Analytical Validation of a Continuum Model for Epitaxial Growth with Elasticity on Vicinal Surfaces
Within the context of heteroepitaxial growth of a film onto a substrate, terraces and steps self-organize according to misfit elasticity forces. Discrete models of this behavior were developed by Duport et al. (J Phys I 5:1317–1350, 1995) and Tersoff et al. (Phys Rev Lett 75:2730–2733, 1995). A continuum limit of these was in turn derived by Xiang (SIAM J Appl Math 63:241–258, 2002) (see also the work of Xiang and Weinan Phys Rev B 69:035409-1–035409-16, 2004; Xu and Xiang SIAM J Appl Math 69:1393–1414, 2009). In this paper we formulate a notion of weak solution to Xiang’s continuum model in terms of a variational inequality that is satisfied by strong solutions. Then we prove the existence of a weak solution.
Unable to display preview. Download preview PDF.
- 1.Butzer, P.L., Nessel, R.J.: Fourier analysis and approximation. In: One-Dimensional Theory, vol 1. Academic Press, New York (1971)Google Scholar
- 2.Duport C., Politi P., Villain J.: Growth instabilities induced by elasticity in a vicinal surface. J. Phys. I 5, 1317–1350 (1995)Google Scholar
- 3.Fonseca, I., Leoni, G.: Modern methods in the calculus of variations: L p spaces. In: Springer Monographs in Mathematics. Springer, New York (2007)Google Scholar
- 4.Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Gauthier-Villars, Paris (1969)Google Scholar
- 7.Xiang, Y., Weinan, E.: Misfit elastic energy and a continuum model for epitaxial growth with elasticity on vicinal surfaces. Phys. Rev. B 69, 035409-1–035409-16 (2004)Google Scholar
- 9.Zeidler, E.: Nonlinear Functional Analysis and Its Applications II/A: Linear Monotone Operators. Springer-Verlag, New York (1990)Google Scholar