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A Poisson–Voronoi-based finite element stress analysis of resonating polysilicon micromachines

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Abstract

The development of reliable miniaturized devices capable of performing sensing, actuation, and computing functions in a robust manner strongly depends on the mechanical performance of arrays of micromachines subjected to alternating stresses during operation. Consequently, to enhance the functionality and longevity of these micromachines, it is essential to accurately determine the structural behavior under both static and dynamic loadings. However, this requires knowledge of the stress distribution and variation due to texture anisotropy effects in critical micromachine regions and insight into the material behavior at micrometer length scales. Accordingly, the objective of this study was to develop an effective computational mechanics framework for performing stress analysis of micromachines at resonance. Dynamic finite element analysis of resonating micromachines was performed to elucidate the effect of material damping on the dynamic response. A Poisson–Voronoi diagram was incorporated in the critical region of finite element models of polysilicon micromachines to mimic the local inhomogeneity and anisotropy of the polycrystalline microstructure. A computationally effective static finite element analysis of the steady-state resonant response, validated by the results of the dynamic analysis, was used to obtain the steady-state stress distribution in the critical region of polysilicon micromachines with different textures. Moreover, a probabilistic stress analysis of hundreds of simulations was carried out to examine the stochastic nature of the grain geometry, size, and orientation and the beam width on the maximum equivalent von Mises stress in the critical micromachine region. The present study provides an efficient computational methodology, which integrates local texture into continuum mechanics modeling, for performing stress analysis of dynamic microdevices exhibiting different polycrystalline microstructures.

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References

  1. Basu, A.K., Basu, A., Ghosh, S., Bhattacharya, S.: MEMS Applications in Electronics and Engineering. AIP Publishing, Melville, NY (2023)

    Google Scholar 

  2. Yeatman, E.M.: Applications of MEMS in power sources and circuits. J. Micromech. Microeng. 17, S184 (2007)

    Google Scholar 

  3. Haque, M.A., Saif, M.T.A.: A review of MEMS-based microscale and nanoscale tensile and bending testing. Exp. Mech. 43, 248–255 (2003)

    Google Scholar 

  4. Kahn, H.: Mechanical properties of micromachined structures. In: Bhushan, B. (ed.) Handbook of Nanotechnology, pp. 1685–1702. Springer, Berlin, Germany (2007)

    Google Scholar 

  5. Sharma, N., Hooda, M., Sharma, S.K.: Synthesis and characterization of LPCVD polysilicon and silicon nitride thin films for MEMS applications. J. Mater. 2014, 954618 (2014)

    Google Scholar 

  6. Ericson, F., Greek, S., Söderkvist, J., Schweitz, J.-Å.: High-sensitivity surface micromachined structures for internal stress and stress gradient evaluation. J. Micromech. Microeng. 7, 30 (1997)

    Google Scholar 

  7. Gaj, A., Le Joncour, L., Baczmański, A., Wroński, S., Panicaud, B., Francois, M., Braham, C., Paradowska, A.: Localization of stresses in polycrystalline grains measured by neutron diffraction and predicted by self-consistent model. Mater. Sci. Forum 681, 103–108 (2011)

    Google Scholar 

  8. Benítez, M.A., Fonseca, L., Esteve, J., Benrakkad, M.S., Morante, J.R., Samitier, J., Schweitz, J.A.: Stress-profile characterization and test-structure analysis of single and double ion-implanted LPCVD polycrystalline silicon. Sens. Actuators A 54, 718–723 (1996)

    Google Scholar 

  9. Karner, S., Blank, O., Rösch, M., Burghammer, M., Zalesak, J., Keckes, J., Todt, J.: X-ray nanodiffraction analysis of residual stresses in polysilicon electrodes of vertical power transistors. Materialia 24, 101484 (2022)

