Jochen Bardong

Thierry Aubert

Natalya Naumenko

Gudrun Bruckner

Sabine Salzmann

Leonhard M Reindl

Experimental and Theoretical Investigations of some Useful Langasite Cuts for High-Temperature SAW Applications

Keywords: high temperature, SAW, SH-SAW, langasite, sensor I. !

Passive high-temperature sensors are a most promising area of use for surface acoustic wave (SAW) devices. Langasite (La3Ga5SiO14, LGS) has been identified as promising piezoelectric material to meet high-temperature SAW challenges. As it is necessary to know the material behavior for an accurate device design, the frequency-temperature behavior of Rayleigh SAW (R-SAW) and shear horizontal SAW (SH-SAW) LGS cuts is investigated on delay line and resonator test structures up to 700°C by RF characterization. In the range of the 434 MHz-ISM band, the (0°, 22°, 90°) SH-SAW cut shows similar thermal behavior as the (0°, 138.5°, 26.7°) R-SAW cut. Associated with the (0°, 22°, 31°) cut, in which SAW present mixed type of polarization, the (0°, 22°, 90°) SH-SAW orientation might allow differential measurements on one single substrate. In the temperature range of 400 to 500°C, delay line test devices using the SH-SAW cut show a considerable drop of signal. Theoretical analysis evidences that this newly described behavior might be a result of anisotropy effects in this cut, occurring in case of any slight misorientation of electrode alignment.

INTRODUCTION

As surface acoustic wave (SAW) devices are very sensitive to external conditions like temperature or pressure, while being passive, they offer the possibility to realize wireless sensors that could operate in high-temperature conditions for long time periods [START_REF] Hornsteiner | Surface acoustic wave sensors for hightemperature applications[END_REF]. During the last decade, many efforts have been made to identify and characterize piezoelectric materials that could be used in such conditions. Indeed, conventional SAW substrates, such as quartz, lithium niobate or lithium tantalate, are all limited to temperatures below 500°C because of intrinsic physical or chemical transformations [START_REF] Damjanovic | Materials for high temperature piezoelectric transducers[END_REF]. Up to now, the most reliable results concerning device stability and reproducibility have been obtained with langasite (La3Ga5SiO14, abbreviate as LGS). Thus, a SAW signal was recently recorded up to 1140°C by Aubert et al. [START_REF] Aubert | Iridium interdigital transducers for high-temperature surface acoustic wave applications[END_REF], while Pereira da Cunha et al. have successfully operated LGS-based SAW devices for several months at 800°C [START_REF] Peireira Da Cunha | High temperature stability of langasite surface acoustic wave devices[END_REF]. So, LGS is currently the best-suited piezoelectric material for high-temperature SAW applications taking into account availability of crystals and different orientations, too. Nowadays, a major task is to find the crystal cuts and orientations appropriate for specific purposes. Up to now, the prediction of cuts fitting dedicated needs is numerically inaccurate as the available LGS material tensor sets are incorrect above 250°C [START_REF] Aubert | Extraction of COM parameters on Pt/LGS for high temperature SAW sensor[END_REF][START_REF] Sakharov | Theoretical and experimental investigation of langasite as material for wireless high temperature SAW sensors[END_REF]. In order to improve existing data sets, the presented study examines experimentally the frequency-temperature laws of three different LGS cuts in the temperature regime between the ambient and 700°C. For this investigation, the following LGS cuts have been chosen, as they provide sufficiently high electromechanical coupling coefficient K 2 and zero power flow angle: the (0°, 138.5°, 26.7°) Rayleigh SAW (R-SAW) cut, which is the most extensively examined LGS cut up to now, the shear horizontal SAW (SH-SAW) cut with Euler angles (0°, 22°, 90°), and the (0°, 22°, 31°) cut which presents mixed type of polarization [START_REF] Sakharov | Theoretical and experimental investigation of langasite as material for wireless high temperature SAW sensors[END_REF][START_REF] Koskela | Surface transverse waves on langasite[END_REF][START_REF] Berkenpas | Pure shear horizontal SAW biosensor on langasite[END_REF][START_REF] Naumenko | Optimal cuts of langasite, La3Ga5SiO14 for SAW devices[END_REF]. All crystals were processed with delay line devices as well as with resonator structures using the same device mask. The pitch is 2.4 µm and 3.3 µm for the delay line and the resonator test structures respectively. A platinum based multilayer with a total thickness h of 120 nm was developed and applied to the crystals to enable the electrodes to withstand elevated temperatures of estimated 900°C.

In the second section, the SAW devices and experimental methods used to examine these substrates are described. During the third section, the experimental results are presented and discussed, whereas in the fourth section, theoretical investigations aiming at the explanation of several experimental observations on the (0°, 22°, 90°) SH-SAW cut are given. The conclusions are presented in the fifth section.

II.

EXPERIMENTAL METHODS

The investigations were done mainly by S-parameter measurements on delay lines. A description of the setup dedicated to in situ RF high-temperature measurements can be found in [START_REF] Bardong | Characterisation setup of SAW devices at high temperatures and ultra high frequencies[END_REF]. The delay lines examined during this study provide four operating frequencies ranging from 140 MHz to 530 MHz by combining two interdigital transducers (IDTs), called here IDT A and B, with different frequency splitting. This allows the use of identical pitches for both IDTs, which reduces the mask complexity. IDT A is designed to work at frequencies of 1/4, 3/4, 5/4 and 7/4, whereas IDT B shows its harmonic frequencies at 3/7, 5/7, 9/7 and 11/7 of the center frequency f0 (Fig. 1). This center frequency f0 corresponds to an imaginative standard IDT with used finger pitch of 2.4 µm and can be located at about 600 MHz, differing slightly from cut to cut according to different propagation velocities of the excited waves [START_REF] Engan | Surface acoustic wave multielectrode transducers[END_REF][START_REF] Campbell | Multi-band SAW filter design using the Remez exchange algorithm[END_REF].

Because of the high mass loading of R-and SH-SAW waves under a platinum based metallization, a large part of the energy of the highest harmonic frequencies, i.e. those above f0, is lost. This resulted in four reliable signals per test device, where each IDT contributed with two harmonics (Fig. 2): 1/4, 4/7, 3/4 and 6/7 of f0, corresponding respectively to relative thicknesses of 0.625%, 1.43%, 1,875% and 2.14%.

The frequency-temperature behavior of the examined LGS cuts was determined by heating the samples continually from room temperature to 900°C (temperature ramp: 100°C/h) in fine vacuum conditions (pressure = 2*10 -4 mbar). Due to this ramp slope, a measurement could be done with 1 Kelvin difference between start and end of a frequency sweep and it was steep enough to shift the device failure to temperatures above 850°C. With a ramp less steep, dewetting processes starting at temperatures above 700°C on Pt based layers, have more time to impact the metallization which leads to stronger signal disruption and device failure at temperatures below 850°C.

Concerning the (0°, 22°, 90°) SH-SAW cut, the measurement results were reproducible, but inconclusive. So, a counter-check using a one-port synchronous resonator structure, with a pitch of 3.3 µm, was added to clarify the behavior of this cut. These resonators were measured in the same way as the delay line test structures mentioned above.

III. MEASUREMENT RESULTS AND DISCUSSION

The two cuts (0°, 138.5°, 26.7°) and (0°, 22°, 31°) give reliable results until temperatures up to 700°C (Fig. 3 and4). At temperatures above this mark, the deviation from the fitting parabola curve is getting significant and is accompanied by an irreversible signal amplitude degradation, which increases with operating frequency. That can be explained by two main effects: one is the starting agglomeration and dewetting of the Pt electrodes from this temperature on [START_REF] Aubert | Behavior of platinum/tantalum as interdigital transducers for SAW devices in high-temperature environments[END_REF]. The other effect is an increasing surface roughness of LGS substrates at high temperatures under low-pressure environments [START_REF] Aubert | Iridium interdigital transducers for high-temperature surface acoustic wave applications[END_REF][START_REF] Bardong | Influence of packaging atmospheres on the durability of high-temperature SAW sensors[END_REF]. Waves with higher frequencies and therefore shorter wavelengths λ have a smaller surface penetration depth which makes them more sensitive to any change in surface and electrode roughness. The cavities described in [START_REF] Bardong | Influence of packaging atmospheres on the durability of high-temperature SAW sensors[END_REF] that can be observed all over the crystal surface appear during the heating process. They grow in the scale of several micrometers and work as diffraction centers for a passing SAW. The diffracting effect rises as the wavelength shrinks and its penetration depth reaches the cavities extent. Fig. 5(a)-(d) give an impression of the degradation effects.

