Micro-electro-mechanical systems (MEMS) are micro-scale sensors and actuators manufactured by modern micro-manufacturing technology (Wang et al., 2009) within multi-disciplinary and multi-physics fields such as machinery, electronics, biology, acoustics, and energy (Bi-Qiang et al., 2006). Compared with conventional sensors, MEMS sensors have great advantages in terms of volume, price, properties, and capacity (Fitzgerald et al., 2021) and are gradually replacing those traditional sensors in many fields such as automotive, biomedical devices, and personal electronics. (Bogue and Review, 2013; Mohd Ghazali et al., 2020).

The microphone is one of the key acoustic-electric transducers which convert acoustic signals into electrical signals (Choe and Bulat, 2005; Arora et al., 2018). Proposed in 1962, electret capacitive microphones (ECMs) are low-cost, high-sensitivity, with a broadband, which have revolutionized the microphone industry (Kapps and Dobbins, 2014; Adorno et al., 2022). The ECMs have created built-in charges due to the existence of an electret layer (i.e., a thin plastic foil that can hold the charge) and are standard solutions for telephones and personal electronic products. The first MEMS microphone was introduced in 1983 (Ozdogan et al., 2020). However, it was not until 2003 when the first silicon-based MEMS microphone became the mass production and Knowles launched the first SiSonic^TM MEMS microphone (Resound, 2014), which pioneered the development of the MEMS market. Compared with the ECMs, MEMS microphones are compatible with integrated circuits (IC) technology and surface mount technology (SMT) (Oh et al., 2021) and have shown much better mechanical shock resistance and anti-interference performance, which have gradually developed into the mainstream of microphones in the commercial market. Today, new types of MEMS microphones are still being developed in the market, and it is reported that the annual output of MEMS microphones is as high as billions (Zhang et al., 2021).

In order to meet the stringent requirements of different industries, a variety of MEMS microphone products have been developed, and the prototype of MEMS microphone for each demand has to be significantly shortened to reduce the development cycle and reduce the development cost. Therefore, an efficient design methodology and techniques are required. Common design methods for MEMS devices include the lumped element method (LEM) and the finite element method (FEM).

In 2017, Rufer et al. proposed an accurate lumped parameter equivalent circuit to simulate various configurations of capacitive microphones (Esteves et al., 2017). Studies have shown that this equivalent model can easily be adapted into other specific microscopic configurations. In 2019, Lee et al. (2019) proposed a dynamic open circuit sensitivity model based on electrical equivalent circuits, which has shown high accuracy and has produced results that are only ∼2% different from the simulation results of FEM. Auliya et al. (2019) proposed a 2D finite element model to analyze the performance of MEMS microphones. The 2D model is used to simplify the calculation by applying axisymmetric conditions, which greatly reduces the application range of the model. In 2021, Shubham et al. (2021) demonstrated a MEMS microphone chip with flexible springs in the diaphragm and developed a lumped element model to predict the mechanical and electrical behavior of the microphone. Experiments show that there is a good correlation between the measured curve and the simulated curve.

The aforementioned equivalent methods have achieved good results in the prediction of the sensitivity and frequency response of the microphones, but there still exist quite different issues, because some parameters in these models require to be obtained by FEM or experiments, which is often inconvenient in the prototype design stage of MEMS microphones. Research and development of MEMS microphones involves multiple thermo-electro-mechanical couplings and various physical and electrical fields. Unfortunately, there is currently no any accurate and efficient three-dimensional (3D) MEMS microphone simulation platform, which can be applied for design of its chip parameters and packaging characteristics.

In this study, we proposed a 3D simulation platform for MEMS microphones using the commercial software COMSOL, and designed a set of parameters to describe the structure of the MEMS microphone. The sensitivity and frequency responses of the microphone were investigated using the proposed platform, and the accuracy of the platform was evaluated by comparing the obtained results with the simulation results using the LEM.

A typical MEMS microphone is composed of a MEMS chip, an ASIC chip, a printed circuit board, and a metal shell with an acoustic port, as illustrated in Figure 1A,B. The MEMS chip mainly includes a rigid backplane and an elastic thin diaphragm (Naderyan et al., 2021), which are normally made of polysilicon materials. There are a large number of air holes in the silicon backplane, which allow air to pass through in order to reduce air damping (Naderyan et al., 2020). The diaphragm also needs vent holes to balance the pressure between the front cavity and the back cavity. In a first approximation, as shown in the basic schematic illustration as shown in Figure 1C, the capacitive MEMS microphone sensor can be modeled as a parallel plate capacitor (Sant et al., 2019). In this model, the DC bias voltage V b i a s is applied to the variable capacitor through a resistance R b i a s with a maximum resistance value. The initial air gap height of the capacitor is d 0 , the effective area is S , the accumulated charge of the capacitor is Q , and the initial capacitance is C 0 . The large time constant makes the capacitor work in a constant charge mode.

