A Top-Down, Three-Scale Numerical Analysis of Wafer-to-Wafer Metallic Bonding

To study the sensitivity to micro-scale imperfections of the strength of a metallic, wafer-to-wafer MEMS bonding, we propose a three-scale numerical (finite element) approach. At the wafer level (macro-scale), accounting for the whole metallic sealing as nonlinear springs connecting the two silicon wafers modelled as thin plates, we link the force transferred by each single MEMS die to the external pressure applied to the wafers. This force is next used as an index for the input pressure at the die level (meso-scale), where the geometry of the metallic rings is accurately described: the local stress field at the interface between the two upper and lower metallic rings is so obtained. Finally, a micro-scale model is used to link the aforementioned local stress fields in the rings, to the bonding strength: representative volumes of the outer surfaces of the rings getting into contact, able to represent in a statistically way the relevant roughness (which is on the order or tens of nanometers at most), are adopted to obtain the relationship between the external pressure and the percentage of the sealed area. This information is exploited to discriminate, in terms of the expected bonding strength, different ring geometries.