1. Some Aspects of Surface Diffusion
Martin Burger
Institut für Numerische und Angewandte Mathematik,
Center for Nonlinear Science CeNoS
Westfälische Willhelms-Universität Münster

2. Some Aspects of Surface Diffusion
Outline
Introduction: Motivation, Applications of Surface Diffusion
Strong anisotropies: Including strong anisotropies, curvature regularization, equilibria, dynamics, numerical simulation
Adatom diffusion: Change from 4th order to 2nd order system, change of equilibria, numerical simulation
Chemotaxis: limiting behaviour of packed cell densities
Some Aspects of Surface Diffusion
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3. Some Aspects of Surface Diffusion
Collaborations
Frank Hausser, Christina Stöcker, Axel Voigt (CAESAR Bonn)
Christian Schmeiser, Yasmin Dolak-Struss (Universität Wien)
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4. Some Aspects of Surface Diffusion
Introduction
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.
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5. Some Aspects of Surface Diffusion
Growth Mechanisms
Various fundamental surface growth mechanisms can determine the dynamics, most important:
Attachment / Detachment of atoms to / from surfaces
Diffusion of adatoms on surfaces
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6. Some Aspects of Surface Diffusion
Growth Mechanisms
Other effects influencing dynamics:
Anisotropy
Bulk diffusion of atoms (phase separation)
Exchange of atoms between surface and bulk
Elastic Relaxation in the bulk
Surface Stresses
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7. Some Aspects of Surface Diffusion
Growth Mechanisms
Other effects influencing dynamics:
Deposition of atoms on surfaces
Effects induced by electromagnetic forces (Electromigration)
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8. Isotropic Surface Diffusion
Simple 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 (ask Harald Garcke)
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9. Level Set / Graph Formulation
Level set function or graph parametrization u of surface determined from
(graph)
(level set) @ t u = d i v ( P r ) = d i v ( r u Q ) P = Q ( I r u ) Q = p 1 + j r u 2 Q = j r u
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10. Some Aspects of Surface Diffusion
Level Set Formulation
We have to deal with fourth-order equation, no maximum principle
No global level set formulation
Efficient computations and proofs still widely open (One of the „major mathematical challenges in materials science“, Jean Taylor, AMS, 2002 / Robert Kohn, SIAM, 2002)
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11. Applications: Nanostructures
SiGe/Si Quantum Dots
Bauer et. al. 99
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12. Applications: Nanostructures
SiGe/Si Quantum Dots
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13. Applications: Nanostructures
InAs/GaAs Quantum Dots
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14. Applications: Nano / Micro
Electromigration of voids in electrical circuits
Nix et. Al. 92
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15. Some Aspects of Surface Diffusion
Applications: Macro
Formation of Basalt Columns:
Giant‘s Causeway
Panska Skala (Northern Ireland)
(Czech Republic)
See:
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16. Energy The energy of the system is composed of various terms:
Total Energy =
(Anisotropic) Surface Energy +
(Anisotropic) Elastic Energy +
Compositional Energy +
.....
We start with first term only
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17. Some Aspects of Surface Diffusion
Surface Energy
Surface energy is given by
Standard model for surface free energy
Some Aspects of Surface Diffusion
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18. Some Aspects of Surface Diffusion
Chemical Potential
Chemical potential m 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
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19. Some Aspects of Surface Diffusion
Surface diffusion appears in many important applications - in particular in material and nano science
Growth of a surface G with velocity
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20. Some Aspects of Surface Diffusion
F ... Deposition flux
Ds .. Diffusion coefficient
W ... Atomic volume
s ... Surface density
k ... Boltzmann constant
T ... Temperature
n ... Unit outer normal
m ... Chemical potential =
energy variation
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21. Analysis and Simulation
Isotropic / Weakly Anisotropic:
Existence results Elliott-Garcke 1996
Numerical simulation Bänsch-Morin-Nocchetto 2003, Deckelnick-Dziuk-Elliott 2004
Anisotropic:
Studies of equilibrium structures, Gurtin 1993, Spencer 2003, Cecil-Osher 2004
Numerical simulation of asymptotic model (obtained from long-wave expansion), Golovin-Davies-Nepomnyaschy 2002 / 2003
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22. Some Aspects of Surface Diffusion
Surface Energy
In several situations, the surface free energy (respectively its one-homogeneous extension) is not convex. Nonconvex energies can result from different reasons:
Special materials with strong anisotropy: Gjostein 1963, Cahn-Hoffmann1974
Strained Vicinal Surfaces: Shenoy-Freund 2003
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23. Some Aspects of Surface Diffusion
Surface Energy
Effective surface free energy of a compressively strained vicinal surface (Shenoy 2004)
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24. Curvature Regularization
In order to regularize problem (and possibly since higher order terms become important in atomistic homogenization), curvature regularization has beeen proposed by several authors (DiCarlo-Gurtin-Podio-Guidugli 1993, Gurtin-Jabbour 2002, Tersoff, Spencer, Rastelli, Von Kähnel 2003)
