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Failure of packaged polysilicon micro-electro-mechanical systems (MEMS) subjected to impacts involves phenomena occurring at several length-scales. In this paper we present a multi-scale finite element approach to properly allow for: (i) the propagation of stress waves inside the package; (ii) the dynamics of the whole MEMS; (iii) the spreading of micro-cracking in the failing part(s) of the sensor. Through Monte Carlo simulations, some effects of polysilicon micro-structure on the failure mode are elucidated.

Localized failures of polysilicon, inertial MEMS induced by accidental drops and impacts are still a major issue in the assessment of micro-system reliability. Shock-induced failures have been recently investigated, both from experimental and theoretical perspectives, in [

Accuracy of the numerical solutions is typically affected by the presence of at least three length-scales (along with the relevant time-scales) in the failure process. At the package length-scale the propagation of stress waves inside the whole device needs to be accurately tracked; at the sensor length-scale the link between the anchor point motion and the sensor dynamics needs to be established, so as to figure out where the stress state may exceed the polysilicon strength and cause localized ruptures; at the polysilicon length-scale possible failure mechanisms, consisting of cracking at grain boundaries (GBs) and within grains, need to be modeled.

At variance with former semi-analytical proposals (like, e.g. [

In this work we furnish a brief description of the multi-scale approach, so as to highlight how accidental drops of micro-systems can be properly modeled; additional details can be found in [

To accurately model the failure of polysilicon MEMS when subjected to shock loadings, three length-scales are allowed for (see

Macro-scale. At this scale stress waves propagating inside the whole package need to be tracked; the typical size of the specimens is on the order of a few millimeters at most.

Meso-scale. At this scale the dynamics of the whole MEMS and the local deformation field in regions close to the anchor points, where the stress field would likely exceed first the polysilicon strength, need to be captured; the size of the specimens is on the order of hundreds of micrometers at most.

Micro-scale. Because of the brittleness of polysilicon, at this scale the nucleation and subsequent propagation of inter- and trans-granular cracks have to be modeled; the size of the specimens is on the order of micrometers.

While at the macro- and meso-scales all the materials in the device can be considered homogeneous, even if somehow anisotropic, at the micro-scale the morphology of the polycrystal, i.e. the shape and orientation of each silicon grain, must be taken in due account to get a precise picture of the failure mode. Polysilicon is extremely brittle at room temperature [

As for length-scale interactions, it is worth mentioning that:

The interaction between macro-scale and meso-scale is negligible in the case here studied. In fact, after the impact this interaction is represented by the tractions transmitted between sensor and package at the anchor points, as resulting from MEMS dynamics; since the mass of the MEMS is three-four orders of magnitude smaller than the package one, the effect of such tractions on the dynamics of the whole device turns out to be negligible.

The interaction between meso-scale and micro-scale should not be disregarded in principle. In fact, nonlinear processes occurring at the micro-scale directly affect the response of the whole MEMS, up to failure. Because of the brittle behavior of silicon, sensor failure usually occurs almost instantaneously after crack inception: as testified by the forthcoming results, this interaction can be therefore disregarded.

Since length-scale interactions are negligible or can be ignored, the multi-scale approach gets simplified and becomes uncoupled (or hierarchical) [

The capability of the proposed approach is here assessed through a case study. Twin uni-axial accelerometers, whose geometry is depicted in _{drop} = 1.5 m.

At the meso-scale the interaction between the vibrating seismic plates and the surrounding fluid has been accounted for through proper damping terms in the equations of motion. This interaction was thoroughly investigated in [_{3} in _{1}-_{2} plane of the two other axes of elastic symmetry of each grain is randomly distributed. The overall elastic polycrystal response thus turns out to be transversely isotropic, depending on the following five independent parameters: the in-plane (i.e. within plane _{1}-_{2}) Young's modulus

As already mentioned, micro-mechanical experimental campaigns [_{1}-_{2} plane of the axes of elastic symmetry of the silicon grains.

Micro-cracking evolution in the polysilicon is here simulated through a cohesive approach, whose predictive capabilities in the dynamic regime were shown e.g. in [^{M}_{n}_{s}_{n}_{s}^{J}

Because of the layout of the modeled sensor, it results that failure occurs because of the percolation of micro-cracks at the joint section between a suspension spring and the anchor point (see the highlighted detail in the sensor picture,

To assess the effect of polycrystal features on MEMS failure, results of Monte Carlo simulations are reported henceforth. According to what already shown in [^{2} to be homogeneously distributed in space, in the former case trans-granular and inter-granular strengths amount to

To partially understand the links between grain orientation and crack pattern at failure, failure modes relevant to two analyses, wherein micro-structure A with perfect GBs has been adopted, are shown in _{cohe} graph).

The previous outcomes suggest that, to thoroughly understand the effects of micro-structure on the failure mode, a statistical analysis is mandatory. Results of Monte Carlo simulations are collected in

As for micro-structure A (top row of

The same trend is shown in

From these outcomes it turns out that the presence of GBs, specially defective ones, close to the reentrant corners at the joint sections between the suspension springs and the anchor, may be detrimental of the shock-carrying capacity of the sensor. Since micro-structural features (like, e.g., the GB network) can not be controlled during production, an assessment of the effects of shocks and drops on polysilicon MEMS is therefore in need of a statistical analysis, like the one here presented.

In this paper a multi-scale, finite element approach has been adopted to assess the effects of polysilicon morphology on the failure mode of an inertial MEMS exposed to shocks. It has been emphasized that, to accurately model such failure process at least three length-scales need to be explored: a macro-scopic one, at the package level; a meso-scopic one, at the sensor level; a micro-scopic one, at the polysilicon level. Thanks to small MEMS inertia and to silicon brittleness, length-scale interactions have been ignored.

Forecasts of cracking pattern at failure have been obtained through Monte Carlo simulations at the micro-scale, at varying polycrystal morphology and in case of perfect/defective grain boundaries. It has been shown that the location of crack inception, and the relevant time elapsed after the impact are affected by micro-structural features. As far as failure is concerned, while the time needed to complete the failure process weakly depends on micro-structural features, the spreading of cracking is highly affected by the network of grain boundaries and by its interaction with the sensor layout, specially close to reentrant corners at the end cross-sections of the suspension springs.

This work has been developed within the frame of MIUR – PRIN07 project

Detail of a shock-induced failure of a suspension spring of a polysilicon MEMS.

Length scales and domains involved in failure modeling, ranging from macro-scale at the package level down to micro-scale at the polycrystal level.

Geometry of the studied uni-axial accelerometers (measures in μm; thickness of the seismic plates is 15 μm).

Adopted effective traction

Micro-structure A, perfect GB case (

Micro-structure A, perfect GB case (

Statistical forecast of the dominant cracks at failure. Top row: micro-structure A; bottom row: micro-structure B. Left column: perfect GBs (

Statistical forecast of the cohesive zone at failure. Top row: micro-structure A; bottom row: micro-structure B. Left column: perfect GBs (