# Enhancing Performance of a MEMS-Based Piezoresistive Pressure Sensor by Groove: Investigation of Groove Design Using Finite Element Method

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Working Principle of MEMS Piezoresistive Pressure Sensors

#### 2.2. Design Considerations

#### 2.3. Groove Designs

## 3. Finite Element Analysis

#### 3.1. Stress Distribution of the Sensor with the Local Groove LG1

#### 3.2. Stress Distribution of the Sensor with the Local Groove LG2

#### 3.3. Stress Distribution of the Sensor with the Annular Groove AG

## 4. Groove Design Comparison

## 5. Functional Forms of Averaged Stress Difference and Maximum Deflection of Sensor with Groove

^{−4}, respectively. Therefore, the functional form of ${\delta}_{max}$ of the sensor with the local groove $LG2-LT-Top$ with $\overline{{l}_{g}}$ = 0.35 can be expressed as

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**MEMS piezoresistive pressure sensor for ultra-low pressure measurements with optimized geometric parameters, proposed by Thawornsathit et al. (2022) [40].

**Figure 2.**Local groove design 1: (

**a**) grooves at longitudinal piezoresistors $\left(LG1-L0\right)$, (

**b**) grooves at transverse piezoresistors $\left(LG1-0T\right)$ and (

**c**) grooves at locations of both longitudinal and transverse piezoresistors $(LG1-LT$ ). (Note: not to scale.)

**Figure 3.**Local groove design 2: (

**a**) grooves on top of diaphragm at longitudinal piezoresistors $\left(LG2-L0-Top\right)$, (

**b**) grooves at bottom of diaphragm at longitudinal piezoresistors $\left(LG2-L0-Bottom\right)$, (

**c**) grooves on top of diaphragm at transverse piezoresistors $\left(LG2-0T-Top\right)$, (

**d**) grooves at bottom of diaphragm at transverse piezoresistors $\left(LG2-0T-Bottom\right)$, (

**e**) grooves on top of diaphragm at both longitudinal and transverse piezoresistors $(LG2-LT-Top$ ) and (

**f**) grooves at bottom of diaphragm at both longitudinal and transverse piezoresistors $(LG2-LT-bottom$ ). (Note: not to scale.)

**Figure 4.**Annular groove: (

**a**) grooves on top of diaphragm $\left(AG-Top\right)$ and (

**b**) grooves at bottom of diaphragm $\left(AG-Bottom\right)$. (Note: not to scale.)

**Figure 5.**Computational domain for finite element analysis: (

**a**) boundary conditions, (

**b**) mesh distribution for LG1, (

**c**) mesh distribution for LG2-Top, (

**d**) mesh distribution for LG2-Bottom, (

**e**) mesh distribution for AG-Top and (

**f**) mesh distribution for AG-Bottom.

**Figure 6.**Equivalent stress distributions at the applied pressure of 5 kPa: (

**a**) sensor without groove and (

**b**) sensor with local groove $LG1-LT$ ($\overline{{d}_{g}}$ = 0.8). (Only one-fourth of the domain is displayed.)

**Figure 7.**Stress difference distributions at the longitudinal piezoresistor without groove and with local groove $LG1-LT$ ($\overline{{d}_{g}}$ = 0.8) at the applied pressure of 5 kPa. (Only one-half of the domain is displayed.)

**Figure 8.**Variations of (

**a**) sensitivity and (

**b**) nonlinearity error with dimensionless groove depth of sensor with local grove $LG1$.

**Figure 9.**Variation of ratio of sensitivity to nonlinearity error $\left(\frac{S}{NL}\right)$ with dimensionless groove depth of sensor with local grove $LG1$.

**Figure 10.**Stress difference distributions at the applied pressure of 5 kPa: (

**a**) transverse piezoresistor and (

**b**) longitudinal piezoresistor for $LG2-LT-Top$ ($\overline{{d}_{g}}$ = 0.8). (Only one-half of the domain is displayed.)

**Figure 11.**Variations of (

**a**) sensitivity and (

**b**) nonlinearity error with dimensionless groove depth of sensor with local grove $LG2$ on the top of diaphragm at $\overline{{l}_{g}}=0.175$.

**Figure 12.**Variations of (

**a**) sensitivity and (

**b**) nonlinearity error with dimensionless groove depth of sensor with local grove $LG2$ at the bottom of diaphragm at $\overline{{l}_{g}}=0.175$.

**Figure 13.**Effects of $\overline{{l}_{g}}$ on (

**a**) sensitivity and (

**b**) nonlinearity error of sensor with local groove $LG2$ on the top of diaphragm.

**Figure 14.**Comparisons of (

**a**) sensitivity and (

**b**) nonlinearity error between sensors with local groove $LG2$ and $\overline{{l}_{g}}$ = 0.35 on top and at bottom of diaphragm.

**Figure 15.**Variation of ratio of sensitivity to nonlinearity error $\left(\frac{S}{NL}\right)$ with dimensionless groove depth of sensor with local grove $LG2$ at $\overline{{l}_{g}}=0.35$.

