# Multi-scale Analysis of MEMS Sensors Subject to Drop Impacts

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mechanical properties of polysilicon films

_{3}(see Figure 1), i.e. perpendicular to the substrate surface.

**S**of elastic moduli (linking the stress vector σ to the strain vector ε through σ =

**S**ε) for single-crystal silicon can be expressed as follows:

_{11}= 165.7 GPa, s

_{12}= 63.9 GPa, s

_{44}= 79.6 GPa.

_{3}but a random orientation of the other two axes of elastic symmetry in the x

_{1}– x

_{2}plane. The overall response of the crystal assembly, to be adopted in meso-scale analyses, can hence be assumed transversely isotropic, the axis of transverse isotropy being coincident with axis x

_{3}. In the relevant homogenized matrix S of elastic moduli five independent parameters appear: the in-plane (namely in the x

_{1}– x

_{2}plane) Young's modulus E and Poisson's ratio v; the out-of-plane (namely along axis x

_{3}) Young's modulus Ē; the shear modulus Ḡ and the Poisson's ratio v̄ linking in-plane and out-of-plane deformation components. According to the crystal structure described above, constants Ē, v̄, Ḡ are assumed coincident with the single crystal ones, i.e. Ē = 130.10 GPa, Ḡ = 79.6 GPa, v̄ = 0.28; on the other hand, in-plane moduli E and v are bounded as described here below.

^{T}stands for transpose and

**T**

_{e}is the orthogonal transformation matrix that defines the variation of the strain vector components while passing from the local privileged crystal reference frame to the overall one, aligned with axes x

_{i},i= 1,2,3.

**T**

_{σ}plays the same role of

**T**

_{ε}when defining the variation of the stress vector components.

_{σ}= 147.1 GPa and E

_{ε}= 158.7 GPa, while the in-plane Poisson's ratio is bounded by v

_{ε}= 0.18, v

_{σ}= 0.22. Due to relatively small difference between the two bounds, henceforth we assume as in-plane elastic moduli the mean value of the two bounds, i.e. E = 152.9 GPa and v = 0.2.

## 3. Multi-scale analysis of inertial polysilicon MEMS: preliminaries

- -
- air viscosity during drop is neglected;
- -
- the impacted (target) surface is assumed flat and rigid;
- -
- contact between the device and the target surface is frictionless;
- -
- fluid-sensor interaction, leading to viscous damping, is neglected.

_{0}≈ 4 GPa; this value corresponds to a failure probability of 63.2% for a sample under uniaxial tensile loading conditions. This reference value σ

_{0}is then deter-ministically compared to the local stress field envelopes in order to assess if and where the MEMS can fail.

## 4. Simulation of MEMS failure caused by accidental drops

_{d}and Poisson's ratio v

_{d}, and the target is assumed perfectly flat and made of an isotropic elastic material with Young's modulus E

_{t}and Poisson's ratio v

_{t}, then an analytical estimate ā of the said acceleration peak felt by the sensor is given by [9]:

_{imp}being the velocity of the die while impacting the target surface, and m its mass. The analyses are here aimed at modeling accidental drops mainly due to mis-handling; hence, drop height has been assigned as h=150 cm. For the studied device, in case of impacts against a rigid target, the reference acceleration ā turns out to be on the order of 10

^{5}g, g = 9.81 m/s

^{2}being the gravity acceleration.

#### 4.1. Macro-scale analysis

_{d}=130 GPa, Poisson's ratio v

_{d}=0.22 and mass density ρ

_{d}= 2330 Kg/m

^{3}.

_{2}. To account for the effects of this tilting, two different collisions with the target surface have been explored, see Figure 3: in the first one (bottom case) the die strikes the target surface with its bottom surface; in the second one (top case) the die strikes the target surface up-side down, leading to a contact with the upper surface of the cap. According to symmetry, tilting around axis x

_{1}has been prevented.