The relative thickness h/λ of the electrodes also demonstrated an increasing influence on the frequency-temperature law with rising temperature. Related to the strengthening with temperature of the IDTs mass loading, this effect is understood and predictable by simulation methods, as it can impact the performance of SAW filters used in telecommunication devices considerably [START_REF] Pastureaud | Prediction of the thermal sensitivity of surface acoustic waves excited under a periodic grating of electrodes[END_REF]. In the case of SAW devices operating in high-temperature environments, it was observed that this phenomenon influences strongly the device behavior for some cuts. For instance, in case of the (0°, 22°, 31°) cut, the observed thermally induced frequency shift between room temperature and 600°C is smaller by 2500 ppm at 440 MHz (h/λ = 2.14%) compared to the observed shift at 130 MHz (h/λ = 0.62%) (Fig. 4). LGS (0°, 22°, 31°) 129 MHz, h/! = 0.625% 294 MHz, h/! = 1.43% 386 MHz, h/! = 1.875% 440 MHz, h/! = 2.14% Thus, to compare the temperature behavior of the different cuts, it is mandatory to use devices with operating frequencies as close as possible to each other and devices of equal design. Therefore, the formerly presented delay lines can be considered as a good possibility to do so.

Unfortunately the analysis of the signals deduced from delay lines on the (0°, 22°, 90°) SH-SAW cut remained inconclusive. In the 400-500°C regime, a clearly reproducible signal drop occurs, disabling the evaluation algorithm to track the signal amplitude, which causes a jump in the trend lines of Fig. 6.

Outside this temperature area, the signal amplitude was analyzable. However, the material degradation effects like dewetting and crystal decomposition on the SH-SAW cut were comparable to those observed on the two other cuts. Compared with the spectrogram of the (0°, 138.5°, 26.7°) R-SAW cut, many highly energized spurious modes can be detected. This indicates that with delay lines and IDTs using spatial sampling, numerous unwanted modes are excited in addition to the SH-SAW wave modes. Some of the observed parasitic modes can be identified as bulk acoustic wave modes (BAW), like those in the big circle in the upper left of Fig. 8. This group of BAW modes shows a characteristic shift in time and frequency from one energy maximum to the next leading to a stairway-like pattern. The BAW are emitted and generated inside the IDT, become reflected between top and bottom of the crystal and transverse the propagation path from the input IDT in a zigzagging manner between crystal surface and bottom towards the receiving IDT. The higher the number of zig-zag reflections, the longer the transmission time and the shorter the wavelength, as the latter is dependent on the angle θ between surface and propagation direction of the BAW (Fig. 9) [START_REF] Morgan | Surface-wave devices for signal processing[END_REF]. Other marked modes visible on Fig. 8 are more complex to interpret, which is beyond the scope of this text. Between 400 and 500°C, spectrograms corresponding to the (0°, 22°, 90°) SH-SAW cut confirm a considerable signal drop that occurs at all observed harmonic frequencies, with some differences in intensity (Fig. 10). The effect has a stronger impact on the designed IDT harmonics than for the spurious modes. The amplitude energy of several IDT harmonics decreases with rising temperature and drops to a comparable level like that of the neighboring parasitic modes at about 400°C. During the thermal translation from 400°C to 500°C, these parasitic modes grow in amplitude whereas that of some designed IDT harmonics falls below noise level. Above 500°C, the main harmonics recover to a signal level above that one of the spurious mode (Fig. 11). Compared to that effect, the spectrograms from devices on (0°, 138.5°, 26.7°) (Fig. 12) and (0°, 22°, 31°) cuts do not show any kind of mode conversion in this temperature regime. In addition, measurements were realized on synchronous resonator structures fabricated on the STW cut. Fig. 13 shows the evolution of the corresponding Bode plot (S 11 magnitude and phase) over the investigated temperature range. Strong insertion losses and phase distortion are caused by the impossibility of calibration. However, it appears that contrariwise to the delay line case, the signal remains fully analyzable up to 500°C. Beyond this temperature, the magnitude decreases gradually, preventing an accurate determination of the evolution of the synchronous frequency (Fig. 13a). Thus, the frequency-temperature behavior of the device was extracted via phase tracking measurements (Fig 13b ), which allows to obtain reproducible and reliable results, removing uncertainty about the behavior of the STW cut between 400 and 500°C (Fig. 14). Small discrepancies between these results and the data coming from Sakharov et al. [START_REF] Sakharov | Characterization of langasite for application in high temperature SAW sensors[END_REF] can be observed. Their origin can be likely attributed to the difference At higher temperatures (580°C and above), the signals of the latter cuts decrease slightly but signal analysis is still possible (Fig. 11), whereas in the STW cut case, the signal artifacts decrease faster in At temperatures above 580°C, the signals of the (0°, 138.5°, 26.7°) and the (0°, 22°, 31°) cuts decrease gradually, according to expected propagation losses, and signal analysis remains possible over the whole temperature sweep (Fig. 13). Despite these new information provided by the use of the spectrograms, the results about the (0°, 22°, 90°) SH-SAW cut were still not easily understandable.

The inconclusive results from the delay line signal analysis on the (0°, 22°, 90°) SH-SAW cut were then completed by additional measurements using synchronous resonator test structures. Fig. 14 shows the evolution of the corresponding Bode plot (S11 magnitude and phase) over the investigated temperature range. Due to the lack of calibration possibilities and total cable lengths of 3 meters leading to strong insertion losses and signal phase distortion effects, a considerable part of the information was lost. However, it was observed that, in contrast to delay line measurements, the resonator signal is stable over the temperature regime between 400 and 500°C. Above this temperature, the signal magnitude decreases gradually, which prevents an accurate determination of the evolution of the synchronous frequency (Fig. 14(a)). Thus, the frequency-temperature behavior of devices was extracted via phase tracking measurements (Fig. 14(b)). Reproducible results were then obtained, removing the uncertainty about the thermal behavior of the (0°, 22°, 90°) SH-SAW cut between 400 and 500°C (Fig. 15). The results are consistent with data from Sakharov et al. [START_REF] Sakharov | Characterization of langasite for application in high temperature SAW sensors[END_REF]. At higher temperatures (580°C and above), the signals of the latter cuts decrease slightly but signal analysis is still possible (Fig. 11), whereas in the STW cut case, the signal artifacts decrease faster in intensity than the main lobes, the latter even recover to some signal level, which allows to track them more efficiently (Fig. 12). Still, the results concerning the STW cut were not satisfying and hard to interpret. A theoretical approach, based on a recent paper [START_REF] Morgan | Surface-wave devices for signal processing[END_REF], considering interactions between STW and spurious SAW modes resulting from slight and unwilled misorientation of IDTs will be discussed in the next section of this paper. LGS cut, and those taken from [START_REF] Sakharov | Characterization of langasite for application in high temperature SAW sensors[END_REF].

With the data derived from the additional resonator measurements on the (0°, 22°, 90°) SH-SAW cut, the frequency-temperature behaviors of all examined cuts at frequencies close by the 434MHz ISMband were compared (Fig. 16). Therefore some prospective applications may be suggested. Thus, the (0°, 22°, 90°) SH-SAW cut and the (0°, 138.5°, 26.7°) cut are characterized by similar temperature Res 45 Res 50 Res 57 Exp [START_REF] Sakharov | Characterization of langasite for application in high temperature SAW sensors[END_REF] coefficient of frequency (TCF) value trends, which are close to zero at room temperature. They can subsequently be considered both for applications requiring thermal compensation (e.g. non-temperature sensing) at ambient temperature assuming an appropriate choice of metal and thicknesses for the electrodes so that the TCF reaches effectively 0 ppm/°C [START_REF] Naumenko | Optimal cuts of langasite, La3Ga5SiO14 for SAW devices[END_REF]. In addition, providing a pure transversal wave mode, the (0°, 22°, 90°) SH-SAW cut appears to be ideal for liquid sensing [START_REF] Berkenpas | Pure shear horizontal SAW biosensor on langasite[END_REF]. Considering temperature sensing applications, the (0°, 22°, Ψ) cut seems to be a top candidate from ambient temperatures to 500°C, as in this temperature range, the orientations Ψ = 31° and Ψ = 90° have different temperature behaviors and could therefore be combined on one die to create a pair of sensors [START_REF] Sakharov | Theoretical and experimental investigation of langasite as material for wireless high temperature SAW sensors[END_REF]. This allows differential measurement methods, required in resonator configurations as the RF link itself may notably affect the resonance frequency, with a minimized-size device [START_REF] Ballandras | Micro-machined, all quartz-package, passive wireless pressure and temperature sensor[END_REF].

IV. THEORETICAL INVESTIGATION OF THE (0°, 22°, 90°) SH-SAW LGS CUT

The experimentally observed unexpected features in the behavior of SH-SAW in the LGS cut (0°, 22°, 90°), described in the previous section, cannot be predicted with known models of SAW propagation in a substrate with IDTs or periodic gratings. Some of the spurious responses, which degrade the performance of experimental delay lines, can be referred to shear horizontal (SH) polarized BAW, which propagates with a velocity slightly higher than that of SH-SAW. However, a more complete understanding of the observed phenomena is possible if the effect of anisotropy on the wave propagation in this cut is taken into account [START_REF] Naumenko | Effect of anisotropy on characteristics and behavior of shear horizontal SAWs in resonators using langasite[END_REF]. This cut belongs to the symmetric orientations described by the Euler angles (0°, θ, 90°), in which two degenerate SAW solutions can exist. One is the piezoelectrically coupled SH-SAW (also known as Bleustein-Gulyaev wave or surface transverse wave (STW) when it propagates in a periodic grating), with SH polarization along X-axis of LGS. The second one is the uncoupled Rayleigh wave polarized in the sagittal plane, i.e.

