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 |Δ 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(t1/2), the averaged gradient is bounded above by O(t1/4), and the averaged energy is bounded below by O(- log t). We then consider a small coefficient ε2 of |Δ h|2 with ε = 1/L and L the linear size of the underlying system, and study the energy asymptotics in the large system limit ε → 0. We show that global minimizers of the free-energy functional exist for each ε > 0, the L2-norm of the gradient of any global minimizer scales as O(1/ε), and the global minimum energy scales as O( log ε). 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.