# Experimental-Numerical Comparison of the Cantilever MEMS Frequency Shift in presence of a Residual Stress Gradient

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fabrication process and specimen's geometry description

- At first a 1000 nm thick thermal field oxide is grown at 975°C in a wet ambient on a silicon wafer, then a nitrogen annealing at the same temperature is performed. A 630 nm thick polysilicon layer employed for resistors and for actuation lines is deposited by LPCVD and subsequently patterned through dry etching.
- A 300 nm thick silicon oxide is deposited at 718°C, also by LPCVD process and via-holes are opened through dry etching.
- A multi-layer metal for signal lines is sputtered and subsequently patterned by dry etching. Temperature profile: Ti(30nm): 400°C; TiN(50 nm): 400°C; Al/Si+Ti(410/60 nm): ambient temperature; TiN(80 nm): 300°C. A 100 nm thick oxide layer is then deposited at 430°C. In order to define via-holes for opening contacts or to uncover the multimetal line oxide removal is defined with a mask by dry etching.
- A 150 nm gold layer is deposited by PVD and patterned through wet etching. A 3 μm thick sacrificial photoresist layer is deposited and patterned.
- A 1.3 μm thick gold layer is electroplated at 52°C employing a chromium-gold PVD adhesion layer, called seed layer. This is the suspended/movable part of the devices.
- The last deposition step is another gold layer deposited at 52°C used to reinforce the anchors and the suspended parts of the structures. Finally, the structure release is obtained by ashing the sacrificial layer through plasma oxygen etching. At the end of the process a sintering is performed at 190°C.

_{2}O

_{3}. This diffusion process can be quite extensive, with complete depletion of chromium adhesive layer and formation of channelled grain boundaries that are occupied with Cr

_{2}O

_{3}and eventually formation of single crystals of Cr

_{2}O

_{3}at surface. The chromium transport may manifest itself in development of undesirable characteristics, such as decrease in electrical conductivity and generation of internal stress. Residual stresses vary along the beam thickness because of the difference on percentage of diffused chromium.

## 3. Measurement methods and experimental results

#### 3.1. Profilometry measurements

#### 3.2. Out of plane vibration measurements

_{G}to the suspended structure and a negative voltage to the ground. This voltage has a dc and an ac component

_{dc}and V

_{ac}are values of the dc and ac components of the applied voltage, f is the excitation frequency and T the time. V

_{ac}was maintained at low values if compared with V

_{dc}, in order to investigate the system behaviour around the electrostatic equilibrium position. This allowed neglecting non-linearity due to the electrostatic/structural coupling while building numerical models.

## 4. Frequency shift: analytical models and residual stress evaluation

#### 4.1. Analytical models

_{0}is the planar constant stress and σ

_{z}the coefficient of linear stress variation in z direction. The general uniaxial residual stress field in a thin film is represented by a polynomial; in this first approximation σ

_{0}represents the cumulative effect of all the symmetric polynomial terms and σ

_{z}represents the influence of the gradient stress anti-symmetric functions [23].

_{b}

^{3}/12, c is the curvature. Assuming that M

_{b}is uniform along the x-axis, the integration of Equation (5) with respect to x is

_{1}= A

_{1}(σ

_{0},σ

_{z}) [23].

_{L}is obtained using Equation (4) in Equation (6)

_{0}is the gap in ideally flat conditions, also known from specimen profilometry. In Figures 7 and 8 a comparison between experimental and analytically calculated profiles of the upper side of the microbeams is presented. In Figure 7 specimens of wafer 1 are disposed by increasing length while in Figure 8 specimens of wafer 2 are disposed by increasing thickness.

#### 4.2. Numerical models

_{0}obtained with the following formula:

^{3}was used [18].

_{1}, γ

_{2}, γ

_{3}are fitting parameters depending on the constraints. In the case of cantilever γ

_{1}=0.07, γ

_{2}=1,γ

_{3}=0.42.

_{c}formulation was encountered by stopping at the second term a Taylor series expansion. It's thus possible to pass from V

_{pi}from Equation (13) to V

_{pi,c}as follows

_{pi}is the pull-in voltage of a flat cantilever, V

_{pi,c}is the pull-in voltage of a curled cantilever, R

_{c}is curvature radius, calculated as the inverse of curvature c.

## 5. Conclusions

## Acknowledgments

## References

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## Abbreviations

c | curvature |

E | elastic modulus |

f | exitation frequency |

f_{c} | corrective coefficient for pull-in tension |

f_{n} | natural frequency |

g | air-gap |

g_{0} | initial gap |

g_{m} | medium gap |

I | inertia moment of beam section |

L | effective length |

L_{c} | total length |

m | mass |

M_{b} | bending moment |

R_{c} | curvature radius |

S | surface |

t | thickness |

V_{ac} | ac voltage |

V_{dc} | dc voltage |

w | width |

u | vertical beam deflection |

z | vertical coordinate along the thickness |

ε_{0} | dielectric constant |

σ_{0} | planar constant stress |

σ_{z} | coefficient of linear stress variation |

Ω | stress gradient |

**Figure 5.**Experimental frequency spectrum of a microbeam with V

_{ac}=4.5V and three different values of DC voltage: 0V: f

_{n}=41640 Hz, Q= 43.4; 20V: f

_{n}=41230 Hz, Q=38.2; 40V: f

_{n}=40110, Q=35.9. Q factor was calculated with the half power method.

