3.2.1. C-Plane Samples

The symmetry of the c-plane geometry implies that [0◦ , 30◦ ] is an irreducible interval of angles *θ* for the SAW wavevector directions. LU measurements were taken for nine different wavevector directions corresponding to angles *θ* in this interval. For each direction, the SAW pulse shapes were recorded at 32 different distances from the source. From these data, the SAW dispersion curves were determined in a frequency range from 50 up to 400 MHz, shown in Figure 9. They exhibit a small, but non-negligible curvature.

Simulated dispersion curves for the nine wavevector directions are also presented in Figure 9 for comparison.

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**Figure 9.** Experimental and theoretical dispersion curves for the different directions θ for Ti/Al0.77Sc0.23N(0001)/Al2O3(0001). In the calculations, the elastic constants from [30] for sapphire and material constants from [5] for AlScN were used. **Figure 9.** Experimental and theoretical dispersion curves for the different directions *θ* for Ti/Al0.77Sc0.23N(0001)/Al2O<sup>3</sup> (0001). In the calculations, the elastic constants from [30] for sapphire and material constants from [5] for AlScN were used. Ti/Al0.77Sc0.23N(0001)/Al2O3(0001). In the calculations, the elastic constants from [30] for sapphire and material constants from [5] for AlScN were used.

θfor

**Figure 9.** Experimental and theoretical dispersion curves for the different directions

#### 3.2.2. a-plane samples 3.2.2. A-Plane Samples 3.2.2. a-plane samples Because of the lower symmetry of this configuration, an irreducible interval of SAW

Because of the lower symmetry of this configuration, an irreducible interval of SAW wavevector directions on this surface is the range of angles θ between 0° and 90°. In Figure 10, experimental dispersion curves are shown for wavevector directions with angles θ from 15° to 70° in steps of 5° and, in addition, for θ = 90°. For angles in the vicinity of 10° or 80°, the detected signals are very weak. The reason is presumably that here the surface displacements in the direction normal to the surface are much smaller than for other wavevector directions and the same frequency, since the SAW penetrates deeply into the substrate. Here, two very weak signals emerge because of mode repulsion (see Section 3.1.1) instead of one strong signal, leading to inaccuracies in the data processing. Because of the lower symmetry of this configuration, an irreducible interval of SAW wavevector directions on this surface is the range of angles *θ* between 0◦ and 90◦ . In Figure 10, experimental dispersion curves are shown for wavevector directions with angles *θ* from 15◦ to 70◦ in steps of 5◦ and, in addition, for *θ* = 90◦ . For angles in the vicinity of 10◦ or 80◦ , the detected signals are very weak. The reason is presumably that here the surface displacements in the direction normal to the surface are much smaller than for other wavevector directions and the same frequency, since the SAW penetrates deeply into the substrate. Here, two very weak signals emerge because of mode repulsion (see Section 3.1.1) instead of one strong signal, leading to inaccuracies in the data processing. wavevector directions on this surface is the range of angles θ between 0° and 90°. In Figure 10, experimental dispersion curves are shown for wavevector directions with angles θ from 15° to 70° in steps of 5° and, in addition, for θ = 90°. For angles in the vicinity of 10° or 80°, the detected signals are very weak. The reason is presumably that here the surface displacements in the direction normal to the surface are much smaller than for other wavevector directions and the same frequency, since the SAW penetrates deeply into the substrate. Here, two very weak signals emerge because of mode repulsion (see Section 3.1.1) instead of one strong signal, leading to inaccuracies in the data processing. The dispersion curves show a modest amount of curvature which is favorable for the

The dispersion curves show a modest amount of curvature which is favorable for the extraction of elastic constants of the AlScN film. The experimental dispersion curves are in good agreement with the results of calculations, carried out with input data from [5]. The dispersion curves show a modest amount of curvature which is favorable for the extraction of elastic constants of the AlScN film. The experimental dispersion curves are in good agreement with the results of calculations, carried out with input data from [5]. extraction of elastic constants of the AlScN film. The experimental dispersion curves are in good agreement with the results of calculations, carried out with input data from [5].

