# Stochastic Effects on the Dynamics of the Resonant Structure of a Lorentz Force MEMS Magnetometer

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## Abstract

**:**

## 1. Introduction

## 2. A Reduced-Order Model of the Resonating Structure

## 3. Effective Elastic Properties of the Polysilicon Film

#### 3.1. Analytical, Deterministic Homogenization

#### 3.2. Numerical, Stochastic Homogenization

## 4. Monte Carlo Analysis: Sensitivity to Imperfections

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Gad-el-Hak, M. (Ed.) The MEMS Handbook; CRC Press: Boca Raton, FL, USA, 2002. [Google Scholar]
- Ko, W. Trends and frontiers of MEMS. Sens. Actuators A-Phys.
**2007**, 136, 62–67. [Google Scholar] [CrossRef] - Corigliano, A.; Ardito, R.; Comi, C.; Frangi, A.; Ghisi, A.; Mariani, S. Mechanics of Microsystems; John Wiley and Sons: Hoboken, NJ, USA, 2018. [Google Scholar]
- Li, M.; Rouf, V.T.; Thompson, M.J.; Horsley, D.A. Three-axis Lorentz-force magnetic sensor for electronic compass applications. J. Microelectromech. Syst.
**2012**, 21, 1002–1010. [Google Scholar] [CrossRef] - Bagherinia, M.; Bruggi, M.; Corigliano, A.; Mariani, S.; Lasalandra, E. Geometry optimization of a Lorentz force, resonating MEMS magnetometer. Microelectron. Reliabil.
**2014**, 54, 1192–1199. [Google Scholar] [CrossRef] - Bagherinia, M.; Corigliano, A.; Mariani, S.; Horsley, D.A.; Li, M.; Lasalandra, E. An efficient earth magnetic field MEMS sensor: Modelling and experimental results. In Proceedings of the MEMS IEEE, San Francisco, CA, USA, 26–30 January 2014; pp. 700–703. [Google Scholar]
- Bagherinia, M.; Bruggi, M.; Corigliano, A.; Mariani, S.; Horsley, D.A.; Li, M.; Lasalandra, E. An efficient earth magnetic field MEMS sensor: Modeling, experimental results and optimization. J. Microelectromech. Syst.
**2015**, 24, 887–895. [Google Scholar] [CrossRef] - Corigliano, A.; De Masi, B.; Frangi, A.; Comi, C.; Villa, A.; Marchi, M. Mechanical characterization of polysilicon through on-chip tensile tests. J. Microelectromech. Syst.
**2004**, 13, 200–219. [Google Scholar] [CrossRef] - Mariani, S.; Ghisi, A.; Corigliano, A.; Zerbini, S. Multi-scale analysis of MEMS sensors subject to drop impacts. Sensors
**2007**, 7, 1817–1833. [Google Scholar] [CrossRef] [PubMed] - Ghisi, A.; Fachin, F.; Mariani, S.; Zerbini, S. Multi-scale analysis of polysilicon MEMS sensors subject to accidental drops: Effect of packaging. Microelectron. Reliabil.
**2009**, 49, 340–349. [Google Scholar] [CrossRef] - Ghisi, A.; Kalicinski, S.; Mariani, S.; De Wolf, I.; Corigliano, A. Polysilicon MEMS accelerometers exposed to shocks: Numerical-experimental investigation. J. Micromech. Microeng.
**2009**, 19, 035023. [Google Scholar] [CrossRef] - Mariani, S.; Ghisi, A.; Corigliano, A.; Zerbini, S. Modeling impact-induced failure of polysilicon MEMS: A multi-scale approach. Sensors
**2009**, 9, 556–567. [Google Scholar] [CrossRef] - Mariani, S.; Martini, R.; Ghisi, A.; Corigliano, A.; Simoni, B. Monte Carlo simulation of micro-cracking in polysilicon MEMS exposed to shocks. Int. J. Fract.
**2011**, 167, 83–101. [Google Scholar] [CrossRef] - Mariani, S.; Martini, R.; Corigliano, A.; Beghi, M. Overall elastic domain of thin polysilicon films. Comput. Mater. Sci.
**2011**, 50, 2993–3004. [Google Scholar] [CrossRef] - Mirzazadeh, R.; Eftekhar Azam, S.; Mariani, S. Micromechanical characterization of polysilicon films through on-chip tests. Sensors
**2016**, 16, 1191. [Google Scholar] [CrossRef] [PubMed] - Mirzazadeh, R.; Mariani, S. Uncertainty quantification of microstructure-governed properties of polysilicon MEMS. Micromachines
**2017**, 8, 248. [Google Scholar] [CrossRef] - Mirzazadeh, R.; Eftekhar Azam, S.; Mariani, S. Mechanical characterization of polysilicon MEMS: A hybrid TMCMC/POD-kriging approach. Sensors
**2018**, 18, 1243. [Google Scholar] [CrossRef] - Mariani, S.; Ghisi, A.; Mirzazadeh, R.; Eftekhar Azam, S. On-Chip testing: A miniaturized lab to assess sub-micron uncertainties in polysilicon MEMS. Micro Nanosyst.
**2018**, 10, 84–93. [Google Scholar] [CrossRef] - Bagherinia, M.; Mariani, S.; Corigliano, A.; Lasalandra, E. Stochastic effects on the dynamics of a resonant MEMS magnetometer: A Monte Carlo investigation. In Proceedings of the 1st International Electronic Conference on Sensors and Applications, Basel, Switzerland, 1–16 June 2014. [Google Scholar]
- Bahreyni, B.; Shafai, C. A micromachined magnetometer with frequency modulation at the output. In Proceedings of the MEMS IEEE 2005, Irvine, CA, USA, 30 October–3 November 2005; pp. 580–583. [Google Scholar]
