Thermal stress in through-silicon-vias: Theory-of-elasticity approach

Simple physically meaningful analytical (mathematical) predictive stress models are developed for typical through-silicon-via (TSV) designs using theory-of-elasticity approach. Two extreme cases of the TSV geometry are addressed: disc-vias (with height-to-diameter ratios below 0.25), when plane stress approximation can be used, and rod-vias (with height-to-diameter ratios above 2.5), when plane strain approximation is applicable. The objectives of the analysis is to evaluate the effect of the size of the opening in the silicon material on the induced pressure at the boundary of the opening, when the TSV structure is heated up, and to assess the possible role of a “surrogate” layer, if any, on the level of the longitudinal interfacial shearing stresses. While the pressure at the Si opening and the maximum interfacial stresses determine the reliability of the silicon material, the longitudinal shearing stress at the Si-copper (Cu) interface is critical from the standpoint of the adhesive and cohesive strength of the TSV design. The numerical example is carried out for different opening radii and for the case when Indium is considered as a suitable buffering material. The calculated data indicate that larger openings in the Si result in lower pressures on it and in lower maximum interfacial stresses, especially if disc-like vias are employed. On the other hand, as has been shown earlier (see Suhir and Savastiuk, 2008) elastic stability of disc-vias should be evaluated and ensured. The developed models can be used also to determine if there is sufficient incentive for employing additional, “surrogate”, materials (i.e., materials not needed from the standpoint of the functional performance of the design) for lower thermally induced stresses.