Modeling of Probabilistic Failure of Polycrystalline Silicon MEMS Structures

Abstract

Microelectromechanical Systems (MEMS) devices typically need to be designed against a very low failure probability, which is on the order of or lower. Experimental determination of the target strength for such a low failure probability requires testing of tens of thousands of specimens, which can be cost prohibitive for the design process. Therefore, understanding the probabilistic failure of MEMS devices is of paramount importance for design. Currently available probabilistic models for predicting the strength statistics of MEMS structures are based on classical Weibull statistics. Significant advances in experimental techniques for measuring the strength of MEMS devices have produced data that have unambiguously demonstrated that the strength distributions consistently deviate from the Weibull distribution. Such deviations can be explained by the fact that the Weibull distributions are derived based on extreme value statistics, which is inapplicable to MEMS devices where the dimensions of the material microstructure are not negligible compared to the characteristic structural dimensions. This paper presents a robust probabilistic model for strength distribution of polycrystalline silicon (poly‐Si) MEMS structures that could be extended to other brittle materials at the microscale. The overall failure probability of the structure is related to the failure probability of each material element along its sidewalls through a weakest‐link statistical model. The failure statistics of the material element is determined by both the intrinsic random material strength as well as the random stress field induced by the sidewall geometry. Different from the classical Weibull statistics, the present model is designed to account for structures consisting of a finite number of material elements, and it predicts a scale effect on their failure statistics. It is shown that the model agrees well with the measured strength distributions of poly‐Si MEMS specimens of different sizes, and the calibrated mean strength of the material element is consistent with the theoretical strength of silicon. Meanwhile, it is shown that the two‐parameter and three‐parameter Weibull distributions cannot provide optimum and consistent fits of the observed size‐dependent strength distributions, and thus have very limited prediction capability. The present model explicitly relates the strength statistics to the size effect on the mean structural strength, and therefore provides an efficient means of determining the failure statistics of MEMS structures.