_{3}(see Figure 2) caused by top and bottom drops, in the interval 0 < t < 100 μs; to highlight the sequence and the magnitude of the peaks, a detail of the records just after the impact, namely for 0 < t < 2 μs, is also depicted. Due to the multiple reflections of the shock waves at the free surfaces of the device, a large amount of peaks show up. The highest peaks are anyway located in short time intervals after the impacts, whereas oscillations appear to be damped for t > 40 μs. By comparing these graphs with the values furnished by Eq. (4), it can be noticed that the analytical estimate fails because of two reasons: it is not able to distinguish between top and bottom drops, namely it is not able to determine which impact configuration is the worst one (obviously, in terms of the highest acceleration peak); it underestimates the acceleration peaks by two orders of magnitude.

#### 4.2. Meso-scale analysis

_{end}= 100 μs in the above mentioned joint sections, as caused by bottom and top impacts respectively. When contact between sensor and die is disregarded, it can be noticed that low frequency variations, with period T

_{low}≈ 13 μs, are superposed to high frequency ones. These latter ones are linked to higher vibration modes of the springs, but they can be spurious artifacts of the simulations. When interaction between sensor and die is allowed for, vibrations turn out to be damped soon after the seismic mass strikes the die surface.

_{3}) lead to different relative movement records registered at points A and D, and at points B and C, thereby causing a coupled bending-torsional vibration of the springs. This is also shown by the Fourier transform of the maximum principal stress at the anchor point, see Figure 9: in the two drop configurations, the same peak in the excitation show up at a frequency corresponding to the fifth vibration mode of the sensor. Figure 10 collects the first six vibration modes of the sensor; here, obviously, the interaction between the sensor and the die has been disregarded to avoid nonlinear effects. As anticipated by plots in Figure 8, the fifth mode produces an out-of-plane bending of the springs coupled to longitudinal torsion.

## 5. Conclusions

## Acknowledgments

## References

- Suhir, E. Is the maximum acceleration an adequate criterion of the dynamic strength of a structural element in an electronic product? IEEE Transactions on Components, Packaging and Manifacturing Technology
**1997**, 20, 513–517. [Google Scholar] - Li, G.; Shemansky, F. Drop test and analysis on micro machined structures. Sensors and Actuators A
**2000**, 85, 280–286. [Google Scholar] - Hauck, T.; Li, G.; McNeill, A.; Knoll, H.; Ebert, M.; Bagdahn, J. Drop simulation and stress analysis of MEMS devices. Ernst, L. J., Zhang, G. Q., Rodgers, P., Meuwissen, M., Marco, S., de Saint Leger, O., Eds.; In Proc. Eurosime06; pp. 203–207. Como (Italy), April 2006. [Google Scholar]
- Cho, S.; Jonnalagadda, K.; Chasiotis, I. Mode I and mixed mode fracture of polysilicon for MEMS. Fatigue and Fracture of Engineering Materials and Structures
**2007**, 30, 21–31. [Google Scholar] - Boroch, R.; Wiaranowski, J.; Mueller-Fiedler, R.; Ebert, M.; Bagdahn, J. Characterization of strength properties of thin polycrystalline silicon films for MEMS applications. Fatigue and Fracture of Engineering Materials and Structures
**2007**, 30, 2–12. [Google Scholar] - Espinosa, H.; Zavattieri, P. A grain level model for the study of failure initiation and evolution in polycrystalline brittle materials. Part I: theory and numerical implementation. Mechanics of Materials
**2003**, 35, 333–364. [Google Scholar] - Corigliano, A.; Cacchione, F.; Frangi, A.; Zerbini, S. Simulation of impact rupture in polysilicon MEMS. Ernst, L. J., Zhang, G. Q., Rodgers, P., Meuwissen, M., Marco, S., de Saint Leger, O., Eds.; In Proc. Eurosime06; pp. 197–202. Como (Italy), April 2006. [Google Scholar]
- Corigliano, A.; Cacchione, F.; Frangi, A.; Zerbini, S. Micro-scale simulation of impact rupture in polysilicon MEMS. Gdoutos, E. E., Ed.; In Proc. ECF16; Alexandropoulos (Greece), July 2006.
- Falcon, E.; Laroche, C.; Fauve, S.; Coste, C. Collision of a 1-D column of beads with a wall. The European Physical Journal B
**1998**, 5, 111–131. [Google Scholar] - Srikar, V.; Senturia, S. The reliability of Microelectromechanical systems (MEMS) in shock environments. Journal of Microelectromechanical Systems
**2002**, 11, 206–214. [Google Scholar] - Zavattieri, P.; Espinosa, H. Grain level analysis of crack initiation and propagation in brittle materials. Acta Materialia
**2001**, 49, 4291–4311. [Google Scholar] - Cho, S.; Chasiotis, I. Elastic properties and representative volume element of polycrystalline silicon for MEMS. Experimental Mechanics
**2007**, 47, 37–49. [Google Scholar] - Zienkiewicz, O. C.; Taylor, R. L. The finite element method: the basisvolume 1, Butterworth-Heinemann Oxford, 5th edition; 2000. [Google Scholar]
- Brantley, W. A. Calculated elastic constants for stress problems associated with semiconductor devices. Journal of Applied Physics
**1973**, 44, 534–535. [Google Scholar] - Mullen, R. L.; Ballarini, R.; Yin, Y.; Heuer, H. Monte Carlo simulation of effective elastic constants of polycristalline thin films. Acta Materialia
**1997**, 45, 2247–2255. [Google Scholar] - Nye, J. F. Physical properties of crystals.; Clarendon: Oxford, 1985. [Google Scholar]
- Chasiotis, I.; Knauss, W. G. The mechanical strength of polysilicon films. Part 2: size effect associated with elliptical and circular perforations. Journal of the Mechanics and Physics of Solids
**2003**, 51, 1551–1572. [Google Scholar] - Weibull, W. A statistical distribution of wide applicability. Journal of Applied Mechanics
**1951**, 18, 293–297. [Google Scholar] - Corigliano, A.; Cacchione, F.; De Masi, B.; Riva, C. On-chip electrostatically actuated bending tests for the mechanical characterization of polysilicon at the micro scale. Meccanica
**2005**, 40, 485–503. [Google Scholar]

