A cumulative fatigue damage model of polysilicon films for MEMS resonator under repeated loadings
Graphical abstract
Introduction
In the past decade, microelectromechanical systems (MEMS) have been fueling the proliferation of the internet of things with a number of successful products including microphones, accelerometers, gyroscopes and bulk acoustic wave filters. Emerging innovations in MEMS from research labs such as zero-power sensors [1], [2], [3], piezoelectric resonant sensors [4], [5], micromachined ultrasonic transducers [6], [7] have shown great potential to keep up the fast-growing trend of MEMS. By its nature, MEMS comprises of multiple physics (such as electrical, mechanical, thermal) [8], which can result in different kinds of failure mechanisms like fatigue, creep, contact wear, short circuit, etc.. These failures may exist in the process of manufacturing or occur later in the stage of usage. Due to the complexity of failure mechanisms and their progression during the lifetime of MEMS devices, achieving high reliability has been one of the bottlenecks historically in the fast-paced commercialization of MEMS-based products. In this context, systematic study of common failure mechanisms of popular MEMS structures would great benefit the development of reliable MEMS devices suitable for robust and long-term operation. Indeed, the investigation of MEMS reliability started with the fatigue analysis of micro-cantilevers made of crystalline silicon [9], [10], [11] owning to the popularity of such a structure and material choice in the early times of MEMS. Micron-scale silicon is more susceptible to fatigue compared to common bulk silicon due to the high surface area to volume ratio [12], [13]. The fatigue behavior of silicon films was first reported by Connally and Brown [14], which is one of the pioneer work in the investigation of resonant frequency shift due to fatigue in silicon. Subsequently, the fatigue mechanisms were studied by many other researchers (e.g., C.L.M., E.A.S., R.O.R [15], [16], [17] and H.K [18], [19], [20]) who have made significant contributions to fatigue mechanisms of micro/nanometer-scale silicon.
Smooth, notched, and pre-cracked specimens, as typical fatigue characterization structures, have been used to evaluate the mechanisms of crack initiation and growth and other fatigue behaviors of thin films [15], [16], [17], [18], [20], [21]. From the scanning electronic microscope (SEM) or high voltage transmission electron microscope (HVTEM) images of the notch after subjected fatigue loadings, a clear dominant crack trajectory can be observed, indicating that the notched structure starts to fail mechanically [15], [16], [17]. Instead, by stopping cyclic loading of fatigue specimens prior to failure and examining them with SEM or HVTEM, several small cracks (~nanoscale) can be seen inside the native oxide layer, on the order of tens of nanometers in length [16], [19]. Based on these observations, fatigue failures of silicon films due to the evolution of these small cracks have been demonstrated, which suggests the cracks nucleation and growth is associated with progressive accumulation of damage [22], [23], [24]. In particular, substantial evidence indicates that time-dependent damage of brittle or ductile materials starts from the initiation of microstructurally small cracks (MSC) [25] or dislocations whose size are smaller than the grain of the material. Numerous such distributed small cracks will further grow to form a dominant crack [26], [27], [28]. MEMS structures based on silicon thin film are more vulnerable to the small cracks even before they evolving into a dominant one because some physical properties (e.g. electrical resistivity and Young's modulus) of the structure change as soon as the small cracks start to grow. Such changes in critical areas of MEMS structures may directly affect the performance of the device [29], [30], [31]. One the other hand, this connection between fatigue damage accumulation and material properties can be exploited for the estimation of remaining lifetime of a MEMS device by monitoring specific performance parameters (e.g. resistance and resonant frequency) that are dependent on the material properties.
