**2. Sensitivity to Imperfections: A Simple Model for MEMS Offset**

The effect of mechanical and geometrical uncertainties at the microscale can be clearly observed in the case of statically-indeterminate movable structures. As a benchmark example, inspired by the geometry of single-axis inertial MEMS devices working as shown in Figure 1 (see also [21–23]), we focus on a simplified scheme, where a proof mass is connected to the substrate through a couple of polycrystalline silicon beams or springs in series. In this case, even if the target design is to have the two stiffnesses *k*<sup>1</sup> and *k*<sup>2</sup> equal, the randomly-varying grain morphology, the over-etch defects [9], and the residual stresses arising from the manufacturing process induce an offset *u* away from the rest position.

**Figure 1.** (**a**) Example and (**b**) structural scheme of single-axis inertial MEMS devices, featuring a proof mass anchored to the die via two springs in series.

Whatever the geometry of the two springs is, the said imperfections cause instead the stiffness values *k*<sup>1</sup> and *k*<sup>2</sup> to differ. The resulting offset *u* that can detrimentally affect the performance indices of the device linked, e.g. to capacitive readout, often proves negligible, but it may become relevant when the dimensions of critical structural details (like e.g. the in-plane spring width) become comparable to the average silicon grain size or to the microfabrication tolerances related to the etching stage. By assuming the proof mass to be a rigid body, in order to compute *u* we assume two sources that induce the (either positive or negative) elongation of the springs: an inelastic deformation *<sup>r</sup>* linked to the residual stress in the polysilicon film; an elastic deformation *<sup>e</sup>* induced by the constraints at the anchor points that prevent any motion at the end points. The latter effect can be formally represented by a force *F* acting on both the springs in series. Due to the said constraints at the anchors, the compatibility equation for this statically-indeterminate system is given by:

$$F\left(\frac{1}{k\_1} + \frac{1}{k\_2}\right) + \epsilon\_r \ (L\_1 + L\_2) = 0.\tag{1}$$

Within the proposed frame, the sources *<sup>r</sup>* and *F* are assumed without any dependence on the out-of-plane direction; possible effects of residual stress gradients are therefore disregarded to simplify the analysis, with a focus only on the in-plane motion of the proof mass. Moreover, the lengths *L*<sup>1</sup> and

*L*<sup>2</sup> of the springs are meant in a very general sense as kinds of effective values, in order to allow also for folded geometries like the one depicted in Figure 1a. Solving Equation (1) for *F*, we end up with:

$$F = -\frac{L\_1 + L\_2}{\frac{1}{k\_1} + \frac{1}{k\_2}} \, \varepsilon\_r$$

and the offset *u* thus reads:

$$
\mu = \frac{\frac{k\_1}{k\_2}L\_1 - L\_2}{1 + \frac{k\_1}{k\_2}} \varepsilon\_r. \tag{3}
$$

For the sake of simplicity, we now assume *L*<sup>1</sup> = *L*<sup>2</sup> = *L*. To avoid in the analysis any dependence on *r*, which, as stated, represents an inelastic effect of the residual stresses and stands as a kind of algorithmic, model-based parameter, Equations (2) and (3) are solved for it, and the results are given next in terms of the ratio *u*/*F*, according to:

$$\frac{\mu}{F} = \frac{1 - \frac{k\_1}{k\_2}}{2 \, k\_1} \tag{4}$$

which depends on the stiffnesses of the springs only. Equation (4) shows that, if the two stiffnesses are equal (i.e. *k*<sup>1</sup> = *k*2), no offset is induced. When, instead, microscale scattering leads to different stiffness values, the offset shows up. To account for such discrepancy between the two values of the spring stiffness, we assume that the scattering induces the values *k*<sup>1</sup> = *k* + Δ*k*<sup>1</sup> and *k*<sup>2</sup> = *k* + Δ*k*2, with Δ*k*<sup>1</sup> = Δ*k*<sup>2</sup> (where *k* can be assumed as the target reference value). Equation (4) is then modified as:

$$\frac{\Delta H}{F} = \frac{1 - \frac{k + \Delta k\_1}{k + \Delta k\_2}}{2\left(k + \Delta k\_1\right)} = \frac{\Delta k\_2 - \Delta k\_1}{2\left(k + \Delta k\_1\right)\left(k + \Delta k\_2\right)}.\tag{5}$$

It is noted that a difference in the stiffness values is not sufficient to get an offset: the force *F* arising from the aforementioned residual stress distribution is also required to trigger the proof mass displacement. This through-the-thickness distribution of the residual stresses is actually a whole problem on its own [24]: though the simple model (4) can be adopted to estimate them via MC techniques like that suggested in [25], provided that measurements are available for a number of statistically-representative devices, in this work we do not aim to address this issue, but instead a quantification of the uncertainty effects linked to the stiffness of structural components. As mentioned in the Introduction, we handle two sources of the scattering of the stiffness values: the heterogeneity of the polycrystalline material and the geometrical imperfections due to the manufacturing process. Both micromechanical sources are specifically discussed in Section 3.

### **3. Characterization of the Uncertainties at the Microscale**

To quantify the uncertainties in the spring stiffness, the two sources linked to the polycrystalline morphology and to the etch defects are dealt with separately. In Section 3.1, an MC procedure is adopted to determine the homogenized elastic properties of the polysilicon film, in terms of microstructure-affected values of Young's modulus *E*, shear modulus *G*, and Poisson's ratio *ν*. Since we considered polycrystalline silicon with a columnar structure, the material was treated as transversely isotropic at the device scale. The homogenized elastic properties obtained with the MC procedure at the micro-scale refer to the plane parallel to the substrate, so that the out-of-plane direction coincides with the grain columns. Due to the FCC crystal lattice of silicon, Young's moduli obtained from the homogenization along two orthogonal directions are always equal, and deviations from isotropy prove small. Next, in Section 3.2, these scattered overall properties are used jointly with the statistics of over-etch to provide an estimate of the scattering in the overall spring stiffness according to beam bending theory.