When the acoustic pressure P   ( 1 Pa ) is applied to the diaphragm to cause its vibration, the capacitance of this capacitor changes with a value of Δ C , when the air gap of the parallel plate is changed with Δ d . Δ C = ε S d 0 − Δ d = ε S d 0 . 1 1 − Δ d / d 0 = C 0 . d 0 d 0 − Δ d . (1)

Since the charge of the parallel plate capacitor is assumed to be a constant, the output voltage Δ V of the capacitor will change with the capacitance value. An ASIC chip can be used to read the voltage on the capacitor plate and process it when needed. Δ V = Q Δ C = V b i a s Δ d d 0 . (2)

The sensitivity of the sensor is the ratio of the specified output to the specified input. Therefore, the sensitivity of the microphone can be defined as S 0 = Δ V P = ( V b i a s d 0 ) Δ d . (3)

In this work, a 3D FEM platform for MEMS microphones was proposed using the commercial COMSOL software, as shown in Figure 2. In this simulation platform, graphics can be used to show the geometry of the microphone model, a top toolbar provides geometry, calculation buttons, and presets for some drawing commands, and a form collection presents the modifiable model parameters. In order to reduce the calculation time, symmetrical boundary conditions are applied on the longitudinal section and only half of the MEMS microphone structure is modeled based on the symmetrical MEMS microphone structure. The study of MEMS microphone’s sensitivity and frequency responses involve multi-physics fields such as “Thermoviscous Acoustics,” “Membrane,“ “Electrostatics,” and “Circuit.“

In addition, a lumped element model was also developed to describe the dynamic behavior of the MEMS microphone and evaluate the sensitivity and frequency responses of the microphone, as shown in Figure 3. Table 1 presents the description of each lumped element used in this model. The equivalent circuit model of our MEMS microphone was simulated in the simulation program with integrated circuit emphasis (SPICE), in which the transformer adopted the ideal transformer model composed of controlled sources. The geometric and material parameters of each part of the MEMS microphone are listed in Table 2.

Figure 4A shows the calculated results obtained by the FEM using COMSOL software and the LEM model, respectively, which show good consistency, demonstrating the efficiency, accuracy, and reliability of the 3D FEM platform. The sensitivity of the MEMS microphone at 1 kHz is ?56.3 dBV and ?56.6 dBV for the FEM model and LEM model, respectively, and the difference is only 0.5 % . For both the FEM model and the LEM model, the change in sensitivity of frequency response is less than ± 3 dBV from 100 Hz to 12 kHz. The resonance frequencies are 21.5 and 19.5 kHz for the FEM model and the LEM model, respectively, indicating the consistency of the frequency results. The LEM model appears to show a relatively higher sensitivity at the resonance frequency. This is because in the lumped element model, the nonlinear dependence of capacitance on the diaphragm displacement is often overestimated by the piston approximation, leading to a significant overestimation of the spatial derivative of capacitance (Anzinger et al., 2021) and a higher sensitivity.

Visualization of calculation results makes designers more intuitive to find the deficiencies in the model, which is also one of the advantages of our new platform. Figure 4b∼d and Figure 5 show the results from the FEM simulation. Figure 4B shows the displacement distributions of the diaphragm under the simulation sound pressure of 1 Pa. The displacement at the center of the diaphragm is the largest, and the displacement on both sides decreases gradually. The displacement of the membrane at different times of 1 kHz is shown in Figure 5. Figure 4C depicts the potential of the symmetrical plane of the microphone motor, and the obtained results clearly show that the holes on the backplane will affect the field. It can be seen that the electric field has a very strong gradient in the area where the electrode is located, and it will be decreased outside that area. Figure 4Dillustrates the temperature oscillation distribution at 1 kHz in the enclosure. Due to the isothermal boundary condition, the amplitude of the temperature oscillation becomes zero at the boundary between the enclosure and the surface of the silicon chip and reaches a peak at the region between the enclosure wall and the silicon chip, which is about 98.9 μ K .

In order to ensure the universal capability of the established finite element platform for MEMS microphones, some parameters of the MEMS microphone were varied in the simulations. Figure 6 describes the influences of air gap height ( d 0 ) , bias voltage ( V b i a s ) and front cavity volume ( V f ) on MEMS microphone frequency responses, respectively. The results obtained from the two methods have shown good consistency.

Figure 6A shows that the dynamic open circuit sensitivity is -58.5 dBV, -56.3 dBV, and -53.4 dBV at the gap heights of 4.7, 3.7, and 2.7 μm, respectively. Figure 6C shows the dynamic open circuit sensitivities of −56.3 dBV, −52.7 dBV, and −48.9 dBV at 3.3, 5.0, and 8.0 V bias, respectively. Results show that reducing air gap height or increasing bias voltage can improve the sensitivity of MEMS microphones, which is compatible with the prediction from Eq. 3. Figure 6E shows that when the volumes of the front cavity are 1.04 m m 3 , 1.24  m m 3 and 1.47  m m 3 , the sensitivity of the microphone is about—56.3 dBV, and the resonant frequencies are 23.0, 21.5, and 20.0 kHz, respectively. It can be seen that the smaller volume of the front cavity will result in smaller frequency responses and higher resonant frequencies of the microphone, mainly because of the acoustic port and the front cavity forming a Helmholtz resonator (Mackey et al., 2019).

Based on the aforementioned analysis, we can obtain the MEMS microphone with a dynamic open circuit sensitivity of −48.9 dBV at an air gap height of 3.7 μm, a bias voltage of 8.0 V, which is much higher than that (−55.4 dBV) in previous work (Lee et al., 2012).

In this study, we introduce a 3D simulation platform for MEMS microphone based on commercial COMSOL software, which can be used to study the influences of microphone geometric or physical parameters on its sensitivity and frequency responses. The results are in good agreement with those calculated by the LEM. The model is built with parameters of the inputs of the sensor, which can be readily changed by the designer. After the parameters have been changed, the model will be automatically reconstructed and meshed. The visualization of research results helps designers more intuitively find the shortcomings of the model, optimize and improve the model, and greatly promote the prototype design stage.