**Figure 10.** Experimental (dots) and theoretical (lines) dispersion curves for various SAW propagation directions for Mo/Al0.77Sc0.23N(11-20)/Al2O3(1-102). For the calculations the constants from [30] and [5] were used. **Figure 10.** Experimental (dots) and theoretical (lines) dispersion curves for various SAW propagation directions for Mo/Al0.77Sc0.23N(11-20)/Al2O3(1-102). For the calculations the constants from [30] and [5] were used. **Figure 10.** Experimental (dots) and theoretical (lines) dispersion curves for various SAW propagation directions for Mo/Al0.77Sc0.23N(11-20)/Al2O<sup>3</sup> (1-102). For the calculations the constants from [5,30] and were used.

### *3.3. Sensitivity Analysis 3.3. Sensitivity analysis*  In order to assess to what extent the agreement between the dispersion curves meas-

In order to assess to what extent the agreement between the dispersion curves measured with the LU method and those simulated with the calculated material constants of [5] can be taken as a confirmation of the latter, the sensitivity of the dispersion curves with respect to changes of each single elastic constant of the AlScN film has been analyzed (Figure 11). With the same material data used for the simulation of the dispersion curves in Figures 9 and 10, we calculated the relative change ∆*v/v* of phase velocity *v* with a 1% increase for each of the five independent elastic constants *c*µν (see Table 3), while leaving the remaining material constants unchanged. Figure 11a,b show results of this calculation for the c-plane geometry and the a-plane geometry, respectively. ∆*v/v* is plotted as a function of wavevector direction at the fixed frequency 400 MHz. This frequency value has been chosen since it corresponds to the upper edge of the frequency interval of the measured dispersion curves for the c-plane samples and is located in the upper third of the frequency band for the a-plane geometry. With increasing frequency, the fraction of the SAW displacement field localized in the AlScN film is expected to rise as well. When comparing the relative velocity changes in Figure 11a,b, one has to account for the slightly different thicknesses of AlScN layer in the two types of samples (1 µm for the c-plane, 860 nm for a-plane samples). ured with the LU method and those simulated with the calculated material constants of [5] can be taken as a confirmation of the latter, the sensitivity of the dispersion curves with respect to changes of each single elastic constant of the AlScN film has been analyzed (Figure 11). With the same material data used for the simulation of the dispersion curves in Figure 9 and Figure 10, we calculated the relative change Δ*v/v* of phase velocity *v* with a 1% increase for each of the five independent elastic constants *c*μν (see Table 3), while leaving the remaining material constants unchanged. Figure 11a and Figure 11b show results of this calculation for the c-plane geometry and the a-plane geometry, respectively. Δ*v/v* is plotted as a function of wavevector direction at the fixed frequency 400 MHz. This frequency value has been chosen since it corresponds to the upper edge of the frequency interval of the measured dispersion curves for the c-plane samples and is located in the upper third of the frequency band for the a-plane geometry. With increasing frequency, the fraction of the SAW displacement field localized in the AlScN film is expected to rise as well. When comparing the relative velocity changes in Figure 11a and Figure 11b, one has to account for the slightly different thicknesses of AlScN layer in the two types of samples (1 µm for the c-plane, 860 nm for a-plane samples).

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**Figure** 11. Calculated relative change of phase velocity with 1% increase of elastic constants for Mo/Al0.77Sc0.23/Al2O3 structures with 1 µm thick Al0.77Sc0.23N film and additional Mo coating with 50 nm thickness, as a function of SAW propagation direction on the surface. (a) case of AlScN(0001)/Al2O3( 0001) and (b) AlScN(11-20)/Al2O3(1-102). **Figure 11.** Calculated relative change of phase velocity with 1% increase of elastic constants for Mo/Al0.77Sc0.23/Al2O<sup>3</sup> structures with 1 µm thick Al0.77Sc0.23N film and additional Mo coating with 50 nm thickness, as a function of SAW propagation direction on the surface. (**a**) case of AlScN(0001)/Al2O<sup>3</sup> (0001) and (**b**) AlScN(11-20)/Al2O<sup>3</sup> (1-102).

