Eptaxial growth without slope selection: energetics, coarsening, and dynamic scaling

Bo Li University of California at San Diego Jian-Guo Liu University of Maryland

Analysis of PDEs mathscidoc:1702.03063

Journal of Nonlinear Science, 14, (5), 429–451, 2004.10
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(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 e2 of |? h|2 with e = 1/L and L the linear size of the underlying system, and study the energy asymptotics in the large system limit e ? 0. We show that global minimizers of the free-energy functional exist for each e>0, the L2-norm of the gradient of any global minimizer scales as O(1/e), and the global minimum energy scales as O( log e). 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.
epitaxial growth, coarsening, interface width, dynamic scaling, energy minimization, equilibrium
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  title={Eptaxial growth without slope selection: energetics, coarsening, and dynamic scaling},
  author={Bo Li, and Jian-Guo Liu},
  booktitle={Journal of Nonlinear Science},
Bo Li, and Jian-Guo Liu. Eptaxial growth without slope selection: energetics, coarsening, and dynamic scaling. 2004. Vol. 14. In Journal of Nonlinear Science. pp.429–451. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170209094913681685366.
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