New Homogenization Method for Equivalent Thermal Expansion Coefficient of Wiring Structures Using Embedded Unit Cells

Thermal mismatch, as a crucial factor affecting the thermal reliability of IC products, is of great importance to control. Due to the presence of multi-level cross-scale structures inside IC products, considering the constraints of computational resources, it is inevitable to homogenize the complicated structures within IC products to simplify the models during finite element simulations. Among these structures, routing structures like RDL, BEOL, and conductive layers in PCBs, exhibit significant heterogeneity, resulting in severe thermal mismatch issues. Therefore, homogenizing the thermal-mechanical performance of wiring structures is essential for controlling thermal mismatch in IC products. This paper proposes a partition homogenization approach based on the embedded unit cells for wiring layers, combined with mesoscopic finite element analysis, to calculate the equivalent thermal expansion coefficient of wiring structures in IC products. In this paper, 30 samples of wiring structures were randomly selected. Each sample was partitioned, and the equivalent thermal expansion coefficients of each partition were calculated using the above homogenization methods. The homogenized partitions were then assembled to establish the corresponding equivalent models. Furthermore, pixelated detailed models were established for each wiring structure sample as a reference, and the thermal expansion simulation results of the equivalent models were compared to validate the rationality of using embedded cells to calculate the equivalent thermal expansion coefficients of wiring partitions. Additionally, compared to the periodic unit cell, using embedded unit cell to homogenized the thermal expansion coefficients of each partition can avoid the assumption of periodic response in classical homogenization theories, better aligning with the non-periodic structural characteristics of wiring layers and providing certain advantages in equivalent accuracy. The entire process was implemented through batch processing scripts, ensuring the efficiency of the partition homogenization process. Finally, the limitations of the partition homogenization method were briefly discussed in this paper.