**1. Introduction**

Because of the microfabrication process, a polysilicon MEMS structure is never found at the exact designed place, but after the production steps, a shift from the rest position is often observed [1,2]. The difference can be so small to be inconsequential for the working conditions, but in particular cases, it can represent a significant disturbance. Since the phenomenon depends on the actual geometry and material involved, it is intrinsically of a stochastic nature. To deal with the consequences of these uncertainties quantitatively, a statistical approach should be therefore followed, by introducing a probabilistic distribution for the relevant variables involved in the manufacturing process.

This variable initial offset creates a series of uncertainties that require workarounds or specific electronics to account for them. The most common remedy is of the type exemplified by [3], where an initial offset due to process mismatch (called "zero-g offset" or "intrinsic sensor offset"), generating a variation in the initial capacitance of the sensor up to 10% and 20% for a three-axis accelerometer, is treated through a compensation circuit. Another calibration (or auto-calibration) procedure was shown in [4], to account for the varying gap geometry of MEMS featuring comb drives due to the fabrication process. More importantly, the mentioned uncertainties affect the stiffness of critical components, spreading it around the designed value: a plethora of methods, a good review of which can be found in [5], has been invented to overcome this problem. Stiffness is in fact a key parameter for defining relevant design variables, for example the resonance frequency: e.g. in [6] it was discussed how, because of the uncertainty on the geometry due to the manufacturing process, it is a challenging objective to obtain a tight sense-drive frequency separation (e.g. 2%) for an MEMS gyroscope, since the stiffness cannot be controlled finely enough.

When one considers the previous examples, it is understandable that uncertainties affecting the initial positions are however related to the ones responsible for the variability from the target status (i.e. not due to drifts or noise) during the MEMS working conditions: they are in fact both generated by the same sources during the manufacturing process. For example, in [7] a list of sensitive parameters, namely key geometrical dimensions (varying about 6–10%) or material properties (varying about 10%) (as taken from [8]) was used to explore how the uncertainty propagates in a mathematical model describing the working conditions of an MEMS energy harvester: it is easy to recognize that the same quantities would affect an offset from the designed configuration as well.

It would be therefore a useful insight to quantify the role of these uncertainties in an effective and manageable way, before moving to the working conditions, where other issues could be involved. Moreover, the corrections necessary to overcome the initial offset can become themselves too costly to be economically convenient, as shown in [3].

Going more into the details, the sources of these uncertainties can be (i) the so-called over-etch generated from the sequence of deposition, masking alignment, photolithography, and the deep reaction-ion etching process [9–11] and (ii) the effect of the material heterogeneity due to the polycrystalline morphology [12–14]: the former affects the geometry layout, while the latter influences the effective material properties. Both the causes tend to be neglected or hastily (and roughly) estimated, when the dimensions of critical MEMS components are significantly larger than a characteristic length of the microstructure, such as the average silicon grain size. However, both of these causes become relevant as far the miniaturization proceeds and the aforementioned critical dimensions shrink.

In the past, the authors devised a numerical approach to foresee the mechanical behavior of a polycrystalline ensemble by carefully representing, through an artificial reconstruction, the network of the grain and grain boundaries [15–18]. The approach is typically used to carry out a homogenization procedure and to establish the conditions necessary to build a Representative Volume Element (RVE) [19,20], but it can also be adopted to study Statistical Volume Elements (SVEs), i.e. to construct a statistical set accounting for all the desired uncertainties (such as the grain topology and the size and orientations of the elastic axis) whenever the characteristics at the micro scale become non-negligible for the mechanical quantities. This condition exactly arises when the miniaturization is put to the current technological limit and the silicon grain size becomes on the order of magnitude of the minimum dimensions of the characteristic structural parts, like the width of the slender suspension beams.

From the perspective of the quantification of the uncertainties due to material heterogeneity, in this work we aim to extract the information necessary to create quantitatively-informed stochastic analytical distributions of the elastic properties, such as the apparent Young's moduli *E* and *G* and Poisson's ratio *ν*, from numerical simulations of the SVEs. The latter represent only a realization of the stochastic variables involved in the definition of the mechanical properties at the upper, device-level scale, different from the more commonly-encountered RVEs that would represent the effective properties in a deterministic sense. However, the knowledge of the statistics of the aforementioned elastic quantities allows rigorously quantifying their scattering around a mean value, and therefore to foresee the mechanical uncertainty transferred to the structural behavior, e.g. the suspension spring's stiffness.

With regard to the geometrical uncertainty due to over-etch fluctuations, different from the commonly-adopted simplification of a constant value for a single device, in this work we explore the effect of a scattering from the mentioned constant value along the supporting beams. In this case, an analytical distribution is (a priori) assumed, and the contribution of variable over-etch on the beam moment of inertia is considered in addition to the previously-mentioned material-generated uncertainty.

The main purpose of our approach is to overcome the computational burden of the polycrystalline morphology analysis, thanks to the analytical distributions, which are therefore good to devise an engineering tool useful for design.

In the following Section 2, an exemplary MEMS configuration possibly leading to an offset is described. Then, the procedure exploiting (i) Monte Carlo (MC) analyses of SVEs aiming to define the elastic mechanical properties of homogenized polysilicon, (ii) the extraction of the mentioned analytical stochastic distributions from the numerical data, and (iii) their employment to build the (statistically-informed) stiffnesses of critical MEMS details is carefully detailed in Section 3. This reasoning is adopted to solve a typical offset problem in Section 4, where a discussion of the outcome is also carried out.
