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 through 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 upper and lower metallic rings is so obtained. Finally, a local (micro-scale) model is used to link the aforementioned local stress field in each die to the bonding strength: representative volumes of the rings getting into contact, accounting in a statistically way for the relevant surface 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 sealed area. This information is exploited to assess the properties of the rings, in terms of expected bonding strength.