Split singularities: Stress field near the edge of a silicon die on a polymer substrate

In current design the circuitry is extended very close to the edges of a silicon die to maximize useful surface area. When the die is bonded to a polymer substrate, with the circuitry facing the polymer, thermal misfit stress concentrates at the die edges and may damage the circuitry. The stress distribution near a die edge is quantified using a combination of asymptotic analysis and finite element calculation. The asymptotic field consists of two modes of singular stresses, scaling with the distance from the edge r as, respectively, r^-λ1; and r^-λ2, where λ₁ > λ₂. It is shown that the more singular field (i.e. the λ₁-singularity) prevails in an exceedingly small zone, smaller than 10^-6 times the die thickness. Once both modes are included, however, the asymptotic field matches the full field in a zone about 10^-1 times the die thickness. This finding resolves several controversies in the literature on electronic packaging. The near-edge stress distribution is presented for various substrate thicknesses, elastic moduli and thermal expansion coefficients. The results can be used to explore design options.