Theory, modeling and characterization of PZT-on-alumina resonant piezo-layers as acoustic-wave mass sensors

Sensors and Actuators A 92 (2001) 182±190

Theory, modeling and characterization of PZT-on-alumina resonant piezo-layers as acoustic-wave mass sensors Vittorio Ferrari*, Daniele Marioli, Andrea Taroni Dipartimento di Elettronica per l'Automazione and Istituto Nazionale Fisica della Materia INFM, UniversitaÁ di Brescia, Via Branze 38, 25123 Brescia, Italy Accepted 18 January 2001

Abstract Lead zirconate titanate (PZT) ®lms screen-printed on alumina substrate form composite piezoelectric resonators that can be exploited as acoustic-wave mass sensors. The paper presents a theoretical treatment of such layered resonators based on the use of the Mason's model to derive the expression of the distributed-parameter electric impedance, followed by an approximation yielding a lumped-element equivalent circuit valid around the sensor fundamental resonance. Expressions for the resonant frequency and mass sensitivity are obtained, which are validated by experimental results relative to PZT thick-®lm sensors. The model is of general applicability, provided that the PZT and substrate operate close to their quarter-wave resonance. # 2001 Elsevier Science B.V. All rights reserved. Keywords: Acoustic-wave sensor; Mass sensor; Equivalent-circuit model; Piezoelectric resonator; Thick-®lm sensor

1. Introduction Bulk acoustic-wave resonant piezo-layers (RPL) made by lead zirconate titanate (PZT) ®lms on alumina substrate are sensitive to an added surface mass and can be exploited for gravimetric chemical sensing [1,2]. In contrast to homogeneous resonators, such as quartz crystal microbalance (QCM) sensors, RPLs comprise a piezoelectric layer acoustically coupled to a non-piezoelectric substrate. Therefore, they are inherently composite resonators. Detailed mathematical treatments of multilayer composite resonators were developed by transfer matrix descriptions [3], perturbation methods [4], and acoustic transmission-line theory [5,6]. Though such treatments are complete and general in scope, their complexity makes the analysis of the device behavior rather involved. On the other hand, for sensor design and use, a simpli®ed reference model based on manageable expressions and equivalent circuits is desirable to gain ®rst-order insight into the fundamental parameters in¯uencing the sensor operation. In this perspective, the present work proposes a sensororiented approach to the analysis of RPL composite resonators which is based on a simple yet descriptive approximate model. The model leads to a lumped-element * Corresponding author. Tel.: ‡39-030-3715469; fax: ‡39-030-380014. E-mail address: [email protected] (V. Ferrari).

equivalent circuit accounting for both the piezoelectric and the substrate layers, and to explicit expressions for the resonant frequency and the mass sensitivity. The theoretical predictions are con®rmed by the presented experimental results. 2. Theory and modeling The RPL structure is made by the superposition of a nonpiezoelectric substrate, speci®cally alumina, a bottom-electrode layer, a PZT layer poled along its thickness, and a topelectrode layer. The overall thickness is small compared to the lateral dimensions. An alternating voltage across the electrodes can set the composite structure in thickness± expansion resonance. A mass load on the top electrode causes a shift in the resonant frequencies. Therefore, the RPL can be exploited as an acoustic-wave mass microbalance sensor. The present analysis is based on the following simplifying assumptions.  The resonator lateral dimensions are indefinitely large, and the wave propagation is assumed to be purely planar along the thickness without boundary effects.  The metal electrodes are considered infinitely thin, therefore they are neglected.  Dissipative effects are not taken into account, therefore a loss-free system is assumed.

0924-4247/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 4 2 4 7 ( 0 1 ) 0 0 5 6 1 - 1

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Eq. (3) can be solved in the harmonic regime taking into account the boundary conditions at the bottom and top faces which, assuming the velocity to be positive for particle motion directed inward to the PZT, are _ ˆ 0† ˆ z_ bottom ; z…z Fig. 1. Simplified monodimensional structure of a resonant piezo-layer sensor with load.

