1. 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. Surface Diffusion Wien, Feb 20062 Outline Introduction Modelling Stages: Atomistic and continuum Small Slopes: Coherent coarse-graining of BCF Joint work with Axel Voigt

3. Surface Diffusion Wien, Feb 20063 Surface diffusion processes 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. Introduction

4. Surface Diffusion Wien, Feb 20064 Various fundamental surface growth mechanisms can determine the dynamics, most important: - Attachment / Detachment of atoms to / from surfaces / steps - Diffusion of adatoms on surfaces / along steps, over steps Growth Mechanisms

5. Surface Diffusion Wien, Feb 20065 From Caflisch et. Al. 1999 Atomistic Models on (Nano-)Surfaces

6. Surface Diffusion Wien, Feb 20066 Other effects influencing dynamics: - Anisotropy - Bulk diffusion of atoms (phase separation) - Elastic Relaxation in the bulk - Surface Stresses - Effects induced by electromagnetic forces Growth Mechanisms

7. Surface Diffusion Wien, Feb 20067 Applications: Nanostructures SiGe/Si Quantum Dots Bauer et. al. 99

8. Surface Diffusion Wien, Feb 20068 Applications: Nanostructures SiGe/Si Quantum Dots

9. Surface Diffusion Wien, Feb 20069 Applications: Nano / Micro Electromigration of voids in electrical circuits Nix et. Al. 92

10. Surface Diffusion Wien, Feb 200610 Applications: Nano / Micro Butterfly shape transition in Ni-based superalloys Colin et. Al. 98

11. Surface Diffusion Wien, Feb 200611 Applications: Macro Formation of Basalt Columns: Giant‘s Causeway Panska Skala (Northern Ireland) (Czech Republic) See: http://physics.peter-kohlert.de/grinfeld.html

12. Surface Diffusion Wien, Feb 200612 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 Atomistic Models on (Nano-)Surfaces

13. Surface Diffusion Wien, Feb 200613 Atomistic simulations (DFT -> MD -> KMC) limited to small / medium scale systems Continuum models for surfaces easy to couple with large scale models Need for Continuum Models

14. Surface Diffusion Wien, Feb 200614 Continuum Surface Diffusion Simple continuum model for surface diffusion in the isotropic case: Normal motion of the surface by minus surface Laplacian of mean curvature Can be derived as limit of Cahn-Hilliard model with degenerate diffusivity Physical conditions for validity difficult to verify

15. Surface Diffusion Wien, Feb 200615 Continuum Surface Diffusion Simple continuum model for surface diffusion in the isotropic case: Normal motion of the surface by minus surface Laplacian of mean curvature Can be derived as limit of Cahn-Hilliard model with degenerate diffusivity Physical conditions for validity difficult to verify

16. Surface Diffusion Wien, Feb 200616 Growth of a surface  with velocity F... Deposition flux, D s.. Diffusion coefficient ... Atomic volume, ... Surface density k... Boltzmann constant, T... Temperature n... Unit outer normal, ... chemical potential Surface Diffusion

17. Wien, Feb 200617 Chemical potential  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 Chemical Potential

18. Surface Diffusion Wien, Feb 200618 Surface energy is given by Standard model for anisotropic surface free energy Surface Energy

19. Surface Diffusion Wien, Feb 200619 Faceting of Thin Films Anisotropic Surface Diffusion mb-Hausser-Stöcker-Voigt-05

20. Surface Diffusion Wien, Feb 200620 Faceting of Crystals Anisotropic surface diffusion

21. Surface Diffusion Wien, Feb 200621 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.. Disadvantages of Continuum Models

22. Surface Diffusion Wien, Feb 200622 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 Small Slope Approximations

23. Surface Diffusion Wien, Feb 200623 Step Interaction Models To understand continuum limit, start with simple 1D models Steps are described by their position X i and their sign s i (+1 for up or -1 for down) Height of a step equals atomic distance a Step height function

24. Surface Diffusion Wien, Feb 200624 Step Interaction Models Energy models for step interaction, e.g. nearest neighbour only Scaling of height to maximal value 1, relative scale  between x and z, monotone steps

25. Surface Diffusion Wien, Feb 200625 Step Interaction Models Simplest dynamics by direct step interaction Dissipative evolution for X

26. Surface Diffusion Wien, Feb 200626 Continuum Limit Introduce piecewise linear function w N on [0,1] with values X k at z=k/N Energy Evolution

27. Surface Diffusion Wien, Feb 200627 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)

28. Surface Diffusion Wien, Feb 200628 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

29. Surface Diffusion Wien, Feb 200629 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)

30. Surface Diffusion Wien, Feb 200630 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

31. Surface Diffusion Wien, Feb 200631 Chemical Potential Chemical potential is the difference between adatom density and equilibrium density From equilibrium boundary conditions for adatoms From adatom diffusion equation (stationary)

32. Surface Diffusion Wien, Feb 200632 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 ?

33. Surface Diffusion Wien, Feb 200633 Continuum Limit Formal 4-th order limit

34. Surface Diffusion Wien, Feb 200634 Gradient Flow Formulation Reformulate BCF-model as dissipative flow Analogous as above, we only need to change metric  appropriate projection operator

35. Surface Diffusion Wien, Feb 200635 Gradient Flow Structure Time-discrete formulation Minimization over manifold for suitable deformation T

36. Surface Diffusion Wien, Feb 200636 Continuum Limit Manifold constraint for continuous time for a velocity V Modified continuum equations

37. Surface Diffusion Wien, Feb 200637 Continuum Limit 4th order vs. modified 4th order

38. Surface Diffusion Wien, Feb 200638 Explicit model for surface diffusion including adatoms Fried-Gurtin 2004, mb 2006 Adatom density , chemical potential , normal velocity V, tangential velocity v, mean curvature , bulk density  Kinetic coefficient b, diffusion coefficient L, deposition term r Example: adatoms

39. Surface Diffusion Wien, Feb 200639 Surface free energy  is a function of the adatom density Chemical potential is the free energy variation Surface energy: Surface Free Energy

40. Surface Diffusion Wien, Feb 200640 Numerical Simulation - Surfaces

41. Surface Diffusion Wien, Feb 200641 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

42. Surface Diffusion Wien, Feb 200642 Download and Contact Papers and Talks: www.indmath.uni-linz.ac.at/people/burger e-mail: martin.burger@jku.at