Warpage analysis of multilayer thin film/substrate systems using the Eigenstrain method

Warpage resulting from the deposition of films on substrates remains a persistent challenge that significantly impacts the application of numerous advanced devices. To precisely evaluate warpage in multilayer thin film/substrate systems, we employ the eigenstrain method in the warpage analysis. An analytical model is developed to predict warpage and residual stress in multilayer film/substrate system based on classical laminate theory. We introduce a novel concept termed “Eigenstress”, which serves as the fundamental cause of warpage and characterizes the influence of the manufacturing process on warpage. Theoretical analysis and simulation demonstrate that warpage is entirely determined by eigenstress and other process-independent parameters. This model also explains that there is little interaction between thin films deposited on the same substrate. An experimental method is studied for measuring eigenstress using a standard plate. The measured eigenstress differs significantly from the corresponding thermal stress. We also propose a critical condition and its governing equation for warpage in multilayer film/substrate systems. Experimental results confirm that adding a compensation layer on a warped substrate can considerably reduce warpage. The warpage model based on “Eigenstress” provides a theoretical foundation for accurately predicting and controlling warpage in multilayer thin film/substrate systems.