*2.5. Dynamic Behavior Measurements*

The natural frequencies of thin film can be measured by several methods. Each technique integrates mechanical excitation setup and optical system. In this study, we used a piezoelectric actuation and a laser system for measuring the film deformation amplitude. The sample was glued using silver paste onto a lead zirconate titanate (PZT) disk and excited by a periodic burst signal at a voltage range between 0.1–10 Vpp. An MSA-500 scanning laser Doppler vibrometry (LDV), from Polytec GmbH (Waldbronn, Germany), was used in order to experimentally investigate the dynamic behavior of the membrane through the identification of its resonance modes. Measurements were performed in air. The diameter of the diaphragm exceeded the aperture angle of the MSA lenses. Consequently, for one acquisition, an area of 890 × 660 μm<sup>2</sup> is measured using the 20× objective. Thus, a stitching method was developed in order to scan the whole membrane surface allowing the determination of its vibrations mode shape. Among all the observed modes, we focused our attention on the first six out-of-plane vibration modes. As an example, Figure 4 shows different measured mode shapes of the 4H-SiC circular membrane.

**Figure 4.** (**a**) Schematic diagram of the resonance frequency method. Vibration mode shapes measured using laser Doppler vibrometry for (**b**) (1, 1); (**c**) (0, 2); and (**d**) (1, 2) modes.

## **3. Results and Discussion**

## *3.1. Bulge Test Results*

As previously explained, the Young's modulus and the residual stress can be determined using the least square fit of Equation (1). The load-deflection response at the membrane centre *h* versus *P* is shown in Figure 5.

**Figure 5.** Dot line: Bulge test results for 4H-SiC diaphragm. Solid line: Theoretical fit.

For the calculations, we fixed the Poisson's ratio value at 0.25, which is the most common value used in the literature. It has been pointed out by several authors that the influence of υ in the bulge test is negligible [34,37]. Indeed, our calculations confirm that a variation of υ from 0.2 to 0.3 impacts the Young's modulus value for less than 10%. The parameters used for fitting the data are listed in Table 2.

**Table 2.** Parameters used for both bulge test and finite element method (FEM) calculations.


Table 3 reports the determined values of *E* and σ0. The calculated Young's modulus values are scattered, depending on the model used. In fact, the main difference between these models is the expression of *C2*. Beams was the first to report a value for this dimensionless coefficient, using the spherical cap model based on a very simple approximation of the real case. However, it can lead to an under estimation of the mechanical property determination [29,38]. Therefore, the models proposed by Pan et al. and Small et al. led to close Young's modulus values, which seems to be normal as both models were adjusted from finite element calculations [20]. Moreover, using the numerical solution proposed by Hohlfelder, we obtained almost the same Young's modulus value. Mitchell et al. explained that the difference in the governing equation, which results in measured values, could vary by as much as 20% [34]. In any case, the calculated Young's modulus values are in reasonably good agreemen<sup>t</sup> with the published results in the literature for silicon carbide thin films. The residual stress value is determined using the linear term in Equation (1). In comparison with *E*, the stress σ0 seems to be less dependent on the model used since *C1* is considered as constant.


**Table 3.** Bulge test results depending on the models used.
