Ultrasonics 39 (2001) 377±382
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Extraction of electromechanical coupling coecient of piezoelectric thin ®lms deposited on substrates Qing-Biao Zhou *, Yue-kai Lu, Shu-Yi Zhang Laboratory of Modern Acoustics, Institute of Acoustics, Nanjing University, Nanjing 210093, People's Republic of China Received 5 September 2000; received in revised form 19 December 2000; accepted 18 January 2001
Abstract A method to extract the electromechanical couplig coecient kt2 of the thickness extensional mode of a piezoelectric thin ®lm deposited on a substrate is presented, in which a two-layer structure is as a composite resonator. Based on the method proposed by Wang et al. (IEEE Trans. UFFC 46 (1999) 1327), we present two alternative approximate equations of kt2 , then kt2 can be calculated 2 directly from the eective electromechanical coupling factor keff of two special modes of the composite resonator. The new method is unnecessary to know the thickness of the piezoelectric ®lm, which is very dicult to accuracy measure for a high frequency composite resonator or transducer. The accuracy and validity of this method are checked by numerical simulation on two typical samples. The results of numerical simulation show that the proposed approximate method is available for a real piezoelectric thin ®lm, if the mechanical loss in the ®lm is less than 5% and the mechanical loss in the substrate is less than 0.8%. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Electromechanical coupling coecient; Piezoelectric thin ®lm; Resonance frequencies
1. Introduction Piezoelectric thin ®lms, such as ZnO, AIN, or PZT ®lm, deposited on proper substrates have been widely applied for fabricating devices, such as high frequency ®lters, resonators, actuators, sensors, etc. For improving the performance of the devices, the electromechanical coupling coecient of the piezoelectric thin ®lm is the most important parameter, which determines the piezoelectric response of the thin ®lm [1]. For conventional piezoelectric thin ®lms, such as PVDF ®lms, several standard measurement techniques have been established to evaluate the electromechanical coupling coecient of the ®lms. But for some piezoelectric dielectric thin ®lms, such as ZnO, AlN and PZT ®lms, are always deposited on a substrate. Furthermore, the properties of some ®lms (such as sol±gel PZT ®lms) are also dependent on the properties of the substrate. Therefore, the electromechanical coupling coecient of the ®lm should be characterized on a composite resonator consisting of the ®lm and substrate. Recently, some data ®tting methods [2,3] and an approximate *
Corresponding author. Fax: +86-25-331-5557/360-5557. E-mail address:
[email protected] (Q.-B. Zhou).
method [4] have been reported to extract the electromechanical coupling coecient kt2 of the thickness extensional mode of a piezoelectric thin ®lm deposited on a proper substrate. The methods reported in Refs. [2,3] are not the direct methods to characterize kt2 , and the mechanical loss in the ®lm is ignored in Ref. [3]. But for a thick ®lm, such as PZT ®lm, the eect of the mechanical loss in the ®lm should be taken into consideration. In Ref. [4], the eect of the mechanical loss in the substrate is ignored, but in practical conditions, the mechanical loss of the substrate cannot be ignored. In addition, the accurate value of the thickness of the ®lm must be known for these methods. But for a high frequency composite resonator or transducer where the thickness of the piezoelectric ®lms is in the range of a few microns, therefore, it is very dicult to accurately measure the thickness of the ®lms. In this paper, an alternative evaluating method is presented to extract the electromechanical coupling coecient kt2 of the piezoelectric thin ®lm deposited on a substrate. Firstly, by analyzing the series and parallel resonance frequencies, and the eective coupling factor 2 keff for each vibration mode of the composite resona2 in frequency tor, one can obtain the distribution of keff range. Based on the related approximate equations [4],
0041-624X/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 1 - 6 2 4 X ( 0 1 ) 0 0 0 6 2 - 2
