaues f N, f T and the effective couping coefficient keff m N + 1 are measured. They are isted in Tabe II. Here the SPRF and keff are potted against frequency, not the mode order as in their definitions. In fact, both are amost identica in the distribution shape and ony differ in the horizonta axis by a factor of f 0, the SPRF of the bare substrate. It is shown that the data of the SPRF are a itte dispersive, and the characteristic vaues of f N, f T are determined by averaging the measurement data. Even so, the periodic shape of the SPRF distribution is very reguar and the resuts are fairy sure. It is interesting to notice that the data of keff m are very smoothy distributed over a wide frequency range and amost no dispersion. As a resut, no data fitting is necessary, and the characteristic vaue of keff m N + 1 can be evauated accuratey. These experiment resuts ceary show the practicaity of this method. The density ρ, the ongitudina veocity V then the eastic constant c D 33, and the eectromechanica couping coefficients kt of the ZnO fim were then cacuated by using the resonant spectrum method 5, 6, and 9. They are isted in Tabe II. The data of the thickness and the parameters of the fused quartz substrate and auminum eectrodes [10], [11] are given in Tabe I. IV. Vaidity and Accuracy of the Resonant Spectrum Method Since the ZnO fim parameters are process reated, there are no standard vaues to compare with the resuts we got from the experiment. We wi investigate the vaid-

5 zhang et a.: characterization of piezoeectric fim in composite resonator 35 TABLE II Characteristic Vaues and Parameters Determined by Experiment on ZnO/fused Quartz Composite Resonators. Experiment Parameters deduced from experiment f N f T keff m N +1 ρ c D 33 V kt Sampe MHz MHz % m N + 1 kg/m N/m m/s % Fig. 4. SPRF top and keff bottom distribution of Sampe I 6 mi ZnO/fused quartz resonator. Fig. 5. SPRF top and keff bottom distribution of Sampe II 9 mi ZnO/fused quartz resonator. ity and accuracy of the method by numerica simuation. Three exampes are simuated. The first two are the sampes discussed in ast section i.e., ZnO fim on fused quartz substrate, which represent the case in which the acoustic impedance of the piezoeectric ayer is greater than that of the substrate. The vaues of the veocity, the density, and the couping coefficient of the ZnO fims are the data determined by experiment in the ast section. In the simuation they are used as the input parameters. The third sampe is a porous PZT fim on a stainess stee pate [1], which represents the case in which the acoustic impedance of the piezoeectric ayer is ess than that of the substrate. The parameters used in the simuation are isted in Tabe I. The parae and series resonant frequencies of these composite resonators are cacuated by soving the maximums and minimums of the resistance from 1. The distribution of the SPRF and keff m of Sampe I and Sampe II are shown in Figs. 4 and 5 by soid dots. For Sampe III,theyareshowninFig.6.Thesimuatedcharacteristic vaues of f N, f T,andkeff m N + 1 are obtained from these figures. Using the equations of the resonant spec-

6 36 ieee transactions on utrasonics, ferroeectrics, and frequency contro, vo. 50, no. 3, march 003 TABLE III Simuation Resuts on the Sampes in Tabe I. ρ c D 33 V k t Sampe kg/m 3 error N/m error m/s error % error I % % % % II % % % % III % % % % input vaues are within 3.5% for a three parameters and c D 33 as we. Assuming the data evauated from the experiment given in Tabe II and isted in Tabe I as the input parameters are the true vaues for Sampe I and Sampe II, the determined vaues by this method are accurate to 3.5% for the sampes used here. For such high frequency devices, a few percent errors are quite acceptabe. V. Effect of the Eectrodes Fig. 6. SPRF top and keff bottom distribution of Sampe III PZT/stainess stee resonator. trum method in Section II and the materia parameters in Tabe I, we obtained the simuation output of three parameters of the piezoeectric fims. The resuts are isted in Tabe III. The errors isted are with reference to the input data shown in Tabe I. Sma imaginary parts are introduced into the veocities of both the piezoeectric fim and the substrate to avoid singuarities. The imaginary parts of the