# Investigation of the Young’s Modulus and the Residual Stress of 4H-SiC Circular Membranes on 4H-SiC Substrates

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{+}4H-SiC substrate and n

^{−}4H-SiC epilayer, enabling the fabrication of freestanding thin films. Moreover, the crystalline quality of the epilayer, and so, the suspended thin-film, is not affected by the process [7]. This method could pave the road for the fabrication of novel SiC-based detectors [6]. However, designing such original film for MEMS applications requires knowing the mechanical properties of the epilayers. Several popular approaches exist to monitor the static behavior of free-standing films: Curvature measurements [8,9], beam-bending testing [10,11], Raman spectroscopy [12,13], nanoindentation [14], and bulge test [15,16]. Moreover, dynamic techniques based on the resonance frequency determination of thin film are also intensively used [17,18]. In order to determinate the mechanical properties (Young’s modulus and residual stress values) of the 4H-SiC film, this study aims to evaluate two experimental techniques: The bulge test and the vibrating method.

## 2. Materials and Methods

#### 2.1. Sample Preparation

^{18}cm

^{−3}n-type substrate (supplied from CREE

^{®}, Durham, NC, USA). The epitaxial layer is around 9 µm thick 10

^{13}cm

^{−3}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 [6,21,22,23]. The process is capable of removing highly doped (≥ 10

^{18}cm

^{−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

^{13}cm

^{−3}doped n

^{−}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 effects related to the internal stress of a film. Indeed, the stress is equi-biaxial, so the loading will not produce any discontinuity.

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

_{0}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. C

_{1}and C

_{2}are dimensionless constants, which depend on the membrane shape. Note that C

_{2}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 coefficients can be estimated, leading to the determination of the Young’s modulus and the residual stress.

_{1}and C

_{2}values, 4 and 8/3, respectively. A more accurate numerical solution indicated that C

_{2}can be expressed as (8/3) × (1.015 − 0.247υ). Table 1 summarizes the reported value of C

_{1}and C

_{2}for circular suspended films from literature. For comparison purposes, C

_{2}values assuming υ = 0.25 are also listed.

#### 2.3. Membrane Vibration

#### 2.3.1. Analytical Description

_{0}is the residual stress, ρ is the density of the 4H-SiC thin-film and α

_{mn}values are derived from the roots of the first order Bessel functions. Zeros of the Bessel functions can be computed and tabulated [17].

#### 2.3.2. Finite Element Method Approach

#### 2.4. Load-Deflection Measurements

^{2}printed circuit board (PCB) holder with a drilled hole in the centre. Several studies have highlighted the importance of the bonding step for the reliability of the deflection measurements. The most common approach to fix the sample is to add adhesive around its edges. In such case, Jayaraman et al. observed that the sample moved during the measurement for a pressure up to 2.8 bars [35]. Mitchell et al. proposed a multi-step bonding method to seal and constrain the sample to the chuck without any displacement of the substrate [34]. Inspired by this method, we deposited an Ablebond 84-3J epoxy adhesive on the PCB and mounted the sample on it. Then, we applied the adhesive around all the edges of the sample to seal and prevent air leakage. Lastly, an annealing step of 1 h at 150 °C was carried out. For the bulge testing, the chip was placed in an airtight square cell, drilled on two lateral faces, in order to inject and measure the air pressure. The membrane is pressurized through its cavity while the front face remains at the atmospheric pressure. So, the sample was characterized under differential pressures, between 0.04 and 4 bars. Pressure regulation and measurements were carried out using both pressure controller and sensors, operating in the range of 0 to 4 bars.

_{atm}) was measured before and after the chip preparation. Values of h are close, around 1.5 µm. So, the impact of the stress induced by the sample preparation seems to be negligible as shown in Figure 3b. The deflection profile was recorded at each stabilized pressure level. A scanning area of 4700 × 4700 µm

^{2}was defined in the centre of the membrane. Thus, we obtained a mapping including the maximum deflection point in the centre and also the edges of the membrane as presented in Figure 3c.

#### 2.5. Dynamic Behavior Measurements

^{2}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.

## 3. Results and Discussion

#### 3.1. Bulge Test Results

_{0}. 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 C

_{2}. 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 agreement 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 σ

_{0}seems to be less dependent on the model used since C

_{1}is considered as constant.