YZ plane of LGS. The velocities of those two modes are generally different.

Any deviation from a symmetric cut, e.g. misorientation caused by technological errors, eliminates degeneracy of SAW solutions and results in a transformation of each mode into the generalized SAW or leaky wave with mixed type of polarization. The electromechanical coupling of the Rayleigh-type mode slowly grows with such a deviation. When misorientation does not exceed 0.5° and the velocities of both modes are substantially different, the influence of the Rayleigh-type SAW on the device performance is negligible. A specific feature of the LGS cut (0°, 22°, 90°) is that both modes propagate here with nearly the same velocity. Therefore, even a small deviation from symmetric orientation is followed by a strong interaction between the modes, which affects the device performance. Fig. 17 shows the velocities of SH-type and Rayleigh-type SAWs arising nearly from the same velocity point when the propagation direction deviates by the angle ψ from the YZ plane towards the Xaxis of LGS. These and further calculations were made with LGS constants reported in [START_REF] Bungo | Analysis of surface acoustic wave properties of the rotated Y-cut langasite substrate[END_REF]. In Fig. 17, metalized surface is considered, which is equivalent to short-circuit electrical boundary conditions on the surface. If ψ=0, the velocities of both SAWs are slightly smaller than VBAW, i.e. the velocity of the bulk acoustic wave propagating along the surface with a wave vector generally tilted into the bulk [START_REF] Naumenko | Effect of anisotropy on characteristics and behavior of shear horizontal SAWs in resonators using langasite[END_REF]. With increasing ψ, SH-SAW holds its quasi-bulk nature. Rayleigh-type SAW transforms into leaky SAW when ψ>0.5°, and its attenuation coefficient grows fast with ψ. With the deposition of a metal film or a periodic metal grating of finite thickness, the velocities of both modes decrease, but the difference between them stays small as long as ψ<0.5°. For example, Fig. 18 shows the variation of effective velocities of both SAW modes in orientation (0°, 22°, 90.1°) with periodic metal grating, as functions of the normalized electrode thickness h'=hel/2p. Platinum electrodes (hPt=100 nm) with titanium sublayer (hTi=10 nm) and a variable pitch of the electrode structures, p=3…10 µm, have been considered. These parameters are consistent with the experimental devices. The effective velocities were calculated in short-circuit (SC) and open-circuit (OC) gratings as VR=2pfR and VA=2pfA, respectively, where fR and fA are resonant and anti-resonant frequencies extracted from numerically derived admittance functions Y(f). The velocities VR and VA refer to SH-type SAW propagating in a periodic grating. Rayleigh-type mode is also trapped by the grating and builds its own reflection stopband. The velocities V1=2pf1 and V2=2pf2 refer to the edges of this stopband, f1 and f2, which are nearly the same for SC and OC electrical conditions, because of the low piezoelectric coupling of the Rayleigh-type SAW. As the influence of electrode thickness is smaller on Rayleigh-type than on SH-SAW, f1(h') and f2(h') dependences cross fR(h') and fA(h') when the normalized electrode thickness increases. Each intersection means that the two SAW modes propagate with equal velocities and the interaction between them grows dramatically. Velocity (m/s) Fig. 17 LGS (0º, 22º, 90º+!) When the working temperature rises, the velocities VR, VA, V1 and V2 all decrease. SH-SAW is characterized by nearly zero TCF in the cut (0°, 22°, 90°) whereas the spurious mode shows stronger dependence on temperature. As a result, the intersections between the velocities of the two SAW modes occur at temperatures which depend on the electrode pitch. For instance, if p=3.2 µm (which corresponds to f ≈ 440 MHz, close to the ISM-band), the admittance looks nearly perfect when T=25°C [Fig. 19(d)].

Angle ! (degrees) " R-SAW V R-SAW V BAW V SH-SAW (a) (b) (c) (d) V 2 V 1 V A V R ! R-SAW
The interaction between the modes occurs at higher temperatures, when fR(T) and fA(T) are crossed by f1(T), as shown in Fig. 20, where misorientation ψ=0.1° is assumed. Fig. 21 illustrates how the lower edge of the Rayleigh-type SAW stopband f1 moves with respect to fR and fA when temperature increases from 25° to 650°C. When T=250°C, then f1≈fA and anti-resonance looks split because of the strong interaction between the modes. At higher temperatures, f1 is confined between fR and fA. A maximum effect of the parasitic mode on the resonant frequency is expected at

(c) (d) V 2 V 1 V A V R ! R-SAW
# R-SAW # R-SAW # R-SAW # R-SAW
Y(f)/Y(0) f 1 f A f R
T≈500°C. Thus, it can be assumed that the signal drop observed with delay lines experimentally between 400 and 500°C is related to this interaction between the SH-SAW and the spurious SAW mode. However, the fact that the phenomenon is largely more intense with these devices than with the investigated resonator structures remains currently unexplained. It should be mentioned that though theoretical analysis of the spurious mode was performed at fixed deviation of propagation direction from symmetric orientation, ψ=0.1°, a further increase of angle ψ up to 0.5° does not give noticeable shift of the found intervals of pitches and temperatures, in which a maximum degradation of the device performance is expected. However, the piezoelectric coupling of the spurious mode increases with ψ.

The presence of a parasitic SAW, traveling with temperature, is experimentally observed on the admittance curves obtained from measurements performed on resonators (Fig. 22). At room temperature, the resonant and anti-resonant frequencies are located in the vicinity of 440 MHz and 442 MHz, respectively, whereas a weak spurious mode can be seen close to 445 MHz. This is in good agreement with the calculations (Fig. 20), as well as the fact that the parasitic SAW is more sensitive to temperature changes than the main mode. Consequently, the Rayleigh mode is located between resonance and antiresonance from about 250°C on, which is also in good agreement with the simulation. Both peaks superpose when temperature is near 440°C, whereas it was expected around 500°C according to the calculations. This discrepancy could be caused by the lack of an accurate set of physical LGS constants for temperatures above 250°C and the ignored temperature dependence of Pt constants. The multiple spurious modes visible at frequencies above the parasitic SAW mode are caused by transversal modes inside the resonator structure, which propagate faster than the main SAW mode [START_REF] Schwelb | Effect of anisotropy on waveguide modes in SAW resonators[END_REF].

V. CONCLUSION (0°, 22°, 90°) SH-SAW, (0°, 138.5°, 26.7°) R-SAW and (0°, 22°, 31°) langasite cuts have been experimentally investigated between room temperature and 900°C, via in situ electrical measurements.

Both (0°, 138.5°, 26.7°) and (0°, 22°, 31) cuts show reliable results up to 700°C. Operating frequencies measured beyond this point are not relevant because of Pt electrode degradation and crystal decomposition. In the case of the SH-SAW cut, an intense drop in amplitude of the observed harmonic frequencies between 400 and 500°C, corresponding with a coincidental rise in amplitude of adjacent spurious modes occurs, which disables peak tracking of the delay line signal in that temperature regime.

The complete frequency-temperature law of this cut was then deduced via measurements conducted on resonator structures. It appears that the thermal behavior of this cut is very close to that of the (0°, 138.5°, 26.7°) orientation, with, in particular, a quasi-null TCF at room temperature. Such a property coupled to the shear-horizontal nature of the SAW propagating along the (0°, 22°, 90°) direction makes this cut an ideal candidate for liquid sensing. Moreover, this cut is also very promising to make resonator-based wireless temperature sensors working in high-temperature environments [START_REF] Mayer | SAW parameters for langasite at high temperatures[END_REF]. Indeed, if coupled to the (0°, 22°, 31°) cut, which has a turnover temperature close to 300°C, it could allow the fabrication of a pair of sensors and thus differential measurements on one single die, which is mandatory in a potential resonator configuration. But it has to be kept into account that any slight IDT misorientation on the (0°, 22°, 90°) cut leads to excitation of a spurious Rayleigh SAW mode with a velocity very close to that of the SH-SAW main mode. Consequently, coupling between these two modes occurs at dedicated temperatures depending on the relative electrode thickness, which leads to a serious degradation of the device performance. To avoid this, electrode thicknesses of hPt/λ>2% shall be considered for implementation [START_REF] Naumenko | Effect of anisotropy on characteristics and behavior of shear horizontal SAWs in resonators using langasite[END_REF] though such thicknesses of Pt layers must be carefully examined considering massloading and wave damping effects. Calculations show that such phenomena should happen at temperatures close to 500°C for devices working in the vicinity of the 434 MHz-ISM band, which might explain the observed signal drop on delay lines in that temperature region.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Electrode pattern of the IDT with N/7 th harmonic splitting

Fig. 3 .Fig. 4 .

 34 Fig. 3. Experimental frequency-temperature laws of (0°, 138.5°, 26.7°) LGS cut

Fig. 5 (

 5 Fig. 5(a): Overview of a dewetted metal layer, 50x enhancement in optical microscope. The IDT structures are still recognizable as are the formerly closed metal layers of this delay line with metalized propagation path (noted as MET).