- Bahreyni, B.; Shafai, C. A Resonant Micromachined Magnetic Field Sensor. IEEE Sens. J.
**2007**, 7, 1326–1334. [Google Scholar] [CrossRef] - Ghosh, S.; Lee, J.E.-Y. A piezoelectric-on-silicon width-extensional mode Lorentz force resonant MEMS magnetometer. Sens. Actuators A Phys.
**2017**, 260, 169–177. [Google Scholar] [CrossRef] - Ko, J.S.; Lee, M.L.; Lee, D.S.; Choi, C.A.; Kim, Y.T. Development and application of laterally driven electromagnetic microactuator. Appl. Phys. Lett.
**2002**, 81, 547–549. [Google Scholar] - Kumar, V.; Mazrouei Sebdani, S.; Pourkamali, S. Sensitivity enhancement of a Lorentz force MEMS magnetometer with frequency modulated output. J. Microelectromech. Syst.
**2017**, 26, 870–878. [Google Scholar] [CrossRef] - Wu, L.; Tian, Z.; Ren, D.; You, Z. A Miniature resonant and torsional magnetometer based on Lorentz force. Micromachines
**2018**, 9, 666. [Google Scholar] [CrossRef] - Aydemir, G.A.; Saranlı, A. Characterization and calibration of MEMS inertial sensors for state and parameter estimation applications. Measurement
**2012**, 45, 1210–1225. [Google Scholar] [CrossRef] - Camps, F.; Harasse, S.; Monin, A. Numerical calibration for 3-axis accelerometers and magnetometers. In Proceedings of the IEEE International Conference on Electro/Information Technology, Windsor, ON, Canada, 7–9 June 2009; pp. 217–221. [Google Scholar]
- LSM6DSOX Datasheet; Technical Report; STMicroelectronics: Geneva, Switzerland, 2019.
- Bagherinia, M. MEMS Sensors for Measuring the Earth Magnetic Field: Mechanical Aspects. Ph.D. Thesis, Politecnico di Milano, Milan, Italy, 2014. [Google Scholar]
- Younis, M.I. MEMS Linear and Nonlinear Statics and Dynamics; Springer: New York, NY, USA, 2011. [Google Scholar]
- Shih, Y.S.; Wu, G.Y.; Chen, E.J.S. Transient vibrations of a simply-supported beam with axial loads and transverse magnetic fields. Mech. Struct. Mech.
**1998**, 26, 115–130. [Google Scholar] [CrossRef] - Xi, X.; Yang, Z.; Meng, L.; Zhu, C. Primary resonance of the current-carrying beam in thermal-magneto-elasticity field. Appl. Mech. Mater.
**2010**, 29–32, 16–21. [Google Scholar] [CrossRef] - Comi, C.; Corigliano, A.; Langfelder, G.; Longoni, A.; Tocchio, A.; Simoni, B. A resonant microaccelerometer with high sensitivity operating in an oscillating circuit. J. Microelectromech. Syst.
**2010**, 19, 1140–1152. [Google Scholar] [CrossRef] - Han, J.S.; Ko, J.S.; Korvink, J.G. Structural optimization of a large-displacement electromagnetic Lorentz force microactuator for optical switching applications. J. Micromech. Microeng.
**2004**, 14, 1585–1596. [Google Scholar] [CrossRef] - Bao, M.; Yang, H. Squeeze film air damping in MEMS. Sens. Actuators A
**2007**, 136, 3–27. [Google Scholar] [CrossRef] - Mariani, S.; Martini, R.; Ghisi, A.; Corigliano, A.; Beghi, M. Overall elastic properties of polysilicon films: A statistical investigation of the effects of polycrystal morphology. Int. J. Multiscale Comput. Eng.
**2011**, 9, 327–346. [Google Scholar] [CrossRef] - Nemat-Nasser, S.; Hori, M. Micromechanics: Overall Properties of Heterogeneous Materials; Volume 37 of Applied Mathematics and Mechanics; North-Holland: Amsterdam, The Netherlands, 1993. [Google Scholar]
- Cho, S.; Chasiotis, I. Elastic properties and representative volume element of polycrystalline silicon for MEMS. Exp. Mech.
**2007**, 47, 37–49. [Google Scholar] [CrossRef] - Sanchez-Palencia, E. Nonhomogeneous Media and Vibration Theory; Volume 127 of Lecture Notes in Physics; Springer: Berlin Heidelberg, Germany, 1980. [Google Scholar]
- Hopcroft, M.A.; Nix, W.D.; Kenny, T.W. What is the Young’s Modulus of Silicon? J. Microelectromech. Syst.
**2010**, 19, 229–238. [Google Scholar] [CrossRef] - McConnell, A.D.; Uma, S. Thermal conductivity of doped polysilicon layers. J. Microelectromech. Syst.
**2001**, 10, 360–369. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) SEM picture of a movable structure of the resonant MEMS magnetometer [19], and (