 The coating layer is uniform and constant in thickness over the top surface of the resonator. Under these assumptions the problem can be analyzed with reference to the simpli®ed monodimensional structure of Fig. 1. The used symbols have the following meanings. The subscripts P, A and L refer to the PZT, alumina substrate and loading layer, respectively. The symbols l, r and Y are the thickness, density and elastic modulus. A is the electrode area. Z o ˆ rv ˆ …rY†1=2 is the characteristic acoustical impedance, and v ˆ …Y=r†1=2 is the acoustic wave speed. b ˆ o=v is the wave propagation constant in a given medium at the angular frequency o. Inside the PZT, the following piezoelectric constitutive equations hold [7]: T ˆ YPD S

hD

(1a)

Eˆ

1 D eS

(1b)

hS ‡

where T, S, E and D are respectively the stress, strain, electric field and electric displacement, which are all functions of both the vertical position z and the time t. YPD and eS are respectively the stiffened elastic modulus, i.e. at zero electric displacement, and the clamped dielectric permittivity, i.e. at zero strain. The coefficient h is equal to e/eS, where e is the piezoelectric stress constant. In addition to Eqs. (1a) and (1b), inside the PZT it is valid the Newton's law dT d2 z dz_ ˆ rP 2 ˆ rP dz dt dt

(2)

where z and z_ ˆ dz=dt are the particle displacement and velocity, respectively. By differentiating Eq. (1a) with respect to the position z and inserting Eq. (2), considering that S ˆ dz=dz, and that for an insulator as PZT it is dD=dz ˆ 0, it follows the longitudinal-wave equation for the velocity d2 z_ dz2

rP d2 z_ d2 z_ 1 d2 z_ ˆ ˆ0 (3) YPD dt2 dz2 …vP †2 dt2 p Since the term YPD =rP represents the acoustic speed vP along the PZT thickness, it can be noticed that the wave propagation parallel to the poling direction is governed by the piezoelectrically-stiffened elastic modulus YPD .

_ ˆ lP † ˆ z…z

z_ top

(4)

If the force F ˆ AT acting on the PZT faces is now considered, by rearranging the solution of Eq. (3) with the conditions (4) and inserting into Eq. (1a), it can be obtained " # z_ top z_ bottom h (5a) ‡ ‡ I Fbottom ˆ jAZoP jo tan…bP lP † sin…bP lP † " # z_ top z_ bottom h Ftop ˆ jAZoP (5b) ‡ ‡ I jo sin…bP lP † tan…bP lP † where the electric current I is given by I ˆ A…dD=dt† ˆ joAD. According to the direct electromechanical analogy, which relates force to voltage and velocity to current, the quantities Zbottom ˆ Fbottom =z_ bottom and Ztop ˆ Ftop =z_ top are the mechanical impedances presented by the PZT at the bottom and top faces, respectively. Considering now the Eq. (1b), the voltage V across the PZT can be expressed as Z lP Z lP lP Vˆ E dz ˆ h S dz ‡ S D e 0 0 i h h_ 1 zbottom ‡ z_ top ‡ I (6) ˆ j o joC0 where C0 ˆ AeS =lP is the clamped electrical capacitance. Therefore, the PZT can be seen as a three-port network with two mechanical ports and one electrical port, respectively, described by Eqs. (5a), (5b) and (6), which can be ®nally summarized in the following matrix form: 3 2 AZoP AZoP h 6 tan …b lP † sin …b lP † 2 3 2 3 o 7 P P 7 z_ 6 Fbottom bottom 7 6 AZ AZoP h 76 _ 6 7 6 7 oP 74 ztop 5 4 Ftop 5 ˆ j6 6 sin …bP lP † tan …bP lP † o 7 7 6 V I 4 h h 1 5 o

o

oC0 (7)

The mathematical expressions in Eq. (7) correspond to the Mason's distributed-parameter electromechanical circuit shown in the dashed portion of Fig. 2, which represents the PZT film as a three-port element [8]. The negative capacity C0 arises because the wave propagation is along the electric field direction [7]. The effect of the alumina substrate and the upper loading layer can be taken into account by connecting the respective mechanical impedances to the appropriate mechanical ports of the PZT circuit. According to the acoustic transmission

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Fig. 2. Distributed-parameter equivalent circuit of a resonant piezo-layer sensor with load. The dashed rectangle includes the Mason's model of the PZT film.

