# Wafer-Level-Based Open-Circuit Sensitivity Model from Theoretical ALEM and Empirical OSCM Parameters for a Capacitive MEMS Acoustic Sensor

^{*}

## Abstract

**:**

## 1. Introduction

## 2. On-Wafer Level Based Open-Circuit Sensitivity Model

#### 2.1. Capacitive MEMS Acoustic Sensors and Its Structure

_{2}/Si

_{3}N

_{4}insulator/Al metal/a SiO

_{2}insulator and the top was composed of TiN/PECVD-Si

_{3}N

_{4}/TiN multi layers. Air gap between the fixed electrode and the moving electrode was implemented by forming a polyimide sacrificial layer between the two electrodes and then removing it by O

_{2}etching method.

#### 2.2. Empirical OSCM Model

_{mea}) is composed of the serial impedance (Z

_{s}) and the parallel admittance (Y

_{o}) of the probe itself and the impedance (Z

_{sensor}) of the entire sensor chip. If Z

_{s}is negligible and Y

_{o}is simply simplified to cable capacitance (C

_{cable}), then the entire sensor chip will have the capacitance of the active area and the capacitance of the pad area. If it can be distinguished, it can be represented by the right equivalent circuit in Figure 2a. Here, the capacitance of the active region is subdivided into an intrinsic component (C

_{int}) and a parasitic component (C

_{par}). As in Figure 2b, the concentration parameters that make up the entire sensor chip can be configured. The measured total capacitance (C

_{tot}) was 2.09 pF. Also, the magnitude of the measured impedance (|Z

_{mea}|) was 0.21 MΩ at 360 kHz. To extract the capacitance of the active area, a membrane-less acoustic sensor chip was fabricated and the capacitance measured, as shown in Figure 2c. The measured cable capacitance (C

_{cable}) was 0.05 pF and the measured pad capacitance (C

_{m-pad}) was 0.28 pF, leading to the pad capacitance (C

_{pad}) of 0.23 pF. Thus, the active area capacitance (C

_{sen}) of the fabricated acoustic sensor was 1.81 pF under a bias condition of 0 V. It is clear that the capacitance value of the pad region can only be determined by extraction because it is an experimentally determined value.

#### 2.3. Analytical ALEM Model

_{sen}equation as

_{0}and g

_{0}are the permittivity of vacuum and the height of air-gap. Here, it can be expressed as a distance with respect to the Z axis of the cylindrical coordinate axis. In addition, the electrostatic force (F

_{ele}) is the force acting on the opposing two electrode plates. The net force, which acts on the two plates placed perpendicular to the Z axis, is the force in the Z axis direction, as in Equation (2). Accordingly, the radial component becomes a parasitic component. In order to model the electrostatic force of the intrinsic component, the fringe capacitance generated by the etching hole should be divided in the axial direction and the radial direction. As shown in Figure 3b, the fringe capacitance can be linearized by the ALEM model [9,15]. In the unit cell, the z-axis direction (axial) becomes an intrinsic component and the radial direction becomes a parasitic component. F

_{ele}can be obtain as

_{sen}is the sum of the intrinsic and parasitic components of the active region and V

_{b}is the DC bias voltage applied to the capacitor. Theoretically, it is very complicated to model the fringe capacitance due to the structural features, but using the ALEM model is advantageous for extracting parasitic components.

#### 2.4. Model Verification

_{int-u}/C

_{sen-u}) representing the ratio of the intrinsic component among the total capacitances was investigated in the modeled unit cell. As shown in Figure 4b, the attenuation coefficient was 90.4% at 1.5 μm electrode spacing and 88.6% at 1.0 μm spacing. These results show that the intrinsic component ratio is reduced by about 10% in the entire active region capacitance due to the etching holes. Therefore, the accurate attenuation factor must be properly reflected for open-circuit sensitivity extraction.

