Abstract : Integrated circuits in electronic chips are etched on thin films of semi-conductors. Shape instabilities may appear during the manufacturing of these films by hetero-epitaxy. This work is devoted to the numerical study of one such instability, known as the Grinfeld instability.
From a modeling point of view, instabilities of films free surfaces fall in the class of free boundary problems and moving interfaces. We study the particular case of motion by mean curvature and its approximation by the phase field method via the Allen-Cahn equation. We propose a finite element discretization of this equation, that allows us to consider several extensions: conservation of the volume, forcing terms, anisotropy.
A numerical study of a variationnal model for the Grinfeld instability is presented, that combines epitaxial growth with elastic interactions in the bulk. This model couples the Allen-Cahn equation to the system of linearized elasticity. The effect of elastic deformations in the substrate can be accounted for in this model.
We also propose a phase field model to study step bunching instabilities on vicinal surfaces of crystals. Our numerical computations are based on an algorithm similar to simulated annealing. This analogy induced us to use phase field approximations to compute global minima in optimization problems.