Laser-Doppler vibrometry (LDV) promises to attain these advantages. It enables the measurement of the piezoelectric coefficient and the deflection of a piezoelectric resonator independent of the temperature of the material. The measurement is contactless and allows the material to vibrate completely undisturbed by the measurement. To determine the piezoelectric coefficient with this approach, it is necessary to measure and examine the displacement during oscillation of the samples. If the vibrational direction is aligned with the laser beam direction and the excitation voltage is known, then the piezoelectric constant follows directly from the measured displacement when the system is operated significantly off resonance. This is where the main advantage becomes apparent. Because the measuring “tip” is a laser beam and thus purely optical, there is no mechanical influence on the sample or the vibration movement. The second major advantage is that the entire measuring technology is located outside the area of measurement, which is in the furnace, and is therefore in the cold zone. This means that the theoretical upper temperature limit of this measuring method is determined by the furnace used.

Measurements can be performed in both the off-resonant and the resonant cases. The first case is perhaps the most direct measurement approach. The difficulty here is that the displacements determined are relatively small, often only a few picometers, but this can still be resolved by such LDV systems (Wulfmeier et al., 2021; Schewe et al., 2023; Kohlmann et al., 2023). In the second case, the displacements are larger and thus also show a better signal-to-noise ratio (SNR). However, in order to determine the piezoelectric constant, the quality factor of the corresponding vibration mode must also be determined. A further advantage of this resonant approach is that it also allows for piezoelectric constants to be determined whose direction of vibration is not in the beam plane.

The following work demonstrates the functionality of this method by investigating the piezoelectric properties of LiTaO₃ at various temperatures up to 400 °C and thus well below its Curie temperature. LiTaO₃ serves as a model system here to prove the validity of this direct non-contact approach. The data obtained are set in context with the available literature values at room temperature, which were determined using various other methods. Since this work is a proof of concept for the suitability of LDV for determining piezoelectric constants, it does not claim to provide a complete analysis of all piezoelectric coefficients occurring in the LiTaO₃ system. Within the scope of this feasibility study, the focus is on the constant d33.

Several studies on the piezoelectric coefficient d33 of lithium tantalate have already been published, as well as a few on the piezoelectric coefficient d15. An overview of the available literature values at room temperature is given in Table 1. The values for d33 show a broad range from 5.7–23.1 pm V⁻¹, while d15 exhibits significantly more constant values of 26.2 ± 0.2 pm V⁻¹. However, it should be mentioned that all three literature values of d15 of which the authors are aware of were derived by the same technique. Here, an indirect method was used to determine the piezoelectric constants from a calculation based on resonance and antiresonance spectra. The wider spread of d33 goes along with a variety of the measurement techniques as well. If we consider the same references as for d15 only, the range is reduced to 5.7–9.2 pm V⁻¹, which is still significantly more divergent than the d15 values but significantly lower, as the higher values obtained using direct-contact measurement methods are no longer taken into account.

In the past, the majority of the measurements were conducted using indirect techniques (marked with an “i” in the last column of the table): in some cases (Warner et al., 1967; Yamada et al., 1969; Smith and Welsh, 1971), d33 was calculated from measurements of the resonance and antiresonance frequencies of the material in conjunction with other material constants (see Xu 2016) or, in the case of Smith and Welsh (1971), based on the mathematical models of McSkimin (1965).

The biggest disadvantage of indirect measurements is that additional material constants and physical models must be known precisely in order to handle or to correct the raw data to derive piezoelectric constants, e.g., from fitting the resonances and antiresonances of impedance spectra in the vicinity of resonance. Here, even small uncertainties can have a major impact on the correct determination of the piezoelectric coefficients.

Direct measurements (marked with “d” in Table 1) do not share this need for additional material parameters. Measurements on LiTaO₃ were performed using piezoelectric dii meters (Wang and Jiang, 2009) or piezoelectric force microscopy (Verma et al., 2021). Even though the disadvantages of indirect measurements do not apply here, nevertheless, other challenges must be taken into account for this measurement approach. In particular, it should be noted that these measurements were made in contact. With the small deflections prevailing here, even the smallest mechanical obstructions, such as the contact pressure of the measuring head or cantilevers, can lead to damping and thus reduced values. These influences are often compensated for directly in the measurement software, but this also carries the risk of overcompensation. In addition, a fairly rigid sample holder is required, which also serves as a reference for the mechanical measurements. The samples are thus very rigidly constrained on a massive plate and, in general, can only move in one direction. As this results in asymmetrical vibrational movements, this has to be compensated for in order to extrapolate the measured sample behavior to an undisturbed sample.

