The multilayer-modified Stoney’s formula for laminated polymer composites on a silicon substrate

Abstract

The thermomechanical behavior of multilayer structures is a subject of perennial interest. Stoney’s formula has long been one of the most important tools for understanding thermomechanical stress for single-layered structures like spin-coated polyimides or deposited metal thin film on substrates. In today’s microelectronics, however, as multilayer substrates have become widely available, the “modified version” of Stoney’s formula for multilayer applications is not only useful but necessary. While the majority of reports in the literature have focused on single-layer analysis, in this study, we examined an extended usage of Stoney’s formula for multilayer analysis. A simple model, the multilayer-modified Stoney’s formula, which predicts the stress contribution of each individual layer is proposed and verified through experiments and numerical analysis. Using various kinds of materials employed in a typical lamination-based multichip module technology, the thermomechanical behavior of the lamination-based multilayer substrates was measured by a laser profilometry during thermal cycling. The measured values were compared with calculated values using the multilayer-modified Stoney’s formula.