In the case of the c-plane samples, the sensitivities of the SAW phase velocity with respect to changes in the elastic constants of the AlScN film are comparatively small, except for the constant *c*11. The isotropy of the hexagonal film in the surface plane and the moderate deviations of the SAW slowness curve from a circle on c-plane sapphire (Figure 3e) are reflected in a very weak dependence of the sensitivities with respect to the wavevector direction (Figure 11a), with the exception of the sensitivity to changes of *c*12, which vanishes at the wavevector direction with Euler angle θ = 30°. In the case of the c-plane samples, the sensitivities of the SAW phase velocity with respect to changes in the elastic constants of the AlScN film are comparatively small, except for the constant *c*11. The isotropy of the hexagonal film in the surface plane and the moderate deviations of the SAW slowness curve from a circle on c-plane sapphire (Figure 3e) are reflected in a very weak dependence of the sensitivities with respect to the wavevector direction (Figure 11a), with the exception of the sensitivity to changes of *c*12, which vanishes at the wavevector direction with Euler angle *θ* = 30◦ .

In the case of the a-plane geometry, the sensitivities exhibit a remarkable dependence on the wavevector direction. This is due to the strong anisotropy of the film in the surface plane, and it is also associated with the strong variation of the SAW mode pattern in the neighborhood of the Euler angles θ, where the SAW slowness curve of r-plane sapphire crosses the intersection curve of the slowness surface of acoustic bulk waves. Moreover, the sensitivities are on average clearly larger than those on the c-plane samples. In the case of the a-plane geometry, the sensitivities exhibit a remarkable dependence on the wavevector direction. This is due to the strong anisotropy of the film in the surface plane, and it is also associated with the strong variation of the SAW mode pattern in the neighborhood of the Euler angles *θ*, where the SAW slowness curve of r-plane sapphire crosses the intersection curve of the slowness surface of acoustic bulk waves. Moreover, the sensitivities are on average clearly larger than those on the c-plane samples.

A feature of particular interest is the relative size of the sensitivities for the directions with Euler angles *θ* in the vicinity of 0◦ on the one hand and in the neighborhood of 90◦ on

the other. In the first range of wavevector directions, the sensitivities with respect to *c*<sup>11</sup> and *c*<sup>12</sup> dominate, while the sensitivities with respect to the other elastic moduli are largely negligible. In the second range, the dispersion curve is mainly sensitive to *c*<sup>33</sup> and *c*<sup>13</sup> and the other elastic constants play a largely negligible role. Knowledge of this behavior should be very helpful for fitting strategies to extract the elastic constants from measured dispersion curves. At wavevector directions with Euler angles *θ* around 45◦ , the relative velocity variations with relative changes of *c*11, *c*33, and *c*<sup>44</sup> are of comparable size. In general, one may notice that for each elastic constant there are ranges of wavevector directions where this constant has a non-negligible influence on the SAW dispersion curves. Figure 12 shows how the sensitivities vary as functions of frequency for the fixed wavevector directions with Euler angles *θ* = 0◦ , 45◦ and 90◦ . The sensitivities with respect to almost all elastic constants increase at higher frequencies because of the increasing localization of the SAW displacement field in the AlScN film. In the case of *θ* = 0◦ , the sensitivities with respect to *c*<sup>11</sup> and *c*<sup>12</sup> dominate over the whole frequency range from zero to 600 MHz, and likewise the sensitivities with respect to *c*<sup>33</sup> and *c*<sup>13</sup> in the case of *θ* = 90◦ . and *c*12 dominate, while the sensitivities with respect to the other elastic moduli are largely negligible. In the second range, the dispersion curve is mainly sensitive to *c*33 and *c*13 and the other elastic constants play a largely negligible role. Knowledge of this behavior should be very helpful for fitting strategies to extract the elastic constants from measured dispersion curves. At wavevector directions with Euler angles θ around 45°, the relative velocity variations with relative changes of *c*11, *c*33, and *c*44 are of comparable size. In general, one may notice that for each elastic constant there are ranges of wavevector directions where this constant has a non-negligible influence on the SAW dispersion curves. Figure 12 shows how the sensitivities vary as functions of frequency for the fixed wavevector directions with Euler angles θ = 0°, 45° and 90°. The sensitivities with respect to almost all elastic constants increase at higher frequencies because of the increasing localization of the SAW displacement field in the AlScN film. In the case of θ = 0°, the sensitivities with respect to *c*11 and *c*12 dominate over the whole frequency range from zero to 600 MHz, and likewise the sensitivities with respect to *c*33 and *c*13 in the case of θ = 90°. The sensitivities of the dispersion curves for SAW in AlScN films on sapphire, dis-