**Figure 16.**Stress difference distributions at the applied pressure of 5 kPa: (

**a**) transverse piezoresistor and (

**b**) longitudinal piezoresistor in case of AG- Top ($\overline{{d}_{g}}$ = 0.8). (Only one-half of the domain displayed).

**Figure 17.**Variations of (

**a**) sensitivity and (

**b**) nonlinearity error with dimensionless groove depth of sensors with annular grove $AG$ created on the top (solid line) and at the bottom (dotted line) of diaphragm.

**Figure 18.**Variations of ratios of sensitivity to nonlinearity error $\left(\frac{S}{NL}\right)$ with dimensionless groove depth of sensors with annular groove $AG$.

**Figure 19.**Variations of (

**a**) output voltage and (

**b**) nonlinearity error with the applied pressures of 1–5 kPa of sensors with three optimal groove designs ($LG1-L0$ with $\overline{{d}_{g}}$ = 0.2, $LG2-LT-Top$ with $\overline{{d}_{g}}$ = 0.6 and $\overline{{l}_{g}}$ = 0.35 and $AG-Top$ with $\overline{{d}_{g}}$ = 0.2) and without groove.

**Figure 20.**Ratios of sensitivity to nonlinearity error $\left(\frac{S}{NL}\right)$ of sensors with three optimal groove designs ($LG1-L0$ with $\overline{{d}_{g}}$ = 0.2, $LG2-LT-Top$ with $\overline{{d}_{g}}$ = 0.6 and $\overline{{l}_{g}}$ = 0.35 and $AG-Top$ with $\overline{{d}_{g}}$ = 0.2) and without a groove.

**Figure 21.**Variations of three dimensionless parameters in the case of sensor with local groove $LG2-LT-Top$ at $\overline{{l}_{g}}=0.35$.

**Figure 22.**Variations of three dimensionless parameters in the case of sensor with annular groove $AG-Top$.

**Figure 23.**(

**a**) Variations of averaged stress differences of longitudinal piezoresistor $\left(\Delta {\sigma}_{1}\right)$ and transverse piezoresistor $\left(\Delta {\sigma}_{2}\right)$ with maximum deflection (${\delta}_{max}$ ) and (

**b**) variations of dimensionless averaged stress differences of longitudinal piezoresistor ($\overline{\Delta {\sigma}_{1}}$ ) and transverse piezoresistor ($\overline{\Delta {\sigma}_{2}}$ ) with dimensionless maximum deflection $\left(\overline{{\delta}_{max}}\right)$ from two datasets (each with 20 points) in each averaged stress difference in the case of sensor with local groove $LG2-LT-Top$ at $\overline{{l}_{g}}=0.35$.

**Figure 24.**(

**a**) Variations of averaged stress differences of longitudinal piezoresistor $\left(\Delta {\sigma}_{1}\right)$ and transverse piezoresistor $\left(\Delta {\sigma}_{2}\right)$ with maximum deflection (${\delta}_{max}$ ) and (

**b**) variations of dimensionless averaged stress differences of longitudinal piezoresistor ($\overline{\Delta {\sigma}_{1}}$ ) and transverse piezoresistor ($\overline{\Delta {\sigma}_{2}}$ ) with dimensionless maximum deflection $\left(\overline{{\delta}_{max}}\right)$ from two datasets (each with 20 points) in each averaged stress difference in the case of sensor with annular groove $AG-Top$.

**Figure 25.**Comparisons of simulation results with (

**a**) functional form of maximum deflection ${\delta}_{max,LG2-LT-Top}$ and (

**b**) functional forms of averaged stress differences $\Delta {\sigma}_{1}{}_{,LG2-LT-Top}$ and $\Delta {\sigma}_{2}{}_{,LG2-LT-Top}$.

**Figure 26.**Comparisons of simulation results with (

**a**) functional form of maximum deflection ${\delta}_{max,AG2-Top}$ and (

**b**) functional forms of averaged stress differences $\Delta {\sigma}_{1}{}_{,AG-Top}$ and $\Delta {\sigma}_{2}{}_{,AG-Top}$.

**Figure 27.**Variations of nonlinearity error with the ratio of the compression stress to the tension stress $\left(\frac{\left|{\sigma}_{l,c}\right|}{\sigma l,t}\right)$ versus $\overline{{d}_{g}}$ in the longitudinal stress direction in the case of (

**a**) $LG1-L0$, (

**b**) $LG2-LT-Top$ at $\overline{{l}_{g}}=0.35$ and (

**c**) $AG-Top$. Variations of nonlinearity error and the ratio of maximum deflection to diaphragm thickness $\left({\delta}_{max}/j\right)$ versus $\overline{{d}_{g}}$ in the case of (

**d**) $LG1-L0$, (

**e**) $LG2-LT-Top$ at $\overline{{l}_{g}}=0.35$, and (

**f**) $AG-Top$.

Configuration | Groove Depth $\left(\overline{{\mathit{d}}_{\mathit{g}}}\right)$ | Groove Length $\left(\overline{{\mathit{l}}_{\mathit{g}}}\right)$ |
---|---|---|