veocities have itte effect on the distribution of the SPRF and the effective couping coefficients. This wi be discussed in Section V. It is shown in Tabe III and the input data given in Tabe I that the difference between the output vaues and In our previous work [8], the mechanica effect of eectrode was ignored. In other words, the thickness of the eectrodes was taken to be zero. In this paper, we have taken the eectrodes into account. But in the derivation of the resonant spectrum method formuae given in the Appendix, a thin eectrode approximation is used. The errors caused by this approximation may be significant for high frequency resonators in which the thickness of the eectrodes is comparabe with the thickness of the piezoeectric fim. Therefore, it is necessary to investigate the avaiabe range of the eectrode thickness imitation and the errors caused by the eectrodes. Fig. 7 shows the SPRF and keff m distributions for various eectrode thicknesses in a ZnO/fused quartz composite resonator as Sampe I. It can be seen that the thicker the eectrode, the greater the keff m. These resuts are quaitativey coincident with 15. Tabe IV ists the simuation resuts for ρ, V,andkt for various eectrode thicknesses. It is shown that the errors of the three parameters increase with the increase of the eectrode thickness. When the thickness of the auminum eectrodes is within 10% of the ZnO fim, the error is ess than 5%, which is quite acceptabe. The resut has a cear mathematica interpretation. As given in the Appendix A7, an approximation: tan γ e γ e, 17 is used in deriving the resonant spectrum method, which means the eectrodes can be considered as a mass oading. For ZnO/quartz composite resonator with auminum eectrodes as simuated here, V e is cose to V. When the eectrode thickness is a tenth of the piezoeectric fim thickness: γ e γ/10 π/10, 18 at the first norma region. The use of approximation of 17 gives an error of 3.4%. With contributions from other approximations, the overa error is about 5%. The condi-

7 zhang et a.: characterization of piezoeectric fim in composite resonator 37 For the 0.4-µm eectrode case in Tabe IV, kt wi be 7.83%, and the error wi be 7.74% when other parameters use the input vaue in Tabe I. If the veocity and density use the simuation resuts in Tabe III, the error of kt wi be 10%. When taking the eectrode effect into account and using 15, we find that the error of kt is ony.1%. The resut shows ceary that effects of the eectrode have to be compensated and our modified formuae are effective. The data given in Tabe IV shows that the resuts obtained from 15 seem better than those from 9. This may be caused from the approximation of m N introducing an opposite error that cances out part of the error from 9. VI. Effect of Mechanica Loss Fig. 7. SPRF top and keff bottom distribution for various eectrode thicknesses in a four-ayer ZnO/fused quartz composite resonator. tion for the eectrode thickness being a tenth of the piezoeectric fim thickness can be used as a rough criterion to achieve good accuracy. As a further comparison, we use the resonant spectrum method formuae with the eectrode effect ignored [8]. Let e1 = e = 0, 15 becomes: kt = 1+ ρ sb sb keff m N In deriving the parae and series resonant frequencies determination and 3, from which the formuae for evauating kt are derived, a the materia parameters are assumed to be rea. This means a materias are assumed ossess. For high frequency devices or porous piezoeectric ceramics, however, the osses are significant in determination of the eectric input impedance of a resonator, from which the resonant frequencies are cacuated. Therefore, it is necessary to investigate the vaidity of the resonant spectrum method when the materias are ossy, in another word, whether 9 and 10 are avaiabe when the resonant frequencies are directy cacuated from impedance 1. The effect of the mechanica osses on evauating kt is investigated with numerica simuation by taking the veocity as a compex vaue. The ratio of the rea part to imaginary part of the veocity is referred to as materia Q vaue. Taking different Q vaues for the piezoeectric fim and the substrate, and directy cacuating the resonant frequency spectra from the eectric impedance of the composite resonator, we evauated