Fig. 5 (

 5 Fig. 5(b): Detail (500x enhancement) of Fig. 5 (a). As can be recognized, the IDT structure and the formerly closed metal layer have been separated into isolated droplets during the annealing process.

Fig. 5 (

 5 Fig. 5(c): Scanning electron micrograph with 20000x enhancement of an IDT finger and LGS surface in between. The surface shows several micro fissures, the finger beginning dewetting as can be recognized in agglomeration and deformation.

Fig. 5 (

 5 Fig. 5(d): Cross-sectional transmission electron micrograph with 40000x enhancement of an IDT finger and its adjacent LGS surrounding. The fissures from Fig. 5 (c) seem to be the top view of deeper cavities. The finger has a width of 1.2 µm so the left cavity is about 500 nm in depth. On top of the surface is a preparation layer necessary to perform the cross-sectional preparation.

Fig. 6 .

 6 Fig. 6. Experimental frequency-temperature laws of SH-SAW (0°, 22°, 90°) LGS cut

Fig. 7 .

 7 Fig. 7. Spectrogram of (0°, 138.5°, 26.7°) LGS cut at 28°C. Parasitic signals are marked by circles.

Fig. 6 .

 6 Fig. 6. Spectrogram of (0°, 138.5°, 26.7°C) LGS cut at 28°C. Parasitic signals are marked by circles.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Spectrogram of SH-SAW (0°, 22°, 90°) LGS cut at 28°C. Parasitic signals are marked by circles.

Fig. 6 .Fig. 7 .Fig. 10 .

 6710 Fig. 6. Spectrogram of (0°, 138.5°, 26.7°C) LGS cut at 28°C. Parasitic signals are marked by circles.

Fig. 12 .

 12 Fig. 12. Spectrogram of STW (0°, 22°, 90°) LGS cut at 580°C.

Fig. 10 .

 10 Fig. 10. Spectrogram of (0°, 138.5°, 26.7°C) LGS cut at 442°C.

Fig. 13 .

 13 Fig. 13. Spectrogram of (0°, 138.5°, 26.7°) LGS cut at 580°C.

Fig. 10 .

 10 Fig. 10. Spectrogram of (0°, 138.5°, 26.7°C) LGS cut at 442°C.

Fig. 11 .

 11 Fig. 11. Spectrogram of (0°, 138.5°, 26.7°C) LGS cut at 580°C.

Fig. 14 .

 14 Fig. 14. Evolution with temperature of S11 magnitude (a) and phase (b) of a typical SAW resonator built on SH-SAW (0°, 22°, 90°) LGS cut. Temperature is given by the color scale on the right. The phase plot is utilized to follow the frequency change with temperature via 0° tracking (see grey strip).

Fig. 15 .

 15 Fig. 15. Comparison between results obtained on 3 typical resonators built on the SH-SAW (0°, 22°, 90°) LGS cut,

Fig. 16 .

 16 Fig. 16. Experimental frequency-temperature laws of the 3 investigated LGS cuts close to the 434 MHz ISM-band.

Fig. 17 .

 17 Fig. 17. Velocities of SH-type (red line) and Rayleigh-type surface waves (blue line) and attenuation of Rayleigh-type leaky SAW (dashed line) in LGS orientations with Euler angles (0º, 22º, 90º+ψ), as functions of the angle ψ. The black line refers to SH-BAW velocity.

  Fig.16

Fig. 18 .

 18 Fig. 18. Effective SH-SAW velocities at resonant (VR) and anti-resonant (VA) frequencies, and velocities V1,2 indicating the edges of Rayleigh-type SAW stopband, ΔR-SAW, in Ti/Pt grating on LGS cut (0º, 22º, 90.1º), as functions of the normalized Pt thickness. The labels (a)-(d) refer to the periods of the grating analyzed in Fig. 19.

  Fig.17

Fig. 19 .

 19 Fig. 19. Admittance of infinite grating with Ti/Pt electrodes on LGS cut (0º, 22º,90.1º), hPt=100 nm, hTi=10 nm, and different pitches of electrode structure: (a)-(d) refer to the normalized thicknesses labeled in Fig. 18. Two parasitic resonances occur at the edges of the stopbands built by Rayleigh-type SAW, ΔR-SAW.

Fig. 20 .

 20 Fig. 20. Calculated resonant (fR) and anti-resonant (fA) frequencies of SH-SAW and resonant frequency of Rayleightype SAW at its lower stopband edge (f1) in Ti/Pt grating with pitch p=3.2 µm, on LGS cut (0º, 22º, 90.1º), as functions of temperature.

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Fig. 19

 19 Fig.19

Fig. 21 .

 21 Fig. 21. Variation of the admittance of resonators with Ti/Pt electrodes and a pitch p=3.2 µm on LGS cut (0º, 22º, 90.1º) with temperature increasing from 25° to 650°C.

Fig. 22 .

 22 Fig. 22. Evolution with temperature of the measured admittance of a typical investigated resonator built on SH-SAW (0°, 22°, 90°) LGS cut. Temperature is given by the color scale on the right.

  Fig.19

  Fig. 20

ACKNOWLEDGEMENTS

The authors would like to thank Alfred Binder, David Eisele and Elena Mayer for helpful discussions and suggestions during the work on this article. The measurements were partly carried out with SAW devices from the EU-SAWHOT project ([FP7/2007-2013]), grant agreement n°[NMP4-SL-2009-247821].


Pascal Nicolay

Thierry Aubert

A numerical method to derive accurate temperature coefficients of material constants from high-temperature SAW measurements. Application to langasite

Keywords: simulated annealing, high-temperature, langasite, material constants, surface acoustic waves I 6) SIMULATED DATA

The design of wireless surface acoustic waves (SAW) sensors for high-temperature applications requires accurate knowledge of the constitutive materials physical properties in the desired temperature range. In particular, it is crucial to use reliable temperature coefficients of the stiffness, piezoelectric, dielectric, and expansion constants of the propagation medium to achieve correct simulations of the considered devices. Currently, the best-suited piezoelectric material for high-temperature SAW applications is langasite (LGS). Unfortunately, the available coefficients do not allow for precise prediction of the temperature dependence of LGS-based SAW devices above 300°C. A novel method, based on a simulated annealing algorithm coupled with a Rayleigh wave simulation program, was developed to find optimal LGS temperature coefficients. This approach has proven to yield accurate results up to at least 800°C.

INTRODUCTION

The need for passive wireless sensors able to operate at very high temperature is steadily growing. Surface acoustic wave (SAW) technology constitutes a promising way to meet this need. The best results so far have been obtained with sensors based on langasite (La3Ga5SiO14; LGS). Recently, a SAW signal was recorded up to 1140°C [START_REF] Aubert | Iridium interdigital transducers for high-temperature surface acoustic wave applications[END_REF] and a first commercial sensor able to operate above 800°C for long-time periods was developed [2]. Accurate simulations at high temperatures are now needed to enable the design of even better SAW sensors. Unfortunately, such simulations are currently impossible to perform as the first and second order temperature coefficients of the material constants (stiffness, piezoelectric, dielectric, and expansion constants) of LGS are not precisely known. Several groups worldwide have spent a lot of efforts measuring the first [START_REF] Nakamura | Elastic constants of langasite and alpha quartz at high temperatures measured by antenna transmission acoustic resonance[END_REF] and second [START_REF] Malocha | Recent measurements of material constants versus temperature for langatate, langanite and langasite[END_REF][START_REF] Bungo | Experimental and theoretical analysis of SAW properties of the langasite substrate with Euler angle (0°,140°,θ)[END_REF][START_REF] Kräusslich | Temperature dependence of lattice parameters of langasite single crystals[END_REF] order temperature coefficients of LGS. It is also possible to extract third order coefficients from the data published in [START_REF] Schulz | Electromechanical properties of langasite resonators at elevated temperatures[END_REF]. However, none of the published sets of coefficients (which significantly differ from each other) allow for a precise prediction of the behavior of SAW on LGS at temperatures higher than 300°C [START_REF] Aubert | Extraction of COM parameters on Pt/LGS for high-temperature SAW sensor[END_REF][START_REF] Sakharov | Characterization of langasite for application in high temperature SAW sensors[END_REF][START_REF] Weihnacht | Langasite: high temperature properties and SAW simulations[END_REF].

To overcome this current limitation, we developed a new computer-assisted method to derive optimized temperature coefficients of LGS material constants. This method yielded excellent results up to at least 800°C and is described in the following section.

II. METHOD PRINCIPLES

The basic idea of the method is to start from the best set of available temperature coefficients and to then tune it in order to optimally fit the available experimental data (i.e. to reduce the distance or error between simulated and experimental data). For this purpose, simulated annealing (SA) was chosen as an algorithm to optimize the set of temperature coefficients because of its ability to avoid local minimums and converge efficiently towards the optimum of a given multivariable function in a large search space [START_REF] Bertsimas | Simulated annealing[END_REF]. In our case, a SA function readily available in the Matlab environment was used [START_REF] Yang | Simulated Annealing for Constrained Optimization[END_REF]. The function was only slightly modified to comply with our needs. The number of variables was changed as well as the "initial guess" calculation formula. The latter was modified in order for the calculation to start from a chosen set of values. The SA function generated new sets of test-coefficients to be used by a specific calculation program (SCP) to compute the corresponding simulated data and the resulting global error.

