Performance Analysis of Simulated Low-Temperature Co-Fired Ceramic Diaphragm Using Finite Element Method ^†

Abstract

In this study, a low-temperature co-fired ceramic (LTCC)-based circular diaphragm design was considered for Fabry–Pérot Interferometer (FPI) pressure sensor applications. The characteristics of the LTCC-based circular diaphragm were analyzed using FEM analysis. The selected thicknesses of the LTCC diaphragms were 50 μm, 75 μm and 100 μm, with diameters of 3 mm, 4 mm and 5 mm, respectively. Our results showed that the sensitivity and frequency response of this structure can be designed flexibly by adjusting the parameters of the ceramic diaphragm size, including the radius and thickness. The key contribution of this work is that it demonstrates the performance of LTCC diaphragms with different sizes, which could be useful for future works.

1. Introduction

MEMS (Microelectromechanical system) pressure sensors have been extensively studied and used for different applications. Miniaturization, a low fabrication cost and high performance are the some of their advantages over their macro-size pressure sensor counterparts [1,2]. The working principle of diaphragm-based MEMS sensors includes deformation of the diaphragm after applying external pressure and the measurement of capacitance or resistance due to this deformation [3,4]. The sensitivity and natural frequency of the diaphragm are important parameters in terms of the mechanical performance of the sensor, and thus, they should be considered in the design of pressure sensors. These parameters are simply calculated by using the material properties and geometrical parameters of the diaphragm [5,6,7]. As mentioned above, diaphragm material selection is a critical issue, in addition to shape or geometry optimization, to improve the mechanical performance of micro-fabricated pressure sensors. Although single-crystal silicon (Si), polysilicon (PolySi), graphene, and Si₃N₄ are the common diaphragm materials [8], LTCC (low temperature co-fired ceramic) is a good candidate as a diaphragm for high temperature applications of (FPI) Fabry–Pérot Interferometer pressure sensors [9,10]. However, there are limited studies in the literature regarding LTCC-based FPI sensor fabrication. One of these studies proposed and experimentally demonstrated the performance of a square diaphragm that has a length, width, and thickness of 10 mm, 10 mm, and 100 μm, respectively, for high-temperature applications [10].

In this study, an LTCC-based circular diaphragm design was considered for FPI pressure sensor applications. The characteristics of the LTCC-based circular diaphragm were analyzed using the finite element method. The sensitivity, resonance frequency and static deflection of the diaphragm were analyzed and evaluated to assess MEMS-based LTCC diaphragm performance. The selected thicknesses of the LTCC diaphragms were 50 μm, 75 μm and 100 μm, with diameters of 3 mm, 4 mm and 5 mm, respectively. The performance of diaphragms, which was assessed using ANSYS, was compared with analytical results. Our results showed that the sensitivity and frequency response of this structure can be designed flexibly by adjusting the parameters of the ceramic diaphragm size, including the radius and thickness. The key contribution of this work is that it shows the performance of circular LTCC diaphragms with different sizes, which could be useful for future works on LTCC-based sensor design and fabrication for different pressure sensor applications at high temperatures.

2. Materials and Methods

2.1. Low-Temperature Co-Fired Ceramic (LTCC)

LTCC has generally been used for the anodic bonding of Si wafers for more compact 3D device integration/packaging processes [11,12,13,14], and it is a good alternative to glass and TGV (Through Glass Via) substrate owing to its good material properties and low-resistance inner wirings embedded inside the layers [15]. It is a multilayer ceramic substrate that consists of an alumina–cordierite ceramic powder and Na₂O-Al₂O₃-B₂O₃-SiO₂ glass powder [15]. The material properties of LTCC substrate were reported in a previous study [16].

2.2. LTCC Diaphragm Design

In this study, the natural frequency and sensitivity of circular LTCC diaphragms are investigated using the finite element method (FEM), and the results are compared with theoretical calculations. For a flat circular diaphragm with a clamped edge, the center deflection caused by pressure is analytically expressed as [17]

$$\omega \left(r=0\right)={\displaystyle \frac{P{a}^4}{64D}}$$

(1)

where P is the pressure, a is the diaphragm radius and D is the flexural rigidity of the diaphragm. The frequency response is another important parameter, and it is given for a circular diaphragm with a clamped edge as below [17]

$$f={\displaystyle \frac{10.2}{2\pi }}\sqrt{{\displaystyle \frac{E}{12(1-v^2)\rho }}}{\displaystyle \frac{t}{a^2}}$$

(2)

Here, t is the thickness, E is the Young’s modulus, and ρ is the density of the diaphragm. Nine different designs were used for the FEM analysis. The selected thicknesses of the LTCC diaphragms were 50 μm, 75 μm and 100 μm, with diameters of 3 mm, 4 mm and 5 mm, respectively.

3. Results and Discussion

LTCC diaphragm deflection values with different sizes under pressure were obtained by FEM and then compared with the analytical results. Firstly, displacement of the diaphragm was simulated (Figure 1). Figure 2 shows the comparison of theoretical and FEM results of center displacement for 100 µm thick diaphragms with diameters of 3 mm, 4 mm and 5 mm as a function of pressure. It can be seen that the theoretical and FEM results are consistent. Sensitivity was obtained by dividing diaphragm deformation by the applied pressure, and these results are summarized in

Table 1. Sensitivity (nm/kPa).

	t (µm) = 50		t (µm) = 75		t (µm) = 100
Diameter (mm)	Theoretical	FEM	Theoretical	FEM	Theoretical	FEM
3	88.79	88.00–88.29	26.31	26–26.17	11.10	11.12
4	280.62	278.33–279.00	83.15	82.33–82.56	35	34.90
5	685.1	680–681.2	203	201–201.8	85.6	84–84.6

. These results indicated that sensitivity increases with the size of the diaphragm (diameter), as expected from Equation (1) (ω ∝ $a^4$). Similarly, sensitivity decreases as a function of diaphragm thickness due to a proportion of ω ∝ 1∕$t^3$ between the center deflection and diaphragm thickness.

The effects of diaphragm thickness and radius on the natural resonant frequency were also studied by using FEM analysis, and the results are presented in

Table 2. Comparison of FEM and theoretical natural frequency (kHz) results of LTCC diaphragm for different thicknesses and diameters.

	t (µm) = 50		t (µm) = 75		t (µm) = 100
Diameter (mm)	Theoretical (kHz)	FEM (kHz)	Theoretical (kHz)	FEM (kHz)	Theoretical (kHz)	FEM (kHz)
3	61.78	62.00	92.67	92.89	123.56	123.32
4	34.75	34.89	52.13	52.34	69.50	69.67
5	22.24	22.32	33.36	33.49	44.48	44.64

. Figure 3 shows the FEM results of resonant frequency for the 75 µm thick diaphragm with a diameter of 4 mm.

As is known from Equation (2), the natural frequency decreases as a function of radius and in proportion to the thickness of the diaphragm. The FEM results are very consistent with the theoretical results as expected.

4. Conclusions

In this work, the effect of diaphragm thickness and radius on the sensitivity and natural frequency of circular LTCC diaphragms was studied. It was evaluated using a finite element model (FEM), and the results were then compared to the analytical results. The main purpose of this study is to provide preliminary results for future works on LTCC-based diaphragm design and fabrication for high-temperature applications. Future work will include a performance analysis (numerical and analytical) of LTCC diaphragms with different geometries (circular, square, hexagon, etc.).