Funding was provided in part by DEPSCoR grant # FA9950-07-1-0519, Petroleum Research Fund Grant ACS PRF# 42747-AC10, and Air Force Office AFO Grant # FA8650-05-D-5807.

Impact of High-Temperature Dielectric and Piezoelectric Behavior on LGT Acoustic Wave

Properties up to 900°C

Peter M. Davulis and Mauricio Pereira da Cunha Dept. of Electrical and Computer Engineering and Laboratory for Surface Science and Technology,

University of Maine, Orono, ME, 04469, U.S.A. mdacunha@eece.maine.edu

Abstract— The langatate (LGT) elastic, dielectric and piezoelectric constants with respective temperature coefficients up to 900°C are reported for the first time. The new set of constants was used to improve the predictions of high-temperature LGT surface acoustic wave (SAW) properties such as phase velocity (vp), temperature coefficient of delay (TCD), and electromechanical coupling (K2) along multiple orientation sweeps up to 900°C. These predictions were then compared to previous calculations, which ignored the temperature dependence of the dielectric and piezoelectric constants, and to measured data up to 900°C, obtained from SAW delay lines fabricated along 6 orientations in the LGT plane (90°, 23°, Ψ). The average discrepancy between predicted and measured vp and TCD responses between 25 and 900°C were reduced by a factor of 4 for vp and 13% for TCD when the temperature dependence of both dielectric and piezoelectric constants are considered. The extracted LGT piezoelectric constants and temperature coefficients show that e11 and e14 change by up to 62% and 77%, respectively, for the entire 25°C to 900°C range when compared to room temperature values. In addition, this paper uncovers the full set of high-temperature LGT elastic, piezoelectric, and dielectric constants and temperature coefficients applicable up to 900°C, including the respective estimated uncertainty.

I. INTRODUCTION There is growing need for frequency control devices and

wireless sensors that operate at very high temperatures, e.g. from 400 to 1000°C, in the aerospace, energy generation, material processing, and oil extraction industries [#1], [#2]. Langasite (LGS) and langatate (LGT) surface acoustic wave (SAW) devices have demonstrated wireless operation above 900°C [#3], [#4]. The high-temperature acoustic wave constants and temperature coefficients are needed for proper acoustic wave (AW) modeling and device design. In particular, the dielectric and piezoelectric temperature dependences must be considered for more accurate identification of turnover temperatures and orientations for high-temperature devices. Previous work by the authors on LGT found that when the dielectric and piezoelectric effects are treated as not changing with temperature there is increased uncertainty in extracting the elastic constants and increased compounded error in determining AW properties, such as velocity, and their temperature dependence [#5], [#6].

In this work, the LGT dielectric and piezoelectric constants are extracted as functions of temperature from room

temperature (RT) up to 900°C. The full set of LGT elastic, dielectric and piezoelectric constants with respective temperature coefficients up to 900°C are reported for the first time. High-temperature capacitor and resonance ultrasound spectroscopy (RUS) measurements were used to extract the LGT dielectric, ε(T), and piezoelectric, e(T), temperature dependence and a revised set of elastic constants considering the ε(T) and e(T) behavior. High-temperature LGT SAW properties, such as phase velocity (vp), temperature coefficient of delay (TCD), and electromechanical coupling (K2), were determined using the new set of constants. Surface acoustic wave delay lines were fabricated along 6 orientations in the LGT plane (90°, 23°, Ψ) and tested data up to 900°C to demonstrate the importance of the dielectric and piezoelectric temperature effect and to validate the constants. The resulting comparisons between predictions and measured SAW propagation properties used to validate the constants, show significant improvement with respect to previous calculations that ignored the temperature dependence of the dielectric and piezoelectric constants, as detailed in this paper.

Section II discusses the LGT dielectric permittivity measurements at high temperature. Section III describes the methodology used to extract the elastic and piezoelectric constants. The extracted elastic and piezoelectric constants and temperature coefficients are discussed in Section IV. Section V reports on the SAW measurements used to validate the high-temperature constants and discusses the impact of the dielectric and piezoelectric temperature dependence. Section V concludes the paper.

