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 Lyapunov or “free energy” functional. This functional consists of the term |∆h|2 that represents the surface diffusion and − log(1 + |∇h|2) that 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 |∆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 wavelength. 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. 2000 MSC: 35K55, 35Q99, 74K35, 82D25. 2003 PACS: 68.35.Ct; 68.43.Jk; 81.15.Aa.