*3.2. Film on Substrate: Al0.77Sc0.23N/Al2O<sup>3</sup>*

In order to acquire information about the elastic constants of AlScN films on sapphire substrates, LU measurements have been carried out for the two systems Al0.77Sc0.23N(0001)/ Al2O3(0001) and Al0.77Sc0.23N(11-20)/Al2O3(1-102) (in the following called "c-plane samples" and "a-plane samples" for short, respectively, Samples 1 and 3 in Table 2).

SAW dispersion curves were simulated with the semi-analytic Greens function method. Input data for the substrate material in these calculations were the set of elastic constants from [30] with the corrected sign of *c*<sup>14</sup> along with the mass density of sapphire from [31] and dielectric constants from [33] (see Section 3.1.1).

The material constants chosen for the isotropic metal layer molybdenum are given in Section 3.1.1.

For the elastic and piezoelectric constants of the Al0.77Sc0.23N films, we used the data obtained in ab initio calculations, based on the density functional theory, by Urban et al. [5]. In [5], an interpolation formula is given for the mass density and for each individual elastic constant *c*µν(x) and piezoelectric constant *e*iµ(x) of Al1-xScxN containing terms of up to second order in the scandium concentration x. The coefficients in these formulas were obtained by a fit to the corresponding material constants from ab initio calculations for a set of scandium concentrations x. As a second variant, Urban et al. suggest a rescaling of their interpolation formulas such that in the limit x to zero, the material constants take the values calculated for pure AlN. The SAW dispersion curves simulated with these two variants of material constants for the Al0.77Sc0.23N films are indistinguishable on the scales of the graphs in Figures 6, 9 and 10.

For both independent dielectric constants *ε*11, *ε*<sup>33</sup> the value 17.56 was used in the calculations. It follows from an interpolation formula of experimental data for *ε*33, provided in [36] (see also [10]), applied to the scandium concentration x = 0.23.