II. LGT DIELECTRIC PROPERTIES UP TO 900°C The LGT dielectric properties at high temperatures were

measured from 25 to 925°C including the effect of losses [#7]. The losses are included in the anaylis by treating the dielectric permittivity as a complex number, ε = ε′ – jε″, where ε′ accounts for the energy storage effect and ε″ describes the polarization losses and by using the electrical conductivity, σ. The losses that originate from σ and ε″ are treated independently, since they have diverse physical mechanisms and different temperature and frequency dependences.

Parallel-plate capacitors were fabricated using high-temperature Pt/Rh/ZrO2 electrodes [#2] on LGT plates oriented along the X- and Y-axes, yielding ε11 and σ11, and along the Z-axis, yielding ε33 and σ33. The electric field

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fringing effect was accounted for by measuring multiple capacitors with circular electrodes of increasing diameters [#7], [#8]. The real and imaginary components of the capacitve admittances, measured from 10 Hz to 1 MHz, were used along with their frequency dependences to extract the ε′, ε″, and σ at each temperature, thus separating the effects of dielectric losses and conductivity from the real permittivity. Table I shows meaured LGT permittivity constants up to 900°C, and updates the results from [#7] by including Y-axis plate measurements.

The capacitance measurements are made below the acoustic resonance of the wafers and yield the dielectric permittivities measured under constant-stress, εT

11 and εT33

according to [#9]. The dielectric permittivities under constant-strain, εS

11 and εS33, are calculated by

εSij = εT

ij – eip sEpq eqj, where eip is the piezoelectric constant

tensor, eqj is the transposed piezoelectric tensor and sEpq is the

elastic compliance tensor under constant electric field. Note that for class 32 crystals εS

33 = εT33.

III. MATERIAL PARAMETER EXTRACTION USING RUS The LGT elastic and piezoelectric parameters are

determined from RT to 900°C using RUS. Six LGT parallelepiped samples were X-ray aligned, cut, ground, and polished at the University of Maine (UMaine) using LGT boules purchased from Fomos (Fomos-Materials, Moscow, Russia) [#5], [#6]. Two different orientations were fabricated with nominal dimensions of (2.9 mm, 12.6 mm, 18.6 mm) with the dimensions along the XYZ crystalline axes, and the other with dimensions along the XZY axes. The RUS measurements were performed both at the UMaine and the High Temperature Materials Laboratory (HTML) of the Oak Ridge National Laboratories [#5], [#6]. Similar Quasar RU Spec (Quasar, Albuquerque, NM) measurement setups were used in both locations to obtain the resonance spectra and custom furnaces were used to heat the samples.

The predicted resonance frequencies were calculated based on Lagrangian minimization, using Legendre polynomials as the basis functions to approximate the displacements and electric potential [#10], [#11]. A non-linear least-squares fitting in MATLAB was used to find the elastic and piezoelectric constants that minimize the difference norm between the measured and calculated frequencies. The uncertainty of the extracted parameters is determined from the inverse of the curvature of the minimized function at the solution and is larger for constants that have less effect on the resonance spectra and thus the cost function [#10]. The

thermal expansion coefficients used in this work were extracted up to 1200°C [#12].

The dielectric permittivity is required for the extraction of piezoelectric constants from RUS [#11], thus the LGT dielectric permittivity results up to 900°C [#7] allow for the determination LGT piezoelectric properties at high temperature. While all ε′, ε″, and σ were extracted, as discussed in Section II, the RUS technique implimented does not include losses, and thus only the real permittivity, ε′, was utilized. The capacitor measurements yielded εT

ij but the RUS calculations require εS

ij, which has to be calculated as indicated in Section II at each step of the fitting process from the measured εT

ij and the trial elastic and piezoelectric constants.

IV. LGT HIGH TEMPERATURE ELASTIC AND PIEZOELECTRIC CONSTANTS RESULTS

A new set of high-temperature (up to 900°C) LGT elastic and piezoelectric parameters were extracted using RUS technique described in Section III. The uncertainty in the extracted elastic constants is lowest for c33, c44, and c66 which have less than 1% relative uncertainty for all the measurement points. The elastic constants c11, c13, and c14 have at worst 1.9%, 2.9%, and 3% uncertainty at 900°C. At room temperature all the elastic constants have uncertainty of 1.5% or better. At room temperature the relative uncertainty of e11 and e14 are 6.8% and 19.1%, respectively, reflecting their reduced effect on the resonance spectra compared to the elastic constants.