An optical heterodyne interferometer as used by Royer and Kmetik (1992) in principle is a direct and non-contact measurement and, thus, the approach most comparable to the LDV used in this work. However, in this case, the sample was not freely oscillating either but was firmly attached to a surface, which required the use of a correction term to determine the correct piezoelectric coefficient. Thus, this was also effectively an indirect method, even though the measurement method would, a priori, allow direct measurements.

Vibrometers are not uncommon for measuring piezoelectric coefficients (Li, 2021). Michelson interferometry is a proven technique for measuring piezoelectric coefficients (Yamaguchi and Hamano, 1979; Zhang et al., 1988; Li et al., 1995; Moilanen and Leppävuori, 2001). Laser-Doppler vibrometry has already been used to support the measurement of piezoelectric coefficients of langasite (La₃Ga₅SiO₁₄) (Ogi et al., 2002) and lithium niobate (LiNbO₃) (Ogi et al., 2003). Here, the LDV measurement data were used to clearly assign the maxima of frequency spectra recorded by other methods to the respective vibration modes. This confirms that laser-Doppler vibrometry is suitable for measuring vibrations of the order of kHz or MHz. However, the authors are not aware of any measurements on LiTaO₃ with the exception of the somewhat indirect measurement mentioned in the previous section (Royer and Kmetik, 1992).

Yamada et al. (1969) calculated temperature-dependent values for d33 and predicted a steady increase with temperature, from 9.2 pm V⁻¹ at room temperature to approximately 16 pm V⁻¹ around 400 °C, which corresponds to the range experimentally characterized in this work. This is followed by an even stronger increase in the values towards the Curie temperature. This behavior is known for materials such as perovskites (Acosta et al., 2017) or hydrogen-bonded ferroelectrics (Ren et al., 2021) but is not reported in other publications on ferroelectric tantalates. Smith and Welsh (1971) calculated the temperature dependence in the range from 0 to 110 °C. However, they only observed a very moderate increase from 5.7 to 6.3 pm V⁻¹, being an average increase of 5.5×10-3 pm (V K)⁻¹ only. To the best of the authors' knowledge, there are no other experimental characterizations or, in particular, any direct characterizations of the temperature dependence of d33 in LiTaO₃. However, de Castilla et al. (2017) used the electrochemical impedance spectroscopy resonance method to investigate the very similar crystal system LiNbO₃ with sample dimensions comparable to those in this work. Since the thickness vibration mode of a very similar crystal was investigated here, it can be assumed that the temperature dependence is similar to that of the LiTaO₃ crystal in this work. In de Castilla's measurements, the piezoelectric coefficient shows a slight increase over temperature, averaging about 1.6×10-5 pm V K⁻¹. In the measurement range investigated in this work up to 400 °C, the specific value increased by 1.2 % from 2.46 to 2.49 pm V⁻¹.

Measurements are carried out either at a constant frequency by performing a lateral scan across the sample surface or at a constant position by varying the excitation frequency.

The first approach is suited especially for displacements far from any resonances. For the samples discussed in this work, measurements were carried out at 55 kHz. Here, the measured displacement equals the piezoelectric constant at the cost of having relatively small displacements and, thus, a low SNR.

The latter approach is generally applied in the vicinity of a resonance frequency. In this work, the thickness vibration of the sample at 5.8 MHz was chosen to demonstrate this. In resonance, the absolute displacements and therefore the SNR are higher as well compared to the off-resonant state. To evaluate the piezoelectric constants now, the resonator's Q factor also has to be taken into account in addition to the measured displacement and the applied excitation voltage. The benefit of characterizing in the resonant state is that both vibrational modes aligned with the laser geometry and out-of-plane movements can be evaluated (not shown in this work).

The following subsections briefly explain how to calculate the sample displacement in both off-resonant and resonant states. In the off-resonant state, this leads to the direct determination of the piezoelectric constants. In the resonant state, a resonance amplification occurs. Frequency sweeps in this range can show the frequency-dependent displacement in the vicinity of resonance. With this, resonance is detectable even without an additional electrical measurement, e.g., by a network analyzer or a frequency response analyzer. In the context of this work, this approach is also used to validate that there are no resonances near the measurement frequencies used for the determination of the piezoelectric constants in the off-resonant state. As the sample geometry and the chosen piezoelectric constant d33 are very well suited for the characterization in the off-resonant state, the resonant state approach was not used for the evaluation of the piezoelectric constants within this work.