the corresponding keff m N + 1. Because kt is definitey determined by keff m N +1orkeff m T + 1 when other materia parameters are taken as constants, ony the keff m distribution is necessary to be simuated for different oss. Fig. 8 shows the eectric input impedance of the ZnO/SiO composite resonators with different materia Q vaues. In Fig. 8b the Q vaues of the two materias are chosen such that the impedance is cose to the experimenta resut [Fig. 8a]. Figs. 8c and 8d are the cases in which the materia Q vaues are much higher and much ower, respectivey. In the high Q case, the imaginary parts of the veocities of the piezoeectric fim and the substrate decrease to 1/10 and 1/ from the nomina vaues in Fig. 8b, respectivey. In the ow Q case, the imaginary parts increase by five times and twice, respectivey. It is shown that the resonant ampitude changes significanty. The smaer the propagation osses, the higher the resonant peaks, and vice versa. The distribution curves of keff m versus the frequency are shown in Fig. 9 for high Q and ow Q cases. It is noticed that, athough the mechanica osses in the piezoeectric fim and substrate have been changed by 50 and 4 times, the distributions of keff m have no significant difference. The vaue at the first norma region keff m N + 1 changes ony 1.6%. Thus, kt of the piezoeectric fim cacuated from keff m N + 1 with the resonant spectrum method is insensitive to the mechanica oss for moderate oss piezoeectric fims. This behavior is in accordance with Naik et a. [4] and the IEEE standard [5]. Other simuations show that, when the oss in the substrate is taken as the nomina vaue, the 50 times variation of the oss in the piezoeectric fim brings a trivia difference to the keff m. This is in accordance with the

8 38 ieee transactions on utrasonics, ferroeectrics, and frequency contro, vo. 50, no. 3, march 003 TABLE IV Method Errors for Various Auminum Eectrode Thicknesses in ZnO/fused Quartz Composite Resonator. Eectrode ρ V c D 33 k t 9 k t 15 thickness kg/m 3 error m/s error N/m error % error % error % % % % % 0. µm % % % % % 0.4 µm % % % % 7.1.1% 0.6 µm % % % % % 0.8 µm % 61..5% 0.4.0% % % Fig. 8. Imaginary part eectric impedances of ZnO/fused quartz resonators with different materia mechanica osses showing a significant effect of the oss on the resonant ampitude. Horizonta axes, frequency in MHz; Vertica axes, imaginary part impedance in Ω. physica nature of a composite resonator that the acoustic wave traves most of the time in the substrate, and the oss of the piezoeectric fim has itte effect. Because the substrate is usuay high quaity materias such as fused quartz or siicon, their acoustic osses are fairy constant and wi not exceed the variation range in the simuation of Fig. 9. Therefore, the effect of mechanica oss in both the piezoeectric fim and the substrate for practica materias may be much ess than what have been shown in the simuation Fig. 9. This feature gives further advantages to the resonant spectrum method for keeping good accuracy for a wide range of practica piezoeectric fims up to moderate high mechanica oss. VII. Concusions The principes of a direct measurement method for piezoeectric fim, named as the resonant spectrum method, are presented briefy. After knowing the resonant spectrum of a composite resonator, three major parameters of piezoeectric fims i.e., the eectromechanica cou-

9 zhang et a.: characterization of piezoeectric fim in composite resonator 39 Fig. 9. keff distribution of ZnO/fused quartz resonators for different materia mechanica osses showing the sma effect of the oss on the keff. ping coefficient, the eastic constant and the density can be evauated by a set of expicit formuae. It has been found that the acoustic impedance ratio of the piezoeectric fim and the substrate dominates the SPRF and keff distributions. For the soft substrate case where z sb < 1, f N of the SPRF is the minimum and f T is the maximum. The keff has maximum at the first norma region and 9 or 15 shoud be used to cacuate the kt. Contrariy, for the hard substrate case where z sb > 1, f N of the SPRF is the maximum and f T is the minimum. The keff has maximum at the first transition region and 10 or 16 shoud be used to cacuate the kt. Measurement on two sampes of ZnO/fused