Multi-Step Mechanical and Thermal Homogenization for the Warpage Estimation of Silicon Wafers

Zhouyi Xiang, Min Chen*, Yonghui Deng, Songhua Huang, Sanli Liu, Ji Li

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


In response to the increasing demand for high-performance capacitors, with a simultaneous emphasis on minimizing their physical size, a common practice involves etching deep vias and coating them with functional layers to enhance operational efficiency. However, these deep vias often cause warpages during the processing stage. This study focuses on the numerical modeling of wafer warpage that occurs during the deposition of three thin layers onto these vias. A multi-step mechanical and thermal homogenization approach is proposed to estimate the warpage of the silicon wafer. The efficiency and accuracy of this numerical homogenization strategy are validated by comparing detailed and homogenized models. The multi-step homogenization method yields more accurate results compared to the conventional direct homogenization method. Theoretical analysis is also conducted to predict the shape of the wafer warpage, and this study further explores the impact of via depth and substrate thickness.

Original languageEnglish
Article number408
Issue number3
Publication statusPublished - Mar 2024


  • integrated capacitor
  • multi-step mechanical homogenization
  • multi-step thermal homogenization
  • wafer warpage


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