line theory, the mechanical impedance of a medium with propagation constant b and characteristic impedance Zo at a distance x from a terminating load impedance ZT is given by [8] Zx ˆ Zo

ZT ‡ jZo tan …bx† Zo ‡ jZT tan …bx†

(8)

If the surrounding air is considered a stress-free boundary (T ˆ 0), then the sides of the alumina and the loading layer opposite to the PZT are terminated by a mechanical short circuit (ZT ˆ 0). Therefore, the complete model of Fig. 2 can be finally obtained. The mechanical impedance at the nodes C±D is

ZCD ˆ

jAZoL tan bL lL  jAZoL bL lL ˆ jAorL lL ˆ jomL

where the symbol || denotes the parallel connection of impedances. The condition of mechanical resonance Z CD ˆ 0 corresponds to Ze ! 1, i.e. to the electrical antiresonance. Conversely, the electrical resonance Ze ˆ 0 occurs for ZCD ˆ N 2 =joC0 . In the following, the fundamental-mode resonance will be considered, with its associated resonant frequency f1 ˆ o1 =2p. In the absence of the load, Eq. (9) simpli®es to A‰ZoP tan P ‡ ZoA tan AŠ j tan P…2 tan P=2 ‡ …ZoA =ZoP †tan A†

(12)

Therefore, an acoustically thin load behaves as a concentrated mass and makes the resonator work in the gravimetric regime [9,10]. Such a circumstance is typically encountered

A‰ZoP tan P ‡ ZoA tan A ‡ ZoL tan L……ZoA =ZoP †tan P tan A 1†Š j tan P…2 tan P=2 ‡ …ZoA =ZoP †tan A ‡ …ZoL =ZoP †tan L†

where P ˆ bP lP ˆ olP =vP ; A ˆ bA lA ˆ olA =vA , and L ˆ bL lL ˆ olL =vL . This reflects into an electrical impedance Ze at the nodes A±B given by    

ZCD V 1 1

Ze ˆ ZAB ˆ ˆ (10) I joC0 N 2 joC0

ZCD ˆ

respective quarter-wave resonance. This is what typically happens for RPLs of PZT thick films on alumina, as it is demonstrated in Fig. 3 which graphically shows the solution of Z CD ˆ 0 for lA ˆ 254 mm and lP ˆ 100 mm. If the load is now taken into account, and it is assumed to be acoustically thin, i.e. no signi®cant phase shift of the propagating wave is developed along the load thickness, it can be written

(9)

in acoustic-wave chemical sensors operated in air, representing the original application of RPL sensors [2,11], in which the load is essentially made by the mass increase of a thin sorption layer sensitive towards the target analytes. As long as mL is suitably small, the first resonance f1 again remains located in the region where both tan P and tan A

(11)

If ZoA and ZoP are of the same order of magnitude and the terms lP =vP and lA =vA differ by less than about 50%, then ZCD firstly vanishes at a frequency f1 where both P and A are close to p/2, i.e. the PZT and the substrate operate about their

Fig. 3. Graphical solution of Z CD ˆ 0 for a PZT film on alumina substrate with respective thickness of 100 and 254 mm.

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diverge. The tangent function around its singularity at p/2 can be approximated by the formula tan X  8X=…p2 4X 2 † [7,8]. After insertion in Eq. (9) and some rearrangements, it can be obtained the following approximated expression of ZCD   p2 4P2 p2 4A2 mL ‡ HZoA ‡ ZCD  A ZoP j8P j8A A    2 D   lP rP lA rA mL 1 p YP p2 YA ‡H ‡ ‡H ˆ A jo ‡ jo 8lP 2 2 A 8lA (13) 2 2 where H ˆ ZoP =ZoA . Eq. (13) interestingly indicates that, around f1, the RPL composed resonator can be approximately considered as a homogeneous resonator with an unloaded effective mass Meff ˆ A…lP rP ‡ HlA rA †=2, and an effective stiffness Keff ˆ …p2 A=8†‰…YPD =lP † ‡ H…YA =lA †Š. Fig. 4 compares the extended acoustical impedance ZCD/A of Eq. (11) with the approximation of Eq. (13), calculated for an unloaded sensor with lP ˆ 100 mm and lA ˆ 254 mm. The model locates the ®rst resonance f1 at 6.9 MHz with a ®tting agreement of about 5%, while the second harmonic at f2, as expected, is not spotted. The electrical impedance Ze can now be derived by substituting Eq. (13) into Eq. (10), leading to the lumpedelement equivalent circuit of Fig. 5. This can be seen as an extension of the Butterworth-Van Dyke (BVD) circuit for homogeneous resonators [12] to the case of a composite resonator. The equivalent circuit values are the following:

C0 ˆ

eS A ; lP

Nˆ

1 AlP rP ; N2 2 H AlA rA ; LA ˆ 2 N 2 LP ˆ

eA ˆ hC0 ; lP mL ; N2 Z2 H ˆ 2oP ZoA

Lm ˆ

CP ˆ N 2 CA ˆ

8lP 2 p AYPD

;

N 2 8lA ; H p2 AYA (14)

Fig. 4. Acoustical impedance at nodes C±D of Fig. 2 for the distributedparameter (- - -) and lumped-element (Ð) models.

Fig. 5. Lumped-element equivalent circuit of a loaded RPL sensor valid around the fundamental resonance.

The unloaded resonant frequency, corresponding to the parallel electrical resonance fp, is then given by r 1 CP CA 1 hpi 1 …LP ‡ LA † ˆ LT CT 2p CP ‡ CA s r 1 …p2 =8†……YPD =lP † ‡ H…YA =lA †† 1 Keff ˆ ˆ 2p …lP rP =2† ‡ H…lA rA =2† 2p Meff

fp ˆ

1 2p

(15) where LT ˆ Meff =N 2 and CT ˆ N 2 =Keff represent the total electrical inductance and capacitance that are the analog equivalents of the mechanical effective mass and compliance participating in the first vibration mode. The resonant frequency does not depend on the electrode area A, in accordance with the assumption of in®nite lateral dimensions. The model consistently describes the limiting case of ZoA ! 1, where the PZT ®lm works as a rigidly-backed resonator with the alumina face acting as a perfectly re¯ecting surface. In fact, if ZoA ! 1 it follows that LT ! LP and CT ! CP , causing fp to become the rigidlybacked resonant frequency fprb ˆ vP =4lP, which is what is expected for a quarter-wave resonator. On the contrary, the model fails in representing the freelysuspended limiting case given by ZoA ˆ 0, which apparently leads to the non-physical condition LT ! 1 and CT ! 0. This, however, is not surprising because the basic assumption on which the lumped-element circuit is derived from Eq. (11), i.e. both tan P and tan A diverge around the resonant frequency, breaks down for ZoA ˆ 0. Therefore, in this case, reference must be made back to Eq. (11), which for ZoA ˆ 0 reduces to ZCD ˆ AZoP =…j2 tan P=2†. If ZCD is expanded around the ®rst resonance correspondent to P=2 ˆ p=2, it follows that LT and CT now reduce to the freely-suspended inductance and capacitance of the PZT, 2 fs 2 given by Lfs P ˆ AlP rP =…8N † ˆ LP =4 and CP ˆ 8N lP = 2 D …p AYP † ˆ CP . This simpli®es the model of Fig. 5 to the

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usual BVD circuit of a free-standing unperturbed resonator, and leads to the freely-suspended resonant frequency fpfs ˆ vP =2lP , corresponding to half-wave resonance. The formulation based on the lumped-element parameters of Eq. (15) suggests a possibility to formally extend the present lossless model to the case when losses due to internal friction within the PZT and alumina substrate are considered. In such a case, the elastic moduli of both PZT and substrate become complex and given by Y ˆ Y 0 ‡ jY 00 where Y 00 ˆ oZ and Z is the material viscosity. Under this assumption, two equivalent resistors Rp and RA appear in series to the equivalent capacitances Cp and CA in the lossless circuit of Fig. 5. The expressions of these resistors are ZP HZ RP ˆ ; RA ˆ 2 0 A (16) 0 N YA CA N 2 YPD CP The overall dissipation in the composite resonator is not lower than that arising from RP and RA and, due to the fairly dishomogeneous and porous nature of PZT thick films, it can be guessed that RP most likely dominates over RA. In addition, other dissipation factors, such as those caused by the electrodes and the radiation losses, must be also considered in real devices. Globally, they determine the quality factor Q of the resonator. Coming back to the lossless model, if now fpL indicates the resonant frequency when the resonator is loaded, the fractional mass sensitivity Sm can be de®ned as Sm ˆ lim