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25. Anisotropic Surface energy
Cubic anisotropy
surface energy becomes non-convex for
e > 1/3
Faceting of the surface
Microstructure possible without curvature term
Equilibria are local energy minimizers only ( n ) = 1 + P 4 j
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26. Some Aspects of Surface Diffusion
Chemical Potential
We obtain
Energy variation corresponds to fourth-order term (due to curvature variation)
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27. Some Aspects of Surface Diffusion
Curvature Term
Derivative
with matrix
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28. Minimizing Movement: SD
SD can be obtained as the limit (t →0) of minimization
subject to
Some Aspects of Surface Diffusion
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29. Minimizing Movement: SD
Level set version:
subject to
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30. Some Aspects of Surface Diffusion
Numerical Solution
Basic idea: Semi-implicit time discretization +
Splitting into two / three second-order equations +
Finite element discretization in space
Natural variables for splitting:
Height u, Mean Curvature k, Chemical potential m
(denoted u, v, w in the following)
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31. Some Aspects of Surface Diffusion
Time Discretization
Based on variational principle, minimizing movement
subject to
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32. Some Aspects of Surface Diffusion
Time Discretization
Quadratic approximation of the convex terms in the energy, linear approximation of the non-convex terms around u(t)
Rewrite local variational problem as minimization over u, v, and w With constraints defining v and w
KKT condition yields indefinite linear system,
Lagrangian variables are multiples of v and w
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33. Spatial Discretization
Discretization of the variational problem in space by piecewise linear finite elements
and P(u) are piecewise constant on the triangularization, all integrals needed for stiffness matrix and right-hand side can be computed exactly
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34. Some Aspects of Surface Diffusion
Discrete Problem
After few manipulations we obtain indefinite linear system for the nodal values
A stiffness matrix from diffusion coefficient 1/Q
B stiffness matrix from diffusion coefficient P/Q
M mass matrix for identity, C mass matrix for 1/Q
Iterative solution by multigrid-precond. GMRES
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35. Some Aspects of Surface Diffusion
SD e = 3.5, a = 0.02, 10t =
Some Aspects of Surface Diffusion
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36. Some Aspects of Surface Diffusion
SD e = 1.5, a = 0.02, 10t =
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37. Some Aspects of Surface Diffusion
Faceting
Graph Simulation: mb JCP 04,
Level Set Simulation: mb-Hausser-Stöcker-Voigt 06
Adaptive FE grid around zero level set
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38. Some Aspects of Surface Diffusion
Faceting
Anisotropic mean curvature flow
Some Aspects of Surface Diffusion
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39. Some Aspects of Surface Diffusion
Faceting of Thin Films
Anisotropic Mean Curvature Anisotropic Surface Diffusion mb 04, mb-Hausser Stöcker-Voigt-05
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40. Faceting of Bulk Crystals
Anisotropic surface diffusion
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41. Some Aspects of Surface Diffusion
Modelling Aspects
Standard surface diffusion models have some strange aspects, in particular for nanostructures / epitaxy:
No kinetic effects
Problems with topology change (atoms can only hop on single surface, not on a second one, even for small distances)
They do not correspond to the atomistic picture
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42. Atomistic Models on (Nano-)Surfaces
Standard Description (e.g. Pimpinelli-Villain):
(Free) Adatoms hop on surfaces
Coupled with attachment detachment kinetics
for the surface atoms on a crystal lattice
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43. Atomistic Models on (Nano-)Surfaces
From Caflisch et. Al. 1999
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44. Some Aspects of Surface Diffusion
Modelling
Need two equations for two coupled processes
Need diffusion equation for adatoms
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45. Some Aspects of Surface Diffusion
Modelling
Explicit model for surface diffusion including adatoms Fried-Gurtin 2004, mb 2006
Adatom density d, chemical potential m, normal velocity V, tangential velocity v, mean curvature k, bulk density r
Kinetic coefficient b, diffusion coefficient L, deposition term r
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46. Some Aspects of Surface Diffusion
Surface Free Energy
Surface free energy y is a function of the adatom density
Chemical potential is the free energy variation
Surface energy:
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47. Some Aspects of Surface Diffusion
Modelling
Relation to standard surface diffusion: convergence as the cost of free adatoms (in the surface free energy tends to infinity)
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48. Some Aspects of Surface Diffusion
Equilibrium Shapes
Equilibrium shapes minimize the surface energy
at constant mass
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49. Some Aspects of Surface Diffusion
Equilibrium Shapes
Equilibrium films: minimum at vanishing adatom density, flat surface. Same as without adatoms.
