Finite-element calculations of elastic fields within flip-chip solder bumps and Cu-pillar bumps under the influences of thermal stresses and joule heating

Abstract

In this paper, finite-element models were developed to calculate the elastic fields within both solder and Cu-pillar bumps under the influences of thermal stresses and Joule heating. First, a steady-state equation of the electric potential is solved to determine the electric field and thus, the effect of Joule heating. Then, the temperature field is obtained by solving a steady-state equation of thermal conduction under the influence of Joule heating. Finally, the elastic displacements are determined by solving the quasi-stationary Cauchy-Navier equations subjecting to the effects of thermal stresses. Using our finite element models, the elastic stresses are obtained and then the von Mises stress is calculated to indicate the magnitude of elastic distortion within the bumps. Modeling results show that: (1) under the influences of external stresses only and with the same magnitude of the applied external stresses, the maximums of Mises stresses in solder bumps are much higher than those in Cu-pillar bumps, moreover, within both solder and Cu-pillar bumps, the shearing stress tends to results in a higher maximum of Mises stress than the tensile stress; (2) for both bumps at service, the thermal stresses becomes dominant, especially at the lower external stresses, and furthermore, within both bumps, the maximum of Mises stresses decreases as the tensile stress increases while it increases as the shearing stresses increases; (3) the maximum of the Mises stresses increases linearly as the average temperature of the bumps increases with that in the Cu-pillar bump a bit higher, and in addition, the increase in current densities has a stronger influence on the maximum of the Mises stresses in the solder bump than that in the Cu-pillar bump.