mL !0

Dfp =fp fpL fp ˆ lim m !0 mL fp mL L

(17)

where, according to Eq. (13) and Fig. 5, fpL corresponds to a total vibrating mass of Meff ‡ mL , associated with a total inductance LT ‡ DLT . For mL =Meff ˆ DLT =LT ! 1, Eq. (15) can be linearized as Dfp =fp  DLT =2LT ˆ mL =2Meff . Then the following expression for Sm results Sm ˆ

1 ˆ 2N 2 LT

1 ˆ 2Meff

1 A…lP rP ‡ HlA rA †

(18)

The same result can alternatively be obtained by using differentials into the definition of Sm. In this way, Eqs. (15) and (17) directly lead to Sm ˆ …dfp =fp dmL † ˆ …1=2Meff †…dMeff =dmL † ˆ …1=2Meff † because, as expressed by Eq. (13), it is dMeff =dmL ˆ 1. Eq. (18) is consistent with the generalized form of the mass sensitivity of acoustic-wave sensors [13], in which Sm is inversely proportional to the resonator effective mass participating in the particular vibration mode. 3. Sensor fabrication The RPL sensors were fabricated by screen printing PZT thick ®lms on 96% alumina substrates using a purposely prepared paste. This was made by mixing a PZT powder

(Ferroperm PZ26) with 20 wt.% of PbO and adding a solution of ethylcellulose and terpineol as the liquid vehicle [1,2]. For the metal electrodes, a PdAg ink (Heraeus C1214D) was used. The following process steps were followed in the manufacturing: 1. Screen printing of the bottom electrode, drying at 1508C for 30 min, firing at 8508C peak temperature for 30 min. 2. Screen printing of the PZT, drying at 1508C for 30 min, firing at 9508C peak temperature for 30 min. Multiple print-dry-fire steps were carried out up to the final film thickness. 3. Same as (1) for the top electrode. 4. Poling of the PZT by applying an electric field of 5 MV/ m at 1508C for 30 min. The alumina substrates had dimensions of 25 mm 25 mm, and thickness lA of 254 mm, as speci®ed by the manufacturer. The planar shape of the sensors was circular with PZT and electrode diameters of 6 and 5.5 mm, respectively. For the PZT ®lms the thickness lP varied among different sensors, taking the values of 90, 95 and 100 mm. The thickness lP were measured with a linear-variable-differential-transformer (LVDT) transducer leading to an estimated uncertainty of 2.5 mm. The thickness of the metal elctrodes was about 10 mm. A SEM photograph of a 100 mm thick sensor is shown in Fig. 6, evidencing a good compactness of the PZT compared to the electrodes and the substrate, and also validating the thickness measurement made with the LVDT. The relevant material parameters are the following: rA ˆ 3:7  103 kg/m3, rP ˆ 5  103 kg/m3, Y A ˆ 330 GPa, YPD ˆ 31:2 GPa. The parameter values for alumina are taken from tabulated data. For the PZT ®lms rP was measured by weighting, and YPD could be derived by a numerical ®t over the fundamental resonance and the second harmonic of the measured impedance Ze(o) against the expressions of Eqs. (10) and (11) for sensors with different PZT thickness. Notably, both rP and YPD are lower than the correspondent parameters for bulk sintered ceramic due to the porous microstructure of PZT thick ®lms [14]. 4. Experimental results 4.1. Resonant frequency The impedance spectrum around the fundamental resonant frequency was measured for all the sensors with a HP4194A impedance analyzer. The plot in Fig. 7, which refers to a 90 mm sensor, shows an example of the typical results obtained. The parallel resonant frequency fp can be assumed to be the frequency where the phase crosses the Ze ˆ 0-axis with a negative slope. This is strictly true for an ideal undamped resonator [15], while in the real case it is a convenient approximation.