A feature of particular interest is the relative size of the sensitivities for the directions

on the other. In the first range of wavevector directions, the sensitivities with respect to *c*<sup>11</sup>

in the vicinity of 0° on the one hand and in the neighborhood of 90°

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with Euler angles

θ

The sensitivities of the dispersion curves for SAW in AlScN films on sapphire, discussed above, may be compared with those for a c-plane Al0.68Sc0.32N film on a Si(001) substrate, presented in [17] for two different wavevector directions. (Note that the data in Figure 3 of [17] refer to a relative change of 10% of the elastic and piezoelectric constants. The thickness of the AlScN(0001)/Si(001) was ~1 µm.) Remarkably, the sensitivities for the c-plane film on silicon are of comparable size to those of the a-plane sample. However, in both wavevector directions in the c-plane film on the silicon substrate, the influence of the elastic constant *c*<sup>11</sup> of AlScN dominates. cussed above, may be compared with those for a c-plane Al0.68Sc0.32N film on a Si(001) substrate, presented in [17] for two different wavevector directions. (Note that the data in Figure 3 of [17] refer to a relative change of 10% of the elastic and piezoelectric constants. The thickness of the AlScN(0001)/Si(001) was ~1 µm.) Remarkably, the sensitivities for the c-plane film on silicon are of comparable size to those of the a-plane sample. However, in both wavevector directions in the c-plane film on the silicon substrate, the influence of the elastic constant *c*11 of AlScN dominates. Figure 3b in [17] clearly shows that the sensitivities of the SAW phase velocities with

Figure 3b in [17] clearly shows that the sensitivities of the SAW phase velocities with respect to the piezoelectric constants (from D. Urban et al. in Reference [5]) are by more than one order of magnitude smaller than the ones with respect to the elastic constants. respect to the piezoelectric constants (from D. Urban et al. in Reference [5]) are by more than one order of magnitude smaller than the ones with respect to the elastic constants.

**Figure** 12. Relative change of phase velocity with 1% increase of elastic constants in the structure AlScN(11-20)/Al2O3(1-102) with 1000 nm thick Al0.77Sc0.23N film and 50 nm molybdenum film as function of frequency for the cases (a) θ = 0°; (b) θ = 45°; (c) θ = 90°. **Figure 12.** Relative change of phase velocity with 1% increase of elastic constants in the structure AlScN(11-20)/Al2O<sup>3</sup> (1-102) with 1000 nm thick Al0.77Sc0.23N film and 50 nm molybdenum film as function of frequency for the cases (**a**) *θ* = 0◦ ; (**b**) *θ* = 45◦ ; (**c**) *θ* = 90◦ .

#### **4. Discussion and Conclusions 4. Discussion and Conclusions**

The main results of the investigations reported in this contribution may be summarized as follows: The main results of the investigations reported in this contribution may be summarized as follows:

• SAW dispersion curves were measured by laser ultrasound for various wavevector directions in c-plane and a-plane Al0.77Sc0.23N films on sapphire substrates. They are in very good agreement with the corresponding theoretical dispersion curves computed with the elastic moduli and piezo-electric • SAW dispersion curves were measured by laser ultrasound for various wavevector directions in c-plane and a-plane Al0.77Sc0.23N films on sapphire substrates. They are in very good agreement with the corresponding theoretical dispersion curves computed with the elastic moduli and piezo-electric constants obtained in ab initio calculations by Urban et al. [5]. The theoretical elastic constants for Al1-xScxN in [5] are given in the form of interpolation formulas quadratic in the parameter x. For the Sc concentrations, x = 0.14 and x = 0.32, the authors of [5] compare their calculated elastic constants with corresponding data determined by Kurz et al. [10] with the help

of SAW resonators. For both concentrations, the agreement is very good (2% deviation on average, less than 5% in the worst case). This confirms the high-quality of the theoretical data in [5] and may also be regarded as an additional, indirect confirmation of the measured SAW dispersion curves presented here.


