High-Fidelity Thermomechanical Modeling of Substrates Using a Monoclinic Homogenization Method

In advanced packaging products, the thermomechanical properties of the package play a crucial role in the manufacturability and reliability. As a key component in advanced packaging, substrates are rapidly evolving towards higher layer counts and finer wiring. The multi-material and cross-scale properties of substrates pose challenges for thermo-mechanical modeling and simulation. The reduced-order modeling by homogenization methods has become a major engineering tool to address these challenges. In this work, a monoclinic homogenization method (MHM) is proposed for multilayer substrates. The method employs a stiffness matrix with 13 independent parameters and a thermal expansion coefficient matrix with 4 independent parameters as homogenized constitutive parameters. By dividing the substrate into a set of mesoscopic representative volume element (RVE) and then using finite element method (FEM) and fast machine learning-based predictions, then the homogenized material constitutive parameters of the RVE are obtained. It is capable of accurately describing the arbitrary anisotropy of the wiring and perforation layers in the in-plane (XY-plane), as well as the symmetry along the thickness direction (Z-direction). A FEM based on uniaxial strain loading to solve the effective stiffness matrix (Ceff) is developed, along with an algorithm to extract the anisotropic equivalent coefficient of thermal expansion matrix (CTEeff). An automated modeling and simulation workflow based on MHM is designed. Thermal warpage simulation shows that MHM has extremely high fidelity according to the pixel-by-pixel reference modeling method, with a deviation of only 3.4% at the maximum warpage point. The simulation accuracy of MHM is improved by a factor of 7.1 compared to the existing orthogonal homogenization method (OHM) with a deviation of 24.4%. MHM constructs less than 1/3000 of the number of model elements of the traditional direct modeling method, which reduces the simulation time by several orders while maintaining the accuracy. The proposed MHM is widely applicable to the thermo-mechanical simulation of advanced packaging assemblies, including organic substrates, through-silicon Interposer, and redistribution layers, with the advantages of both high fidelity and high efficiency.