    Google Scholar 

  10. Hirsekorn, S.: Elastic properties of polycrystals—a review. Text. Microstruct. 12, 1–14 (1990)

    Google Scholar 

  11. Jensen, B.D., de Boer, M.P., Masters, N.D., Bitsie, F., LaVan, D.A.: Interferometry of actuated microcantilevers to determine material properties and test structure nonidealities in MEMS. J. Microelectromech. Syst. 10, 336–346 (2001)

    Google Scholar 

  12. Corigliano, A., De Masi, B., Frangi, A., Comi, C., Villa, A., Marchi, M.: Mechanical characterization of polysilicon through on-chip tensile tests. J. Microlectromech. Syst. 13, 200–219 (2004)

    Google Scholar 

  13. Jayaraman, S., Edwards, R.L., Hemker, K.J.: Relating mechanical testing and microstructural features of polysilicon thin films. J. Mater. Res. 14, 688–697 (1999)

    Google Scholar 

  14. Ghisi, A., Mariani, S.: Effect of imperfections due to material heterogeneity on the offset of polysilicon MEMS structures. Sensors 19, 3256 (2019)

    Google Scholar 

  15. Sharpe, W.N., Jr., Jackson, K., Coles, G., LeVan, D.A.: Young’s modulus and fracture strength of three polysilicons. Mater. Res. Soc. Symp. Proc. 657, EE5.5 (2001)

    Google Scholar 

  16. DelRio, F.W., Cook, R.F., Boyce, B.L.: Fracture strength of micro- and nano-scale silicon components. Appl. Phys. Rev. 2, 021303 (2015)

    Google Scholar 

  17. Vayrette, R., Galceran, M., Coulombier, M., Godet, S., Raskin, J.P., Pardoen, T.: Size dependent fracture strength and cracking mechanisms in freestanding polycrystalline silicon films with nanoscale thickness. Eng. Fract. Mech. 168, 190–203 (2016)

    Google Scholar 

  18. Mirzazadeh, R., Aldo Ghisi, A., Mariani, S.: Statistical investigation of the mechanical and geometrical properties of polysilicon films through on-chip tests. Micromachines 9, 53 (2018)

    Google Scholar 

  19. McKinstry, H.A., Shull, H.E., Buessem, W.R.: Localized stress in elastic polycrystalline materials. Nucl. Metall. 20, 695–705 (1976)

    Google Scholar 

  20. McKinstry, H.A., Shull, H.E., Buessem, W.R.: Stress and strain fields in idealized polycrystalline materials and the prediction of fracture initiation sites. Nucl. Metall. 20, 929–941 (1976)

    Google Scholar 

  21. Tvergaard, V., Hutchinson, J.W.: Microcracking in ceramics induced by thermal expansion or elastic anisotropy. J. Am. Ceram. Soc. 71, 157–166 (1988)

    Google Scholar 

  22. Kumar, S., Kurtz, S.K.: Simulation of material microstructure using a 3D Voronoi tesselation: calculation of effective thermal expansion coefficient of polycrystalline materials. Acta Metall. Mater. 42, 3917–3927 (1994)

    Google Scholar 

  23. Kumar, S., Kurtz, S.K., Agarwala, V.K.: Micro-stress distribution within polycrystalline aggregate. Acta Mech. 114, 203–216 (1996)

    MATH  Google Scholar 

  24. Mullen, R.L., Ballarini, R., Yin, Y., Heuer, A.H.: Monte Carlo simulation of effective elastic constants of polycrystalline thin films. Acta Mater. 45, 2247–2255 (1997)

    Google Scholar 

  25. Ballarini, R., Mullen, R.L., Heuer, A.H.: The effects of heterogeneity and anisotropy on the size effect in cracked polycrystalline films. Int. J. Fract. 95, 19–39 (1999)

    Google Scholar 

  26. Li, X.-D.: K variations and anisotropy: microstructure effect and numerical predictions. ASME J. Eng. Mater. Technol. 125, 65–74 (2003)