The error was fed back into the SA function, thus enabling the optimization process to run adequately.

The SCP was also developed in Matlab environment (R2011b). Once provided with the stiffness, piezoelectric, dielectric, and expansion constants as well as the respective temperature coefficients, the SCP allowed for calculating the temperature coefficient of delay (TCD) and the fractional frequency change (FFC) for Rayleigh modes whatever the cut, propagation direction and temperature. The SCP was based on the well-known Campbell and Jones method [START_REF] Campbell | A method for estimating optimal crystal cuts and propagation directions for excitation of piezoelectric surface waves[END_REF]. A full description of the calculation steps can also be found elsewhere [START_REF] Royer | Elastic Waves in Solids[END_REF].

To ensure good convergence towards the global minimum, it is important to use relevant experimental data. These data must be accurate and diverse. Ideally, they are measurements of various physical quantities, each of them dependent on the coefficients to be optimized. For instance, SAW velocities (or the associated FFC) can be measured on different crystal cuts. Accordingly, we used the FFC measured in a wide temperature range for five different LGS crystal cuts [START_REF] Aubert | Extraction of COM parameters on Pt/LGS for high-temperature SAW sensor[END_REF][START_REF] Bardong | Experimental and theoretical investigations of some useful langasite cuts for high-temperature SAW applications[END_REF][START_REF] Pereira Da Cunha | Recent advances in harsh environment acoustic wave sensors for contemporary applications[END_REF]. The Euler angles of the cuts were (0,22,31), (0,90,0), (0,138.5,26.6), (0,144,24), and (0,22,90). The FFC was provided for Rayleigh modes on the first four cuts, and for shear horizontal (SH) mode on the fifth cut. It was decided to use the first three cuts as targets for the optimizer and the two remaining cuts to subsequently test and validate the optimized set of coefficients. Note that the selected experimental data were obtained using low-thickness transducers (h/λ ~ 0.6%). Consequently, the electrodes had a negligible influence on the temperature behavior of the considered devices. In our case, this was an important point for proper derivation of accurate temperature coefficients of LGS material constants. Indeed, the electrodes were not simulated in the SCP.

To define the optimization starting point, the SCP was used to test all available sets as well as combinations of them. The best results were obtained using the combination of stiffness, piezoelectric, and dielectric constants (and temperature coefficients) from Bungo et al. [START_REF] Bungo | Experimental and theoretical analysis of SAW properties of the langasite substrate with Euler angle (0°,140°,θ)[END_REF] with the expansion coefficients from Krausslich et al. [START_REF] Kräusslich | Temperature dependence of lattice parameters of langasite single crystals[END_REF] (see Fig. 1). As mentioned before, the SCP was used to (1) compute the FFC of the three target cuts in the [20-800°C] range for each new set of coefficients, (2) compare the simulated and experimental results, and (3) compute the error. To reduce computation time, the following special property of the TCD on LGS cuts was used: it was experimentally observed that the TCD evolution with temperature is linear on LGS cuts. As a consequence, the FFC is parabolic. It is therefore possible to compute the whole FFC plot in a desired temperature range by calculating only a very limited number of TCD values in the range of interest, then linearly fitting the results and properly integrating the obtained linear equation. In our case, one point was calculated every 100°C, from -50 to +850°C. The computation was made on a standard commercial FUJITSU Desktop, equipped with an Intel Core i5-2500 CPU, clocked at 3.3GHz. About thirty seconds were needed per iteration (i.e. to calculate the three cuts in the whole temperature range for each new set of test-coefficients). At the end of each iteration, a mean squared error (MSE) was calculated for each of the three cuts. The total error to be minimized was simply taken as the sum of the three MSE(s). The error was steadily and quickly reduced over the first 500 iterations (see thick black line in Fig. 2). The algorithm managed to further reduce the error over the next 2000 iterations. It finally reached a plateau after 3000 iterations. Besides, a first useful step before running the optimization was to slightly change the different coefficients and then use the SCP to evaluate the effect on the simulated results. This helped selecting the most impacting coefficients to be optimized and thus further reducing the computing effort. If needed, the less impacting coefficients might be optimized in a second step, to fine-tune the results. Practically, the optimization was performed on the first and second order temperature coefficients of the independent stiffness constants (TCS) only -c11, c13, c14, c33, c44 and c66 in the case of LGS. Indeed, it was observed that the temperature coefficients of the piezoelectric and dielectric constants do not significantly alter the FFC in the [20 -800°C] temperature range. It was also observed that an optimization conducted on both the stiffness constants and expansion coefficients do not yield significantly better results than the ones conducted on the stiffness coefficients alone, although the convergence is a bit faster (see thin red line in Fig. 2). Since it is not necessary to modify the expansion coefficients to obtain a good reduction of the error, it was decided not to alter these coefficients at all.

III. RESULTS AND DISCUSSION

The obtained coefficients are presented in Table I Note that the only significant discrepancy between the optimized first order coefficients and the original ones from Bungo et al. is for c66. However, for the second-order coefficients, the difference is much higher. As the constants and higher-order coefficients constitute a coherent whole, it is possible that in Bungo et al., a significant measurement error on c66 has generated a cascade of errors in the second order coefficients. This will be analyzed further in future work.

As can be seen in Fig. 3, there is an excellent agreement between the experimental data and the simulated results obtained with the new set of coefficients for the cuts (0,22,31) and (0,138.5,26.6). This certainly can be seen as a direct consequence of the optimization procedure. Indeed, both presented cuts were used as targets for the optimizer.

The results presented in Fig. 4 for the cuts (0,144,24) and (0,22,90) definitely confirm the quality of the obtained constants. Both plots were indeed calculated a posteriori for cuts that were not considered in the optimization process. As no Rayleigh wave can propagate along the (0,22,90) direction, another software (COM/FEMSDA) developed by K. Hashimoto [START_REF] Hashimoto | Surface Acoustic Wave Devices in Telecommunications[END_REF] was slightly modified and used to compute the SH-SAW characteristics in the desired temperature range. Again, simulations fitted almost perfectly the experimental results. However, the agreement between experimental and simulated results is not as good in the (0,90,0) case, although this cut was also taken as an optimization target (see Fig. 5). This cut actually was the limiting factor that prevented the optimizer to further reduce the error towards zero. It is suggested that the lack of perfect convergence might be linked to the imperfection of the experimental data themselves.

This will be verified in future work. A confirmation of the latter hypothesis would indicate that the optimization procedure could also be used to check a posteriori the quality of experimental data obtained for new cuts. Besides, the scope of the method used to derive this new set of temperature coefficients can be much larger. It basically makes it possible to easily derive accurate temperature coefficients from imperfect measurements, once the measurements are numerous and diverse enough. This might be of great interest for other scientific communities having to deal with high-temperature materials and applications.

Future work will include the search of optimized parameters for other SAW crystal materials as well as multilayered SAW substrates. The convergence process will also be thoroughly studied. Amongst others, it will be checked whether the optimizer converges towards the same global minimum when starting from another set of initial parameters.

Fig. 1 .

 1 Fig. 1. Simulated results for LGS cuts, obtained with different sets of published material constants (Csts) and temperature coefficients of expansion (TCE) -(a) (0,22,31), (b) (0,90,0).

Fig. 2 .

 2 Fig. 2. Error reduction vs. number of iterations -thick black line: optimization of the TCS(s) only; thin red line: simultaneous optimization of the TCS(s) and TCE(s).

Fig. 3 .

 3 Fig.3. Experimental results vs. simulated results obtained with the optimized set of TCS(s), for the cuts (0,22,31) and (0,138.5,26.6). Both cuts were used as targets for the

Fig. 4 .

 4 Fig. 4. Experimental results vs. simulated results obtained a posteriori with the optimized set of TCS(s), for the cuts (0,144,24) (a) and (0,22,90) (b).

Fig. 5 .

 5 Fig.5. Experimental results vs. simulated results obtained with the optimized set of TCS(s) for the cut (0,90,0).

10 10 [N/m 2 ] e → [C/m 2 ] ρ → [kg/m 3 ]

  . They are optimized temperature coefficients of the Bungo et al. stiffness constants. They must be used in conjunction with the Bungo et al. values for the piezoelectric and permittivity parameters (constants and temperature coefficients), as well as the Krausslich et al. values for the temperature coefficients of expansion. All of these coefficients are also presented in TableI.Tab. I. Optimized set of LGS material constants. Stiffness, piezoelectric, permittivity and density constants are from Bungo et al.[START_REF] Bungo | Experimental and theoretical analysis of SAW properties of the langasite substrate with Euler angle (0°,140°,θ)[END_REF], as well as temperature coefficients for piezoelectric and permittivity parameters. TCS were determined within this study. Temperature coefficients of expansion are from Krausslich et al.[START_REF] Kräusslich | Temperature dependence of lattice parameters of langasite single crystals[END_REF].