The room temperature values of the elastic and piezoelectric constants are listed in Table II along with their respective temperature coefficients up to 900°C. A second-order polynomial is fit to the elastic and piezoelectric data, separately extracted from each crystal sample and measurement temperature. The extracted RUS constants in the temperature range 600°C to 800°C were found to have increased uncertainty and variation in the constants between temperatures and were excluded from the best-fit. At these

TABLE I. LGT REAL DIELECTRIC PERMITTIVITIES AND TEMPERATURE COEFFICIENTS UNDER CONSTANT-STRESS, ε′R_IJ

T

Temp. Range [°C]

Tref [°C]

ε′R_Tref [--]

TC1 [10-3 °C-1]

TC2 [10-6 °C-2]

TC3 [10-9 °C-3]

ε′R_11 25 to 500

25 19.62 ± 0.43

0.121 ± 0.089

-1.380 ± 0.3.15

4.298 ± 0.441

500 to 900

500 23.67 ± 1.94

1.440 ± 0.127

-6.385 ± 0.504

7.728 ± 0.825

ε′R_33 25 to 900

25 80.44 ± 0.97

-1.574 ± 0.045

2.091 ± 0.075

-1.066 ± 0.046

TABLE II. LGT ELASTIC AND PIEZOELECTRIC CONSTANTS AND TEMPERATURE COEFFICIENTS FROM 25°C TO 900°C

Elastic constants at 25°C [109 Pa]

TC1 [10-6 °C-1]

TC2 [10-9 °C-2]

C11 192.1 ± 0.7 -91.2 ± 21.7 -25.2 ± 23.4 C12 111.1 ± 0.7 a -- -- C13 102.9 ± 0.5 -119.7 ± 29.6 8.9 ± 32.0 C14 13.81 ± 0.02 -243.4 ± 10.3 0.9 ± 11.9 C33 264.1 ± 0.5 -106.6 ± 10.0 -20.7 ± 10.9 C44 50.99 ± 0.02 3.4 ± 2.3 -70.8 ± 2.7 C66 40.51 ± 0.02 5.5 ± 2.9 -73.7 ± 3.1

Piezoelectric constants at 25°C [C/m2]

e11 -0.377 ± 0.023 1339 ± 392 -2341 ± 500 e14 0.165 ± 0.009 -1741 ± 330 2991 ± 375

a C12(T) is calculated from C12 = C11 − 2C66

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temperatures resonance modes measured for RUS have significantly weakened responses, causing many modes vanish below noise level and others to be measured with less accuracy. As the temperature increases so does the magnitude of the resonance peaks, improving the RUS fitting above 800°C. The fitting correlation coefficient for the elastic constants is lowest for c13, R2 = 0.91, and c11, R2 = 0.95, whereas the other elastic constants were fit with R2 greater than 0.99. The piezoelectric constants, e11 and e14, were fit with R2 values of 0.65 and 0.88, respectively; reflecting the larger variation in these constants with temperature.

The elastic constants determined in this work considering ε(T) and e(T) are compared in Figs. 1a and 1b to those from [#6], which were determined from the temperature coefficients of the piezoelectrically-stiffened elastic constants at elevated temperatures, offset by the difference of the elastic constants extracted at room temperature with piezoelectricity. As can be seen from Fig. 1, there is a significant reduction in the uncertainty for the unstiffened elastic constants when compared to the uncertainty of the piezoelectrically-stiffened elastic constants from [#6], also included in the plots. In particular, the difference between the stiffened (corrected at RT) and unstiffened c11 starts at 0.4% 25°C and increases to 2.2% around 400°C, indicating the relevance to consider ε(T) and e(T) in the extraction of the elastics constants.

The extracted LGT piezoelectric parameters are shown in Fig. 1c up to 900°C. It is found that the magnitude of e11 increases by up to 20% of its room temperature value before starting to decrease with temperature and is reduced by 62% at 900°C. On the other hand, e14 follows and opposite trend, first decreasing with temperature and ultimately increasing by 77% of the room temperature value at 900°.