    Google Scholar 

  27. Lazar, E.A., Mason, J.K., MacPherson, R.D., Srolovitz, D.J.: Complete topology of cells, grains, and bubbles in three dimensional microstructures. Phys. Rev. Lett. 109, 095505 (2012)

    Google Scholar 

  28. Vittorietti, M., Kok, P.J.J., Sietsma, J., Jongbloed, G.: Accurate representation of the distributions of the 3D Poisson-Voronoi typical cell geometrical features. Comput. Mater. Sci. 166, 111–118 (2019)

    Google Scholar 

  29. Ghazvinian, E., Diederichs, M.S., Quey, R.: 3D random Voronoi grain-based models for simulation of brittle rock damage and fabric-guided micro-fracturing. J. Rock Mech. Geotech. Eng. 6, 506–521 (2014)

    Google Scholar 

  30. Wang, Y., Ballarini, R., Rodin, G.J.: Crack-tip parameters in polycrystalline plates with soft grain boundaries. J. Eng. Mech. 134, 100–109 (2008)

    Google Scholar 

  31. Bourne, D.P., Kok, P.J.J., Roper, S.M., Spanjer, W.D.T.: Laguerre tessellations and polycrystalline microstructures: a fast algorithm for generating grains of given volumes. Philos. Mag. 100, 2677–2707 (2020)

    Google Scholar 

  32. Reischig, P., Ludwig, W.: Three-dimensional reconstruction of intragranular strain and orientation in polycrystals by near-field X-ray diffraction. Curr. Opin. Solid State Mater. Sci. 24, 100851 (2020)

    Google Scholar 

  33. Quey, R., Renversade, L.: Optimal polyhedral description of 3D polycrystals: method and application to statistical and synchrotron X-ray diffraction data. Comput. Methods Appl. Mech. Eng. 330, 308–333 (2018)

    MathSciNet  MATH  Google Scholar 

  34. Heller, L., Shayanfard, P., Quey, R., Sittner, P., Karafiatova, I., Pawlas, Z.: Elastic grain interactions in a polycrystalline wire analyzed by a numerical model reconstructed from 3D-XRD data. In: 4th International Congress on 3D Materials Science, Elsimore, Denmark (2018)

  35. White, C.D., Xu, R., Sun, X., Komvopoulos, K.: Dynamic MEMS devices for multi-axial fatigue and elastic modulus measurement. Proc. SPIE 4980, 63–74 (2003)

    Google Scholar 

  36. White, C., Xu, R., Sun, X., Komvopoulos, K.: Characterization of microscale material behavior with MEMS resonators. Proc. Nanotechnol. Conf. 1, 494–497 (2003)

    Google Scholar 

  37. Greek, S., Johansson, S.A.I.: Tensile testing of thin film microstructures. Proc. SPIE 3224, 344–351 (1997)

    Google Scholar 

  38. Sharpe, W.N., Turner, K.T., Jr., Edwards, R.L.: Tensile testing of polysilicon. Exp. Mech. 39, 162–170 (1999)

    Google Scholar 

  39. Krulevitch, P., Nguyen, T.D., Johnson, G.C., Howe, R.T., Wenk, H.R., Gronsky, R.: LPCVD polycrystalline silicon thin films: the evolution of structure, texture and stress. Mater. Res. Soc. Proc. 202, 167–172 (1990)

    Google Scholar 

  40. Krulevitch, P.A.: Micromechanical investigations of silicon and Ni–Ti–Cu thin films. Ph.D. thesis, Department of Mechanical Engineering, University of California, Berkeley, CA (1994)

  41. Hibbitt, Karlsson, and Sorensen: ABAQUS/Standard: User's Manual, 5.8 version, Pawtucket, RI (1998)

  42. Madou, M.J.: Fundamentals of Microfabrication: The Science of Miniaturization, 2nd edn. CRC Press, Boca Raton, FL (2002)

    Google Scholar 

  43. Nagasima, N., Kubota, N.: Structures of Si films chemically vapor-deposited on amorphous SiO2 substrates. Jpn. J. Appl. Phys. 14, 1105–1112 (1975)

