A high-precision analytical model of transient thermal stresses of through-silicon vias in 3-D integrated circuits

Abstract

In three-dimensional integrated circuits (3-D ICs), through-silicon vias (TSVs) are subject to significant thermal stress due to mismatches in the coefficients of thermal expansion (CTE) among constituent materials, which severely compromises device reliability. Grounded in thermoelastic theory, this paper presents a high-precision analytical model for predicting transient thermal stress in TSVs. First, an analytical solution for the transient temperature field within TSVs is derived using heat conduction theory in a cylindrical coordinate system. Subsequently, by incorporating the thermoelastic displacement potential and Love's displacement function, a closed-form analytical model for transient thermal stress is formulated. The model further accounts for axial third-kind thermal boundary conditions and incorporates temperature-dependent material properties to enhance physical realism. Validation against COMSOL multiphysics simulations demonstrates that the maximum relative error and maximum absolute error in the predicted transient temperature field remain below 1.2 % and 0.4 K, respectively, while the corresponding errors for von Mises stress are within 1.6 % and 0.8 MPa. Notably, the computational efficiency of the proposed model exceeds that of the finite element method (FEM) by approximately two orders of magnitude. Moreover, it exhibits strong adaptability across a wide range of material combinations and geometric configurations. We further employ the analytical model to investigate the effect of thermal stress on carrier mobility in TSVs and the stress interaction between adjacent TSVs. In comparison with classical 2-D Lamé solutions and 3-D Kane-Mindlin theory, the proposed model achieves superior accuracy without relying on empirical correction factors and effectively captures the spatial distribution of thermal stress in TSVs under transient thermal conditions. This study confirms that the developed analytical model offers exceptional accuracy and efficiency in characterizing transient thermal stress distributions in 3-D IC TSVs, thereby providing a robust and precise tool for thermomechanical reliability assessment and design optimization.