Fatigue mechanisms of silicon films have been investigated extensively so far, in which two main mechanisms have been identified: fatigue in silicon and fatigue in reaction layer [32]. To date, several models have been developed to quantify the fatigue of the silicon films induced by the two mechanisms. In particular, most of the existing models (e.g., by Ikehara [33] and Baragetti [34]) are established based on fracture mechanics theory where a fracture intensity factor k is introduced to qualitatively evaluate the main crack extension, which is assumed as the equivalence of small cracks growth. In this case, the observed deterioration in driving force can be used to predict the cracks growth. However, the growth of cracks only represents an intermediate indicator to the fatigue damage in the material. Instead, macroscale responses such as the change of resonant frequency or electric resistance are more directly related to the performance of the devices that are made of the material experiencing fatigue damage. In this context, using a damage quantity to represent the degradation degree of a material would be advantageous since a relationship between damage variable and macroscale responses (e.g. resonant frequency) can be readily established using the continuum damage mechanics (CDM) theory. CDM is a method of achieving relatively high resolution modelling of a distributed damage process without the expense of modelling the micromechanics around each individual defect [35], [36], [37]. In this case, the fatigue damage effectively acts as a bridge connecting the cracks growth with the macroscale responses. Differently from previous models established based on fracture mechanics method, in this work, a fatigue model for polysilicon material based on CDM is introduced to quantitatively evaluate the fatigue damage in MEMS actuators for the first time. By using the damage quantity to describe the fatigue performance a connection between the microcrack evolutions with macroscale responses is established. In particular, the proposed model directly links the microscopic damage process to a readily measurable macroscopic structural response, resonance frequency, for the prediction of remain lifetime of a structure made of polysilicon. Another advantage of using CDM is that finite element method can be implemented with the developed damage model to evaluate the damage quantity of the whole structure, which can be used to evaluate reliability of the most vulnerable spots in the structure. In this paper, a case study of a notched microcantilever beam as a fatigue characterization structure is presented, where both analytical modeling and finite element method (FEM) are used for the fatigue analysis of the structure. The proposed model is validated through the comparison with FEM and experimental results.
Section snippets
Definition of fatigue damage variable due to environmentally- and cyclically-assisted cracking in silicon
Muhlstein et al. [16] and Allameh et al. [38] reported that the fatigue of silicon thin films is attributed to the growth of subcritical cracking within the surface oxide layer. The subcritical cracking usually results from a stress-enhanced environmental interaction between the solid and moisture, although the driving force is smaller than the critical stress intensity. Surface oxide formation on mechanically induced subcritical cracks can lead to wedging effects that increase the applied
Determination of model parameters and model verification
In order to apply the developed fatigue damage model to polysilicon films, the model parameters including, , ,,should be determined. Polysilicon transverse rotating resonator is a classic structure for fatigue analysis, which has been widely employed in fatigue tests. To complement model fitting, experimental data from Muhlstein et al. [15], [16], [17] is used to fit for determining model parameters, and the unknown parameters of the developed model can be solved by least square method.
Finite element model and the modal analysis
A two-dimensional analysis is performed to study the in-plane dynamics, while a three-dimensional model is only used to identify the out-of-plane mode shapes and natural frequencies. The two-dimensional analysis is performed using triangular, plane stress elements, and the mesh quality of model as shown in Fig. 6 where all elements quality are in the range 0.5 ~ 1.
Small deformations and elastic isotropy are assumed, fixed constraint at the base of the notched beam, and the elastic modulus
Applications of the developed model on evaluating fatigue performance
The proposed damage model has been validated through the predictions result of cracks growth in Fig. 4 and the deterioration of resonant frequency obtained by finite element simulations in Fig. 9, Fig. 10. Except for the validations, these results can demonstrate that using damage can represent the failure process due to the change of resonant frequency. In order to prove that the proposed damage model has general applications and can predict the damage area more accurately, in which Fig. 10
Conclusion
In this paper, we developed a fatigue damage model to evaluate the fatigue performance and predict the fatigue lifetime of polysilicon films. The proposed model is able to estimate the growth of microcracks and the change of resonant frequency as a function of increasing loading cycles. Finite element analysis based on the proposed fatigue damage model can further reveal the progression of damage at localized areas during the fatigue loading. The result of finite element simulation shows that
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work is partially supported by the Graduate Innovation Project of Jiangsu Province, China (KYCX18_0060) and the State Scholarship Fund provided by China Scholarship Council (CSC201906090082).
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