The Asaro–Tiller–Grinfeld instability revisited

Abstract

Deposition of a thin film on a solid substrate in the presence of a misfit leads to a growth instability that favors three-dimensional (3D) morphology of the free surface. The amount of the misfit and the conditions of the film deposition (molecular beam epitaxy) lead to an elastic problem, where surface energy has the same order of magnitude as the bulk energy. The instability occurs at a critical thickness of the film. The value of the critical thickness is shown to be given by the competition between the bulk and surface effects. We investigate (via a Fourier method) the Asaro–Tiller–Grinfeld instability for cubic materials and in the presence of an arbitrary misfit. We solve the problem in the general case and we specialize our results to recover values which are in good agreement with experimental data in the case of a In 1− x Ga x As alloy. We consider in a 3D framework sinusoidal perturbations of the free boundary at arbitrary orientations with respect to crystallographic axes. Thus, we are able to minimize the sum of the bulk and surface energies with respect to the orientation and therefore to predict qualitative aspects of the surface morphology.