quartz resonators was carried out, and the resuts are reasonabe. Simuation resuts show that, for thin eectrode in which the eectrode can be considered as mass oading, the effects introduced by the eectrode can be compensated by the modified formuae derived in this paper. As a resut, the errors are ess than 5% if the thickness of the eectrode is not more than 10% of that of the piezoeectric fim. The effect of the mechanica osses in the substrate and piezoeectric fim on the accuracy of this method has been investigated by numerica simuation, and it is shown that the effect is sma for practica composite resonators. This method can find wide appication in piezoeectric fim characterization. An obvious advantage of the resonant spectrum method is the directness. Usuay, the parameters of the piezoeectric fims are determined by fitting the data from their eectric characteristic measurements, for exampe, the eectric input impedance. This method, aso based on the eectric impedance measurement, can cacuate three parameters directy. Another advantage of the method is that ony the distributions of resonant frequencies are of interest. Therefore, the caibration of the measurement system is not critica, even though the accuracy of the caibration makes a considerabe difference in the measurement of eectric impedance. This feature is convenient for some appications in which accurate caibration is difficut. In addition, the kt deduced by this method is not sensitive to the oss of the piezoeectric fim, this impies that this method can appy to moderate high oss, ow Q factor piezoeectric fim characterization. The formuae presented in this paper are aso usefu for designing overmoded resonator fiter [1], [13]. The bandwidth of such fiter is mainy determined by the effective eectromechanica couping coefficient keff. By knowing the materia eectromechanica couping coefficient kt,thekeff vaue of the mode in norma or transition region can be evauated directy from the formuae given in Section II. A suitabe keff vaue can be optimized by choosing proper materia parameters and thickness of the composite resonator. There are some imits on this method. The eectrodes of the resonator have to be very thin compared to the piezoeectric fim. This imits the appication of this method to a frequency range under 1 GHz. In addition, there have to be enough resonant modes in one period of SPRF to use the approximation 15 and 16. This requires a arge thickness ratio of the substrate to the piezoeectric fim. As a resut, the inaccuracy of the substrate parameters has more impact on the accuracy of the resonant spectrum method. Appendix A Derivation of the Resonant Spectrum Method A. The Eectric Input Impedance of a Composite Resonator The acoustic impedance of each ayer in a composite resonator can be described using Sittig s matrix presentation [14]. A schematic of a four-ayer composite resonator is shown in Fig. 10. The acoustic impedance of the top eectrode, presenting at the eft side of the piezoeectric ayer, is given by: Z 1 = F 0 u 0 = jz e1 tan γ e1, A1 where F and u represent force and dispacement veocity, respectivey, Z e1 = Sρ e1 V e1 is the acoustic impedance of the top eectrode, γ e1 = ω e1 /V e1 is the phase deay in the top eectrode. The acoustic impedance of the midde eectrode and the substrate, presenting at the right side of the piezoeectric ayer, is given by: Z = F 1 Z sb tan γ sb + Z e tan γ e = j, u 1 1 Z sb /Z e tanγ e tan γ sb A where γ e = ω e /V e, γ sb = ω sb /V sb are the phase deay in the midde eectrode and the substrate, respectivey; Z e = Sρ e V e, Z sb = Sρ sb V sb are the acoustic impedances, ρ e, ρ sb are the densities, V e, V sb are the veocity of this two ayers, respectivey.