	Units:		1 st order (ppm/°C)	2 nd order (ppb/°C 2 )
	c → Material constants
		@ 20°C
			This study	Bungo et al.	This study	Bungo et al.
	c11	18.89	-65	-57.54	-39	-82.65
	c13	10.15	-84	-81.61	-86	-68.1
	c14	1.442	-304	-307.5	88	95.34
	c33	26.83	-105	-104.0	-55	-89.61
	c44	5.33	-63	-57.39	-80	-130.5
	c66	4.237	-29	-7.576	-23	-119.4
	e11	-0.4371	469.8		-428.5
	e14	0.1039	-713.8		1594
	ε11/ε0	19.05	134.5		118
	ε33/ε0	51.81	-787.0		658.6
	α11	-	5.20		6.7
	α33	-	3.72		1.1
	ρ	5764

Acknowledgement

This work was partly funded within the COMET program, coordinated by the Austrian Research Promotion Agency (FFG).


domain: phys spi

text:
Pascal Nicolay

Thierry Aubert

A promising method to derive the temperature coefficients of material constants of SAW and BAW materials. First application to LGS

Keywords: LGS, temperature coefficients, high temperature, optimization, Simulated Annealing, Rayleigh modes I

A promising method to derive the temperature coefficients of material constants of SAW and BAW materials. first application to LGS

INTRODUCTION

The design of high-temperature Surface Acoustic Wave (SAW) sensors requires prior knowledge of the first and second order temperature coefficients of the substrate material constants (TC1 and TC2).

However, direct measurement of material constants is very challenging at high temperature. Concerning

LGS which is one of the most promising SAW substrates for high temperature applications, available TC1 and TC2 sets are reliable up to 300°C at best [START_REF] Weihnacht | Langasite: high temperature properties and SAW simulations[END_REF]. Therefore, it is not possible to perform accurate predictions above this temperature.

If direct characterization of material constants is difficult to achieve, indirect determination might be a good solution. For instance, acoustic waves of various kinds can be used to probe the material and extract the needed information. Indeed, changes in velocity and therefore velocity-related parameters (frequency, time delay, phase…) depend directly on the TC1 and TC2 coefficients. This approach is becoming more and more appealing as recent technical achievements made it possible to characterize SAW properties up to at least 900°C [START_REF] Pereira Da Cunha | Recent advances in harsh environment acoustic wave sensors for contemporary applications[END_REF][START_REF] Aubert | Iridium interdigital transducers for high-temperature surface acoustic wave applications[END_REF]. In this case, one possible way to perform the extraction is a curve fitting procedure.

The SAW properties are simulated first in a wide temperature range then the TC1 and TC2 used in the simulation are adjusted ('optimized') to fit the computed results to the experimental data ('target curves').

The first difficulty of the optimization procedure is that many coefficients are involved and must be simultaneously tuned. For the stiffness coefficients of LGS alone, twelve coefficients must be considered (six TC1 and six TC2). If the TC1 and TC2 of the piezoelectric constants and expansion coefficients are to be considered too, this number rises to twenty. To fit adequately these coefficients, we would need at least an equal number of 'non-correlated' experimental curves. By 'non-correlated' curves, we mean experimental curves with different dependency patterns on the various parameters to be optimized. It is necessary to use non-correlated curves to converge towards the best solution (i.e. the best set of TC1 and TC2 coefficients) the same way as it is necessary to use N non-correlated equations to solve a system of linear equations with N unknown values. Non-correlated curves can notably be obtained using different cuts, different propagation directions or different wave modes. Additionally, a given coefficient will be well optimized only if several of the non-correlated target curves share a strong dependence on it. For instance, if none of the required target curves depends strongly on C11, it won't be possible to optimize this coefficient or at least not with a good enough accuracy. If only one curve depends strongly on C11, the optimization will be improved (especially if the curve depends solely on C11) but it won't be possible to benefit from the global optimization process to improve the fitting accuracy on C11. That is, C11 will be obtained from one curve only which implies a rather poor accuracy due the measurement uncertainties on the target curve itself. If two or more non-correlated target curves share a strong dependency on C11, the optimization process will improve the accuracy. Indeed, each of the curves will define a given range of possible values for C11 (i.e. the error bar around the calculated value) but all of these possible ranges will only overlap in a much smaller region, delimiting the actual and much improved range of possible values for C11. This feature of a system of target curves will be called 'multiplicity' in the following chapters.

Following this rather intuitive analysis, it appears that a set of coefficients will be well optimized only if enough 'non-correlated' target curves are available, with several of them sharing a strong dependence on each of the coefficients. This will serve as a guideline in the following chapters, where solutions to improve the performances of the optimizer as well as the accuracy of the obtained LGS coefficients will be discussed. This also makes it clear that if only a handful of target curves are available, the optimization process won't yield a complete set of accurate optimized coefficients. Several distinct TC1 and TC2 sets might even emerge as possible solutions to the fitting procedure. However, there are still ways to obtain good results in this case. As will be shown in the next chapters, a good way to avoid converging towards a wrong solution might be first to reduce the number of variables and second, to start from a solution that is already as close as possible to the optimum. The latter implies looking first for the best possible set of available coefficients then use it as a starting point for the optimization.

Besides, it is also very important to select target data that were not (or as little as possible) influenced by parameters other than the ones taken into account in the simulations. For instance, if SAW curves are used as target, it is important to use data obtained with very thin IDTs. Indeed, thicker IDTs have a clear influence on the Fractional Frequency Changes (FFC) curves, especially at high temperature or for very sensitive wave modes and orientations. This was shown for instance for Pt IDTs on LGS, which have been experimentally investigated on the orientations (0°,138.5°,26.6°) and (0°,22°,31°) up to 700°C. Big discrepancies between the frequency-temperature laws for devices made with different Pt relative thicknesses were observed, in particular on the (0°,22°,31°) orientation. First simulations show that this phenomenon could be at least partly related to large thermoelastic effects occurring in Pt electrodes at high temperature when the temperature rises [START_REF] Aubert | Thermoelastic effects in Pt IDTs. Impact on the behavior of hightemperature LGS-based SAW devices[END_REF].

The second difficulty of the fitting procedure is the optimization phase itself. If twelve to twenty coefficients have to be considered simultaneously, it is not reasonable to test all of the possible combinations to ultimately choose the best one. This would consume way too much time. A search algorithm has to be used instead. A promising solution is the Simulated Annealing (SA) algorithm which is well known for its ability to find the global minimum of a multivariable function [START_REF] Goffe | Global optimization of statistical functions with simulated annealing[END_REF].

The approach described above has been tested first on LGS using FFC target curves on three distinct cuts -(0°, 22°, 31°), (0°, 90°, 0°) and (0°, 138.5°, 26.6°) -and a full set of starting constants and associated TCs extracted from [START_REF] Bungo | Experimental and theoretical analysis of SAW properties of the langasite substrate with Euler angle (0°,140°,θ)[END_REF] and [START_REF] Kräusslich | Temperature dependence of lattice parameters of langasite single crystals[END_REF]. The optimization was performed on the six TC1 and six TC2 of the stiffness constants only. This means that the stiffness constants themselves (i.e. Cij at room temperature) were not optimized and therefore kept constant all along the procedure.

The optimized coefficients have been subsequently used to successfully predict the FFCs of two test cuts -(0°, 144°, 24°), (0°, 22°, 90°) -in a wide temperature range (up to 800°C) [START_REF] Nicolay | A numerical method to derive accurate temperature coefficients of material constants from high-temperature SAW measurements. Application to langasite[END_REF]. It is stressed that these two test cuts were not taken into account during the optimization process. They were used afterwards to check the 'quality' of the optimized set of TCs. The constants and optimized coefficients are presented in Table I.

TAB. I. LGS STIFFNESS CONSTANTS [6] AND OPTIMIZED TC [8]

Constant @ 20°C The main goals of the present paper are first to explain in more details the inner-working of the optimizer and provide a thorough analysis of the obtained results, second to expose the limitations of the method, third to provide solutions to improve the quality of the results (i.e. the accuracy of the optimized coefficients).

The complete optimization algorithm, including preparation steps, is presented in chapter 2. The main optimization parameters are described. The results of four optimization runs are compared. The convergences towards a minimal error follow very similar tracks and provide fitting curves of the same quality. However, if all the runs do converge towards the same solution for some of the coefficients, the dispersion for other coefficients is quite high. This constitutes a first limitation of the procedure. In chapter 3, we analyze the reasons for the dispersion and show that the dispersed coefficients were not optimized at all. We provide a simple procedure and graphic tool to support the analysis and predict which coefficients are going to be well optimized and which are not. A solution to minimize the dispersion is suggested, discussed and tested. It combines acoustic measurements of different kinds (SAW and BAW). In chapter 4, we check the algorithm's ability to converge towards the optimum solution when starting from a distant initial solution, built out of published data. It is shown that the algorithm does not converge towards the best optimum in that case, which constitutes a further limitation of the procedure. Complementary test results show that in the particular configuration used to derive the coefficients of Table I that is with the three already-mentioned SAW curves as target, the optimizer is able to re-converge towards the best identified solution as long as the 'measurement error' on the various coefficients (i.e. the differences between the starting values and the optimized values from Table I) is kept below a given limit. Beyond this limit, the optimizer starts converging towards alternative solutions. This definitely validates the capability of the optimizer to find a solution from any starting point. It also confirms that a well-defined and complete set of target curves must be used to ensure that the optimizer finds the right solution (i.e. TCs 'equal' to the natural ones). If such a set is not available, the optimizer will converge towards the right solution only if the starting point is not located too far from it. In chapter 5, an updated procedure is suggested to overcome the limitations presented before. We also propose to add complementary calculation modules to optimize the TCs of the piezoelectric constants. This module would make use of other kind of experimental measurements and simulation methods to compute the error to minimize.