**Figure 8.**Upper profile and frequency shift variation of cantilever beams of set 4 (nominal thickness 1.8 um), set 5 (nominal thickness 3 um) and set 6 (nominal thickness 4.8 um).

g (μm) | t (μm) | L (μm) | u (μm) | c (μm^{-1}) | Ω (MPa/μm) | |
---|---|---|---|---|---|---|

specimen1 | ||||||

set1 | 3.0 | 1.78 | 242.4 | 3.81 | 1.30E-04 | 12.8 |

set2 | 3.0 | 1.68 | 245.0 | 5.30 | 1.77E-04 | 17.4 |

set3 | 3.0 | 1.75 | 240.0 | 4.67 | 1.62E-04 | 16.0 |

mean value | 3.0 | 1.73±0.05 | 242.5±2.5 | 4.59±0.74 | 1.56E-04 | 15.4±2.3 |

specimen2 | ||||||

set1 | 3.0 | 1.84 | 288.5 | 13.43 | 3.23E-04 | 31.8 |

set2 | 3.0 | 1.71 | 289.4 | 13.61 | 3.25E-04 | 32.0 |

set3 | 3.0 | 1.72 | 293.3 | 12.23 | 2.84E-04 | 28.0 |

mean value | 3.0 | 1.76±0.06 | 290.6±2.7 | 13.09±0.69 | 3.11E-04 | 30.6±2.0 |

specimen3 | ||||||

set1 | 3.0 | 1.91 | 340.1 | 19.14 | 3.31E-04 | 32.6 |

set2 | 3.0 | 1.59 | 343.8 | 18.85 | 3.19E-04 | 31.4 |

set3 | 3.0 | 1.55 | 340.1 | 14.11 | 2.44E-04 | 24.0 |

mean value | 3.0 | 1.68±0.18 | 341.3±1.8 | 17.37±2.51 | 2.98E-04 | 29.3±4.3 |

g (μm) | t (μm) | L (μm) | u (μm) | c (μm^{-1}) | Ω (MPa/μm) | |
---|---|---|---|---|---|---|

specimen1 | ||||||

set4 | 3.0 | 1.78 | 192.8 | 4.16 | 2.23E-04 | 22.0 |

set5 | 3.0 | 1.77 | 191.4 | 4.65 | 2.53E-04 | 24.9 |

set6 | 3.0 | 1.74 | 189.6 | 4.23 | 2.35E-04 | 23.1 |

mean value | 3.0 | 1.76±0.02 | 191.3±1.6 | 4.35 | 2.37E-04 | 23.3 |

specimen2 | ||||||

set4 | 3.0 | 2.41 | 191.9 | 2.69 | 1.46E-04 | 14.4 |

set5 | 3.0 | 2.26 | 192.1 | 2.10 | 1.14E-04 | 11.2 |

set6 | 3.0 | 2,31 | 192.1 | 2.55 | 1.38E-04 | 13.6 |

mean value | 3.0 | 2.33±0.07 | 192.0±0.2 | 2.45 | 1.32E-04 | 13.7 |

specimen3 | ||||||

set4 | 3.0 | 4.31 | 189.9 | 0 | 0 | 0 |

set5 | 3.0 | 4.40 | 191.5 | 0 | 0 | 0 |

set6 | 3.0 | 4.37 | 190.5 | 0 | 0 | 0 |

mean value | 3.0 | 4.36±0.04 | 190.6±0.8 | 0 | 0 | 0 |

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**MDPI and ACS Style**

Ballestra, A.; Somà, A.; Pavanello, R.
Experimental-Numerical Comparison of the Cantilever MEMS Frequency Shift in presence of a Residual Stress Gradient. *Sensors* **2008**, *8*, 767-783.
https://doi.org/10.3390/s8020767

**AMA Style**

Ballestra A, Somà A, Pavanello R.
Experimental-Numerical Comparison of the Cantilever MEMS Frequency Shift in presence of a Residual Stress Gradient. *Sensors*. 2008; 8(2):767-783.
https://doi.org/10.3390/s8020767

**Chicago/Turabian Style**

Ballestra, Alberto, Aurelio Somà, and Renato Pavanello.
2008. "Experimental-Numerical Comparison of the Cantilever MEMS Frequency Shift in presence of a Residual Stress Gradient" *Sensors* 8, no. 2: 767-783.
https://doi.org/10.3390/s8020767