Equivalent mechanical properties of through silicon via interposers – A unit model approach

Abstract

A closed-form description of the equivalent mechanical properties of through-silicon via (TSV) interposers is given in this paper. The goal is to homogenize a unit TSV model for global analysis of interposer design and optimization. The equivalence between the heterogeneous TSV model and its homogenized counterpart is in the sense that, both models have an identical radial deformation at the boundary when subjected to the same loadings. The equivalent mechanical properties derived herein are: the in-plane equivalent Young’s modulus, bulk modulus, and coefficient of thermal expansion (CTE); the out-of-plane Young’s modulus, rigidity modulus, CTE, and the Poisson’s ratio. These equivalent properties form a transversely isotropic description of the homogenized TSV model, and can be readily used in finite element or computer-aided design package for globally characterizing the thermomechanical response of the interposer, or simulating its interaction with the annexed layers in the package. These equivalent properties are primarily developed for interposers with vias of uniform diameter through the thickness. When applying to interposers with tapered vias, the predicted results will be perturbed to an extent that is determined collectively by the aspect ratio and tilt angle of the tapered vias. The analytical work, which does not consider the thin liner layer, is compared to some known results that had considered the liner layer, to understand the extent of approximation.