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we present two alternative equations of kt2 which do not require to know the value of the thickness of the ®lm, 2 and then kt2 can be calculated directly from keff of two special modes of the composite resonator. By numerical simulation on two typical samples, the accuracy of the alternative approximate equations is checked under the consideration of that the composite resonator is lossless, and the validity of the approximate equations is also con®rmed by considering the mechanical losses in the ®lm and substrate in practical situation. To obtain a clear insight into the eects of the parameters of the thin ®lm and substrate, the mechanical eects of electrodes are ignored. This limitation corresponds to the case of not very high frequency resonator. 2. The principles of the measurement method 2.1. The original method proposed in Ref. [4] The composite resonator consisting of a piezoelectric thin ®lm deposited on a substrate is shown in Fig. 1. In order to employ one-dimensional propagation assumption, the diameter of the resonator must be much larger than the thickness of the ®lm and substrate. For this kind of resonators, the input electrical impedance can be obtained from Sittig's model [5] by ignoring the eects of electrodes. It is shown as follows: 1 kt2 2 tan
c=2 zr tan n
cb Zin 1
1 jxc0 c
1 zr tan
cb = tan
c where, Zin is the electrical impedance; x 2pf , f is the frequency, c0 e233 A=l is the clamped capacitance of the resonator; kt2 h233 =es33 cD 33 is the electromechanical coupling coecient of the thickness extensional mode of the piezoelectric thin ®lm [6], h33 , es33 , and cD 33 are piezoelectric stress constant, dielectric constant and stiness constant of the ®lm, respectively; c 2pf l=c, and cb 2pf lb =cb , zr Zb =Z0 , Z0 qc and Zb qb cb are the acoustic impedance of thin ®lm and substrate, respectively; q, qb , c and cb are the densities and the longitudinal wave velocities in the thickness direction of the thin ®lm and the substrate; l and lb are the thickness of
thin ®lm and the substrate; A is the area of the top electrode. When an a.c. electrical power source is applied to the electrodes, the composite resonator will generate a timevarying electrical ®eld within the piezoelectric ®lm. This ®eld excites an acoustic wave propagating in thickness direction assuming the piezoelectric ®lm is properly oriented to the electrical ®eld. A part of the acoustic wave is re¯ected at the interface of the ®lm and substrate, and another part is transmitted into the substrate. For the acoustic wave in the composite resonator, there are a series of resonance with almost equal frequency interval due to the resonance of the substrate, and an envelope resonance due to the resonance of the ®lm [3,4]. These resonance characteristics contain a lot of information about the properties of the ®lm and substrate, which consist of the composite resonator. Therefore, it is possible to characterize the material parameters of the ®lm and substrate, especially to extract kt2 of the ®lm. For a lossless composite resonator, the series and parallel resonance frequency equations are obtained from Eq. (1) by setting the amplitude of the input impedance Zin to zero and in®nity. For series resonance, jZin j 0, then tan
c zr tan
cb
kt2 =c2 tan
c=2 zr tan
cb tan
c
2
and for parallel resonance, jZin j ! 1, then tan
c zr tan
cb 0
3
By solving two transcendental algebraic equations, the series and parallel resonance frequencies of the composite resonator can be obtained. The distribution of the resonance frequency is determined by the mechanical properties of the ®lm and substrate. The interval of parallel resonance frequency is almost equal because the thickness of the substrate is much larger than that of the ®lm, but it is modi®ed slightly by the eect of the ®lm. For each parallel resonance frequency, there is a corresponding series resonance frequency. Comparing with the resonance frequencies of a single piezoelectric plate, where only a pair of parallel and series resonance frequencies is appeared for the funda-
Fig. 1. Con®guration of the composite resonator consisting of the piezoelectric thin ®lm and the substrate.