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Fig. 6. SEM photograph of a RPL sensor made by a PZT thick film screen printed on alumina substrate. The insert shows a close-up of the PZT microstructure.

For the sensor of Fig. 7 the measured quality factor Q at fp, de®ned as Q ˆ …fp =2†…dF=dfp † where F is the phase in radiants, is larger than 60, but higher values are possible [2]. The plot in Fig. 8 refers to a 100 mm sensor and shows an enlarged portion of the frequency scale to include also the

second harmonic f2. It can be observed that Q at f2 is lower than at the fundamental frequency f1, as demonstrated by the reduced phase step around f2. This same behavior was observed for all of the sensors. For the sensors with lP of 90, 95 and 100 mm the measured fundamental resonant frequencies were 7.16, 6.91 and 6.85 MHz, respectively, which agree with the predictions of Eq. (15) within about 5%. 4.2. Derivation of the piezoelectric parameters of PZT Making use of the HP4194A analyzer, the measured impedance spectrum of Fig. 7 was ®tted to the BVD equivalent circuit of Fig. 9, allowing to extract the parameter values reported in the ®gure. By comparison with the model of Fig. 5 and using the parameter expressions given in Eq. (14), it can be obtained

Fig. 7. Measured impedance around the first resonance of a RPL sensor with a PZT thickness of 90 mm.

Fig. 8. Measured impedance of a 90 mm RPL sensor vs. frequency including the fundamental and second harmonic resonances.

L ˆ LT ˆ

Meff l2P …lP rP ‡ HlA rA † ˆ N2 2e2 A

(19)

Fig. 9. BVD equivalent circuit and parameter values extracted from the fitting of the impedance spectrum of Fig. 7.

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Eq. (19) can then be solved for the piezoelectric constant e. Considering that A ˆ 2:37  10 5 m2, and H ˆ 0:127, it results e ˆ 7:64 As/m2. From the relationship e ˆ dYPD it is then possible to calculate the piezoelectric coef®cient d, resulting in d ˆ 245  10 12 As/N. The above values of e and d agree in order of magnitude with the typical values of PZT, even if they are lower than those obtainable in bulk sintered ceramic [16]. From the value of Cb in Fig. 9, equal to C0 in the model of Fig. 5, the clamped permittivity can be obtained given by eS ˆ C0 lP =A ˆ 4:82  10 9 F/m, corresponding to a relative permittivity of about 545. For a freely-suspended resonator, when the frequency is decreased much below resonance the permittivity is expected to increase, as the piezoelectric element passes from a clamped condition to an increasingly free condition. As a limit, the low-frequency capacitance C0lf should be governed by the stress-free permittivity eT ˆ eS =…1 k2 †, where k2 is the effective electromechanical coupling factor relative to the resonator con®guration [17]. On the contrary, for the RPL sensor, the measured values of C0lf at 1 kHz was found to be 1.22 nF which is substantially equal to C0. A reasonable explanation of this fact could be that, due to the presence of the substrate, the PZT ®lm unavoidably experiences lateral clamping and is never completely stress-free. A three-dimensional analysis, such as in [18], accompanied by further experimental data could possibly provide clari®cation on this issue. As a general observation, the high value of the dielectric permittivity of RPLs made of PZT, as opposed to quartz microbalances, implies quite large values of C0. This has the advantage of making the resonant frequency fp minimally affected by the parasitic capacitances in parallel to the sensor, and therefore, it increases the accuracy with which an electronic oscillator can track fp, especially when sensor and electronics are separated by some distance. 4.3. Mass sensitivity The mass sensitivity of the sensors was measured by casting with a micropipette prescribed volumes of a ®xed-titre solution of silicone grease in ether, waiting until solvent evaporation, and measuring the correspondent frequency [1,2]. The procedure ensured that the resulting ®lms coated a sensor area A of (2:37  10 5  5%) m2. It should be noted that the quoted uncertainty on the covered area A does not merely arise from a dimensional issue, but rather accounts for an estimation of the effect due the lateral ®niteness of the real resonator. This causes the vibration amplitude to decrease toward the PZT boundary, and as opposed to what predicted from the simpli®ed monodimensional model, ultimately produces a radial dependence of the mass sensitivity which is quantitatively unknown. As the adopted deposition method provides control of the overall loading mass only, and not of the surface mass density, the

