3.1.1. Elastic Constants of Sapphire Substrates

As mentioned in Section 2, for extraction of the elastic properties of AlScN films with thicknesses smaller than 1 µm from SAW dispersion curves in a frequency range below 600 MHz, the elastic moduli and the density of the substrate have to be known with high precision. In Table 3, data from the literature are compiled for Al2O3. In the case of *c*12, the values of the different sources are at variance by more than 3%, and the value of *c*<sup>14</sup> by more than 7%, in addition to the problem of finding the correct sign for this constant, which was resolved in [27,28].


**Table 3.** Elastic constants *c*µν for sapphire, GPa.

\* Calculated from compliances. In all calculations we took the same density *ρ* = 3982 kg/m<sup>3</sup> and dielectric permittivities *ε*<sup>11</sup> = 9.34 and *ε*<sup>33</sup> = 11.54 from [33].

LU measurements were carried out on two samples with sapphire substrates in two different orientations, Al2O3(0001) (Sample 1 in Table 2) and Al2O3(1-102) (Sample 2 in Table 2). The surfaces of both samples were coated with a Mo layer of ~50 nm thickness. Sample 1 had an additional AlScN film of 1 µm thickness between the substrate and the Mo layer. Figure 3b shows the dependence of surface acoustic pulses in the Al2O3(1-102) sample, coated with a Mo layer only (Sample 2 in Table 2), detected at a fixed distance from the line source, on the wavevector direction (surface slope at observation point as a function of the arrival time and angle *θ*). In Figure 4, SAW dispersion curves obtained from the LU measurements on Sample 2 in Table 2 are shown for five different wavevector directions on the surface. Because of the small thickness of the metal layer in comparison to the SAW wavelength, the dispersion curves are essentially straight lines. The SAW phase velocity for the uncoated sapphire surface is obtained by extrapolating these straight lines to zero frequency. In this way, the dependence of the SAW phase velocity on the wavevector direction and hence the slowness curve of SAWs on this surface of sapphire can be determined experimentally.

Figure 3d shows the results of simulations of the local surface displacement *u*<sup>3</sup> for Al2O3(1-102) as a response to a surface traction (1). Note that the quantity measured by probe-beam deflection in our LU setup is the local surface slope, i.e., the directional derivative of *u*<sup>3</sup> along the SAW wavevector. In a homogeneous medium with planar surface, this response depends on the frequency *ω*/(2*π*) and wavelength 2*π*/|*k*| of the excitation (surface traction in (1)) via the ratio *ω*/|*k*| only (phase velocity in Figure 3c,d). The SAW phase velocities correspond to the maxima of |*u*3| (bright curves emerging in Figure 3c,d). Comparison with Figure 3b confirms the expected inverse behavior of the SAW phase velocity and the delay time of the SAW pulses. *Micromachines* **2022**, *13*, x FOR PEER REVIEW 7 of 18 phase velocity for the uncoated sapphire surface is obtained by extrapolating these straight lines to zero frequency. In this way, the dependence of the SAW phase velocity on the wavevector direction and hence the slowness curve of SAWs on this surface of sapphire can be determined experimentally.

**Figure 3.** Dependencies on SAW wavevector direction for the sapphire cuts: c-plane Al2O3(0001) **(a,c,e)** with Euler angles (0°, 0°, θ) and r-plane Al2O3(1-102) **(b,d,f)** with Euler angles (60°, 57.6°, θ). Measured signal on the surface of sample 1 in Table 2 **(a)** and of sample 2 in Table 2 **(b)**. Measurements taken at various angles using a rotary positioning table; **(c)** and **(d)** show the calculated SAW phase velocities on the surface of pure sapphire crystal as functions of Euler angle θ; **(e)** and **(f)** present the intersections of the slowness surface of bulk acoustic waves with the surface plane (quasi-longitudinal sheet (BAW L), two quasi-shear sheets (BAW S1 and BAW S2)) and the slowness curve for Rayleigh waves (SAW) on pure sapphire as functions of the Euler angle θ. **Figure 3.** Dependencies on SAW wavevector direction for the sapphire cuts: c-plane Al2O<sup>3</sup> (0001) (**a**,**c**,**e**) with Euler angles (0◦ , 0◦ , *θ*) and r-plane Al2O<sup>3</sup> (1-102) (**b**,**d**,**f**) with Euler angles (60◦ , 57.6◦ , *θ*). Measured signal on the surface of sample 1 in Table 2 (**a**) and of sample 2 in Table 2 (**b**). Measurements taken at various angles using a rotary positioning table; (**c**,**d**) show the calculated SAW phase velocities on the surface of pure sapphire crystal as functions of Euler angle *θ*; (**e**,**f**) present the intersections of the slowness surface of bulk acoustic waves with the surface plane (quasi-longitudinal sheet (BAW L), two quasi-shear sheets (BAW S1 and BAW S2)) and the slowness curve for Rayleigh waves (SAW) on pure sapphire as functions of the Euler angle *θ*.