    Google Scholar 

  44. Kamins, T.I.: Structure and properties of LPCVD silicon films. J. Electrochem. Soc. 127, 686–690 (1980)

    Google Scholar 

  45. Kahn, H., He, A.Q., Heuer, A.H.: Homogeneous nucleation during crystallization of amorphous silicon produced by low-pressure chemical vapour deposition. Philos. Mag. A 82, 137–165 (2002)

    Google Scholar 

  46. Maier-Schneider, D., Köprülülü, A., Holm, S., Obermeier, E.: Elastic properties and microstructure of LPCVD polysilicon films. J. Micromech. Microeng. 6, 436–446 (1996)

    Google Scholar 

  47. Okabe, A., Boots, B., Sugihara, K., Chiu, S.N.: Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd edn. Wiley, Chichester, England (2000)

    MATH  Google Scholar 

  48. DiCenzo, S.B., Wertheim, G.K.: Monte Carlo calculation of the size distribution of supported clusters. Phys. Rev. B 39, 6792–6796 (1989)

    Google Scholar 

  49. Tipper, J.C.: A straightforward iterative algorithm for the planar Voronoi diagram. Inf. Process. Lett. 34, 155–160 (1990)

    MathSciNet  MATH  Google Scholar 

  50. Tipper, J.C.: FORTRAN programs to construct the planar Voronoi diagram. Comput. Geosci. 17, 597–632 (1991)

    Google Scholar 

  51. Pampuch, R.: Constitution and Properties of Ceramic Materials. Elsevier, New York, NY (1990)

    Google Scholar 

  52. Ting, T.C.T.: Invariants of anisotropic elastic constants. Q. J. Mech. Appl. Math. 40, 431–448 (1987)

    MathSciNet  MATH  Google Scholar 

  53. Tang, W.C., Tu-Cuong, H., Nguyen, H., Howe, R.T.: Laterally driven polysilicon resonant microstructures. Sens. Actuators 20, 25–32 (1989)

    Google Scholar 

  54. Candler, R.N., Park, W.-T., Hopcroft, M., Kim, B., Kenny, T.W.: Hydrogen diffusion and pressure control of encapsulated MEMS resonators. In: 13th International Conference on Solid-State Sensors, Actuators and Microsystems. Digest of Technical Papers. TRANSDUCERS '05 1, 920–923 (2005)

  55. Buser, R.A., De Rooij, N.F.: Very high Q-factor resonators in monocrystalline silicon. Sens. Actuators A 21–23, 323–327 (1990)

    Google Scholar 

  56. Muhlstein, C.L., Brown, S.B., Ritchie, R.O.: High-cycle fatigue and durability of polycrystalline silicon thin films in ambient air. Sens. Actuators A 94, 177–188 (2001)

    Google Scholar 

  57. Muhlstein, C.L., Stach, E.A., Ritchie, R.O.: Mechanism of fatigue in micron-scale films of polycrystalline silicon for microelectromechanical systems. Appl. Phys. Lett. 80, 1532–1534 (2002)

    Google Scholar 

  58. Massey, F.J., Jr.: The Kolmogorov–Smirnov test for goodness of fit. J. Am. Stat. Assoc. 46, 68–78 (1951)

    MATH  Google Scholar 

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Acknowledgements

This work was funded by the National Science Foundation (NSF) under Grant No. DMI-9872324 and the Defense Advanced Research Projects Agency (DARPA) under Grant No. DABT63-98-1-0011.

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  1. Department of Mechanical Engineering, University of California, Berkeley, CA, 94720, USA

    Rui Xu & Kyriakos Komvopoulos

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Xu, R., Komvopoulos, K. A Poisson–Voronoi-based finite element stress analysis of resonating polysilicon micromachines. Acta Mech 234, 6705–6721 (2023). https://doi.org/10.1007/s00707-023-03775-0

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