$LG1-L0$ | 0.2, 0.4, 0.6 and 0.8 | 0.35 |

$LG1-0T$ | 0.2, 0.4, 0.6 and 0.8 | 0.35 |

$LG1-LT$ | 0.2, 0.4, 0.6 and 0.8 | 0.35 |

$LG2-L0-Top$ | 0.2, 0.4, 0.6 and 0.8 | 0.175 and 0.35 |

$LG2-0T-Top$ | 0.2, 0.4, 0.6 and 0.8 | 0.175 and 0.35 |

$LG2-LT-Top$ | 0.2, 0.4, 0.6 and 0.8 | 0.175 and 0.35 |

$LG2-L0-Bottom$ | 0.2, 0.4, 0.6 and 0.8 | 0.175 and 0.35 |

$LG2-0T-Bottom$ | 0.2, 0.4, 0.6 and 0.8 | 0.175 and 0.35 |

$LG2-LT-Bottom$ | 0.2, 0.4, 0.6 and 0.8 | 0.175 and 0.35 |

$AG-Top$ | 0.2, 0.4, 0.6 and 0.8 | 8.0 |

$AG-Bottom$ | 0.2, 0.4, 0.6 and 0.8 | 8.0 |

**Table 2.**Comparisons between sensitivity and nonlinearity of the sensors with three optimal groove designs and those of the sensor without groove.

Groove Design | $\mathit{S}$ [mV/V/kPa] | $\mathit{N}\mathit{L}{}_{\mathit{m}\mathit{a}\mathit{x}}$ [% FSS] |
---|---|---|

$LG1-L0$ $\mathrm{with}\overline{{d}_{g}}$ = 0.2 | 6.707 | 0.075 |

Decrease (−1.4%) | Decrease (−32%) | |

$LG2-LT-Top$ $\mathrm{with}\overline{{d}_{g}}$$=0.6\mathrm{and}\overline{{l}_{g}}$ = 0.35 | 7.547 | 0.099 |

Increase (11%) | Decrease (−10%) | |

$AG-Top$ $\mathrm{with}\overline{{d}_{g}}$ = 0.2 | 7.774 | 0.071 |

Increase (14%) | Decrease (−35%) | |

Without a groove, according to Thawornsathit et al., 2022 [41] | 6.8 | 0.11 |

MEMS Piezoresistive Pressure Sensor | Pressure Range (kPa) | Diaphragm Width | S (mV/V/kPa) | NL (% FSS) | S/NL |
---|---|---|---|---|---|

Tran et al. (2018b) [20] (Local groove) | 0–5 | 2900 μm | 6.93 | 0.23 | 30.15 |

Li et al. (2020) [21] (Annular groove) | 0–6.895 | 3600 μm | 4.48 | 0.25 | 17.92 |

Sahay et al. (2021) [38] (Annular groove) | 0–5 | 3600 μm | 4.061 | 0.15 | 27.07 |

$\mathrm{Present}\mathrm{work},LG1-L0$$(\overline{{l}_{g}}=0.35$$\mathrm{and}\overline{{d}_{g}}=0.2$) | 1–5 | 2900 μm | 6.707 | 0.075 | 89.43 |

$\mathrm{Present}\mathrm{work},LG2-LT-Top$$(\overline{{l}_{g}}=0.35$$\mathrm{and}\overline{{d}_{g}}=0.6$) | 1–5 | 2900 μm | 7.547 | 0.099 | 76.23 |

$\mathrm{Present}\mathrm{work},AG-Top$$(\overline{{l}_{g}}=8$$\mathrm{and}\overline{{d}_{g}}=0.2$) | 1–5 | 2900 μm | 7.774 | 0.071 | 109.49 |

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**MDPI and ACS Style**

Thawornsathit, P.; Juntasaro, E.; Rattanasonti, H.; Pengpad, P.; Saejok, K.; Leepattarapongpan, C.; Chaowicharat, E.; Jeamsaksiri, W.
Enhancing Performance of a MEMS-Based Piezoresistive Pressure Sensor by Groove: Investigation of Groove Design Using Finite Element Method. *Micromachines* **2022**, *13*, 2247.
https://doi.org/10.3390/mi13122247

**AMA Style**

Thawornsathit P, Juntasaro E, Rattanasonti H, Pengpad P, Saejok K, Leepattarapongpan C, Chaowicharat E, Jeamsaksiri W.
Enhancing Performance of a MEMS-Based Piezoresistive Pressure Sensor by Groove: Investigation of Groove Design Using Finite Element Method. *Micromachines*. 2022; 13(12):2247.
https://doi.org/10.3390/mi13122247

**Chicago/Turabian Style**

Thawornsathit, Phongsakorn, Ekachai Juntasaro, Hwanjit Rattanasonti, Putapon Pengpad, Karoon Saejok, Chana Leepattarapongpan, Ekalak Chaowicharat, and Wutthinan Jeamsaksiri.
2022. "Enhancing Performance of a MEMS-Based Piezoresistive Pressure Sensor by Groove: Investigation of Groove Design Using Finite Element Method" *Micromachines* 13, no. 12: 2247.
https://doi.org/10.3390/mi13122247