# A New Macro-Model of Gas Flow and Parameter Extraction for a CMOS-MEMS Vacuum Sensor

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Working Principle of a Thermal-Type Vacuum Sensor

#### 2.1. Working Principle

_{s}, G

_{g}, and G

_{r}are the thermal conductance of solid, gas, and radiation, respectively. Equation (2) shows the dependence of temperature rise ΔT on the pressure as follows:

_{0}is the heating power of the sensor.

_{B}, and T are the dynamic viscosity, the pressure, Boltzmann Constant, and the thermodynamic temperature of gas molecules, and m is the gas relative molecular mass.

^{2}K

^{−1}was obtained for the largest gap of 21 µm. However, the effective thermal conductivity f the gap of 220 nm is as low as 1.2 × 10

^{−3}Wm

^{−1}K

^{−1}. Both of them indicate that the size effect of gaseous heat transport is significant in such microscale devices. The air gap in MEMS devices usually varies from 100 nm to micrometers, boundary effects will affect heat transfer, and the characteristic length is in the micrometer range. The mean free path for air under 1 atm and room temperature is about 80 nm [32].

_{g}of gas molecules under atmospheric pressure P can be expressed as follows [33,34,35,36,37,38]:

_{0}is the thermal conductivity of gas under atmospheric pressure. α and γ are the adaptation coefficient and the specific heat ratio. k

_{g}is one of the main characteristic parameters in this study. From Equations (3)–(6), one can see that the thermal conductivity k

_{g}of gas not only varies with the type of gas but is also related to the state parameters of the gas (temperature, pressure, etc.).

_{g}(P) of gas molecules under different pressures. Therefore, to build the macro-model, the simulation of the thermal-type vacuum sensor was based on the speculation of appropriate characteristic lengths under different vacuum environments at the beginning. The process is shown in Figure 2. These thermal conductivities k

_{g}(P) of gas molecules under different pressures were substituted into the simulation software to simulate and analyze the temperature distribution in the structure of the proposed thermal-type vacuum sensor.

_{th}is proportional to the temperature difference ΔT from the thermopile as follows:

_{th}from the thermal-type element would be described in the vacuum measurement results. The flow chart of the extraction of characteristic length is illustrated in Figure 2 and described below.

#### 2.2. Proposed Thermal-Type Vacuum Sensor

#### 2.3. A Simplified Theoretical Description of Heat Transfer for Thermal-Type Vacuum Sensor

_{s}is the dissipated thermal power, k

_{s}denotes the thermal conductivity of the structure, A is the cross-section area of the membrane through which heat is transferred, T

_{h}and T

_{c}are the temperatures of the hot junction and cold junction, and L

_{s}refers to the distance between the hot junction and cold junction. To consider more complex structures of multiple layers to conduct heat, a simplified theoretical description of heat transfer for the thermal-type vacuum sensor is revealed: When the thermoelectric elements are stacked by different materials in the CMOS-MEMS process, the total thermal solid conductance G

_{s}of the elements can be expressed as follows:

_{eq}can be applied to the heat transfer simulation as the thermal conductivity of the structure for the macro-model. The materials used in the thermoelectric element in this study include Al, N-poly, SiO

_{2}, and Si

_{3}N

_{4}, and their thermal conductivities are 237, 31.4, 1.25, 16 W/mk, respectively. After calculation, the equivalent thermal conductivity of the heater and thermocouple are about 4 W/mk and 8.2 W/mk, respectively.

## 3. Vacuum Measurement

#### 3.1. Experiment Setup

#### 3.2. Signal Acquisition from Thermal-Type Vacuum Sensor

## 4. Building of Macro-Model of Vacuum Measurement and Verification

_{g}under pressure P dominates the response of the proposed thermal-type vacuum sensor described as follows [39,40,41,42]:

_{a}is a parameter related to the free molecular conductivity of gas under the ambient temperature, which depends on the type of gas and the specific properties of the materials used in the sensor and the configuration in detail. A

_{s}refers to the floating-membrane area. P

_{t}denotes the transition pressure associated with the transfer of heat, which is the critical pressure used to characterize the mechanisms underlying the thermal conductivity of the gas in the chamber [36].

