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Model Hierarchies for Surface Diffusion

Model Hierarchies for Surface Diffusion. Martin Burger. Johannes Kepler University Linz SFB Numerical-Symbolic-Geometric Scientific Computing Radon Institute for Computational & Applied Mathematics. Outline. Introduction Modelling Stages: Atomistic and continuum

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Model Hierarchies for Surface Diffusion

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  1. Model Hierarchies for Surface Diffusion Martin Burger Johannes Kepler University Linz SFB Numerical-Symbolic-Geometric Scientific Computing Radon Institute for Computational & Applied Mathematics

  2. Outline • Introduction • Modelling Stages: Atomistic and continuum • Small Slopes: Coherent coarse-graining of BCF Joint work with Axel Voigt Surface Diffusion Wien, Feb 2006

  3. Introduction • Surface diffusionprocesses appear in various materials science applications, in particular in the (self-assembled) growth of nanostructures • Schematic description: particles are deposited on a surface and become adsorbed (adatoms). They diffuse around the surface and can be bound to the surface. Vice versa, unbinding and desorption happens. Surface Diffusion Wien, Feb 2006

  4. Growth Mechanisms • Various fundamentalsurface growth mechanismscan determine the dynamics, most important: • Attachment / Detachment of atoms to / from surfaces / steps • Diffusionof adatoms on surfaces / along steps, over steps Surface Diffusion Wien, Feb 2006

  5. Atomistic Models on (Nano-)Surfaces • From Caflisch et. Al. 1999 Surface Diffusion Wien, Feb 2006

  6. Growth Mechanisms • Other effects influencing dynamics: • Anisotropy • Bulk diffusion of atoms (phase separation) • Elastic Relaxationin the bulk • Surface Stresses • Effects induced by electromagnetic forces Surface Diffusion Wien, Feb 2006

  7. Applications: Nanostructures • SiGe/Si Quantum Dots • Bauer et. al. 99 Surface Diffusion Wien, Feb 2006

  8. Applications: Nanostructures • SiGe/Si Quantum Dots Surface Diffusion Wien, Feb 2006

  9. Applications: Nano / Micro • Electromigration of voids in electrical circuits Nix et. Al. 92 Surface Diffusion Wien, Feb 2006

  10. Applications: Nano / Micro • Butterfly shape transition in Ni-based superalloys Colin et. Al. 98 Surface Diffusion Wien, Feb 2006

  11. Applications: Macro • Formation of Basalt Columns: Giant‘s Causeway Panska Skala (Northern Ireland) (Czech Republic) See: http://physics.peter-kohlert.de/grinfeld.html Surface Diffusion Wien, Feb 2006

  12. Atomistic Models on (Nano-)Surfaces • Standard Description (e.g. Pimpinelli-Villain): • (Free) Atoms hop on surfaces • Coupled with attachment-detachment kinetics for the surface atoms on a crystal lattice • Hopping and binding parameters obtained from quantum energy calculations Surface Diffusion Wien, Feb 2006

  13. Need for Continuum Models • Atomistic simulations (DFT -> MD -> KMC) limited to small / medium scale systems • Continuum models for surfaces easy to couple with large scale models Surface Diffusion Wien, Feb 2006

  14. Continuum Surface Diffusion • Simple continuum model for surface diffusion in the isotropic case: Normal motion of the surface by minus surfaceLaplacian of mean curvature • Can be derived as limit of Cahn-Hilliard model with degenerate diffusivity • Physical conditions for validity difficult to verify Surface Diffusion Wien, Feb 2006

  15. Continuum Surface Diffusion • Simple continuum model for surface diffusion in the isotropic case: Normal motion of the surface by minus surfaceLaplacian of mean curvature • Can be derived as limit of Cahn-Hilliard model with degenerate diffusivity • Physical conditions for validity difficult to verify Surface Diffusion Wien, Feb 2006

  16. Surface Diffusion • Growth of a surface G with velocity • F ... Deposition flux, Ds .. Diffusion coefficient • W ... Atomic volume, s ... Surface density • k ... Boltzmann constant, T ... Temperature • n ... Unit outer normal, m ... chemical potential Surface Diffusion Wien, Feb 2006

  17. Chemical Potential • Chemical potentialm is the change of energy when adding / removing single atoms • In a continuum model, the chemical potential can be represented as a surface gradient of the energy (obtained as the variation of total energy with respect to the surface) • For surfaces represented by a graph, the chemical potential is the functional derivative of the energy Surface Diffusion Wien, Feb 2006

  18. Surface Energy • Surface energy is given by • Standard model for anisotropic surface free energy Surface Diffusion Wien, Feb 2006

