Model for stress generated upon contact of neighboring islands on the surface of a substrate

Abstract

In the early stage of growth of a metal film on a substrate by the Volmer–Weber mechanism, a tensile stress in the film is observed to arise at about the point in the process when islands of deposited material begin to coalesce. The mechanism commonly proposed as the origin of this tensile stress is that the coalescing islands deform in order to form a relatively low energy grain boundary, at the expense of some surface energy by surface area reduction, and that this proceeds until a stress is generated that has magnitude sufficient to prevent further area reduction. Several models have been proposed for this process, but the inferred tensile stress estimates have been much larger than observed stress magnitudes in many cases. The purpose here is to introduce a model for the process based on the theory of contact of elastic solids with cohesion. A description of the process is developed on this premise for one-dimensional, two-dimensional, and three-dimensional states of deformation of coalescing islands. It is found that the latter case leads to an estimate of film stress generally consistent with observations. The features of the model for different dimensionalities are compared and contrasted with each other, as well as with other models which have been proposed for this process.