10 330 ieee transactions on utrasonics, ferroeectrics, and frequency contro, vo. 50, no. 3, march 003 Fig. 10. The matrix mode to derive the eectric input impedance of a four-ayer composite resonator. The eectric input impedance of such a four-ayer composite resonator is given by: Z in = U I = 1 [ 1 k t jωc 0 γ ] z 1 + z sinγ + j1 cos γ, z 1 + z cosγ + j1 + z 1 z sinγ A3 where z 1 = Z 1 /Z 0 and z = Z /Z 0 are the normaized acoustic impedance of the top eectrode ayer and the midde eectrode/substrate combination. B. The First Norma Region. f N and k t Parae resonances correspond to maximums of the resistance of a composite resonator. If we ignore the imaginary parts of materia parameters, parae resonances correspond infinite impedance Z in, which gives a determinative equation for the parae resonant frequencies: z 1 + z cosγ + j1 + z 1 z sinγ =0. A4 At the center of the first norma region, the m N +1 order resonant mode corresponds to: γ π, γ sb m N π, A5 where m N is the mode order of the bare substrate pate resonator at the center of the first norma region. So we have approximate expressions as: tan γ γ π, tan γ sb γ sb m N π. A6 Because the eectrodes usuay are much thinner than the piezoeectric fim, this means γ e1 1andγ e 1at the first norma region, thus: Therefore, tan γ e1 γ e1, tan γ e γ e. A7 z 1 j Z e1 Z 0 γ e1 = jz e1 γ e1, A8 and z sb γ m N π+z e γ e z j 1 z sb /z e γ sb m N πγ e j [z sb γ m N π+z e γ e ]. A9 Substitute approximations A5 A9 into A4, we get A10 see next page. The m N + 1-order parae resonant frequency of the composite resonator is: f P m N +1= ze1 e1 V e1 m N z sb +1 + z sb sb V sb + zee V e + V. A11 The space between two parae resonant frequencies at the first norma region is: f N = 1 z e1 e1 V e1 + z sb sb V sb z sb + zee V e + V = 1 ρ sb V sb ρ e1 e1 + ρ sb sb + ρ e e + = f 0 1+ ρ A1 1 e1 e1 + ρ e e +, ρ sb sb where f 0 = V sb sb is the parae resonance frequency spacing of a bare substrate pate. On the other hand, the series resonant frequencies 3 give: kt = γ[z 1 + z +j1 + z 1 z tanγ] z 1 + z tanγ +jsec γ 1, A13 at series resonant frequencies. As proven before, both z 1 and z are first-order sma quantities at a resonance in the first norma region. Therefore, z 1 z 1andz 1 + z γ π 1. Ignoring these second-order sma quantities, we get an approximate expression for kt given in A14 see next page. Substitute A11 into the A14 to get A15 see next page. If we introduce an effective couping coefficient keff as: k eff = π 4 f s f p fp f s f p, A16

11 zhang et a.: characterization of piezoeectric fim in composite resonator 331 jz e1 γ e1 + j[z sb γ m N π+z e γ e ] + j1 + jz e1 γ e1 j[z sb γ m N π+z e γ e ]γ π =0, [ ] πf e1 πfsb πf e z e1 + z sb m N π + z e V e1 V sb V e { [ ]} πf e1 πfsb πf e πf + 1 z e1 z sb m N π + z e V e1 V sb V e V π =0 A10 kt = γ 4j [z 1 + z +jγ π] = 1 πf s 4 V [πf s z e1 e1 V e1 + z sb sb V sb + z e e V e + V ] A14 z sb m N +1π kt = π V f e1 sb e sf p f s z e1 + z sb + z e + V e1 V sb V e V = π 4 f s f p f p f s 1 + m N z sb f p 1+ ρe1e1+ρee+ρ sb sb A15 the couping coefficient k t kt = 1 + m N z sb can be expressed as: 1+ ρe1e1+ρee+ρ sb sb k eff. A17 In some specia cases, the kt equation can be further simpified. As shown in A5, m N is the resonant mode order of the substrate pate at the center of the first norma region. For the case of a four-ayer composite resonator, the composite resonator can be considered as a top resonator composed of a piezoeectric fim, a top eectrode, and part of a midde eectrode deposited on a composite substrate composed of the other part of the midde eectrode and a substrate pate. As an approximation, we spit haf of the midde eectrode into the top resonator and the other haf into the composite substrate. The center frequency of the first norma region corresponds to the resonant frequency of the top resonator: f c = V + ρ e1 e1 / ρ + 1 ρ e e / ρ. A18 The fundamenta resonant frequency of the composite substrate is: f sb = V sb sb + 1 ρ e e / ρsb. A19 We define R as the ratio of resonant frequency of the top resonator to the fundamenta resonant frequency of the composite substrate: R = f c f sb = V sb + 1 ρ / e e ρsb / V sb + ρe1 e1 ρ + 1 ρ /. e e ρ m N is the vaue of R rounded to the nearest integer: m N = the nearest integer of R. A0 A1 It shoud be noted that the resonant mode order of the composite resonator at the center of the first norma region, is m N + 1, rather than m N. If R is cose to an integer, or is arge enough that the fraction part can be ignored, m N R: 1+m N z sb = + ρ e1 e1 + ρ e e + ρ sb sb + ρ e1 e1 + 1 ρ e e. A Substitute A into A17, we get a simpified expression for the eectromechanica couping coefficient: kt = + ρ e1 e1 + ρ e e + ρ sb sb + ρ e1 e1 + 1 ρ k e e eff m N +1. A3 C. The First Transition Region, f T and k t The first transition region occurs where γ = π/. Because, in a resonator, the tota phase deay has to be an integer mutipication of π, the phase shift of the substrate wi yied another π/. We assume that: γ = π/+, γ sb =m T +1/π + δ, A4