An important parameter for a piezoelectric resonator is its quality factor Q. It indicates how efficiently the resonator stores mechanical or electrical energy relative to the energy dissipated (lost) per oscillation cycle due to losses. Besides this stored-energy definition, the bandwidth definition is quite common as it is easily accessible by fitting admittance or conductance spectra: 7Q=fRFWHM. Here, fR is the resonance frequency and FWHM the full width at half maximum. Further, the Lorentzian function is used here as a very suitable approximation of the physical solution, which is a Bessel function (Chen et al., 2020). The advantage of the Lorentz fit lies in the direct determination of the characteristic parameters without explicit modeling of all circuit elements. In particular, the FWHM is directly accessible from the fit (Göpel et al., 1994; Bund and Schwitzgebel, 1998).

Figure 2a depicts the amplitude when varying the excitation frequency in the vicinity of the resonance frequency fR (thickness mode vibration) taken at the center spot of the sample. It is clearly visible that the LDV amplitude, i.e., the total displacement in the Z direction for the given geometry (excitation || displacement), shows a clearly visible resonant amplification around fR, which decays back to the base level at an interval of approx. 2–3 kHz. The amplitude is symmetrical to fR and can be fitted very well with a Lorentz function, justifying the approach of applying the usual model of a conductance peak of a linearly damped piezoelectric resonator (Sauerwald et al., 2011) for determining, e.g., the Q factor.

The LiTaO₃ resonator characterized in this series of experiments has a quality factor of approximately 1700. This Lorentz fit can also be used to estimate the undisturbed region, which encompasses all frequency ranges in which the displacement peak signal has dropped to a level that is no more than 1 % above the baseline such that the resonance no longer makes a significant contribution. For the thickness oscillation shown in Fig. 2a, this is a distance of ± 18.8 kHz apart from the resonance frequency at 5.83086 MHz. Outside this range, the behavior of the sample can then be regarded as off resonant again.

To demonstrate that a characterization described by the Lorentz function is also suitable for the mechanically determined resonance characterization using LDV, comparative electrical characterization of the resonators was performed using a network analyzer (NWA). For this, the excitation voltage and resulting current response were continuously (every 10–100 ns) tracked in parallel with the LDV measurements. In the vicinity of resonance, the current applied to the sample also changes when the excitation voltage is kept constant. In a piezoelectric resonator, the resonance frequency results in a minimization of electrical impedance, as the inductive and capacitive components cancel each other out due to a vanishing phase shift. At a constant applied voltage, this results in an increased current flow according to Ohm's law. This effect reflects the efficient coupling between electrical excitation and mechanical vibration in the resonant case. The ratio of excitation voltage to current response represents the resistance shift over frequency in the vicinity and is thus comparable to an impedance spectroscopy pattern. The corresponding measurement curves from LDV (excitation voltage divided by current response) and from NWA are shown in Fig. 2b. The figure highlights the strong agreement between the resonance minima determined by both approaches. The higher noise level in the LDV data is attributed to longer acquisition times (approximately 2 h per scan) and thus increased sensitivity to environmental fluctuations compared to the rapid NWA measurement (2–3 s).

When taking measurements sufficiently off the resonance frequency, a fundamental question is whether the resolution of the measuring device, which is of the order a few picometers for the investigated frequency range (Wulfmeier et al., 2021; Schewe et al., 2023; Kohlmann et al., 2024), is sufficient to resolve the displacements. At 10 V excitation voltage, the determined displacements are in the range of several hundred picometers and thus still exhibit an SNR of at least better than 5, mostly being in the range of 10 to 20, which is more than sufficient for a precise determination of the piezoelectric constants.

Figure 3 shows the individual piezoelectric coefficients distributed over the surface of the sample. As expected, the displacement is homogeneous across the entire surface at a sufficient distance from the edge of the electrode. This fact verifies the approach. Setup and measurement conditions do not influence the displacement at the center of the sample, i.e., the volume of interest for characterizing the piezoelectric constants. For a more detailed discussion on the possible influence on LDV measurements of additional vibration or deformation modes, piezoelectric coupling, boundary conditions, and the contribution of electrode-free regions, see Appendices E, F, and G.

