Viscoelastic stress relaxation in film/substrate systems—Kelvin model

Abstract

Although the Maxwell model was widely applied to analyze viscoelastic behavior of film/substrate systems, Rafferty et al. [Appl. Phys. Lett. 54, 151 (1989)] found that the structural relaxation of silicon oxide films on silicon wafers during thermal annealing could be explained by the Kelvin model but not the Maxwell model. This result motivated us to investigate the relaxation of residual stresses due to viscoelastic deformation in a film/substrate system using the Kelvin model. The system can be a viscoelastic film on an elastic substrate or an elastic film on a viscoelastic substrate. The viscoelastic solutions of both cases can be obtained from the elastic solution using the Laplace transform. The stress relaxation rate increases with decreasing film-to-substrate thickness ratio for a viscoelastic film deposited on an elastic substrate. However, it shows the opposite trend for an elastic film deposited on a viscoelastic substrate. The present results are compared with those obtained from the Maxwell model.