**Figure 4.** Experimental dispersion curves for the directions θ = 15°, 18°, 21°, 24°, 27° on r-plane sapphire with a 50 nm molybdenum film on top (Sample 2 in Table 2). Extrapolation of fitted straight lines to zero frequency gives the values of phase velocity (black crosses) on pure sapphire. **Figure 4.** Experimental dispersion curves for the directions *θ* = 15◦ , 18◦ , 21◦ , 24◦ , 27◦ on r-plane sapphire with a 50 nm molybdenum film on top (Sample 2 in Table 2). Extrapolation of fitted straight lines to zero frequency gives the values of phase velocity (black crosses) on pure sapphire.

Figure 3d shows the results of simulations of the local surface displacement ଷ for Al2O3(1-102) as a response to a surface traction (1). Note that the quantity measured by probe-beam deflection in our LU setup is the local surface slope, i.e., the directional derivative of ଷ along the SAW wavevector. In a homogeneous medium with planar surface, this response depends on the frequency /(2) and wavelength 2/|| of the excitation (surface traction in (1)) via the ratio /|| only (phase velocity in Figure 3c,d). The SAW phase velocities correspond to the maxima of |ଷ| (bright curves emerging in Figure 3c,d). Comparison with Figure 3b confirms the expected inverse behavior of the SAW phase velocity and the delay time of the SAW pulses. The weakness of the detected signal in certain intervals of angle θ, for example in the neighborhood of θ = 12° and θ = 78° in Figure 3d, can be explained by the smallness of the out-of-plane displacement amplitude |ଷ| in these angular intervals. The unusual features in the dependence of the SAW phase velocity on the wavevector direction in the case of the r-plane surface are related to a phenomenon visible in Figure 3f. Here, the slowness curve of SAW propagating on the r-plane surface of sapphire is shown together with the intersection curves of the surface and the three sheets of the slowness surface of bulk The weakness of the detected signal in certain intervals of angle *θ*, for example in the neighborhood of *θ* = 12◦ and *θ* = 78◦ in Figure 3d, can be explained by the smallness of the out-of-plane displacement amplitude |*u*3| in these angular intervals. The unusual features in the dependence of the SAW phase velocity on the wavevector direction in the case of the r-plane surface are related to a phenomenon visible in Figure 3f. Here, the slowness curve of SAW propagating on the r-plane surface of sapphire is shown together with the intersection curves of the surface and the three sheets of the slowness surface of bulk acoustic waves (quasi-longitudinal, fast quasi-shear, and slow quasi-shear). In the angular regions corresponding to wavevector directions with faint signals (two at the same wavevector direction) and comparatively strong variation of the SAW phase velocity, the SAW slowness curve and the intersection curve of the sheet of slow quasi-shear bulk waves approach, and a repulsion of these two modes can be seen. Here, a transition occurs from the usual situation of the SAW velocity being the lowest phase velocity of acoustic modes for a given wavevector direction parallel to the surface, to an interval of wavevector directions where the phase velocity of SAW is larger than that of the slow quasi-shear bulk waves. This effect does not occur in the case of Al2O3(0001) (Figure 3c,e).

acoustic waves (quasi-longitudinal, fast quasi-shear, and slow quasi-shear). In the angular regions corresponding to wavevector directions with faint signals (two at the same wavevector direction) and comparatively strong variation of the SAW phase velocity, the SAW slowness curve and the intersection curve of the sheet of slow quasi-shear bulk waves approach, and a repulsion of these two modes can be seen. Here, a transition occurs from the usual situation of the SAW velocity being the lowest phase velocity of acoustic modes for a given wavevector direction parallel to the surface, to an interval of wavevector directions where the phase velocity of SAW is larger than that of the slow quasi-shear bulk waves. This effect does not occur in the case of Al2O3(0001) (Figure 3c,e). The experimental results for the SAW phase velocities in Al2O3(1-102) were then compared with the results of calculations performed with the semi-analytic Greens function The experimental results for the SAW phase velocities in Al2O3(1-102) were then compared with the results of calculations performed with the semi-analytic Greens function method. Input data for these calculations are the sets of elastic moduli listed in Table 3. For the density of sapphire, the value *ρ* = 3982 kg/m<sup>3</sup> [31] was used, and for the dielectric constants we used the values *ε*<sup>11</sup> = 9.34 and *ε*<sup>33</sup> = 11.54 [33] in all our calculations. The off-cut angle of 3◦ was accounted for. It was found that the set of elastic constants reported by Gladden et al. [27] and also the constants provided by Gieske and Barsch [30], the latter after correction of the sign of *c*14, fit best to the LU data and lead to very good agreement between calculated and measured phase velocities. A comparison of calculated phase velocities with three different input sets and measured values as functions of wavevector direction is provided in Figure 5.