_{t}, the gas thermal conductance is proportional to the pressure P in Equation (11). At the same time, the pressure closes to higher vacuum. The gas thermal conductance will have the smallest value and the heat transfer in the thermoelectric element is dominated by the solid thermal conduction. Therefore, the temperature difference across the thermocouple will be higher. On the other hand, when P >> P

_{t}, the gas thermal conductance can be regarded as a constant, i.e., the pressure of gas does not affect the gas thermal conductance. The gas thermal conductance will have a maximum value, so the temperature difference on the thermocouple under heating operation is lower [5,40]. Therefore, the pressure resolution of the floating-membrane sensor could be determined by the transition pressures [42]. After the vacuum measurement and simulation, the transition pressures were extracted by comparison with each other, where the complete steps are shown in Figure 2 and described as follows.

#### 4.1. Simulation of Temperature under Different Characteristic Lengths

_{g}is derived from these characteristic lengths and substituted into the macro-model of the sensor. After the simulation by ANSYS, then the relationships between temperature difference and pressures at various characteristic lengths were established as shown in Figure 6.

#### 4.2. Parameter Extraction and Verification of Macro-Model

_{t}was extracted as 2396 mTorr shown in Figure 8.

#### 4.3. Validity of Macro-Model for Different Heating Powers

_{h}were compared, and the transfer function relationship between the function temperature and the output voltage is formulated as follows:

_{th}is the output voltage measured by the CMOS-MEMS thermal-type vacuum sensor, and ΔT is the temperature difference simulated by the macro-model of the thermal-type vacuum sensor. a, b are the slope and intercept of the linear equation, respectively, and they are functions of heating power as shown in Equation (12). Then according to Equation (12), the results of simulation and measurement with heating power of 0.144 mW, 0.414 mW, and 1.055 mW were compared to establish the fitting equations between a, b and the heating power, which are shown in Figure 13 and Figure 14, respectively. Based on these operations, a macro-model was established through simulation of the thermal-type vacuum sensor, which can be used to predict the measurement results of different heating powers. The flow of these procedures is shown in Figure 10 with blue arrows.