Q.-B. Zhou et al. / Ultrasonics 39 (2001) 377±382
mental mode, and then kt2 is explicitly given by two frequencies p fs p fs p2 fs
fp fs 2 kt
4 tan 2 fp 2 fp fp 4 fp where, fs and fp are the series and parallel resonance frequencies of the fundamental mode of the single piezoelectric plate, respectively. A similar de®nition of the eective coupling factor for the mth resonance mode is given by
m
m 2
m f f p s p fs 2 keff
m
5
m 4 fp
m fp 2 where, keff
m is eective coupling factor, fs
m and fp
m are the series and parallel resonance frequencies of the mth mode resonance, respectively. If the composite resonator is lossless, the electromechanical coupling coecient kt2 can be obtained from Eq. (2) as following:
kt2 c
tan
c zr tan
cb 2 tan
c=2 zr tan
cb tan
c
6
where, f fs
m . In principle, by measuring fs
m for any one mode m, and substituting it into Eq. (6), the value of kt2 can be calculated. But kt2 cannot be obtained satisfactorily only by the series resonance frequency measurement, because Eq. (6) is deduced under the conditions of non-losses in the ®lm and substrate. Eq. (6) can amplify the error of measured series resonance frequency. Practically, the losses exist in the thin ®lm and substrate, the measured series resonance frequency is aected by the mechanical loss of the resonator, which can introduce the error to the series frequency comparing with the condition of non-losses in the ®lm and substrate. On the other hand, Eq. (6) can be employed at two special mode c p=2 and c p, where the error is not ampli®ed. When c p, then tan
c ! 0, it is referred to as the center of the ®rst normal region. When c p=2, then tan
c ! 1, it is referred to as the center of the ®rst transition region. Using Eq. (5), Wang et al. [4] presented two explicit formulae, relating kt2 to the eective coupling factor of the centers of the ®rst normal and transition region, can be derived as below: 2 kt2
1 qb lb =qlkeff
mN
kt2
7
lb qc2 1 2 k
mT 1 l qb c2b C eff qc 2pl fp
mT 1 qb c b c 2plb fs
mT cb
C12
8 p 2
mT
1 2
379
center of the ®rst normal region and ®rst transition region respectively. C is a correction factor that is not necessarily near unity when mT is not very large. 2.2. An alternative evaluating method From Eqs. (7) and (8), one can see that when density, thickness, longitudinal wave velocity of the ®lm and the 2 substrate, and keff distribution are known, then kt2 can be calculated from Eqs. (7) and (8) directly. However, for a high frequency composite resonator or transducer where the thickness of the piezoelectric ®lm is in the range of a few microns, it is very dicult to accurately measure the thickness of the ®lm. Therefore, it is necessary to present an alternative method to evaluate the value of kt2 , which does not require to knowing the thickness of the piezoelectric ®lms. By taking the approximations mN clb =
cb l mT
1=2clb =
cb l
10 1
11
then substituting mN and mT into Eqs. (7) and (8), omitting the 1=C, it is easy to ®nd the following relations
1 qb lb =ql
1 mN zr and lb qc2 1 1
2mT 1=zr l qb c2b
12
13
Eqs. (7) and (8) can be expressed in another from: 2
mN kt2
1 mN zr keff
14
2
mT kt2 1
2mT 1=zr keff
15
Above formulae are more suitable in the case of that the thickness of the ®lm is unknown or dicult to be 2 2 measured. Because keff
mN or keff
mT is the peak of the 2 distribution of keff (see the following sections), there2 fore, when the distribution of keff is obtained from the measured input impedance data and mN or mT are also known, then kt2 can be calculated from Eqs. (14) and (15) directly. It means that one can determine kt2 only requiring the acoustic impedance ratio of the ®lm and sub2 strate is known, and the distribution of keff is obtained from the measured input impedance. In the following sections, the accuracy and validity of this approximate will be discussed by numerical simulation.
