3.1.2. Off-Cut Angle in Al2O3(1-102) Substrates

In order to find out to what accuracy the LU method is able to determine the off-cut angle *χ* of the r-plane sapphire substrate, the SAW phase velocity was calculated as a function of wavevector direction for the values *χ* = −3 ◦ , 0◦ , +3◦ . A comparison of the calculated data with the measured SAW phase velocities in Figure 7 proves that an offset angle of ±3 ◦ is clearly detectable. It also reveals in which regions of wavevector directions weak for

gles χ

gles χ

θ

weak for

near 0°.

θ

χ

angle

angle

(solid and dashed lines for

χ

(solid and dashed lines for

the phase velocity is particularly sensitive to the off-cut. The wavevector directions around *θ* = 90◦ are particularly suitable for the detection of off-cut angles larger or equal to 3◦ , whereas directions with *θ* near 0◦ seem to be more suitable for values of the off-cut angle smaller than −3 ◦ . However, the detected signals of the SAW pulses were comparatively weak for *θ* near 0◦ . whereas directions with θ near 0° seem to be more suitable for values of the off-cut angle smaller than –3°. However, the detected signals of the SAW pulses were comparatively near 0°.

near 0° seem to be more suitable for values of the off-cut angle

= 0°, respectively). The latter were obtained with material

= –3°, 0°, +3°. A comparison of the calcu-

= –3°, 0°, +3°. A comparison of the calcu-

= 0°, respectively). The latter were obtained with material

θ

θ

χ

In order to find out to what accuracy the LU method is able to determine the off-cut

of the r-plane sapphire substrate, the SAW phase velocity was calculated as a

lated data with the measured SAW phase velocities in Figure 7 proves that an offset angle of ±3° is clearly detectable. It also reveals in which regions of wavevector directions the phase velocity is particularly sensitive to the off-cut. The wavevector directions around

smaller than –3°. However, the detected signals of the SAW pulses were comparatively

In order to find out to what accuracy the LU method is able to determine the off-cut

of the r-plane sapphire substrate, the SAW phase velocity was calculated as a

χ

χ

lated data with the measured SAW phase velocities in Figure 7 proves that an offset angle

phase velocity is particularly sensitive to the off-cut. The wavevector directions around

= 90° are particularly suitable for the detection of off-cut angles larger or equal to 3°,

*Micromachines* **2022**, *13*, x FOR PEER REVIEW 10 of 18

θ

= 30° and for

constants of sapphire from different authors (see Table 3).

θ

θ

*Micromachines* **2022**, *13*, x FOR PEER REVIEW 10 of 18

= 30° and for

θ

3.1.2. Off-cut angle in Al2O3(1-102) substrates

function of wavevector direction for the values

constants of sapphire from different authors (see Table 3).

3.1.2. Off-cut angle in Al2O3(1-102) substrates

function of wavevector direction for the values

θ

**Figure 7.** Experimental values of the SAW phase velocities for various propagation directions on rplane sapphire (black crosses) and corresponding calculated curves, taking into account off-cut an-= ±3°. Here the elastic constants from [30] were used for calculations. **Figure 7.** Experimental values of the SAW phase velocities for various propagation directions on r-plane sapphire (black crosses) and corresponding calculated curves, taking into account off-cut angles *χ* = ±3 ◦ . Here the elastic constants from [30] were used for calculations. = ±3°. Here the elastic constants from [30] were used for calculations. Figure 8 shows calculated SAW phase velocities for the correct off-cut angle χ = –3° and four additional values of . The experimental data for the phase velocities shown in

Figure 8 shows calculated SAW phase velocities for the correct off-cut angle χ = –3° and four additional values of χ. The experimental data for the phase velocities shown in this figure are downshifted by 3 m/s as an attempt to correct a systematic error in our measurements due to a misalignment between the pump beam and the moving direction Figure 8 shows calculated SAW phase velocities for the correct off-cut angle *χ* = −3 ◦ and four additional values of *χ*. The experimental data for the phase velocities shown in this figure are downshifted by 3 m/s as an attempt to correct a systematic error in our measurements due to a misalignment between the pump beam and the moving direction of the translation stage mentioned in Section 2.3. this figure are downshifted by 3 m/s as an attempt to correct a systematic error in our measurements due to a misalignment between the pump beam and the moving direction of the translation stage mentioned in Section 2.3.

**Figure 8.** Influence of the off-cut angle χ on the angular dependence of the SAW phase velocity on Al2O3(1-102). Here, the experimental results are downshifted by 3 m/s. **Figure 8.** Influence of the off-cut angle *χ* on the angular dependence of the SAW phase velocity on Al2O<sup>3</sup> (1-102). Here, the experimental results are downshifted by 3 m/s.

**Figure 8.** Influence of the off-cut angle χ on the angular dependence of the SAW phase velocity on Al2O3(1-102). Here, the experimental results are downshifted by 3 m/s. The comparison of the data in Figure 8 suggests that the off-cut angle of the r-plane geometry can be determined by LU with an accuracy of almost 1◦ with an optimized alignment of the optical and mechanical components. Obviously the accuracy reachable in the determination of the off-cut angle by LU strongly depends on the precision to which the elastic constants and the density of sapphire are known. A 0.1 % change of the mass density translates into a phase velocity shift of approximately 3 m/s.

> We note here that the SAW slowness curve on the r'-plane in Al2O<sup>3</sup> (Euler angles (0◦ , 57.6◦ , *θ*)) differs substantially from that on the r-plane (Euler angles (60◦ , 57.6◦ , *θ*)),

because the elastic constant *c*<sup>14</sup> has a non-negligible magnitude in comparison to the other elastic constants (Table 3). In the case of hexagonal symmetry, the two planes would be equivalent [35]. On the r'-plane, the SAW slowness curve does not cross an intersection curve of the bulk wave slowness surface with the crystal surface. Therefore, r-plane and r'-plane samples can easily and quickly be distinguished by an LU measurement.

The correct sign of *c*<sup>14</sup> has been clarified by ab initio calculations and new experiments ([27,28] and references therein). A change of sign of c<sup>14</sup> causes an interchange of the elastic properties of r-plane and r'-plane.