_{c}. Furthermore, the measured values V

_{th}from the output of the circuit are compared to verify the effectiveness of the macro-model, where the steps are shown with the green arrows in Figure 10. The results are summarized as shown in Table 1, Table 2 and Table 3.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Wilfert, S.; Edelmann, C. Miniaturized vacuum gauges. J. Vac. Sci. Technol. A
**2004**, 22, 309–320. [Google Scholar] [CrossRef] - Chen, S.J.; Wu, Y.C. Active thermoelectric vacuum sensor based on frequency modulation. Micromachines
**2020**, 11, 15. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chen, S.J.; Chen, B. Research on a CMOS-MEMS infrared sensor with reduced graphene oxide. Sensors
**2020**, 20, 4007. [Google Scholar] [CrossRef] [PubMed] - Ma, B.; Liang, P.Z.; Chen, S.J.; Cheng, Z.X. Coupled-field analysis of a silicon-thermopile-based vacuum sensor. Nanotechnol. Precis. Eng.
**2008**, 6, 336–342. [Google Scholar] - Weng, P.K.; Shie, J.S. Micro-Pirani vacuum gauge. Rev. Sci. Instrum.
**1994**, 65, 492–499. [Google Scholar] [CrossRef] - Shie, J.S.; Chou, B.C.; Chen, Y.M. High performance Pirani vacuum gauge. J. Vac. Sci. Technol. A
**1995**, 13, 2972–2979. [Google Scholar] [CrossRef] [Green Version] - Chen, C.N.; Chen, C.C. Thermal Type Vacuum Gauge. U.S. Patent App. 15/070,276, 21 September 2017. [Google Scholar]
- Chen, C.N.; Huang, W.C. A CMOS-MEMS thermopile with low thermal conductance and a near-perfect emissivity in the 8–14-μm wavelength range. IEEE Electron Device Lett.
**2010**, 32, 96–98. [Google Scholar] [CrossRef] - Wang, X.; Liu, C.; Zhang, Z.; Liu, S.; Luo, X. A micro-machined Pirani gauge for vacuum measurement of ultra-small sized vacuum packaging. Sens. Actuator A Phys.
**2010**, 161, 108–113. [Google Scholar] [CrossRef] - Puigcorbe, J.; Vogel, D.; Michel, B.; Vila, A.; Gracia, I.; Cane, C.; Morante, J. Thermal and mechanical analysis of micromachined gas sensors. J. Micromech. Microeng.
**2003**, 13, 548. [Google Scholar] [CrossRef] - Mahdavifar, A.; Aguilar, R.; Peng, Z.; Hesketh, P.J.; Findlay, M.; Stetter, J.R.; Hunter, G.W. Simulation and fabrication of an ultra-low power miniature microbridge thermal conductivity gas sensor. J. Electrochem. Soc.
**2014**, 161, B55. [Google Scholar] [CrossRef] - He, F.; Huang, Q.-A.; Qin, M. A silicon directly bonded capacitive absolute pressure sensor. Sens. Actuator A Phys.
**2007**, 135, 507–514. [Google Scholar] [CrossRef] - Shen, C.H.; Chen, S.J.; Guo, Y.T. A novel infrared temperature measurement with dual mode modulation of thermopile sensor. Sensors
**2019**, 19, 336. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Graf, A.; Arndt, M.; Sauer, M.; Gerlach, G. Review of micromachined thermopiles for infrared detection. Meas. Sci. Technol.
**2007**, 18, R59. [Google Scholar] [CrossRef] - Van Herwaarden, A.; Sarro, P.; Meijer, H. Integrated vacuum sensor. Sens. Actuators
**1985**, 8, 187–196. [Google Scholar] [CrossRef] - Van Herwaarden, A.; Sarro, P. Performance of integrated thermopile vacuum sensors. J. Phys. E
**1988**, 21, 1162. [Google Scholar] [CrossRef] - Van Herwaarden, A.; Van Duyn, D.; Van Oudheusden, B.; Sarro, P. Integrated thermopile sensors. Sens. Actuator A Phys.
**1990**, 22, 621–630. [Google Scholar] [CrossRef] - Folkmer, B.; Siber, A.; Bley, W.G.; Sandmaier, H.; Lang, W. Improved simulation for strongly coupled micro-electro-mechanical systems: Resonant vacuum sensor optimization. Sens. Actuator A Phys.
**1999**, 74, 190–192. [Google Scholar] [CrossRef] - Todd, S.T.; Xie, H. An electrothermomechanical lumped element model of an electrothermal bimorph actuator. J. Microelectromech. Syst.
**2008**, 17, 213–225. [Google Scholar] [CrossRef] - Niessner, M.; Schrag, G.; Iannacci, J.; Wachutka, G. Macromodel-based simulation and measurement of the dynamic pull-in of viscously damped RF-MEMS switches. Sens. Actuator A Phys.
**2011**, 172, 269–279. [Google Scholar] [CrossRef] - Mele, L.; Rossi, T.; Riccio, M.; Iervolino, E.; Santagata, F.; Irace, A.; Breglio, G.; Creemer, J.; Sarro, P. Electro-thermal analysis of MEMS microhotplates for the optimization of temperature uniformity. Procedia Manuf.
