Analytical model for intrinsic residual stress effects and out-of-plane deflections in free-standing thick films

Abstract

An analytical model for the influence of residual stress on the out-of-plane deflection in a free-standing thick diamond film (the bowing phenomenon) is presented. The variation in residual stress with film thickness is usually believed to cause the bowing. In this study, the stress variation is assumed to be produced by a gradual increase in substrate deformation resulting from layer-by-layer deposition of the film. The model uses the infinitesimal plate-bending theory to describe the layer-by-layer film growth more exactly, considering the two deformation modes of contraction or expansion and bending. To verify the suggested model, thick diamond films were fabricated on Si, Mo, and W substrates of varying thicknesses by microwave plasma assisted chemical vapor deposition. The model’s predictions on bowing, based on the intrinsic stress value measured by the curvature method, were in good agreement with the bowing curvature of the as-released films measured by a profilometer. This confirms that the bowing of thick films depends on the intrinsic stress variation of the film associated with a gradual increase in substrate deformation. A method of eliminating bowing by depositing different layers with different intrinsic stresses is discussed.