## 3. Dynamic Open-Circuit Sensitivity Modeling on Wafer Level

#### 3.1. Modeling Process

#### 3.2. Static Characterization

_{O}) is evaluated by the bias voltage (V

_{b}), the air gap (g

_{b}) at a bias, the modelled diaphragm area (A

_{mod}), the modelled spring constant (k

_{mod}) and the damping constant (C

_{int}/C

_{chi}), where C

_{chi}= C

_{int}+ C

_{par}+ C

_{pad}. This can be expressed [1,3,9] by

_{tot}) was 2.08 pF and the air gap was 1.5 μm at 0 V bias condition. In addition, the effective radius of 299 μm was extracted from the diaphragm radius of 325 μm for the first time through the ALEM model. Figure 6b is a graph showing the electrode gap versus the applied voltage for the modeling acoustic sensor as the effective radius. When a 12 V voltage was applied, a collapse occurred at 42% of the initial electrode spacing, leading to the collapse point (g

_{col}). Using the measured pull-in voltage, the effective spring constant of the diaphragm can be extracted. The point where the restoring force of the diaphragm is the same as the magnitude of the electrostatic force applied to the two positive plates is the collapse point. Therefore, the spring constant can be extracted by putting the restoring force and the electrostatic force inside the diaphragm into an equation. The restoring force (F

_{res}) acts perpendicularly on the diaphragm due to the small deflection of a diaphragm compared to its side-length, and it is given at a clamped circular diaphragm by [20,21]

_{mod}is the modelled residual stress of the diaphragm, ρ

_{mod}is the modelled radius of the capacitor, D is the flexural rigidity, and α is the empirical parameter dependent. On the other hand, assuming that the back-plate and the diaphragm are parallel, the electro-static force (F

_{ele}) can be represented by the differential of the effective capacitance in the nominal direction on the two plates from the electro-static energy. F

_{ele}can be represented [20,21] by

_{ele}equation as Taylor series [21], we will see the following result by

_{res}is equal to F

_{ele}on the condition of z = g

_{col}. The extracted effective spring constant was 190 N/m. As shown in Figure 6c, the modeled capacitance (C

_{sen}) at the electrode gap of 1.0 μm was 2.63 pF and the intrinsic capacitance (C

_{int}) was 2.40 pF. The capacitance of the sensor chip (C

_{chi}) was 2.86 pF at the air gap of 1.0 μm, and the attenuation coefficient was decreased, resulting in 0.84. It was confirmed that the pad capacitance of 0.23 pF acts as a parasitic component with the attenuation coefficient reduced. The parasitic component of the pad should be minimized when designing the sensor. The static open-circuit sensitivity at 1.1 μm electrode spacing was modelled as 11.3 mV/Pa.

#### 3.3. Dynamic Open-Circuit Sensitivity

_{O}) at the wafer level. Figure 7 shows an analogous electrical equivalent circuit for a capacitive MEMS acoustic sensor. It can be divided into three domains: the acoustic, the mechanical, and the electric domain. The conversion of each domain is divided by the transformer ratio. The effective conversion area in the acoustic-mechanical domain is the transformer ratio, while in the electro-mechanical domain, the effective charge per unit distance is the transformer ratio. Each lumped parameter can be easily extracted without using ROIC, which can be a useful tool to check the characteristics, especially at the sensor design stage. Dynamic So is defined to the ratio of the electrical output voltage (V

_{out}) to the input sound pressure (P) [11] as

_{a}(ω) is the volume velocity [m

^{3}/s], M

_{r}is the mass of the air close to the diaphragm (kg/m

^{4}), R

_{r}(ω) is the radiation resistance of the diaphragm (kg/s·m

^{4}), R

_{g}is the viscous resistance of the air-gap (kg/s·m

^{4}), R

_{h}is the viscous resistance of the acoustic holes (kg/s·m

^{4}), C

_{bc}is the compliance of the back chamber (m

^{3}/Pa), P

_{a}(ω) is the diaphragm portion of the input pressure that is converted to the related force in the mechanical domain (N/m

^{2}), A

_{tran}is the acoustic transduction factor (m

^{2}), F

_{m}(ω) is the force transformed to the mechanical domain (N), U

_{m}(ω) is the velocity (m/s), M

_{m}is the mass of the diaphragm (kg), C

_{m}is the compliance of the diaphragm (m/N), Fe(ω) is the intrinsic capacitance portion of the converted force (N), V

_{e}(ω) is the voltage transformed to the electrical domain (V), Γ the electrical transduction factor (C/m), i(ω) is the electrical current (A), and V

_{out}(ω) is the output voltage (V). In each domain, the main parameters are expressed by

_{O}, these equations should be linked and subtracted, leading to the transfer function of the output voltage as a function of the input pressure. Subsequently, in the acoustic domain, the mass of the air close to the diaphragm and the radiation resistance of the diaphragm are expressed [24] by

_{0}is the density of the air, ρ

_{tran}is the transduction radius of the diaphragm, ω is the sound circular frequency, and c is the sound velocity. R