**b**) scheme of the structure with the notation adopted.

**Figure 2.**Exemplary regularized micro-structures of (

**a**) 2 × 2 μm and (

**b**) 3 × 3 μm SVEs, where grains are depicted with different colors.

**Figure 3.**Stochastic homogenization, effect of the SVE size on the cumulative distribution functions of the bounds on the homogenized in-plane Young’s modulus $E$ of a polysilicon film featuring ${s}_{g}=0.5$ μm: (

**a**) 2 ×2 μm SVE, (

**b**) 3 × 3 μm SVE.

**Figure 4.**Sensitivity analysis: dependence of ${\mathcal{V}}_{\mathrm{max}}$ (

**a**) on the polysilicon effective Young’s modulus $E$, and (

**b**) on the over-etch depth $o$.

**Figure 5.**Sensitivity analysis: dependence on the over-etch depth $o$ of the ratio between the temperature raise $\mathsf{\Delta}T$ at mid-span, as induced by Joule effect, and its critical value $\mathsf{\Delta}{T}_{cr}$.

**Figure 6.**Sensitivity analysis: dependence of the natural frequency ${f}_{1}$ (

**a**) on the polysilicon effective Young’s modulus $E$, and (

**b**) on the over-etch depth $o$.

**Figure 7.**Monte Carlo simulation, $h=2$ μm: scattering in the value of ${\mathcal{V}}_{\mathrm{max}}$, and comparison with the trends induced by (

**a**) $E$ and (

**b**) $o$ alone.

**Figure 8.**Monte Carlo simulation, $h=3$ μm: scattering in the value of ${\mathcal{V}}_{\mathrm{max}}$, and comparison with the trends induced by (

**a**) $E$ and (

**b**) $o$ alone.

**Figure 9.**Monte Carlo simulation: cumulative distribution functions of ${\mathcal{V}}_{\mathrm{max}}$ for (

**a**) $h=2$ μm and (

**b**) $h=3$ μm.

**Figure 10.**Monte Carlo simulation: cumulative distribution functions of the sensitivity ${\varphi}_{\mathrm{s}}$ for (

**a**) $h=2$ μm and (

**b**) $h=3$ μm.

**Table 1.**Stochastic homogenization: effect of the SVE size on the mean ${E}_{m}$ and standard deviation ${E}_{s}$ of the scattered in-plane polysilicon Young’s modulus.