The complete deflection without mechanical disturbances and, thus, also the correct undisturbed piezoelectric coefficient can be determined. However, slight deviations in the individual measurement data are evident, likely representing edge effects due to fringing fields. Consequently, the peripheral areas are omitted when averaging across the measurement scan lines, as indicated in Fig. 1e. As a consequence, in order to find a good average of the coefficients and to avoid interference with, e.g., edge effects, the averaging for the determination of the piezoelectric coefficient is limited to the central ∼ 60 % of the electrode radius.

These mean values of d33, averaged over this central sample area, are displayed in Fig. 4 for each temperature step. The uncertainties do not allow a reliable statement to be made as to whether there is a slight increase in d33 of LiTaO₃ over temperature or not in the range from 21 to 400 °C. A constant value for d33 would still be in accordance with the statistical deviation, just as a linear fit that allows for slight temperature dependence would be. The latter is depicted in Fig. 4 and shows a slight increase, which, in terms of its magnitude, is comparable to the literature data predicted by Smith and Welsh (1971) and the values measured by de Castilla et al. (2017) on LiNbO₃.

As already discussed in Sect. 2.1, Table 1, the literature values at room temperature differ considerably, ranging from 5.7 to 23.1 pm V⁻¹. They thus vary by a factor of more than 4. The data obtained in this study are within this range, at 12.1 pm V⁻¹ at 21 °C and up to 15.1 pm V⁻¹ at elevated temperatures, and are therefore plausible. No clear reason for the wide variation in the literature data could be found. Since the available references often do not provide any information on this, it may also be due to the samples themselves, and each measurement is correct for the given characterization. Possible factors that could influence the piezoelectric constant include the stoichiometry (congruent, near stoichiometric, stoichiometric) and impurities or doping with elements, e.g., Mg or F (Ye et al., 1988). Other factors that might mainly influence the effective piezoelectric constant or the material's polarization are grain boundaries and domains (Gopalan et al., 2007; Meng et al., 2022).

It is striking that the indirectly determined values are consistently lower than the literature values obtained from direct methods involving mechanical contact with the sample. The respective mean values are 7.9 ± 1.6 pm V⁻¹ for the indirect and 16.7 ± 6.3 pm V⁻¹ for the direct methods.

The data obtained from the direct-measurement methods, however, were acquired in contact with the sample. Here, the influence of the measuring equipment can be overestimated, especially for small measured values, and leads to values that are too high (Berg et al., 2003; Vlachová et al., 2015).

In contrast to all other literature values, the values obtained in this study were determined both directly and without contact. Thus, general measurement deficiencies, as described above, should play a minor role here. This is also supported by the fact that our values fall between the indirect- and direct-contact literature data.

A literature comparison of the quality factor determined experimentally in this work is not applicable, as previous investigations of the Q factor of various vibrations of lithium tantalate and lithium niobate have focused on thin layers (Jacob et al., 2003; Jacob et al., 2004; Kadota et al., 2011, 2021; Lu et al., 2021) and thus describe vibrations in the gigahertz range. As far as is known, there are no studies of Q for lithium tantalate in the kilohertz or megahertz range, nor are there any characterizations of Q using interferometry or laser-Doppler vibrometry.

In this study, lithium tantalate (LiTaO₃, LT) is used as a model material for demonstrating the measurement technique. At elevated temperatures, however, LT is generally no longer suitable for use as a sensor or actuator (Turner et al., 1994) due to its low Curie temperature of 600–630 °C (Levinstein et al., 1966; Glass, 1968; Johnston and Kaminow, 1968; Samuelsen and Grande, 1976). Lithium niobate (LiNbO₃, LN), in contrast, has a significantly higher Curie temperature of around 1140 °C (Smolenskii et al., 1966); however, it suffers from stability problems at higher temperatures above 700 °C (Weidenfelder et al., 2012), particularly due to chemical decomposition (Damjanovic, 1998), which could be stabilized by adding LT. In the form of a solid solution, lithium niobate tantalate (LNT) single crystals have the potential to overcome the disadvantages of both individual materials, and LNT is expected to be usable at elevated temperatures above 600 °C (Suhak et al., 2021; Yakhnevych et al. 2024).