**Figure 1.**Sketch of the polysilicon film, showing a columnar grain assembly with growth direction aligned with axis x

_{3}. The reference frame for each single crystal represents the relevant orientation of the local privileged directions (or axes of elastic symmetry).

**Figure 3.**Snapshots of the bouncing die, taken every 20 μs after the impact event: (left) bottom drop; (right) top drop.

**Figure 4.**Comparison between top and bottom acceleration histories (a) in the full analysis range 0 < t < 100 μs and (b) in the short-time interval 0 < t < 2 μs after the impact.

**Figure 6.**Bottom drop: stress envelopes (a) at the spring-anchor joint sections and (b) at the spring-plate joint sections.

**Figure 7.**Top drop: stress envelopes (a) at the spring-anchor joint sections and (b) at the spring-plate joint sections.

**Figure 10.**First vibration modes of the sensor (displacements are amplified in the plots to help understanding the deformation type).

© 2007 by MDPI ( http://www.mdpi.org). Reproduction is permitted for noncommercial purposes.

## Share and Cite

**MDPI and ACS Style**

Mariani, S.; Ghisi, A.; Corigliano, A.; Zerbini, S.
Multi-scale Analysis of MEMS Sensors Subject to Drop Impacts. *Sensors* **2007**, *7*, 1817-1833.
https://doi.org/10.3390/s7081817

**AMA Style**

Mariani S, Ghisi A, Corigliano A, Zerbini S.
Multi-scale Analysis of MEMS Sensors Subject to Drop Impacts. *Sensors*. 2007; 7(9):1817-1833.
https://doi.org/10.3390/s7081817

**Chicago/Turabian Style**

Mariani, Stefano, Aldo Ghisi, Alberto Corigliano, and Sarah Zerbini.
2007. "Multi-scale Analysis of MEMS Sensors Subject to Drop Impacts" *Sensors* 7, no. 9: 1817-1833.
https://doi.org/10.3390/s7081817