SVE Size | Uniform Stress BCs | Uniform Strain BCs | ||
---|---|---|---|---|

${E}_{m}$ (GPa) | ${E}_{s}$ (GPa) | ${E}_{m}$ (GPa) | ${E}_{s}$ (GPa) | |

2 × 2 μm^{2} | 148.1 | 5.4 | 150.0 | 5.5 |

3 × 3 μm^{2} | 148.4 | 3.3 | 149.9 | 3.3 |

**Table 2.**Values of the geometrical, physical, actuation, and sensing parameters adopted in the analysis.

Property | Value |
---|---|

beam length $L$ (μm) | $575$ |

beam width $h$ (μm) | $2-3$ |

beam thickness $b$ (μm) | $22$ |

gap $g$ (μm) | $2$ |

mass density $\varrho $ (kg/m^{3}), [40] | $2330$ |

thermal expansion coefficient $\alpha $ (K^{−1}), [40] | $2.5\times {10}^{-6}$ |

thermal conductivity ${K}_{H}$ (W/mK), [41] | $25$ |

resistivity $\psi $ (Ωm), [41] | ${10}^{-5}$ |

permittivity ${\epsilon}_{0}$ (F/m) | $8.85\times {10}^{-12}$ |

air viscosity $\mu $ (Ns/m^{2}), [35] | $5\times {10}^{-8}$ |

actuation current $i$ (mA) | $1$ |

bias voltage $V$ (V) | $2$ |

magnetic field intensity $B$ (μT), [4] | $50$ |

**Table 3.**Monte Carlo simulation: effect of the target in-plane film width $h$ and of the type of BCs on the values of mean ${\mathcal{V}}_{\mathrm{max},\mathrm{m}}$ and standard deviation ${\mathcal{V}}_{\mathrm{max},\mathrm{s}}$ of the scattered amplitude of oscillations.

Uniform Stress BCs | Uniform Strain BCs | |||
---|---|---|---|---|

$h$ (μm) | ${\mathcal{V}}_{\mathrm{max},\mathrm{m}}$ (μm) | ${\mathcal{V}}_{\mathrm{max},\mathrm{s}}$ (μm) | ${\mathcal{V}}_{\mathrm{max},\mathrm{m}}$ (μm) | ${\mathcal{V}}_{\mathrm{max},\mathrm{s}}$ (μm) |

2 | 0.028 | 0.004 | 0.028 | 0.004 |

3 | 0.017 | 0.003 | 0.016 | 0.003 |

**Table 4.**Monte Carlo simulation: effect of the target in-plane film width $h$ and of the type of BCs on the values of mean ${\varphi}_{\mathrm{s},\mathrm{m}}$ and standard deviation ${\varphi}_{\mathrm{s},\mathrm{s}}$ of the scattered device sensitivity.

Uniform Stress BCs | Uniform Strain BCs | |||
---|---|---|---|---|

$h$ (μm) | ${\varphi}_{\mathrm{s},\mathrm{m}}$ (pF/T) | ${\varphi}_{\mathrm{s},\mathrm{s}}$ (pF/T) | ${\varphi}_{\mathrm{s},\mathrm{m}}$ (pF/T) | ${\varphi}_{\mathrm{s},\mathrm{s}}$ (pF/T) |

2 | 30.89 | 1.90 | 30.67 | 1.93 |

3 | 18.36 | 1.38 | 18.28 | 1.38 |

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**MDPI and ACS Style**

Bagherinia, M.; Mariani, S. Stochastic Effects on the Dynamics of the Resonant Structure of a Lorentz Force MEMS Magnetometer. *Actuators* **2019**, *8*, 36.
https://doi.org/10.3390/act8020036

**AMA Style**

Bagherinia M, Mariani S. Stochastic Effects on the Dynamics of the Resonant Structure of a Lorentz Force MEMS Magnetometer. *Actuators*. 2019; 8(2):36.
https://doi.org/10.3390/act8020036

**Chicago/Turabian Style**

Bagherinia, Mehrdad, and Stefano Mariani. 2019. "Stochastic Effects on the Dynamics of the Resonant Structure of a Lorentz Force MEMS Magnetometer" *Actuators* 8, no. 2: 36.
https://doi.org/10.3390/act8020036