_{r}has a frequency-dependent value. M

_{r}was 322 kg/m

^{4}at 10 V. The viscous resistance of the back-plate holes [2] is given by

_{b}is the thickness of the back-plate, υ is the viscosity of air, and r

_{b}is the radius of the back-plate etching hole. R

_{h}was 1.51 × 10

^{8}kg/s·m

^{4}at 10 V. As the air-gap can be considered as a purely resistive element, the viscous resistance of the air-gap [2], can be determined by

_{p}is the surface fraction occupied by the acoustic holes. R

_{g}was determined to be 1.88 × 10

^{9}kg/s·m

^{4}at 10 V. It is obvious that g

_{b}acts as the critical parameter to determine R

_{g}. Also, C

_{bc}is considered as the air resistance of the chamber, which is given by [24]

_{cham}is the volume of the back chamber. C

_{bc}was calculated to be 7.06 × 10

^{−14}m

^{3}/Pa at 10 V.

_{d}is the equivalent density of the diaphragm and the value of A

_{tran}/3 is calculated by considering the area as a lumped diaphragm. M

_{m}was modelled to be 2.52 × 10

^{−10}kg at 10 V. The compliance of the diaphragm (C

_{m}) can be modeled by

_{tran}is the transduction spring constant of the diaphragm, which is equal to k

_{mod}/3 due to the assumption of the lumped diaphragm. C

_{m}was modelled to be 1.58 × 10

^{−2}m/N at 10 V. Figure 8a shows the modeled dynamic S

_{O}with bias conditions. The dynamic open-circuit sensitivities were −42.9 dBV/Pa, −40.6 dBV/Pa, and −39.0 dBV/Pa at 7.5 V, 9 V, and 10 V bias conditions, respectively. Also, the resonant frequencies (f

_{r}) were 136.1 kHz, 120.6 kHz, and 83.1 kHz, respectively, at 7.5 V, 9 V, and 10 V bias conditions. From these results, it is confirmed that the damping coefficient decreases as the electrode gap increases, but the sensitivity increases under the condition of the pull-in voltage. Figure 8b shows the effect of parasitic capacitance. The dynamic open circuit sensitivities were −39.0 dBV/Pa, −40.1 dBV/Pa, and −41.2 dBV/Pa, respectively, when the parasitic components were 0.43 pF, 0.80 pF and 1.2 pF at 10 V bias condition, reflecting a decrease in sensitivity. Since the parasitic component of 0.4 pF reduces the sensitivity of about 1 dB, it can be understood that the structure should be determined in the sensor design stage so that the parasitic component can be suppressed as much as possible. Table 1 shows the comparison data of this research with current models.

- Determine C
_{tot}and V_{p}by measuring the voltage-capacitance relation of the DUT with an impedance analyzer. - Determine the air gap (g
_{0}) by measuring the height of the diaphragm with a 3D surface analyzer. - Extract C
_{pad}of sensor chip with OSCM model. - Determine C
_{int}and C_{par}of the active area with the ALEM model. - Model the spring constant (k
_{mod}) and the capacitor area (A_{mod}) by extracting the voltage-electrode spacing relationship from the ALEM model and pull-in measurement results. - Obtain a static open-circuit sensitivity (S
_{O}) at an arbitrary air gap (g_{b}). - To model the dynamic open-circuit sensitivity, we assume a distributed diaphragm as a lumped diaphragm, and determine the lumped parameters (R
_{r}, M_{r}, R_{h}, R_{g}, C_{bc}, M_{m}, C_{m}, C_{int}, C_{par}, and C_{pad}). Owing to the lumped diaphragm, the area of M_{m}is A_{tran}/3 and the magnitude of the spring constant (k_{tran}) in the dynamic range is k_{mod}/3. - The acoustic transduction factor (A
_{tran}) is modeled as the determined static S_{O}is equal to dynamic S_{O}at the condition of ω = 0. - The electrical transduction factor (Γ) is determined as C
_{int·}V_{b}/g_{b}. - Finally, dynamic S
_{O}is modeled with extracted lumped parameters.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic views and images of a capacitive MEMS acoustic sensor fabricated with a TiN/Al/TiN diaphragm: (

**a**) top view; (

**b**) cross section diagram and a SEM image.

**Figure 2.**(

**a**) The original circuit including the probe effect and the approximately circuit only consisting of capacitances; (

**b**) structure-based circuit of capacitance for a capacitive MEMS acoustic sensor and its impedance measurement results; (

**c**) structure-based circuit of capacitance for a sensor chip only containing the pad area and its impedance measurement results.

