Abstract
A finite element framework is presented for the Burton-Cabrera-Frank (BCF) equation. The model is a 2 + 1-dimensional step flow model, discrete in the height but continuous in the lateral directions. The problem consists of adatom diffusion equations on terraces of different atomic height; boundary conditions at steps (terrace boundaries); and a normal velocity law for the motion of such boundaries determined by a two-sided flux, together with one-dimensional edge-diffusion. Two types of boundary conditions, modeling either diffusion limited growth or growth governed by attachment-detachment kinetics at the steps, are considered. We review the basic ideas of the algorithms, already described in [1, 2] and extent it to incorporate anisotropy of the step free energy, the edge mobility and the kinetic coefficients (attachment-detachment rates). The problem is solved using two independent meshes: a two-dimensional mesh for the adatom diffusion and a one-dimensional mesh for the step dynamics governed by an anisotropic geometric evolution law. Finally results on the anisotropic growth of single layer islands are presented.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
E. Bänsch, F. Haußer, O. Lakkis, B. Li, A. Voigt, Finite element method for epitaxial growth with attachment-detachment kinetics J. of Comput. Phys. 194 (2004), 409–434.
E. Bänsch, F. Haußer, A. Voigt, Finite Element Method for Epitaxial Growth with thermodynamic boundary conditions SIAM J. Sci. Comput. (2005), (to appear).
W.K. Burton, N. Cabrera, F.C. Frank, The growth of crystals and the equilibrium of their surfaces Phil. Trans. Roy. Soc. London Ser. A 243 (1951), 299–358.
J. Krug, (this volume).
M.F. Gyure, C. Ratsch, B. Merriman, R.E. Caflisch, S. Osher, J. Zinck, D. Vvedensky, Level set method for the simulation of epitaxial phenomena. Phys. Rev. E 58 (1998), R6931.
R.E. Caflisch, M.F. Gyure, B. Merriman, S. Osher, C. Ratsch, D. Vvedensky, J. Zink, Island dynamics and the level set method for epitaxial growth. Appl. Math. Lett. 12 (1999), 13–22.
S. Chen, B. Merriman, M. Kang, R.E. Caflisch, C. Ratsch, L.-T. Cheng, M. Gyure, R.P. Fedkiw, C. Anderson, S. Osher, Level Set Method for Thin Film Epitaxial Growth. J. Comp. Phys. 167 (2001), 475–500.
M. Peterson, C. Ratsch, R.E. Caflisch, A. Zangwill, Level set approach to reversible epitaxial growth. Phys. Rev. E 64 (2001), 061602.
C. Ratsch, M.F. Gyure, R.E. Caflisch, F. Gibou, M. Peterson, M. Kang, J. Garcia, D.D. Vvedensky, Level-set method for island dynamics in epitaxial growth. Phys. Rev. B 65 (2002), 195403.
D.L. Chopp, A level-set method for simulating island coarsening. J. Comp. Phys. 162 (2000), 104–122.
F. Liu, H. Metiu, Stability and kinetics of step motion on crystal surfaces Phys. Rev. E 49 (1997), 2601–2616.
A. Karma, M. Plapp, Spiral surface growth without desorption Phys. Rev. Lett. 81 (1998), 4444–4447.
A. Rätz, A. Voigt, Phase-Field Model for Island Dynamics Appl. Anal. 83 (2004), 1015–1025.
O. Pierre-Louis, Phase-field models for step flow Phys. Rev. E 68 (2003), 020601.
F. Otto. P. Penzler, A. Rätz, T. Rump, A. Voigt, A diffuse-interface approximation for step flow in epitaxial growth. Nonlin. 17 (2004), 477–491.
F. Otto. P. Penzler, T. Rump, (this volume).
A. Rätz, A. Voigt, A Phase-Field Model for attachment-detachment kinetics in epitaxial growth. Euro. J. Appl. Math. (2004), (to appear).
P. Smereka, Semi-implicit level set methods for motion by mean curvature and surface diffusion J. Sci. Comp. 19 (2003), 439–456.
U. Clarenz, F. Haußer, M. Rumpf, A. Voigt, U. Weikard, (this volume).
A. Rätz, A. Voigt, (this volume).
A. Schmidt, Computation of three dimensional dendrites with finite elements J. of Comput. Phys. 125 (1996), 293–312.
A.-K. Tornberg, Interface tracking methods with applications to multiphase flow PhD thesis, NADA, KTH, Stockholm, Sweden (2000).
E. Bänsch, P. Morin, R.H. Nochetto, A Finite Element Method for surface diffusion: the parametric case J. of Comput. Phys. 203 (2005), 321–343.
G. Dziuk, An algorithm for evolutionary surfaces Numer. Math. 58 (1991), 603–611.
A. Schmidt, K.G. Siebert, Albert — software for scientific computations and applications Acta Math. Univ. Comenianae 70 (2001), 205–122.
F. Haußer, A. Voigt, (this volume).
F. Haußer, A. Voigt, Finite element method for epitaxial island growth. J. Crys. Growth 266 (2004), 381–387.
M. Michely, J. Krug, Island, Mounds and Atoms. Springer, 2004
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Haußer, F., Voigt, A. (2005). A Finite Element Framework for Burton-Cabrera-Frank Equation. In: Voigt, A. (eds) Multiscale Modeling in Epitaxial Growth. ISNM International Series of Numerical Mathematics, vol 149. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7343-1_7
Download citation
DOI: https://doi.org/10.1007/3-7643-7343-1_7
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7208-8
Online ISBN: 978-3-7643-7343-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)