II. THE OPTIMIZATION PROCEDURE

A. The method

The complete procedure is sketched in Fig. 1. First, FFC curves are retrieved in the literature for as many cuts as possible, as well as sets of material constants and coefficients. A simulation tool is used to compute the FFC(s) for each set as well as combinations of subparts of sets. This allows for selecting the best starting set of parameters (X0) and computing a starting error value Z0. Then, the parameters to be optimized are chosen (we chose to perform the optimization on the TC1 and TC2 of the stiffness constants only). The optimization procedure starts with the pseudo-random generation of a new set of parameters (X)

'around' X0. In our case, the new set was generated using the Matlab randn(n) function, which draws values from the standard normal distribution. The chosen standard deviation was 0.5ppm/K for TC1 and 2.5ppb/K 2 for TC2. The FFC(s) are then computed for each of the selected curves and a new error Z is calculated. In the next step, Z is compared to Z0. If Z is smaller than Z0, the old 'best' set is replaced by the new one and a new cycle begins. If Z is bigger than Z0, the decision of acceptance or refusal of the new solution is based on the Metropolis criterion [START_REF] Goffe | Global optimization of statistical functions with simulated annealing[END_REF]. This criterion makes the acceptance of the new (worse) solution all the more likely than 'temperature' is high and Z-Z0 is small. 'Temperature' is used here to describe a parameter that steadily decreases with time, over the whole optimization process. The term 'temperature' was chosen by analogy with the physical process of thermal annealing, which is mimicked by the SA algorithm. The 'temperature' is lowered over time, which progressively decreases the probability to accept a worse solution. The strength of the SA algorithm precisely resides in its ability to keep exploring worse solutions to avoid being stuck in local minima. The optimization stops when the error cannot be further reduced. Concerning the simulation tool, it is chosen with respect to the modes that have to be computed (SAW, BAW, LSAW, etc…). In our case, we selected target FFC curves involving only Rayleigh modes. To compute these modes, we used a self-made tool based on the well-known

Campbell and Jones method [START_REF] Campbell | A method for estimating optimal crystal cuts and propagation directions for excitation of piezoelectric surface waves[END_REF].

B. The results

Four optimizations were performed on a Fujitsu desktop computer (3.3GHz CPU, 4GB RAM). Each run took approximately 24 hours for an average of 2500 iterations. The starting solution and optimization parameters were identical to the ones described in [START_REF] Nicolay | A numerical method to derive accurate temperature coefficients of material constants from high-temperature SAW measurements. Application to langasite[END_REF]. Only the standard deviation (s) was different (five times bigger). All the convergence patterns (including the reference pattern published in [START_REF] Nicolay | A numerical method to derive accurate temperature coefficients of material constants from high-temperature SAW measurements. Application to langasite[END_REF]) are presented in Fig. 2. The patterns are all very similar and converge towards the same minimum value (~ 0.35).

However, it is clear that a bigger s allowed for a significant decrease in convergence time. The FFC(s) obtained with each of the four optimized sets of coefficients are presented in Fig. 3, for two of the target curves. The 'fitting quality' is just as good in each case. The FFC of the (0°, 90°, 0°) cut is not represented here for clarity reasons, the FFC(s) of the (0°, 22°, 31°) and (0°, 90°, 0°) cuts being almost identical. Finally, the variations of the TC values before and after optimization are presented in Fig. 4, for each of the runs. First, it can be seen that some coefficients 'moved' more than others. Second, it appears that the optimization results are very close to each other for some of the coefficients (C66, C44) but strongly dispersed for others (C11, C13). This is an important result which demonstrates one of the main limitations of the optimization procedure. Indeed, if several distinct values can be obtained for some of the coefficients, these values just cannot be trusted. The scale is in ppm/K for TC1 and ppb/K 2 for TC2 values.

III. RESULTS ANALYSIS

The reason for the observed dispersion is that the Rayleigh modes propagating on the three target cuts depend very little on the coefficients C11, C33 and C13. The dispersion is due to the stochastic exploration of the solution space for these constants. The 'optimized' TC1 and TC2 for these constants result from Brownian motions around the starting values (see Fig. 5). However, the very accurate prediction of the SAW behavior on two test cuts as reported in [START_REF] Nicolay | A numerical method to derive accurate temperature coefficients of material constants from high-temperature SAW measurements. Application to langasite[END_REF] show that the optimized values C66, C44 and C14 probably can be trusted. The predictions were good because the observed modes do not significantly depend on C11, C33 and C13 for these two cuts.

A sensitivity map for the three target curves is presented in Fig. 6. The sensitivity of the error Z to the variation of each of the twelve coefficients TC1 and TC2 is quantified for each of the target curves.

The plotted values correspond to the Z variation for a change of +1ppm/K for TC1 and +1ppb/K 2 for TC2 values. We can see that the error depends strongly on C66, quite a lot on C44 and C14 and only very little on C11, C33 and C13, for each of the represented curves. Moreover the three profiles are similar which makes the three target curves rather poorly complementary to each other. The sensitivity map is a useful tool to start with prior to the optimization. It makes it possible to visualize and estimate the two important features described in the Introduction section which are the degrees of correlation (how different are the sensitivity patterns?) and multiplicity (how many target curves do I have showing a strong sensitivity on a given coefficient?). To ensure that all of the chosen coefficients will be optimized, the area(s) enclosed by the sensitivity polygons must be maximized and the polygon shapes must differ as much as possible from one target curve to the other.

In the present case, the three target curves appear to be correlated and global multiplicity is bad (there is no sensitive target curve for C11, C33 and C13) with the consequence that C11, C33 and C13 cannot be well optimized. Another consequence is that the starting point has to be close to the optimum in order to optimize with a good-enough accuracy the three other coefficients. Indeed, a high degree of correlation makes it easier to converge towards alternative solutions (by analogy, if a complete system of linear equations comprises two identical equations, the number of solutions becomes infinite). However, if the starting point is close to the global minimum, we can still hope finding it and this is probably what happened here.

More non-correlated target curves with higher multiplicity on C11, C33 and C13 are obviously needed to optimize these coefficients. The first possibility would be to look for specific cuts and propagation directions whose Rayleigh modes would depend mainly on these constants. But BAW measurements could be simpler to use and very practical. Longitudinal BAW modes (L-BAW) in the directions X (1,0,0) and Z (0,0,1) are respectively dependent on C11 only and C33 only. Using these two directions plus one additional direction (1,0,1) to increase the overall dependency on C13 would yield the sensitivity map presented in Fig. 7. The sum of the individual sensitivity plots is represented in Fig. 8, before and after consideration of the additional three L-BAW modes. When the three L-BAW modes are included, all of the TCs have a non-negligible effect on the global error which is mandatory for the optimization. However, it appears that C13 is still a weak point in this configuration. It is not the purpose of this work to find the best possible configurations to ensure the best possible optimization but this is clearly a very important issue that will have to be addressed in future work. The goal will be to define the minimum number of cuts, orientations and wave-modes (including SH-SAW, pseudo-SAW, SH-BAW, Lamb modes etc…) to characterize in order to ensure an as perfect as possible convergence towards the best possible TCs values. As a first step in this direction, the configuration described in Fig. 7 and 8 was tested. The simulation tool was upgraded to simulate longitudinal and transverse BAW modes. We were not able to find the required target curves in the literature. Therefore, we created them manually, from scratch. That is, we generated parabolic curves without any physical sense, our goal being only to check the convergence pattern of the optimizer in the new configuration described above (i.e. 6 target curves). The convergence patterns for C11, C33 and C13 are showed in Fig. 9, for TC1 and TC2. In each case, the evolution follows a well-defined path which confirms that all of the coefficients are being actually optimized. This is a promising result which confirms the interest of the method although improvements and further testing are still required.

IV. ALGORITHM 'ROBUSTNESS'

In the previous section, we showed that the algorithm optimizes all of the parameters when provided with an adequate set of target curves. Besides, it is also interesting to evaluate the algorithm's ability to converge towards the global optimum when starting from a more distant set of parameters. In other words, we would like to check if the optimizer still retrieves the values from Table I when starting from a very different set of values. To perform this complementary study, we first used starting constants and coefficients from [START_REF] Malocha | Recent measurements of material constants versus temperature for langatate, langanite and langasite[END_REF]. The results of the optimization run (1740 iterations, standard deviation parameters: 0.5ppm/K for TC1 and 2.5ppb/K 2 for TC2) are presented in Fig. 10. The target curves were again (0°, 22°, 31°), (0°, 90°, 0°) and (0°, 138.5°, 26.6°). No 'dummy' BAW targets were used here. The algorithm did converge towards a lower error (Z=1.06) but the results were not satisfying. The algorithm got stuck in a local minimum and did not manage to evolve towards a better solution. To allow the algorithm for finding a better solution, the optimization was re-run with increased excursion parameters. This yielded no better results. The maximum standard deviation parameters tested here were 50ppm/K for TC1 and 250ppb/K 2 for TC2. In this extreme case, no convergence occurred at all. The optimizer kept oscillating between huge error values, never managing to find any possible convergence way. before optimization (using coefficients from [START_REF] Malocha | Recent measurements of material constants versus temperature for langatate, langanite and langasite[END_REF]). Thick line(s): after optimization.