The size of the sensitivities at around 400 MHz leads us to the following conclusion. Assuming that a relative change of velocity by 6 m/s is resolvable in the LU experiment, a change of 2% in the elastic constants *c*<sup>11</sup> of *c*<sup>33</sup> of the AlScN should be detectable. With an efficient use of the anisotropy of both the substrate and the film and with sufficiently small attainable SAW wavelengths on the scale of the film thickness (i.e., a sufficiently large frequency range), a determination of the elastic constants of AlScN films should be achievable. Even in the long-wavelength limit, when the dispersion curves are essentially straight lines, certain combinations of material constants can be extracted.

If the minimal achievable SAW wavelength is larger or of the order of the film thickness, such that the SAW displacement field penetrates deeply into the substrate, the elastic properties of the substrate as well as its orientation (e.g., off-cut angles) have to be known to a high precision. On the other hand, if the frequency range can be considerably extended to higher values, higher-order guided acoustic modes can be used in addition to the lowest SAW to gain information about the elastic moduli of the film [11].

The sensitivity analysis in [17] confirms that the piezoelectric constants have a very small influence on the SAW dispersion curves if compared to the elastic constants. However, depending on the required accuracy in the determination of the elastic moduli, auxiliary measurements with an alternative technique, such as the resonator method [10,11] will be needed.

The measured SAW dispersion curves for c-plane and a-plane Al0.77Sc0.23N compare very favorably with dispersion curves calculated with the data for the elastic and piezoelectric constants in [5] for all SAW wavevector directions. These data were obtained in ab initio calculations treating the atomic positions as static, which means that phononic thermal contributions are disregarded. Very rough estimates on the basis of data for the temperature dependence of Young's modulus of AlN ceramics [37] and predictions of the temperature dependence of the elastic moduli of AlN with highly simplifying assumptions [38] suggest that the thermal contributions at room temperature to the elastic moduli are smaller than 1% of their total value. The sensitivities of the SAW dispersion curves, discussed above, imply that such small variations cannot be resolved with our current laser ultrasound setup.

In conclusion, the results for AlScN films on sapphire substrates, presented in this work, confirm that laser ultrasound can be applied as a viable tool for the determination of elastic properties of anisotropic, including piezoelectric films on anisotropic substrates. This requires measurements of SAW dispersion curves for various propagation directions, which can be pre-selected by simulations and a detailed sensitivity analysis. In comparison to isotropic films, anisotropy poses an additional challenge because of an increased number of independent elastic constants. At the same time, the anisotropy of the film and of

the substrate offers the possibility of gaining additional information on different elastic constants from certain different SAW propagation directions.

**Author Contributions:** Conceptualization, E.A.M., A.D., A.P.M., A.Ž.; methodology, A.M.L., E.A.M., P.D.P.; software, A.M.L., E.A.M., A.P.M., P.D.P.; validation, E.A.M.; formal analysis, E.A.M., O.R.; investigation, E.A.M., O.R.; resources, A.D., A.N., A.Ž.; data curation, E.A.M., O.R.; writing, A.P.M., E.A.M., A.D., A.N., A.Ž.; visualization, E.A.M., A.M.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Fraunhofer Society, project no. 005-601187. A.N. was funded by the COMET Centre ASSIC Austrian Smart Systems Integration Research Center, which is funded by BMK, BMDW, and the Austrian provinces of Carinthia and Styria, within the framework of COMET—Competence Centers for Excellent Technologies. The COMET program is run by FFG.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data are available from the corresponding authors upon reasonable request.

**Acknowledgments:** The authors thank Lutz Kirste and the Structural analysis team at Fraunhofer IAF for their help with the X-ray diffraction experiments and Niclas Feil and Oliver Ambacher for fruitful discussions.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