method. Input data for these calculations are the sets of elastic moduli listed in Table 3. For the density of sapphire, the value *ρ* = 3982 kg/m3 [31] was used, and for the dielectric constants we used the values ଵଵ = 9.34 and ଷଷ = 11.54 [33] in all our calculations. The off-cut angle of 3° was accounted for. It was found that the set of elastic constants reported by Gladden et al. [27] and also the constants provided by Gieske and Barsch [30], the latter after correction of the sign of *c*14, fit best to the LU data and lead to very good agreement This finding is confirmed by measurements and calculations of SAW dispersion curves for 1000 nm thick Al0.77Sc0.23N(0001)/Al2O3(0001) coated with ~50 nm Mo (Sample 1 in Table 2). Two different wavevector directions were considered. The dispersion due to the presence of the AlScN film leads to a broadening of the detected SAW pulse shapes (Figure 3a), and the dispersion curves in Figure 6 are no longer straight lines. In the calculations, the elastic, piezoelectric, and dielectric constants of [5] were used. The material constants of the molybdenum layer, *c*<sup>11</sup> = 440 GPa, *c*<sup>44</sup> = 126 GPa, and density *ρ* = 10,280 kg/m<sup>3</sup> , were taken from [34]. A comparison in Figure 6 shows very good agreement between experimental dispersion curves and those calculated with the elastic constants of sapphire taken from [30] with the corrected sign of *c*14. kg/m3, were taken from [34]. A comparison in Figure 6 shows very good agreement between experimental dispersion curves and those calculated with the elastic constants of sapphire taken from [30] with the corrected sign of *c*14.

ρ

ρ

= 10280

= 10280

constants of the molybdenum layer, *c*11 = 440 GPa, *c*44 = 126 GPa, and density

tween experimental dispersion curves and those calculated with the elastic constants of

between calculated and measured phase velocities. A comparison of calculated phase velocities with three different input sets and measured values as functions of wavevector

This finding is confirmed by measurements and calculations of SAW dispersion curves for 1000 nm thick Al0.77Sc0.23N(0001)/Al2O3(0001) coated with ~50 nm Mo (Sample 1 in Table 2). Two different wavevector directions were considered. The dispersion due to the presence of the AlScN film leads to a broadening of the detected SAW pulse shapes (Figure 3a), and the dispersion curves in Figure 6 are no longer straight lines. In the calculations, the elastic, piezoelectric, and dielectric constants of [5] were used. The material

*Micromachines* **2022**, *13*, x FOR PEER REVIEW 9 of 18

constants of the molybdenum layer, *c*11 = 440 GPa, *c*44 = 126 GPa, and density

between calculated and measured phase velocities. A comparison of calculated phase velocities with three different input sets and measured values as functions of wavevector

This finding is confirmed by measurements and calculations of SAW dispersion curves for 1000 nm thick Al0.77Sc0.23N(0001)/Al2O3(0001) coated with ~50 nm Mo (Sample 1 in Table 2). Two different wavevector directions were considered. The dispersion due to the presence of the AlScN film leads to a broadening of the detected SAW pulse shapes (Figure 3a), and the dispersion curves in Figure 6 are no longer straight lines. In the calculations, the elastic, piezoelectric, and dielectric constants of [5] were used. The material

direction is provided in Figure 5.

sapphire taken from [30] with the corrected sign of *c*14.

direction is provided in Figure 5.

*Micromachines* **2022**, *13*, x FOR PEER REVIEW 9 of 18

**Figure 5.** Theoretical dependencies of SAW phase velocities on wavevector direction for r-plane sapphire (solid and dashed lines) calculated using different sets of elastic constants taken from the literature (see Table 3), and compared with experimental data (black crosses). **Figure 5.** Theoretical dependencies of SAW phase velocities on wavevector direction for r-plane sapphire (solid and dashed lines) calculated using different sets of elastic constants taken from the literature (see Table 3), and compared with experimental data (black crosses). sapphire (solid and dashed lines) calculated using different sets of elastic constants taken from the literature (see Table 3), and compared with experimental data (black crosses).

**Figure 5.** Theoretical dependencies of SAW phase velocities on wavevector direction for r-plane

**Figure 6**. Experimentally obtained dispersion curves for the directions θ = 0° (red circles) and 30° (black circles) on the structure consisting of c-plane sapphire with a 1 µm thick Al0.77Sc0.23N film and a 50 nm molybdenum film on top, compared with corresponding simulated dispersion curves **Figure 6.** Experimentally obtained dispersion curves for the directions *θ* = 0◦ (red circles) and 30◦ (black circles) on the structure consisting of c-plane sapphire with a 1 µm thick Al0.77Sc0.23N film and a 50 nm molybdenum film on top, compared with corresponding simulated dispersion curves (solid and dashed lines for *θ* = 30◦ and for *θ* = 0◦ , respectively). The latter were obtained with material constants of sapphire from different authors (see Table 3).