Fig. 10. Measured frequency shift vs. mass load for RPL sensors with different PZT thickness.

integral effect over the coated area is what is actually observed therefore limiting the accuracy that can be attributed to the effective value of A. Assuming for the density and wave velocity of the coating ®lms the approximated values of 1  103 kg/m3 and 1  103 m/s, respectively, according to Eq. (12) a load of up to 100 mg can be considered acoustically thin at 7 MHz within an error of <2%, and the lumped-parameter model of Fig. 5 can be therefore applied. Similarly, for the same loading range, the linearized form of the mass sensitivity given in Eq. (18) can be used to predict the loaded resonant frequency with negligible errors. Fig. 10 shows the results obtained by measuring fp with the impedance analyzer for three different sensors at varying the mass load. The trends are linear in good agreement with the predictions of Eq. (18). The sensitivity is comparable to that of QCM sensors at the same frequency [12]. The data are ordered as expected with respect to the PZT thickness lP, as demonstrated by Fig. 11 which plots the inverse of the fractional mass sensitivity Sm as a function of lP, and compares the measured and the theoretically predicted values. The error bars on the theoretical values account for the estimated uncertainty in the PZT thickness (2.5 mm on the X-axis), and in the coated area A (5% on the Y-axis). Within such intervals the agreement is met. The fact that the measured sensitivity is lower than predicted can

Fig. 11. Comparison between the measured (*) and the theoretically predicted (*) values of the inverse of the fractional mass sensitivity versus the PZT thickness.

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An analysis of acoustic-wave mass sensors made by thickness-mode composite resonators of PZT layers on non-piezoelectric substrate has been developed. For sensors in which both PZT and substrate operate close to their quarter-wave resonance, a lumped-element model valid around the fundamental resonant frequency has been derived. The theoretical predictions of resonant frequency and mass sensitivity approximate well the measurement results for sensors made by PZT thick ®lms on alumina, and are expected to be extendible to sensors made in different technologies. The proposed model can be readily used in the design of gravimetric chemical sensors based on RPLs, and to derive from resonance measurements the mechanical and piezoelectric parameters of substrate-supported piezoelectric ®lms.

[3] H. Nowotny, E. Benes, General one-dimensional treatment of the layered piezoelectric resonator with two electrodes, J. Acou. Soc. Am. 82 (2) (1987) 513±521. [4] Z. Wang, D.N. Cheeke, C.K. Jen, Perturbation method fo analyzing mass sensitivity of planar multilayer acoustic sensors, IEEE Trans. UFFC 43 (5) (1996) 844±851. [5] V. Edwards Granstaff, S.J. Martin, Characterization of a thicknessshear mode quartz resonator with multiple nonpiezoelectric layers, J. Appl. Phys. 75 (3) (1994) 1319±1329. [6] K.M. Lakin, G.R. Kline, K.T. McCarron, High-Q microwave acoustic resonators and filters, IEEE Trans. Microwave Theory Tech. 41 (12) (1993) 2139±2146. [7] T. Ikeda, Fundamentals of Piezoelectricity, Oxford University Press, Oxford, 1989. [8] W.P. Mason, Electromechanical Transducers and Wave Filters, Van Nostrand, New York, 1948. [9] R. Lucklum, P. Hauptmann, Mass sensitivity, viscoelasticity and acoustic amplification of acoustic-wave microsensors, in: Proceedings of the Eurosensors XIII, The Hague, 12±15 September 1999, pp. 453±456. [10] R.W. Cernosek, S.J. Martin, A.R. Hillman, H.L. Bandey, Comparison of lumped-element and transmission-line models for thickness-shearmode quartz resonator sensors, IEEE Trans. UFFC 45 (5) (1998) 1399±1407. [11] V. Ferrari, D. Marioli, A. Taroni, E. Ranucci, P. Ferruti, Gravimetric chemical sensors in thick film technology with hybrid electronics, in: Proceedings of SAA'96 Ð National Meeting on Sensors for Advanced Applications, Vol. 54, Brescia, 16±17 May 1996, SIF, Bologna, 1997, pp. 3±10. [12] D.S. Ballantine, R.M. White, S.J. Martin, A.J. Ricco, E.T. Zellers, G.C. Fryre, H. Wohltjen, Acoustic Wave Sensors, Academic press, San Diego, 1997. [13] Z. Wang, J.D.N. Cheeke, C.K. Chen, Unified approach to analyse mass sensitivities of acoustic gravimetric sensors, Electr. Lett. 26 (18) (1990) 1511±1513. [14] G. De Cicco, B. Morten, M. Prudenziati, Piezoelectric thick-film sensors, in: M. Prudenziati (Ed.), Thick Film Sensors, Elsevier, Amsterdam, 1994, pp. 209±228. [15] IEEE Standard on Piezoelectricity, ANSI/IEEE Std 176±1987, Institute of Electrical and Electronic Engineers, New York, 1987. [16] J.W. Waanders, Piezoelectric Ceramics, Philips Components, Ehindhoven, 1991. [17] M. Onoe, H. Jumonji, Useful formulas for piezoelectric ceramic resonators and their application to measurements of parameters, J. Acou. Soc. Am. 41 (4) (1967) 974±980. [18] M. Brissaud, Characterization of piezoceramics, IEEE Trans. on UFFC 38 (6) (1991) 603±617.