Nevertheless, even if the optimizer is unable to converge towards the correct solution when supplied with a too distant set of starting parameters, it is interesting to check the maximum distance until which it is still able to converge adequately. To perform the necessary calculations, we generated multiple sets of starting parameters with an increasing distance to the optimized coefficients presented in Table I.

The starting sets were generated using the randn(n) function of Matlab to calculate random variations of each of the TCs prior to the optimization (i.e. TC_initial(Cij) = TC_TableI_(Cij) + random_number_ij, where random_number_ij is obtained using the randn(n) Matlab function with a given standard deviation s). Several sets were generated for a given standard deviation. The standard deviation was progressively I, the latter still have to be considered the best ones (at least for C66, C44 and C14). Indeed, they were obtained using a much better starting set of TCs meaning that the optimizer started form a location that was close enough to the global optimum to find it.

All these results will serve as basis for deeper analysis and improvement of the method. They already indicate that an adequate set of target curves should allow the optimizer for retrieving the optimum values even if the measurements are not accurate. This would advocate in favor of the following and promising method to derive accurate TCs with much less effort than previously required: 1. Look in the literature for a complete set of constants at room temperature; 2. Perform quick and dirty measurements up to 900°C, according to a well-defined and customized experimental planning; 3. Use numerical optimization supported for instance by Simulated Annealing to derive the TCs with a good enough accuracy. The method might be dubbed 'QDMO', after Quick and Dirty Measurements plus Optimization.

V. FUTURE STEPS

Considering the developments presented before, the first obvious step to improve the method will be to define the right set of target curves to be used to ensure that the optimizer always finds the global optimum or best set of TCs, whatever the starting point. Among others, this set will depend on the crystallographic structure.

A second point concerns the TCs of the piezoelectric constants. In essence, the QDMO method is based on complementary experimental results which are used together as a global target for an optimizer.

The optimizer reduces the distance between experimental and numerical results by adjusting some preselected coefficients, therefore tuning them to what is hopefully their actual and natural value. If SAW and BAW measurements are certainly good choices to help tuning the TCs of the stiffness constants, another set of measurements could serve as good complementary targets to help tune simultaneously the TCs of the piezoelectric coefficients. Following the logic used before when L-BAW were suggested as good complements to help tuning C11 and C33, direct piezoelectric coefficient measurements might be used to help tuning e11 and e14. A special piezoelectric module could be added to the calculation core of the optimizer, allowing it to extract an error from the comparison between experimental and numerical results. A possible approach would be to use Laser Vibrometry characterization of piezoelectric materials, as described in [START_REF] Hernando | Simulation and laser vibrometry characterization of piezoelectric AlN thin films[END_REF]. In a few words, piezoelectric constants are extracted from (Laser-measured) vibration patterns using FEM computation to fit the numerical model to the experimental data. This could be advantageously implemented in the error calculation loop. Moreover, the non-contact nature of the measurement method (although electrodes still have to be used and bonded) makes this solution very attractive for high-temperature characterization.

Finally, it will be important to check the validity of the QDMO method against a set of accurate TCs determined in a more classical and direct way. The complete sets of constants and coefficients published recently for LGT [START_REF] Davulis | A full set of langatate high-temperature acoustic wave constants: Elastic, piezoelectric, dielectric constants up to 900°C[END_REF] might very well constitute the perfect reference for testing the whole method. The goal will be to start from a small number of new SAW and BAW measurements (to be performed up to 900°C) then use SA Optimization to derive new TCs then compare these TCs to the published ones. If the difference is small, the validity of the QDMO method will be definitely confirmed.

VI. CONCLUSION

We showed that a SA algorithm coupled with a Rayleigh modes simulation tool might be used for accurate determination of temperature coefficients of material constants (LGS in the present case). The algorithm is able to fit experimental curves and extract the coefficients from the fitting results. However, we demonstrated that a complete optimization of all the coefficients of interest would require a bigger number of specific and non-correlated experimental plots. A graphic tool (sensitivity map) was suggested as a potential help to assess beforehand whether the optimization will actually converge towards a trustable solution for all of these coefficients or not. Future work will include looking for suitable complementary cuts and propagation modes to further optimize the whole set of TC1 and TC2 coefficients for LGS. The algorithm will also be used to extract optimized temperature coefficients for other potentially interesting substrates, including LGT.

Fig. 1 .

 1 Fig. 1. The optimization procedure.

Fig. 2 .

 2 Fig. 2. Error evolution vs. iteration during optimization process. Inset: zoom on the first 100 iterations.

Fig. 3 .

 3 Fig. 3. Starting point (green) and optimization result (blue) on distinct experimental LGS Target Curves (red). Above -(0°,22°,31°); below -(0°,138.5°,26.6°).

Fig. 4 .

 4 Fig. 4. (TC_final-TC_initial) for each of the optimized coefficients.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Convergence paths for C66 (red) and C13 (green). Top: TC1 values; bottom: TC2 values. 'Accepted solution'refers to the number of new sets (X) accepted as solutions by the SA algorithm, during the optimization.

Fig. 7 .

 7 Fig. 7. Sensitivity map, with SAW and L-BAW target curves.

Fig. 8 .

 8 Fig. 8. Global Sensitivity maps with (green) and without (red) L-BAW targets.

Fig. 9 .

 9 Fig. 9. Convergence patterns for C11, C33 and C31 when L-BAW and SAW curves are used as targets. Well-defined convergence patterns are visible for each of the coefficients. Top: TC1 values; bottom: TC2 values.

Fig. 10 .

 10 Fig. 10. Optimization from a distant starting point, for (0°, 22°, 31°) (above) and (0°, 138.5°, 26.6°) (below). Thin line(s):

  a few correlated curves are used as targets. It is stressed that even if at least one completely different set of TCs did allow for fitting the target and test curves as well as the TCs from Table

Fig. 11 .

 11 Fig. 11. Comparison Before/After Optimization for two runs with different starting sets of coefficients. The parameter indices correspond to the TC1s of C11(1), C33(2), C44(3), C66(4), C13(5), C14(6) and to the TC2s of C11(7), C33(8), C44(9), C66(10), C13(11), C14(12).

Fig. 12 .

 12 Fig. 12. Optimization from a distant starting point generated with s = 35, for (0°, 22°, 31°) (above) and (0°, 138.5°,

ppm/°C) TC2(ppb/°C 2 )

	10 10 [N/m 2 ] TC1(C11 18.89	-65	-39
	C33	26.83	-105	-55
	C44	5.33		-63	-80
	C66	4.237		-29	-23
	C13	10.15		-84	-86
	C14	1.442	-304	88

VII. ACKNOWLEDGMENT

This work was partly funded within the COMET program, coordinated by the Austrian Research Promotion Agency (FFG).

increased from 5 to 35. The optimizations were performed in the initial configuration, using only the three SAW curves as targets. It appeared that the optimizer managed to re-converge towards the optimum values from Table I (or very close ones) up to s~15. Above this value, it started to converge towards very different sets of parameters. However, the optimizer always managed to find sets of values which allowed for an almost perfect fit of the target and test curves, even for higher s values. In Fig. 11, we can see four sets of parameters which were the start and end values of two distinct optimization runs. The first of these two runs was used to obtain the data published in Table I. These data constitute our reference set (green line). The starting values (i.e. the Bungo coefficients) are represented in red. The second run was performed with s = 35. The corresponding starting values are indicated in magenta and the result of the optimization run in blue. A comparison between the red and magenta lines gives the distance between the two starting sets. By definition, this distance increases with s. To improve the readability of the figure, the reference set was subtracted to the three other sets. Therefore, the reference set is a green horizontal line on the X axis. A blue line close to the green one would mean that the run converged again towards the data from Table I. As mentioned before, this was observed for optimizations performed with s values smaller than 15 (see thin blue line in Fig. 11 for one example of optimized set obtained with s=5).

However, the blue line obtained for s=35 is far from the green line, meaning that the optimizer converged towards a completely different set of TCs. In Fig. 12, we present the fits obtained at the end of the 's = 35' optimization using this new set of TCs. The result is almost as good as what was obtained using the coefficients from Table I (see Fig. 3).

First of all, this confirms the ability of the optimizer to converge towards a consistent solution even if the starting values are very inaccurate. Second, this confirms that three correlated target curves are not enough to enable the optimizer for discriminating between equivalent neighboring solutions (local optimums) and therefore converge towards the global optimum. In our 's=35' example, the starting point was far enough to prevent the optimizer from re-finding the best solution. This gives a first measure of the error that could be tolerated on the starting values to ensure that the optimizer will still find the right