12 33 ieee transactions on utrasonics, ferroeectrics, and frequency contro, vo. 50, no. 3, march 003 where m T is the mode order of the bare substrate pate near the center of the first transition region, and δ are sma quantities. Therefore, we get approximate expressions at the first transition region as: sin γ 1, cos γ, tan γ sb 1/δ. A5 By taking these approximations into the expression of z,weget: z = j Z 0 Z sb tan γ sb + Z e tan γ e 1 Z sb /Z e tanγ sb tan γ e j z sb 1 δ + z eγ e 1+ z sb γ e z e δ δ j + γ 1 e. z sb z e A6 Take the approximations A8, A5, and A6 into the parae resonance frequency A4, we get: δ jz e1 γ e1 j + γ 1 e z sb + j 1 jjz e1 γ e1 z e δ z sb + γ e z e 1 1=0. In the first bracket, the first term is a sma quantity, and the second term is a arge quantity. As an approximation, we ignore the first term and after some simpification, we may get: + δ + γ e + z e1 γ e1 =0. A7 z sb z e Therefore, we can get the parae resonant frequency at the first transition region: f p m T +1= 1 m T +1/+z sb /. z sb V + sb z sb V sb + e + z e1 e1 V e1 z ev e A8 So the spacing of the parae resonance frequencies at the first transition region is: f T = 1 1 z sb sb = f 0 1+ ρ sbv sb ρv V + z sb V sb + e z ev e + z e1 e1 V e1 + ρ sbvsb e sb ρ e V + ρ sbρ e1 Vsb sb ρ V e 1 e1. sb A9 As at the first norma region, the series resonance frequencies at the first transition region are determined by 3. Since: δ z j + γ 1 e 1andz 1 = jz e1γ e1 1, z sb z e A30 3 at the first transition region can be expressed as: Therefore, k t z + j1 + z 1 z = k t γ [z + j1 + ]. A31 can be cacuated from: kt = γ z + j1 + z 1 z [z + j1 + ] = γ z + j1 + z 1 z z Γ = 1+ V [1 + m T +1/z sb ] sb z sb V sb + e z ev e + z e1 e1 V e1 k eff Γ. A3 Γ is a correction factor, which is in the order of unity: 1 + Γ=1+j z 1+ πf s π v =1 ρ 0 V πfs ρ sb Vsb m T +1/π ρ sb V sb + πf s e ρ e V e. A33 This correction factor stands for the difference between the first transition region center and the m T +1mode series resonance frequency. Again, as at the first norma region, m T is the vaue of R 1/ rounded to the nearest integer: M T = the nearest integer of R 1. A34 When R is cose to an odd integer or is arge enough, m T R 1, see A35 see next page. Because the eectrodes usuay are very thin compared to the substrate, the eectromechanica couping coefficient expression at the first transition region can be simpified by substituting A35 into A3 to get A36 see next page. It has to be noted that, even in the case that R is an odd integer or arge enough, Γ 1 is not necessariy hod. Acknowedgment The authors woud ike to thank Dr. F. S. Hickerne at Motoroa for providing the sampes. Y. Zhang is very gratefu to Concordia University for the Concordia University Feowship award. References [1] L. Zuo, M. Sayer, and C.-K. Jen, So-ge fabricated thick piezoeectric utrasonic transducers for potentia appications in industria materia processes, in Proc. IEEE Utrason. Symp., 1997, pp

13 zhang et a.: characterization of piezoeectric fim in composite resonator m T +1/z sb 1+R/z sb = 1+ ρe1e1 + ρv sb ρ sb V sb ρe1e1 + 1 ρ V ρ sb V sb ρ e e +1 ρe e. A35 k t = 1+ ρe1 e1 + ρv ρ sb V sb sb + 1 ρ V 1+ ρe1e ρe e ρ sb V sb ρ e e keff m T +1. A36 Γ [] R.M.Mabon,D.J.Wash,andD.K.Winsow, Zinc-oxide fim microwave acoustic transducers, App. Phys. Lett., vo. 10, no. 1, pp. 9 10, [3] F. S. Hickerne, Measurement techniques for evauating piezoeectric thin fims, in Proc. IEEE Utrason. Symp., 1996, pp [4] B. S. Naik, J. J. Lutsky, R. Rief, and C. D. Sodini, Eectromechanica couping constant extraction of thin-fim piezoeectric materias using a buk acoustic wave resonator, IEEE Trans. Utrason., Ferroeect., Freq. Contr., vo. 45, no. 1, pp , [5] IEEE Standard on Piezoeectricity ANSI/IEEE Std , IEEE Trans. Utrason., Ferroeect., Freq. Contr., vo. 43, no. 5, pp , [6] A. J. Bahr and I. N. Court, Determination of the eectromechanica couping coefficient of thin-fim cadmium suphide, J. App. Phys., vo. 39, no. 6, pp , [7] A. H. Meitzer and E. K. Sittig, Characterization of piezoeectric transducers used in utrasonic devices operating