Li, B; Liu, J-G, *Epitaxial growth without slope selection: Energetics, coarsening, and dynamic scaling*,
Journal of Nonlinear Science, vol. 14 no. 5
(2004),
pp. 429-451 [doi] .
**Abstract:**

*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(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 ε 2 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 L 2-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. © 2005 Springer.*