**2011**, 25, 387–390. [Google Scholar] [CrossRef] [Green Version] - Chen, X.; Wu, Z. Review on macromodels of MEMS sensors and actuators. Microsyst. Technol.
**2017**, 23, 4319–4332. [Google Scholar] [CrossRef] - Kaczynski, J.; Ranacher, C.; Fleury, C. Computationally efficient model for viscous damping in perforated MEMS structures. Sens. Actuator A Phys.
**2020**, 314, 112201. [Google Scholar] [CrossRef] - Shah, M.A.; Lee, D.-G.; Hur, S. Design and characteristic analysis of a MEMS piezo-driven recirculating inkjet printhead using lumped element modeling. Micromachines
**2019**, 10, 757. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mishra, S.; Balasubramaniam, R.; Chandra, S. Finite element analysis and experimental validation of suppression of span in optical MEMS pressure sensors. Microsyst. Technol.
**2019**, 25, 3691–3701. [Google Scholar] [CrossRef] - To, A.C.; Liu, W.K.; Olson, G.B.; Belytschko, T.; Chen, W.; Shephard, M.S.; Chung, Y.W.; Ghanem, R.; Voorhees, P.W.; Seidman, D.N.; et al. Materials integrity in microsystems: A framework for a petascale predictive-science-based multiscale modeling and simulation system. Comput. Mech.
**2008**, 42, 485–510. [Google Scholar] [CrossRef] - Devienne, F. Low density heat transfer. In Advances in Heat Transfer; Elsevier: Amsterdam, The Netherlands, 1965; Volume 2, pp. 271–356. [Google Scholar]
- Roy, S.; Raju, R.; Chuang, H.F.; Cruden, B.A.; Meyyappan, M. Modeling gas flow through microchannels and nanopores. J. Appl. Phys.
**2003**, 93, 4870–4879. [Google Scholar] [CrossRef] - Barber, R.; Emerson, D. The influence of Knudsen number on the hydrodynamic development length within parallel plate micro-channels. WIT Trans. Eng. Sci.
**2002**, 36, 12. [Google Scholar] - Springer, G.S. Heat transfer in rarefied gases. In Advances in Heat Transfer; Elsevier: Amsterdam, The Netherlands, 1971; Volume 7, pp. 163–218. [Google Scholar]
- Roth, A. Vacuum Technology, 3rd ed.; Elsevier: Amsterdam, The Netherlands, 1990; pp. 17–61. [Google Scholar]
- Huang, Z.; Wang, J.; Bai, S.; Guan, J.; Zhang, F.; Tang, Z. Size Effect of Heat Transport in Microscale Gas Gap. IEEE Trans. Ind. Electron.
**2017**, 64, 7387–7391. [Google Scholar] [CrossRef] - Beskok, A.; Karniadakis, G.E. Report: A model for flows in channels, pipes, and ducts at micro and nano scales. Microscale Thermophys. Eng.
**1999**, 3, 43–77. [Google Scholar] - Dongari, N.; Agrawal, A. Modeling of Navier–Stokes equations for high Knudsen number gas flows. Int. J. Heat Mass Transf.
**2012**, 55, 4352–4358. [Google Scholar] [CrossRef] - Zhang, Q.; Su, Y.; Wang, W.; Lu, M.; Sheng, G. Gas transport behaviors in shale nanopores based on multiple mechanisms and macroscale modeling. Int. J. Heat Mass Transf.
**2018**, 125, 845–857. [Google Scholar] [CrossRef] - Harley, J.C.; Huang, Y.; Bau, H.H.; Zemel, J.N. Gas Flow in Micr-Channels. Gas
**1994**, 9, 2–1994. [Google Scholar] - Reichenauer, G.; Heinemann, U.; Ebert, H.-P. Relationship between pore size and the gas pressure dependence of the gaseous thermal conductivity. Colloids Surf. A Physicochem. Eng. Asp.
**2007**, 300, 204–210. [Google Scholar] [CrossRef] - Han, M.; Liang, X.G.; Tang, Z. Size effect on heat transfer in micro gas sensors. Sens. Actuator A Phys.
**2005**, 120, 397–402. [Google Scholar] [CrossRef] - Chen, C.N. Fully quantitative characterization of CMOS–MEMS polysilicon/titanium thermopile infrared sensors. Sens. Actuators B Chem.
**2012**, 161, 892–900. [Google Scholar] [CrossRef] - Chou, B.C.; Chen, Y.M.; OuYang, M.; Shie, J.-S. A sensitive Pirani vacuum sensor and the electrothermal SPICE modelling. Sens. Actuator A Phys.
**1996**, 53, 273–277. [Google Scholar] [CrossRef] - Chen, C.N. Characterization of gas conductance of a thermal device with a V-groove cavity. IEEE Electron Device Lett.
**2011**, 33, 275–277. [Google Scholar] [CrossRef] - Van Herwaarden, A.W. Thermal Vacuum Sensors Based on Integrated Silicon Thermopiles. Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands, 1987. [Google Scholar]

