Abstract
Epitaxy, a special form of crystal growth, is a technically relevant process for the production of thin films and layers. It can generate microstructures of different morphologies, such as steps, spirals or pyramids. These microstructures are influenced by elastic effects in the epitaxial layer. There are different epitaxial techniques, one being liquid phase epitaxy. Thereby, single particles are deposited out of a supersaturated liquid solution on a substrate where they contribute to the growth process. This article studies a two-scale model including elasticity, introduced in Eck et al. (Eur Phys J Special Topics 177:5–21, 2009) and extended in Eck et al. (2006). It consists of a macroscopic Navier–Stokes system and a macroscopic convection–diffusion equation for the transport of matter in the liquid, and a microscopic problem that combines a phase field approximation of a Burton–Cabrera–Frank model for the evolution of the epitaxial layer, a Stokes system for the fluid flow near the layer and an elasticity system for the elastic deformation of the solid film. Suitable conditions couple the single parts of the model. As the main result, existence and uniqueness of a solution are proven in suitable function spaces. Furthermore, an iterative solving procedure is proposed, which reflects, on the one hand, the strategy of the proof of the main result via fixed point arguments and, on the other hand, can be the basis for a numerical algorithm.
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References
Adams R.A., Fournier J.F.: Sobolev Spaces. Academic Press, Elsevier (2003)
Alt H.-W.: Lineare Funktionalanalysis (5. Auflage). Springer, Berlin (2006)
Burton W.K., Cabrera N., Frank F.C.: The growth of crystals and the equilibrium structure of their surfaces. Philos. Trans. R. Soc. 243, 299–358 (1951)
Caginalp G.: An analysis of a phase field model of a free boundary. Arch. Ration. Mech. Anal. 92, 205–245 (1986)
Dorsch W., Christiansen S., Albrecht M., Hansson P.O., Bauser E., Strunk H.P.: Early growth stages of Ge0.85Si0.15 on Si(001) from Bi solution. Surf. Sci. 331-333, 896–901 (1995)
Chalupecky V., Eck Ch., Emmerich H.: Computation of nonlinear multiscale coupling effects in liquid phase epitaxy. Eur. Phys. J. Special Topics 149, 1–17 (2007)
Eck Ch., Emmerich H.: A two-scale model for liquid-phase epitaxy. Math. Methods Appl. Sci. 32(1), 12–40 (2009)
Eck Ch., Emmerich H.: Homogenization and two-scale models for liquid phase epitaxy. Eur. Phys. J. Special Topics 177, 5–21 (2009)
Eck, C., Emmerich, H.: Liquid-phase epitaxy with elasticity. Preprint 197, DFG SPP 1095 (2006)
Eck C., Kutter M., Sändig A.-M., Rohde Ch.: A two scale model for liquid phase epitaxy with elasticity: an iterative procedure. Z. Angew. Math. Mech. 93, 745–761 (2013)
Emmerich H.: Modeling elastic effects in epitaxial growth. Contin. Mech. Thermodyn. 15, 197–215 (2003)
Girault V., Raviart P.-A.: Finite Element Methods for Navier–Stokes Equations, Theory and Algorithms. Springer, Berlin (1986)
Karma A., Plapp M.: Spiral surface growth without desorption. Phys. Rev. Lett. 81, 4444 (1998)
Kutter, M.: A Two Scale Model for Liquid Phase Epitaxy with Elasticity. Dissertation, University of Stuttgart (2015)
Liu F., Metiu H.: Stability and kinetics of step motion on crystal surfaces. Phys. Rev. E 49, 2601–2616 (1994)
Lo T.S., Kohn R.V.: A new approach to the continuum modeling of epitaxial growth: slope selection, coarsening and the role of uphill current. Phys. D 161, 237–257 (2002)
Lunardi A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Basel (1995)
Maz’ya V., Rossmann J.: Elliptic Equations in Polyhedral Domains, vol. 162. American Mathematical Society, Mathematical Surveys and Monographs, Providence (2010)
Otto F., Penzler P., Rätz A., Rump T., Voigt A.: A diffuse-interface approximation for step flow in epitaxial growth. Nonlinearity 17, 477–491 (2004)
Pazy A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)
Redeker M., Eck Ch.: A fast and accurate adaptive solution strategy for two-scale models with continuous inter-scale dependencies. J. Comput. Phys. 240, 268–283 (2013)
Renardy M., Rogers R.C.: An Introduction to Partial Differential Equations, 2nd edn. Springer, New York (2004)
Russo G., Smereka P.: Computation of strained epitaxial growth in three dimensions by kinetic Monte Carlo. J. Comput. Phys. 214, 809–828 (2006)
Shanahan L.L., Spencer B.J.: A codimension-two free boundary problem for the equilibrium shapes of a small three-dimensional island in an epitaxially strained solid film. Interfaces Free Boundaries 4, 1–25 (2002)
Small M.B., Ghez R., Giess E.A.: Liquid phase epitaxy. In: Hurle, D.T.J. (ed.) Handbook of Crystal Growth, Vol. 3, pp. 223–253. North-Holland, Amsterdam (1994)
Temam R.: Navier–Stokes Equations. North-Holland, Amsterdam (1977)
Villain J.: Continuum models of crystal-growth from atomic-beams with and without desorption. J. Phys. I 1, 19–42 (1991)
Zeidler E.: Nonlinear Functional Analysis and its Applications II/A, Linear Monotone Operators. Springer, New York (1990)
Xiang Y.: Derivation of a continuum model for epitaxial growth with elasticity on vicinal surfaces. SIAM J. Appl. Math. 63(1), 241–258 (2002)
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Communicated by Ralf Müller.
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Kutter, M., Rohde, C. & Sändig, AM. Well-posedness of a two-scale model for liquid phase epitaxy with elasticity. Continuum Mech. Thermodyn. 29, 989–1016 (2017). https://doi.org/10.1007/s00161-015-0462-1
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DOI: https://doi.org/10.1007/s00161-015-0462-1