above 0.1 GHz, J. App. Phys., vo. 40, no. 11, pp , [8] Z. Wang, Y. Zhang, and J. D. N. Cheeke, Characterization of eectromechanica couping coefficient of piezoeectric fim using composite resonators, IEEE Trans. Utrason., Ferroeect., Freq. Contr., vo. 46, no. 5, pp , [9] Y. Zhang, Z. Wang, J. D. N. Cheeke, and F. S. Hickerne, Direct characterization of ZnO fims in composite resonators with the resonance spectrum method, in Proc. IEEE Utrason. Symp., 1999, pp [10] F. S. Hickerne, persona communication, [11] B. A. Aud, Appendix : Properties of materias, in Acoustic Fieds and Waves in Soids. vo. 1, New York: Wiey, [1] K. M. Lakin, G. R. Kine, and K. T. McCarron, High-Q microwave acoustic resonators and fiters, IEEE Trans. Microwave Theory Tech., vo. 41, no. 1, pp , [13] Y. Zhang, Resonant spectrum method to characterize piezoeectric fims in composite resonators, Ph.D. dissertation, Concordia University, Montrea, Quebec, Feb. 00. [14] E. K. Sittig, Design and technoogy of piezoeectric transducers for frequencies above 100 MHz, Physica Acoustics, vo. IX, W. P. Mason and R. N. Thurston, Eds. New York: Academic, pp. 1 75, 197. Yuxing Zhang was born in Changzhou, China, in He received the B.Eng. degree in eectric engineering from the University of Science and Technoogy of China, Hefei, China, in 1987; the M.S. degree in acoustics from the Institute of Acoustics, Chinese Academy of Sciences, Beijing, China, in 1991; and the Ph.D. in appied physics from Concordia University, Montrea, Canada, in 00. From 1991 to 1996, he was a research assistant then research feow with the Utrasonic Eectronics Laboratory, the Institute of Acoustics, Chinese Academy of Sciences, where he was invoved in research and design of SAW sensors, SAW programmabe matched fiters, SAW dispersive deay ines, and SAW Chirp-Z-Transform spectrometers. From 1997 to 00, he was a doctora student ater part-time at the Physics Department of Concordia University, where he worked on deveopment of SAW sensors and characterization of piezoeectric fims. Since 1999, Dr. Zhang has been with the Microeectronics Group and the Wireess Technoogy Lab, Norte Networks Inc., Ottawa, Canada. His current interests are on design of SAW fiters and radio frequency integrated circuits, and their appications in wireess systems. Zuoqing Wang received the B.S. and the M.S. degrees from the Department of Physics, Nanjing University, Nanjing, China, in 196 and 1965, respectivey. From 1966 to 1988, he worked on microwave acoustics, acousto-optica interactions and their appications, piezoeectric transducer anaysis, and surface acoustic wave devices as an Associate Researcher and Associate Research Professor at the Institute of Acoustics, Chinese Academy of Sciences, Beijing, China. From September 1988 to January 001, his research was invoved in acoustic sensors, Lamb-waves propagating in mutiayer structures, circumferentia waves, and piezoeectric fim characterization as a visiting scientist at the Department of Eectrica Engineering in McGi University, Montrea, Quebec, the Department of Physics in Sherbrooke University, Sherbrooke, Quebec, and the Department of Physics in Concordia University, Montrea, Quebec. Since March 001, Dr. Wang has been engaged in research on quartz crysta resonator technoogies as a senior researcher in TXC Corporation, Taiwan, R.O. China. J. David N. Cheeke A 87 SM 91 received the Bacheors and the Masters degree in Engineering Physics from UBC, Vancouver, British Coombia, in 1959 and 1961, respectivey, foowed by the Ph.D. in Low Temperature Physics from Nottingham University, Engand, in He then joined the Low Temperature Laboratory, CNRS Grenobe, France, and was aso a professor at the Universite de Grenobe, France. In 1975 he moved to the Universite de Sherbrooke, Sherbrooke, Quebec, where he set up an utrasonics aboratory speciaizing in physica acoustics, acoustic microscopy, and acoustic sensors. In 1990 he joined the Physics Department at Concordia University, Montrea, Quebec, where he is head of an utrasonics aboratory and was Chair of the Department from 199 to 000. He has pubished over 10 papers on various aspects of utrasonics. He is a senior member of the IEEE and an Associate Editor of the IEEE Transactions on Utrasonics, Ferroeectrics and Frequency Contro.