3. Numerical simulation
p
9
where, mN roundclb =cb l, mT round
clb =cb l 1= 2,
mN 1 and
mT 1 are the mode order of the
3.1. Parameters of samples used in the simulation 1. Sample #1: a PZT ®lm deposited on a stainless steel substrate with parameters: kt2 0:04, c 2:4 km=s, q 7:0 g=cm3 , l 0:05 mm; es33 250,
380
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A 5:0 mm2 ; Vb 5:9 km=s, qb 7:8 g=cm3 , b 2:4 mm, zr > 1. 2. Sample #2: a PZT ®lm deposited on a aluminum plate with the parameters: kt2 0:09, c 4:60 km=s, q 7:60 g=cm3 , l 0:050 mm; es33 480, A 2:0 mm2 ; Vb 5:65 km=s, qb 2:7 g=cm3 , b 1:8 mm, zr < 1.
``quarter wave-length resonator'' in the case of zr > 1 (sample #1), and as the ``half wave-length resonator'' in the case of zr < 1 (sample #2). From Fig. 2, one can obtain mT for sample #1, and mN for sample #2, then kt2 can be extracted using Eqs. (15) and (14), respectively. Numeric calculation display that the error introduced by the approximate method is less than 1.0% for samples #1 and #2, so the approximate method is accurate.
3.2. Accuracy of the approximate method
3.3. Validity of the approximate method
Using Eqs. (2)±(4), the series and parallel resonance frequencies, and the distribution of the eective coupling 2 factor keff in frequency range of the composite resonator can be obtained. Where, the losses of the ®lm and substrate are not considered. The distribution of the eective coupling factor of the samples #1 and #2 are shown in Fig. 2. We can see that the peak of the eective coupling factor is located at the frequency f c=4l for the sample #1 (zr > 1) and f c=2l for the sample #2 (zr < 1). It seems that the piezoelectric ®lm is acting as a
As shown in Section 3.1, all the parameters of the materials are real, it corresponds to the case that the composite resonator is lossless. In practice, some kinds of material losses in the ®lm and substrate, such as dielectric loss, piezoelectric loss and mechanical loss, are always inevitable. Therefore, some parameters should be complex. The impedance equation (1) is still valid for complex parameters. However, for the complex parameters, the resonance frequencies and the eective coupling factor are determined directly from Eq. (1) rather than from Eqs. (2) and (3); thus the validity of the method has to be identi®ed. Here, only the mechanical losses in the ®lm and substrate are considered. The velocities of the ®lm and substrate are taken as complex. The mechanical loss can be indicated by the ratio of the imaginary part to the real part of the velocity. In the simulation, the ratios are taken as 0.1%, 0.5%, 1.0%, 2.0%, 5.0%, 10.0% for the ®lm, and 0.1%, 0.2%, 0.5%, 0.8%, 1.0% for the substrate of the samples #1 and #2. The typical input impedance response of the composite resonator (sample #1) calculated from Eq. (1) is shown in Fig. 3. The response shows a general decrease in the input impedance due to the capacitive eect, and a series of equally resonances due to the substrate eect, also an envelop of these resonance due to the ®lm eect. It also shows that the eect of the mechanical loss in the substrate is much larger than the eect of the mechanical loss in the ®lm. It is due to the thickness of the substrate is much larger than that of the ®lm. For a composite resonator with losses, Eqs. (2) and (3) are invalid to determine the resonance frequencies, which must be calculated directly from the input impedance curve as shown in Fig. 3. Based on the IEEE standard [7], the parallel resonance frequencies are determined when the real part of the impedance takes a maximum, and the series resonance frequencies are determined when the real part of the admittance takes a maximum. These de®nitions are independent of the lumped-parameter equivalent circuit. When the series and parallel resonance frequencies of each mode are obtained, the corresponding eective coupling factor can be calculated from Eq. (5). The in¯uences of mechanical losses in the ®lm and substrate to the distribution of the eective coupling factor (for samples #1 and #2) are displayed in Figs. 4 and 5.
Fig. 2. The eective coupling factor distribution of the composite resonator described in Fig. 1 considering the case of no-losses in the ®lm and substrate: (a) for the sample #1
zr > 1 and (b) for sample #2
zr < 1.
