Critical effective radius for holes in thin films: Energetic and dynamic considerations

Abstract

Questions regarding the stability of holes and arrays of holes in solid thin films have attracted much attention over the past few decades since an absence of holes is necessary for certain devices to operate properly and a presence of holes is needed in various industrial applications. Here, we study the energetic and dynamic stability of a single axisymmetric grain with a hole at its center, under the assumption that the exterior surface evolves by surface diffusion. Our energetic considerations enable us to formulate a criterion in terms of a critical effective hole radius, which distinguishes between energetically stable and unstable steady state hole configurations and which, somewhat surprisingly, is independent of the contact angle at the substrate and should be readily measurable in experiments. The set of steady states for the system is characterized in terms of admissible nodoidal surfaces, whose dynamic stability is studied via numerical simulation of the full non-linear dynamic problem for zero-volume perturbations. Our dynamic stability study confirms and extends our conclusions based on energetic considerations. Our results, moreover, confirm and extend the classical results of Srolovitz and Safran [J. Appl. Phys. 60, 247–254 (1986); J. Appl. Phys. 60, 255–260 (1986)] and Wong et al. [J. Appl. Phys. 81, 6091–6099 (1997); Acta Mater. 45, 2477–2484 (1997)]. Furthermore, our studies of the steady states and their stability contribute to our understanding of various phenomena observed in experiments: void formation, hillock formation, hole induction and propagation, ligament formation and evolution, blistering prior to film rupture, etc. Importantly, our study shows that in order to relate theory with experiments, careful monitoring of spatial variations in the mean curvature in experiments is required.