V. HIGH TEMPERATURE SAW BEHAVIOR VERIFICATION The SAW high-temperature frequency response and TCD

were calculated in this work were compared to those of 6 SAW delay lines fabricated along LGT orientations (90°, 23°, Ψ) with Ψ = 0, 13, 48, 77, 119, and 123°. These devices have also been used in [#8] for the verification of the high-temperature stiffened elastic constants and temperature coefficients [#6]. The fabricated interdigital transducers have periodicity given by wavelength λ = 32 μm and utilized 110 nm thick Pt/Rh/ZrO2 electrodes [#2] to enable stable high-temperature operation above 900°C. The delay lines were heated in a box furnace from 25 to 900°C in 50°C steps. The S21 (transmission) frequency response was measured using an Agilent 8753ES network analyzer (Agilent Technologies, Santa Clara, CA). When necessary, the frequency response was time gated using the inverse Fourier transform to reduce the effect of electromagnetic feed through, SAW triple transit, and other acoustic spurious modes.

The calculated and measured SAW free-surface phase velocities for the plane (90°, 23°, Ψ) are compared in Fig. 2 at 400°C for the 6 orientations identified in the previous paragraph. The calculated vp are within 0.2% of the measured values at 400°C and within 1.0% at 900°C for the 6

(a)

(b)

(c)

Fig. 1. LGT elastic stiffness constants (a) C33, C11, and C13; (b) C44, C66, and C14 and (c) elastic constants e11 and e14. Black solid line: best-fit with ε(T) & e(T). Gray solid line: best-fit with ε(no T) & e(no T) [#6]. Black dashed line: best-fit with ε(T) & e(T). Gray dashed line: best-fit with ε(no T) & e(no T) [#6]. Black circles: extracted data points ε(T) & e(T). Dark gray crosses: data points excluded from the fitting.

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Fig. 2. LGT SAW vp along orientations (90°, 23°, Ψ). Black circle: measured vp for Ψ = 0, 13, 48, 77, 119, 123°. Dark gray, solid curve: vp with ε(T) and e(T) from Table II, including data points (+) at the measured Ψ with errorbars. Light gray, dashed: vp with ε(no T) and e(no T) including data points (□) at the measured Ψ with errorbars [#6].

Fig. 3. LGT SAW coupling, K2, along orientations (90°, 23°, Ψ) for temperatures 25, 400, 600, 900°C, respectively the solid-black, dashed-

gray, dashed-black, solid-gray curves.

experimentally investigated orientations. For all orientations the measured and predicted SAW operation frequencies and velocities agree within 1.0% of the measured quantity in the worst case and 0.2% on average over the temperature range of 25 to 900°C. The calculated TCD is on average within 9.6 ppm/°C of the measured value for the six tested orientations over the entire investigation temperature range. The new set of predictions made considering ε(T) and e(T) reduced the average discrepancy with respect to phase velocity measurements by 77% and for the TCD by 13%, when compared to the results in [#6].

Fig. 3 shows the SAW coupling, K2, calculated up to 900°C for the plane (90°, 23°, Ψ). As can be seen, K2 rapidly decreases with temperature above 400°C.

The impact of the ε(T) and e(T) (Table II) is investigated by comparing SAW predicted turn-over temperatures with the predictions from [#6], where the temperature variation of ε and e were neglected. Four of the fabricated orientations, Ψ = 0, 13, 48, and 77° have measured and predicted SAW turn over temperatures in the studied 25 to 900°C temperature range, more specifically TCD=0 between 280 and 413°C for

these four orientations. Neglecting the temperature variation of ε and e, the predicted turn-over temperatures were on average 48°C, or 17%, less than the measured turn-over temperature. Considering ε(T) and e(T) reduced the average error to 18°C or 5% above the measured turn-over temperature, a 63% improvement.

VI. CONCLUSIONS A new set of high-temperature (up to 900°C) LGT acoustic

wave constants extracted by RUS technique is given, which considers the temperature dependence of both dielectric and piezoelectric constants. The use of these new constants and temperature coefficients resulted in the reduction of the average discrepancies between predictions and measurements of SAW propagation properties at high temperature: for phase velocity 77%; for TCD by 13%, and for the turn-over temperature an improvement of 63%. The reported results verify the accuracy of the determined constants by comparison with SAW measured properties and the importance of including the dielectric and piezoelectric temperature behavior for accurate acoustic wave property predictions and device design at high temperatures.

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[3] R. Fachberger, E. Riha, E. Born, and P. Pongratz, “Homogeneity of langasite and langatate wafers for acoustic wave applications,” Proc. 2003 IEEE Int’l Ultrason. Symp., pp. 100-109.

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2077 2011 IEEE International Ultrasonics Symposium Proceedings