  19. Faceting of Thin Films Anisotropic Surface Diffusion mb-Hausser-Stöcker-Voigt-05 Surface Diffusion Wien, Feb 2006

  20. Faceting of Crystals • Anisotropic surface diffusion Surface Diffusion Wien, Feb 2006

  21. Disadvantages of Continuum Models • Parameters (anisotropy, diffusion coefficients, ..) not known at continuum level • Relation to atomistic models not obvious • Several effects not included in standard continuum models: Ehrlich-Schwoebel barriers, nucleation, adatom diffusion, step interaction .. Surface Diffusion Wien, Feb 2006

  22. Small Slope Approximations • Large distance between steps in z-direction • Diffusion of adatoms mainly in (x,y)-plane • Introduce intermediate model step: continuous in (x,y)-direction, discrete in z-direction Surface Diffusion Wien, Feb 2006

  23. Step Interaction Models • To understand continuum limit, start with simple 1D models • Steps are described by their position Xi and their sign si (+1 for up or -1 for down) • Height of a step equals atomic distance a • Step height function Surface Diffusion Wien, Feb 2006

  24. Step Interaction Models • Energy models for step interaction, e.g. nearest neighbour only • Scaling of height to maximal value 1, relative scale b between x and z, monotone steps Surface Diffusion Wien, Feb 2006

  25. Step Interaction Models • Simplest dynamics by direct step interaction • Dissipative evolution for X Surface Diffusion Wien, Feb 2006

  26. Continuum Limit • Introduce piecewise linear function wN on [0,1] with values Xk at z=k/N • Energy • Evolution Surface Diffusion Wien, Feb 2006

  27. Continuum Height Function • Function w is inverse of height function u • Continuum equation by change of variables • Transport equation in the limit, gradient flow in the Wasserstein metric of probability measures (u equals distribution function) Surface Diffusion Wien, Feb 2006

  28. Continuum Height Function • Transport equation in the limit, gradient flow in the Wasserstein metric of probability measures (u equals distribution function) • Rigorous convergence to continuum: standard numerical analysis problem • Max / Min of the height function do not change (obvious for discrete, maximum principle for continuum). Large flat areas remain flat Surface Diffusion Wien, Feb 2006

  29. Non-monotone Step Trains • Treatment with inverse function not possible • Models can still be formulated as metric gradient flow on manifolds of measures • Manifold defined by structure of the initial value (number of hills and valleys) Surface Diffusion Wien, Feb 2006

  30. BCF Models • In practice, more interesting class are BCF-type models(Burton-Cabrera-Frank 54) • Micro-scale simulations by level set methods etc (Caflisch et. al. 1999-2003) • Simplest BCF-model Surface Diffusion Wien, Feb 2006

  31. Chemical Potential • Chemical potential is the difference between adatom density and equilibrium density • From equilibrium boundary conditions for adatoms • From adatom diffusion equation (stationary) Surface Diffusion Wien, Feb 2006

  32. Continuum Limit • Two additional spatial derivatives lead to formal 4-th order limit (Pimpinelli-Villain 97, Krug 2004, Krug-Tonchev-Stoyanov-Pimpinelli 2005) • 4-th order equations destroy various properties of the microscale model (flat regions stay never flat, global max / min not conserved ..) • Is this formal limit correct ? Surface Diffusion Wien, Feb 2006

  33. Continuum Limit • Formal 4-th order limit Surface Diffusion Wien, Feb 2006

  34. Gradient Flow Formulation • Reformulate BCF-model as dissipative flow • Analogous as above, we only need to change metric • P appropriate projection operator Surface Diffusion Wien, Feb 2006

  35. Gradient Flow Structure • Time-discrete formulation • Minimization over manifold for suitable deformation T Surface Diffusion Wien, Feb 2006

  36. Continuum Limit • Manifold constraint for continuous time for a velocity V • Modified continuum equations Surface Diffusion Wien, Feb 2006

  37. Continuum Limit • 4th order vs. modified 4th order Surface Diffusion Wien, Feb 2006

  38. Example: adatoms • Explicit model for surface diffusion including adatoms Fried-Gurtin 2004, mb 2006 • Adatom densityd, chemical potentialm, normal velocity V, tangential velocityv, mean curvaturek, bulk densityr • Kinetic coefficient b, diffusion coefficient L, deposition term r Surface Diffusion Wien, Feb 2006

  39. Surface Free Energy • Surface free energy y is a function of the adatom density • Chemical potential is the free energy variation • Surface energy: Surface Diffusion Wien, Feb 2006

  40. Numerical Simulation - Surfaces Surface Diffusion Wien, Feb 2006

  41. Outlook • Limiting procedure analogous for more complicated and realistic BCF-models, various effects incorporated in continuum. Direct relation of parameters to BCF models • Relation of parameters from BCF to atomistic models • Possibility for multiscale schemes: continuum simulation of surface evolution, local atomistic computations of parameters Surface Diffusion Wien, Feb 2006

  42. Download and Contact • Papers and Talks: www.indmath.uni-linz.ac.at/people/burger • e-mail: martin.burger@jku.at Surface Diffusion Wien, Feb 2006

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