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Continuum Limit of a Mesoscopic Model with Elasticity of Step Motion on Vicinal Surfaces

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Abstract

This work considers the rigorous derivation of continuum models of step motion starting from a mesoscopic Burton–Cabrera–Frank-type model following the Xiang’s work (Xiang in SIAM J Appl Math 63(1):241–258, 2002). We prove that as the lattice parameter goes to zero, for a finite time interval, a modified discrete model converges to the strong solution of the limiting PDE with first-order convergence rate.

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Notes

  1. Compared to Xiang (2002), we drop all the physical constants that are mathematically unimportant.

  2. For the convenience of calculation, we set the coefficients slightly different from Dal Maso et al. (2014). Moreover, instead of taking h to be increasing as in Dal Maso et al. (2014), we take h to be decreasing corresponding to physical interpretation of h being the height of the vicinal surface, which is the same convention as Xiang (2002), Xiang and E (2004).

References

  • Al Hajj Shehadeh: H., Kohn, R.V., Weare, J.: The evolution of a crystal surface: analysis of a one-dimensional step train connecting two facets in the ADL regime. Phys. D 240(21), 1771–1784 (2011)

    Article  MATH  Google Scholar 

  • Burton, W.K., Cabrera, N., Frank, F.C.: The growth of crystals and the equilibrium structure of their surfaces. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 243(866), 299–358 (1951)

    Google Scholar 

  • Butzer, P.L., Nessel, R.J.: Fourier Analysis and Approximation, vol. 40. Academic Press, London (2011)

    MATH  Google Scholar 

  • Dal Maso, G., Fonseca, I., Leoni, G.: Analytical validation of a continuum model for epitaxial growth with elasticity on vicinal surfaces. Arch. Ration. Mech. Anal. 212(3), 1037–1064 (2014)

  • Duport, C., Nozieres, P., Villain, J.: New instability in molecular beam epitaxy. Phys. Rev. Lett. 74, 134–137 (1995a)

    Article  Google Scholar 

  • Duport, C., Politi, P., Villain, J.: Growth instabilities induced by elasticity in a vicinal surface. J. Phys. I Fr. 5, 1317–1350 (1995b)

    Article  Google Scholar 

  • E, W., Yip, N.K.: Continuum theory of epitaxial crystal growth. I. J. Stat. Phys. 104(1–2), 221–253 (2001)

  • Evans, L.C.: Partial Differential Equations (Graduate Studies in Mathematics), vol. 19. American Mathematical Society, Providence (1998)

    Google Scholar 

  • Fonseca, I., Leoni, G., Lu, X.Y.: Regularity in time for weak solutions of a continuum model for epitaxial growth with elasticity on vicinal surfaces. Commun. Partial Differ. Equ. 40(10), 1942–1957 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Funaki, T., Spohn, H.: Motion by mean curvature from the Ginzburg-Landau interface model. Commun. Math. Phys. 185, 1–36 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • Grinfeld, M.: Instability of the separation boundary between a nonhydrostatically stressed elastic body and a melt. Sov. Phys. Dokl. 31, 831–834 (1986)

    Google Scholar 

  • Guo, M.Z., Papanicolaou, G.C., Varadhan, S.R.S.: Nonlinear diffusion limit for a system with nearest neighbor interactions. Commun. Math. Phys. 118, 31–59 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  • Liu, F., Tersoff, J., Lagally, M.: Self-organization of steps in growth of strained films on vicinal substrates. Phys. Rev. Lett. 80(6), 1268 (1998)

    Article  Google Scholar 

  • Lu, J., Liu, J.-G., Margetis, D.: Emergence of step flow from an atomistic scheme of epitaxial growth in 1+1 dimensions. Phys. Rev. E 91(3), 032403 (2015)

    Article  Google Scholar 

  • Luo, T., Xiang, Y., Yip, N.K.: Energy scaling and asymptotic properties of step bunching in epitaxial growth with elasticity effects. Multiscale Model. Simul. 14(2), 737–771 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Luo, T., Xiang, Y., Yip, N.K.: Private communication

  • Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow, vol. 27. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  • Margetis, D., Kohn, R.V.: Continuum relaxation of interacting steps on crystal surfaces in \(2+1\) dimensions. Multiscale Model. Simul. 5(3), 729–758 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Margetis, D., Nakamura, K.: From crystal steps to continuum laws: behavior near large facets in one dimension. Phys. D 240, 1100–1110 (2011)

    Article  MATH  Google Scholar 

  • Marzuola, J.L., Weare, J.: Relaxation of a family of broken-bond crystal-surface models. Phys. Rev. E 88, 032403 (2013)

    Article  Google Scholar 

  • Nishikawa, T.: Hydrodynamic limit for the Ginzburg-Landau \(\nabla \phi \) interface model with a conservation law. J. Math. Sci. Univ. Tokyo 9, 481–519 (2002)

    MathSciNet  MATH  Google Scholar 

  • Pimpinelli, A., Villain, J.: Physics of Crystal Growth. Cambridge University Press, New York (1998)

    Book  Google Scholar 

  • Shenoy, V., Freund, L.: A continuum description of the energetics and evolution of stepped surfaces in strained nanostructures. J. Mech. Phys. Solids 50(9), 1817–1841 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Sidi, A., Israeli, M.: Quadrature methods for periodic singular and weakly singular Fredholm integral equations. J. Sci. Comput. 3(2), 201–231 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  • Srolovitz, D.J.: On the stability of surfaces of stressed solids. Acta Metall. 37(2), 621–625 (1989)

    Article  Google Scholar 

  • Tang, L.-H.: Flattening of grooves: from step dynamics to continuum theory. In: Duxbury, P.M., Pence, T.J. (eds.) Dynamics of Crystal Surfaces and Interfaces, vol. 169. Springer, New York (1997)

    Google Scholar 

  • Tersoff, J., Phang, Y., Zhang, Z., Lagally, M.: Step-bunching instability of vicinal surfaces under stress. Phys. Rev. Lett. 75(14), 2730 (1995)

    Article  Google Scholar 

  • Weeks, J.D., Gilmer, G.H.: Dynamics of crystal growth. In: Prigogine, I., Rice, S.A. (eds.) Advances in Chemical Physics, vol. 40, pp. 157–228. Wiley, New York (1979)

    Chapter  Google Scholar 

  • Xiang, Y.: Derivation of a continuum model for epitaxial growth with elasticity on vicinal surface. SIAM J. Appl. Math. 63(1), 241–258 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Xiang, Y., E, W., Misfit elastic energy and a continuum model for epitaxial growth with elasticity on vicinal surfaces. Phys. Rev. B 69(3), 035409 (2004)

  • Xu, H., Xiang, Y.: Derivation of a continuum model for the long-range elastic interaction on stepped epitaxial surfaces in \(2+1\) dimensions. SIAM J. Appl. Math. 69(5), 1393–1414 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Yau, H.-T.: Relative entropy and hydrodynamics of Ginzburg-Landau models. Lett. Math. Phys. 22, 63–80 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  • Zangwill, A.: Physics at Surfaces. Cambridge University Press, New York (1988)

    Book  Google Scholar 

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Acknowledgements

We would like to thank the support by the National Science Foundation under Grants DMS-1514826 (JGL), DMS-1454939 (JL), and also through the research network KI-Net RNMS11-07444. We thank Dionisios Margetis, Jeremy Marzuola, Yang Xiang and Aaron N.K. Yip for helpful discussions.

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China

    Yuan Gao

  2. Department of Mathematics, Duke University, Box 90320, Durham, NC, 27708, USA

    Yuan Gao, Jian-Guo Liu & Jianfeng Lu

  3. Department of Physics, Duke University, Box 90320, Durham, NC, 27708, USA

    Yuan Gao, Jian-Guo Liu & Jianfeng Lu

  4. Department of Chemistry, Duke University, Box 90320, Durham, NC, 27708, USA

    Jianfeng Lu

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  1. Yuan Gao
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  2. Jian-Guo Liu
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Correspondence to Yuan Gao.

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Communicated by Robert V. Kohn.

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Gao, Y., Liu, JG. & Lu, J. Continuum Limit of a Mesoscopic Model with Elasticity of Step Motion on Vicinal Surfaces. J Nonlinear Sci 27, 873–926 (2017). https://doi.org/10.1007/s00332-016-9354-1

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