Q.-B. Zhou et al. / Ultrasonics 39 (2001) 377±382
Fig. 3. Amplitude of the input impedance of the composite resonator (sample #1): (a) related to 1.0% mechanical loss in the ®lm; (b) related to 0.5% mechanical loss in the substrate.
For sample #1, kt2 can be obtained form Eq. (15), for sample #2, it can be obtained from Eq. (14). The results show that the error of kt2 introduced by this approximate method is less than 3.0% when the mechanical loss in the ®lm is less than 5.0%, and the error is less than 3.0% when the mechanical loss in the substrate is less than 1.0% for sample #1. The error of kt2 introduced by this approximate method is less than 2.0% when the mechanical loss in the ®lm is less than 5.0%, and the error is less than 2.0% when the mechanical loss in the substrate is less than 1.0% for sample #2. From Figs. 4 and 5, we can also see that the eect of the mechanical loss in the substrate is much larger than that of the ®lm to the distribution of the eective coupling factor. It is reasonable, because the thickness of the substrate is much larger than that of the ®lm. Therefore, the accuracy of the approximate method can be improved if a low loss substrate is chosen. Besides
381
Fig. 4. The eective coupling factor distribution considering the mechanical losses in the ®lm and substrate for sample #1: (a) corresponding to 5.0% loss in the ®lm; (b) corresponding to 1.0% loss in substrate.
that, decreasing the thickness also can improve the accuracy the approximate method.
4. Discussions and conclusions In this paper, an alternative evaluating method is presented to extract the thickness extensional mode electromechanical coupling coecient kt2 of a thin ®lm deposited on a substrate. Based on the method proposed by Wang et al. [4], two alternative approximate Eqs. (14) and (15) are presented. The new method is unnecessary to know the thickness of the piezoelectric ®lm, which is very dicult to accuracy measure for a high frequency composite resonator or transducer. The electromechanical coupling coecient kt2 can be expressed 2 explicitly by the eective coupling factors keff
mN and 2 keff
mT of two special modes of the composite resonator and the ratio of the acoustic impedance of the ®lm and substrate. One can extract kt2 of the ®lm using
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Q.-B. Zhou et al. / Ultrasonics 39 (2001) 377±382
this approximate method is less than 3.0% when the mechanical loss in the ®lm is less than 5.0%, the error is less than 3.0% when the mechanical loss in the substrate is less than 1.0%. For sample #2, the error of kt2 introduced by this approximate method is less than 2.0% when the mechanical loss in the ®lm is less than 5.0%, the error is less than 2.0% when the mechanical loss in the substrate is less than 1.0%. It means that the presented method is available for real piezoelectric thin ®lms if the mechanical loss in the ®lm is less than 5%, and a substrate with small mechanical loss (<0.8%) is chosen. In addition, whatever the mechanical loss of the ®lm and the substrate is considered or not, the value of the eective coupling factor at the peak of the distribution is almost the same in the simulation. It is also demonstrate the accuracy and validity of the presented approximation method. Acknowledgements The authors would like to thank Professor Z. Wang of the Department of Physics, Concordia University, for this inspiring discussions.
References Fig. 5. The eective coupling factor distribution considering the mechanical losses in the ®lm and substrate for sample #2: (a) corresponding to 5.0% loss in the ®lm; (b) corresponding to 1.0% loss in substrate.
approximate equations if the ratio of the acoustic impedance of ®lm and substrate is known, and the input impedance curve can be measured by experiment. The accuracy and validity of this method are checked on two typical samples by numerical simulation. Simulation results show that the case of zr > 1, the Eq. (15) is more suitable to extract kt2 , and the Eq. (14) is more suitable for the case of zr < 1. The error of kt2 introduced by this method is less than 1.0% when the samples #1 and #2 are lossless. The eect of the mechanical losses in the ®lm and substrate to the accuracy of this method are also considered. For sample #1, the error of kt2 introduced by
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