Epitaxial Growth Without Slope Selection: Energetics,Coarsening, and Dynamic Scaling

We study a continuum model for epitaxial growth of thin films in which the slope of mound structure of film surface increases. This model is a diffusion equation for the surface height profile h which is assumed to satisfy the periodic boundary condition. The equation happens to possess a Liapunov or free-energy functional. This functional consists of the term | h|², which represents the surface diffusion, and - log (1 + | h|²), which describes the effect of kinetic asymmetry in the adatom attachment-detachment. We first prove for large time t that the interface width---the standard deviation of the height profile---is bounded above by O(t^1/2), the averaged gradient is bounded above by O(t^1/4), and the averaged energy is bounded below by O(- log t). We then consider a small coefficient ² of | h|² with = 1/L and L the linear size of the underlying system, and study the energy asymptotics in the large system limit 0. We show that global minimizers of the free-energy functional exist for each > 0, the L²-norm of the gradient of any global minimizer scales as O(1/), and the global minimum energy scales as O( log ). The existence of global energy minimizers and a scaling argument are used to construct a sequence of equilibrium solutions with different wavelengths. Finally, we apply our minimum energy estimates to derive bounds in terms of the linear system size L for the saturation interface width and the corresponding saturation time.