**Figure 3.**(

**a**) Diagram of a capacitive MEMS acoustic sensor composed of unit cells; (

**b**) diagram for the capacitive acoustic sensor roughly divided into an intrinsic and a parasitic component by the ALEM model.

**Figure 4.**(

**a**) Electric field distribution concept diagram of the inner part composing a unit cell and potential distribution by FEM simulation; (

**b**) ALEM model and FEM simulation result and error result for the air gap-capacitance relation, and the total and the intrinsic capacitance for the air gap–capacitance relationship and its attenuation coefficient results.

**Figure 5.**Flowchart for modeling dynamic open circuit sensitivity at the wafer level of capacitive MEMS acoustic sensors based on the ALEM and OSCM models.

**Figure 6.**(

**a**) Diaphragm and back-plate height of capacitive MEMS acoustic sensor measured by 3D surface analyzer; (

**b**) the pull-in voltage according to the capacitance-voltage measurement and the air gap on the applied voltage of a modeled capacitive MEMS acoustic sensor; (

**c**) modelled sensor chip capacitances, active area capacitances, intrinsic capacitance for the air gap and the intrinsic component ratio of the sensor chip to the air gap and the intrinsic component ratio of the active capacitance.

**Figure 7.**The electrical equivalent circuit consisting of three (acoustic-, mechanical-, and electrical-domain) regions for dynamic open-circuit sensitivity of a capacitive MEMS acoustic sensor.

**Figure 8.**(

**a**) Modelled dynamic open-circuit sensitivities based on bias voltages (7.5 V, 9.0 V, 10.0 V) of a capacitive MEMS acoustic sensor; (

**b**) modelled dynamic open circuit sensitivities according to parasitic capacitances (0.43 pF, 0.80 pF, 1.20 pF) at 10.0 V bias condition of a capacitive MEMS acoustic sensor.

Model 1 [22] | Model 2 [24] | Model 3 [26] | This Work | |
---|---|---|---|---|

Circuit model | 3 domain-based circuit | Mixed domain-based circuit | Mixed domain-based circuit | 3 domain circuit inserted with VCVS |

Fringe field effect | Not mentioned | Not mentioned | Not mentioned | Included |

Attenuation coefficient | Partially considered | Not considered | Partially considered | Fully considered |

Evaluation | - | With ROIC | With ROIC | On wafer level |

Features | Only proposed | Too simplified | Simplified | One test sample needed |

Parameters and Characteristics | Values |
---|---|

Intrinsic capacitance (C_{int}) at the bias of 10 V | 2.26 pF |

Parasitic capacitance (C_{par}) at the bias of 10 V | 0.20 pF |

Pad capacitance (C_{pad}) | 0.23 pF |

Air gap (g_{b}) at the bias of 10 V | 1.1 um |

Modelled capacitor area (A_{mod}) | 2.81 × 10^{−7} m^{2} |

Modelled spring constant (k_{mod}) | 190 N/m |

Acoustic transduction factor (A_{tran}) at the bias of 10 V | 3.69 × 10^{−7} m^{2} |

Electrical transduction factor (Γ) at the bias of 10 V | 2.05 × 10^{−5} C/m |

Dynamic open-circuit sensitivity (S_{o}) at the bias 10 V | −39.0 dBV/Pa |

First resonance frequency (f_{0}) at the bias 10 V | 83.1 kHz |

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**MDPI and ACS Style**

Lee, J.; Im, J.-P.; Kim, J.-H.; Lim, S.-Y.; Moon, S.-E. Wafer-Level-Based Open-Circuit Sensitivity Model from Theoretical ALEM and Empirical OSCM Parameters for a Capacitive MEMS Acoustic Sensor. *Sensors* **2019**, *19*, 488.
https://doi.org/10.3390/s19030488

**AMA Style**

Lee J, Im J-P, Kim J-H, Lim S-Y, Moon S-E. Wafer-Level-Based Open-Circuit Sensitivity Model from Theoretical ALEM and Empirical OSCM Parameters for a Capacitive MEMS Acoustic Sensor. *Sensors*. 2019; 19(3):488.
https://doi.org/10.3390/s19030488

**Chicago/Turabian Style**

Lee, Jaewoo, Jong-Pil Im, Jeong-Hun Kim, Sol-Yee Lim, and Seung-Eon Moon. 2019. "Wafer-Level-Based Open-Circuit Sensitivity Model from Theoretical ALEM and Empirical OSCM Parameters for a Capacitive MEMS Acoustic Sensor" *Sensors* 19, no. 3: 488.
https://doi.org/10.3390/s19030488