Acknowledgements

Biographies

be partly due to the metal electrodes which increase the sensor effective mass. 4.4. Temperature effect The above experimental results were all obtained at 238C. It was shown previously [2] that the resonant frequency is dependent on temperature with a coef®cient that, for sensors with lA ˆ 254 mm and lP  100 mm, is in the order of 70 ppm/8C at room temperature. Further investigations are needed to assess how such a variation depends on the ratio lP/lA and, possibly, on the sensor average temperature itself. The present model, according to Eq. (15), is expected to be helpful in quantifying how the contributions of the distinct parameters of PZT and substrate concur to the overall temperature coef®cient of frequency, and, as a consequence, it may possibly suggest a proper choice of lA and lP in order to reduce such a thermal dependence. 5. Conclusions

The contribution by P. Ghignatti and D. Lauri in the laboratory activity is greatly acknowledged. The research project is co-funded by the Italian MURST (1999). References [1] V. Ferrari, D. Marioli, A. Taroni, Thick film resonant piezo-layers as new gravimetric sensors, Meas. Sci. Technol. 8 (1) (1997) 42±48. [2] V. Ferrari, D. Marioli, A. Taroni, E. Ranucci, Multisensor array of mass microbalances for chemical detection based on resonant piezolayers of screen-printed PZT, in: Proceedings of the Eurosensors XIII, The Hague, 12±15 September 1999, pp. 949±952, Sens. Actuators B68, (2000) 81±87.

Vittorio Ferrari was born in Milan, Italy, in 1962. In 1988, he obtained the Laurea degree in Physics at the University of Milan. In 1993, he received the Research Doctorate degree in Electronic Instrumentation from the University of Brescia, Italy. He is currently Researcher and Assistant Professor at the Faculty of Engineering of the University of Brescia, where he teaches Electronics. His research activity deals with electronic measuring instrumentation, sensors for physical and chemical quantities, and the related signal-conditioning electronics. In particular, he is presently involved with piezoelectric acoustic-wave sensors in thick-film technology, design of oscillator circuits and frequency-output signal conditioners. Daniele Marioli was born in Brescia, Italy, in 1946. He obtained the Electrical Engineering degree in 1969. From 1984 to 1989 he was an Associate Professor in Applied Electronics and since 1989 he has been a Full Professor of Electronics at the University of Brescia. His main field of

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activity is the design and experimentation of analog electronic circuits for the processing of electrical signals from transducers, with particular regard to S/N ratio optimization. Andrea Taroni was born in 1942. He received the degree in Physical Science from the University of Bologna, Italy, in 1966. He was an

Associate Professor at the University of Modena from 1971 to 1986. Since 1986 he has been Full Professor of Electrical Measurements at the University of Brescia. He has done extensive research in the field of sensors for physical quantities and electronic instrumentation, both developing original devices and practical applications. He is author of more than 100 scientific papers.