**Figure 6.**Simulation for temperature difference ΔT vs. pressure under different characteristic lengths.

**Figure 11.**Temperature different from the macro-model of sensor vs. pressure with the different heating power by simulation.

**Figure 12.**Output voltage of the sensor vs. pressure with the different heating power by measurement.

Pressure (mTorr) | ΔT_{s} (k) | V_{c} (mV) | V_{th} (mV) | Error (%) |
---|---|---|---|---|

10 | 1.036 | 0.5049 | 0.5037 | 0.23% |

100 | 1.019 | 0.4983 | 0.4960 | 0.45% |

1000 | 0.9 | 0.4524 | 0.4506 | 0.40% |

10,000 | 0.67 | 0.3635 | 0.3640 | 0.13% |

100,000 | 0.596 | 0.3349 | 0.3329 | 0.60% |

760,000 | 0.585 | 0.3307 | 0.3286 | 0.64% |

Pressure (mTorr) | ΔT_{s} (k) | V_{c} (mV) | V_{th} (mV) | Error (%) |
---|---|---|---|---|

10 | 2.811 | 1.3841 | 1.3723 | 0.86% |

100 | 2.765 | 1.3660 | 1.3524 | 1.00% |

1000 | 2.443 | 1.2394 | 1.2244 | 1.23% |

10,000 | 1.819 | 0.9941 | 0.9886 | 0.56% |

100,000 | 1.617 | 0.9147 | 0.9038 | 1.20% |

760,000 | 1.589 | 0.9037 | 0.8892 | 1.63% |

Pressure (mTorr) | ΔT_{s} (k) | V_{c} (mV) | V_{th} (mV) | Error (%) |
---|---|---|---|---|

10 | 7.106 | 3.4470 | 3.5165 | 1.98% |

100 | 6.989 | 3.4014 | 3.4638 | 1.80% |

1000 | 6.176 | 3.0844 | 3.1441 | 1.90% |

10,000 | 4.598 | 2.4691 | 2.5311 | 2.45% |

100,000 | 4.087 | 2.2699 | 2.3055 | 1.54% |

760,000 | 4.017 | 2.2426 | 2.2729 | 1.34% |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chen, S.-J.; Wu, Y.-C.
A New Macro-Model of Gas Flow and Parameter Extraction for a CMOS-MEMS Vacuum Sensor. *Symmetry* **2020**, *12*, 1604.
https://doi.org/10.3390/sym12101604

**AMA Style**

Chen S-J, Wu Y-C.
A New Macro-Model of Gas Flow and Parameter Extraction for a CMOS-MEMS Vacuum Sensor. *Symmetry*. 2020; 12(10):1604.
https://doi.org/10.3390/sym12101604

**Chicago/Turabian Style**

Chen, Shu-Jung, and Yung-Chuan Wu.
2020. "A New Macro-Model of Gas Flow and Parameter Extraction for a CMOS-MEMS Vacuum Sensor" *Symmetry* 12, no. 10: 1604.
https://doi.org/10.3390/sym12101604