Equilibrium crystals: Wulff shape with vanishing adatom density is NEVER an equilibrium !
Isotropic equilibrium has nonzero adatom density and smaller radius than Wulff shape
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50. Equilibrium Crystals (Isotropic)
Model free energy
Parameter g measures the cost of free adatoms
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51. Equilibrium Crystals (Isotropic)
Equilibrium radius
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52. Some Aspects of Surface Diffusion
Surface Energy
Different regimes for surface energy:
Convex for small adatom densities and shapes close to equilibrium
Nonconvex for large adatom densities and shapes far away from equilibrium. The surface energy is consequently not lower semicontinuous
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53. Some Aspects of Surface Diffusion
Numerical Simulation
Some Aspects of Surface Diffusion
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54. Some Aspects of Surface Diffusion
Numerical Simulation
Some Aspects of Surface Diffusion
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55. Some Aspects of Surface Diffusion
Numerical Simulation
Some Aspects of Surface Diffusion
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56. Some Aspects of Surface Diffusion
Numerical Simulation
Flat initial shape, nonhomogeneous deposition
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57. Numerical Simulation - Surfaces
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58. Numerical Simulation - Surfaces
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59. Some Aspects of Surface Diffusion
Chemotaxis
Keller-Segel Model with small diffusion and logistic sensitivity
Sensitivity function for quorum sensing derived by Painter and Hillen 2003 from microscopic model
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60. Some Aspects of Surface Diffusion
Chemotaxis
Keller-Segel Model with small diffusion and logistic sensitivity: Plateau formation
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61. Some Aspects of Surface Diffusion
Chemotaxis
Keller-Segel Model with small diffusion and logistic sensitivity: Plateau motion
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62. Some Aspects of Surface Diffusion
Chemotaxis
Keller-Segel Model with small diffusion and logistic sensitivity
Asymptotics at hyperbolic time-scale
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63. Some Aspects of Surface Diffusion
Chemotaxis
Limit is a nonlinear, nonlocal conservation law: we need entropy solutions
Entropy inequality
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64. Some Aspects of Surface Diffusion
Chemotaxis
Stationary solutions
These are entropy solutions iff
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65. Some Aspects of Surface Diffusion
Chemotaxis
Asymptotics for large time by time rescaling
Look for limiting solutions
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66. Some Aspects of Surface Diffusion
Chemotaxis
Asymptotic expansion in interfacial layer (as for Cahn-Hilliard)
Note: entropy condition
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67. Some Aspects of Surface Diffusion
Chemotaxis
We obtain a surface diffusion law with diffusivity and chemical potential
Corresponding energy functional D = 2 @ n S = S 2 [ ]
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68. Some Aspects of Surface Diffusion
Chemotaxis
Flow is volume conserving
Flow has energy dissipation property
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69. Some Aspects of Surface Diffusion
Chemotaxis
Stability of stationary solutions can be studied based on second (shape) variations on the energy functional
Stability condition for normal perturbation
Instability without entropy condition !
Otherwise high-frequency stability, possible low-frequency instability
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70. Some Aspects of Surface Diffusion
Chemotaxis
Instability
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71. Some Aspects of Surface Diffusion
Chemotaxis
Instability
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72. Some Aspects of Surface Diffusion
Chemotaxis
Instability
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73. Some Aspects of Surface Diffusion
Chemotaxis
Surface Diffusion
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74. Some Aspects of Surface Diffusion
Chemotaxis
Surface Diffusion
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75. Some Aspects of Surface Diffusion
Chemotaxis
Surface Diffusion, 3D
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76. Some Aspects of Surface Diffusion
Download and Contact
Papers and Talks:
Anisotropy: mb, JCP mb-Hausser-Stöcker-Voigt JCP 2007
Adatoms: mb, Comm. Math. Sci. 2006
Chemotaxis: mb-DiFrancesco-DolakStruss, SIMA mb-DolakStruss-Schmeiser, Preprint, 2006
Some Aspects of Surface Diffusion
Erlangen, February 2007