#### 3.2. Vibrometry Results

#### 3.3. Finite Element Computations

_{0}between 15 and 45 MPa. Figure 7b presents the simulations results for the first resonance frequency. The couple (E, σ

_{0}) that provides the best fit with the measured resonance frequency was obtained for E = 400 GPa and σ

_{0}= 30 MPa. Consequently, we used these mechanical parameters to calculate the five other resonance frequencies. We reported the results in Figure 7a. The resonance frequencies obtained by FEM simulations using σ

_{0}= 30 MPa seem to be in good agreement with the measured resonance frequencies. Thus, it can be assumed that the residual stress value mainly governs the mechanical behavior of the membrane.

#### 3.4. Etching Profile Determination

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**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.

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

**Figure 3.**(

**a**) Schematic of bulge test apparatus; (

**b**) deflection of the circular 4H-SiC membrane, before and after, sample mounting; (

**c**) typical topography used to measure the diaphragm deflection with LSM measurement.

**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.

**Figure 6.**Measured spectrum of vibration of the 4H-SiC membrane associated with the corresponding mode shapes.

**Figure 7.**(

**a**) Dashed lines: Computed resonance frequencies obtained with FEM calculations using the bulge test results. Solid line: Adjusted FEM calculations with the couple (E, σ

_{0}). Square symbols: Measured resonance frequencies determined with the vibrometry method; (

**b**) calculated resonance frequency depending on E and σ

_{0}. Dot line: Measured resonance frequency for the (0, 1) mode. Symbol lines: Calculated resonance frequencies depending on the residual stress and Young’s modulus values.

**Figure 8.**(

**a**) LSM cross-section image of the etching profile; (

**b**) LSM image of the membrane-undercut boundary.

Models | C_{1} | C_{2} | C_{2}(υ = 0.25) | Approach |
---|---|---|---|---|

Lin [28] | 4.0 | (7 − υ)/3 | 2.25 | Energy minimization |

Beams [27] | 4.0 | 8/3 | 2.67 | Spherical cap |

Small et al. [29] | 4.0 | (8/3) × (1 − 0.241 × υ) | 2.51 | Finite Element Method |

Pan et al. [30] | 4.0 | (8/3)/(1.026 + 0.233 × υ) | 2.46 | Finite Element Method |

Hohlfelder et al. [31] | 4.0 | (8/3) × (1.015 − 0.247 × υ) | 2.54 | Number approximation |

Timoshenko et al. [32] | - | (8/3) × 0.976/(1 + υ) | 2.08 | Energy minimization |

Parameters | Values |
---|---|

A (Pa/m) | 3.0 × 10^{8} |

B (Pa/m^{3}) | 4.6 × 10^{17} |

Density (kg/m^{3}) | 3210 |

Poisson’s ratio | 0.25 |

Membrane radius (µm) | 2250 |

Membrane thickness (µm) | 8.8 |

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**MDPI and ACS Style**

Ben Messaoud, J.; Michaud, J.-F.; Certon, D.; Camarda, M.; Piluso, N.; Colin, L.; Barcella, F.; Alquier, D.
Investigation of the Young’s Modulus and the Residual Stress of 4H-SiC Circular Membranes on 4H-SiC Substrates. *Micromachines* **2019**, *10*, 801.
https://doi.org/10.3390/mi10120801

**AMA Style**

Ben Messaoud J, Michaud J-F, Certon D, Camarda M, Piluso N, Colin L, Barcella F, Alquier D.
Investigation of the Young’s Modulus and the Residual Stress of 4H-SiC Circular Membranes on 4H-SiC Substrates. *Micromachines*. 2019; 10(12):801.
https://doi.org/10.3390/mi10120801

**Chicago/Turabian Style**

Ben Messaoud, Jaweb, Jean-François Michaud, Dominique Certon, Massimo Camarda, Nicolò Piluso, Laurent Colin, Flavien Barcella, and Daniel Alquier.
2019. "Investigation of the Young’s Modulus and the Residual Stress of 4H-SiC Circular Membranes on 4H-SiC Substrates" *Micromachines* 10, no. 12: 801.
https://doi.org/10.3390/mi10120801