*2.1. Sample Preparation*

The 4H-SiC membrane was fabricated starting from a wafer with homo-epitaxial layer grown on 375 μm thick 10<sup>18</sup> cm<sup>−</sup><sup>3</sup> n-type substrate (supplied from CREE ®, Durham, NC, USA). The epitaxial layer is around 9 μm thick 10<sup>13</sup> cm<sup>−</sup><sup>3</sup> n-type. The substrate removal was obtained by electrochemical etching. ECE is an oxidation/oxide-removal process obtained by dipping the SiC wafer in a hydrofluoridric acid-based solution and electrically supply holes for the oxidation through a 100 nm aluminium back metal contact [6,21–23]. The process is capable of removing highly doped (≥ 10<sup>18</sup> cm<sup>−</sup>3) p-type and n-type layers but is selective towards low-doped n-type layers (selectivity > 1000:1 with respect to the 5 × 10<sup>13</sup> cm<sup>−</sup><sup>3</sup> doped n<sup>−</sup> layer). Hence, this process allows the full removal of the highly doped substrate and the local release of the epitaxial layers. The realized membranes have thicknesses and uniformities determined by the epitaxial layer. The 4H-SiC suspended film with circular shape was

fabricated at the Paul Scherrer Institute [6]. In addition, the circular shape is the most appropriate geometry in order to study the e ffects related to the internal stress of a film. Indeed, the stress is equi-biaxial, so the loading will not produce any discontinuity.

A Lext OLS4100 Laser Scanning Microscope (LSM), from Olympus Corporation (Shinjuku-ku, Tokyo, Japan), was used in order to observe the full membrane using the stitching mode. Figure 1a shows the image of the membrane shape after ECE process. One can observe a slight membrane asymmetric geometry, which is due to the fabrication process, i.e., to a non-uniform under-etching of the substrate with respect to the etching mask. Therefore, we assumed that the membrane can be reasonably assimilated as a circle. The diameter measurements were performed in several directions and an average diameter value was evaluated at 4.5 mm. Thickness determination was performed using a Strata DB235 Focused Ion Beam (FIB), from Thermo Fisher Scientific (Waltham, MA, USA). To do that, we injected silver paste through the cavity of the membrane in order to fix and sti ffen the suspended film to prevent any vibrations during the observations. After that, the FIB cross-section images were carried out in order to obtain a direct measurement of the film thickness as shown in Figure 1b. Thus, a membrane thickness of 8.8 μm was measured and a thickness variation of ± 0.2 μm was evaluated. As expected, the thin film has a well-controlled thickness because the 4H-SiC epilayer acts as an etching stop.

**Figure 1.** (**a**) Laser Scanning Microscope (LSM) image of the 4H-SiC membrane using stitching mode, obtained after the electrochemical etching (ECE) process. Circular insert added to show that the 4H-SiC membrane can be assimilated as a circle. (**b**) Focused ion beam (FIB) cross-section image allowing the membrane thickness determination.

#### *2.2. Circular Membrane Deflection under Uniform Pressure*

The bulge test method consists in submitting a membrane to uniform external pressure in order to observe its load-deflection behavior. At low value, the mechanical return forces of the membrane are mainly due to the residual stress. Whereas, when the pressure value increases, the return forces from plate sti ffness govern the displacement. The deflection of a clamped membrane is measured according to the applied pressure. The most common pressure-deflection relationship of a pre-stressed circular membrane can be expressed as [24]:

$$P(h) = C\_1 \frac{t \sigma\_0}{a^2} h + C\_2 \frac{t}{a^4} \left(\frac{E}{1-\upsilon}\right) h^3 = Ah + Bh\_\prime^3 \tag{1}$$

where *P* is the applied pressure, *t* is the membrane thickness, σ0 is the residual stress, *a* is the membrane radius, *h* is the maximum bulge deflection at the centre of the membrane, *E* is the Young's modulus, and υ is the Poisson's ratio of the 4H-SiC thin-film. *C1* and *C2* are dimensionless constants, which depend on the membrane shape. Note that *C2* is also a function of the Poisson's ratio. A schematic representation of a circular diaphragm is shown in Figure 2. By fitting the applied pressure as a

function of the measured deflection, *A* and *B* coe fficients can be estimated, leading to the determination of the Young's modulus and the residual stress.

**Figure 2.** Schematic cross-sectional membrane with initial in-plane tension under uniform pressure *P*.

The models describing the load-deflection behavior of a circular plate as a function of the pressure have been extensively discussed. Several authors examined the large deflection behavior for the pure plate case, originally described by Nádai and Way [25,26]. Beams was the first to report an experimental model using bulge test to measure the mechanical properties of thin films deposited on substrates [27]. For a circular membrane, Beams determined *C1* and *C2* values, 4 and 8/3, respectively. A more accurate numerical solution indicated that *C2* can be expressed as (8/3) × (1.015 − 0.247υ). Table 1 summarizes the reported value of *C1* and *C2* for circular suspended films from literature. For comparison purposes, *C2* values assuming υ = 0.25 are also listed.

**Table 1.** C1 and C2 coe fficients for circular membranes.

