Model Hierarchies for Surface Diffusion

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Model Hierarchies for Surface Diffusion

• Martin Burger

Johannes Kepler University Linz SFB
Numerical-Symbolic-Geometric Scientific
Computing Radon Institute for Computational
Applied Mathematics
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Outline

• Introduction
• Modelling Stages Atomistic and continuum
• Small Slopes Coherent coarse-graining of BCF
• Joint work with Axel Voigt

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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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Growth Mechanisms

• 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

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Atomistic Models on (Nano-)Surfaces

• From Caflisch et. Al. 1999

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Growth Mechanisms

• Other effects influencing dynamics
• Anisotropy
• Bulk diffusion of atoms (phase separation)
• Elastic Relaxation in the bulk
• Surface Stresses
• Effects induced by electromagnetic forces

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Applications Nanostructures

• SiGe/Si Quantum Dots
• Bauer et. al. 99

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Applications Nanostructures

• SiGe/Si Quantum Dots

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Applications Nano / Micro

• Electromigration of voids in electrical circuits
• Nix et. Al. 92

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Applications Nano / Micro

• Butterfly shape transition in Ni-based
  superalloys
• Colin et. Al. 98

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Applications Macro

• Formation of Basalt Columns
• Giants Causeway
• Panska Skala (Northern Ireland)
• (Czech Republic)
• See http//physics.peter-kohlert.de/grinfeld.htm
  l

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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

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Need for Continuum Models

• Atomistic simulations (DFT -gt MD -gt KMC) limited
  to small / medium scale systems
• Continuum models for surfaces easy to couple
  with large scale models

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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

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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

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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

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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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Surface Energy

• Surface energy is given by
• Standard model for anisotropic surface free
  energy

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Faceting of Thin Films

• Anisotropic Surface Diffusion mb-Hausser-Stöcker-
  Voigt-05

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Faceting of Crystals

• Anisotropic surface diffusion

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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 ..

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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

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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

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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

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Step Interaction Models

• Simplest dynamics by direct step interaction
• Dissipative evolution for X

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Continuum Limit

• Introduce piecewise linear function w N on 0,1
  with values Xk at zk/N
• Energy
• Evolution

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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)

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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

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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)

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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

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Chemical Potential

• Chemical potential is the difference between
  adatom density and equilibrium density
• From equilibrium boundary conditions for adatoms
• From adatom diffusion equation (stationary)

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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 ?

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Continuum Limit

• Formal 4-th order limit

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Gradient Flow Formulation

• Reformulate BCF-model as dissipative flow
• Analogous as above, we only need to change metric
  
• P appropriate projection operator

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Gradient Flow Structure

• Time-discrete formulation
• Minimization over manifold
• for suitable deformation T

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Continuum Limit

• Manifold constraint for continuous time
• for a velocity V
• Modified continuum equations

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Continuum Limit

• 4th order vs. modified 4th order

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Example adatoms

• 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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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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Numerical Simulation - Surfaces
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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

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Download and Contact

• Papers and Talks
• www.indmath.uni-linz.ac.at/people/burger
• e-mail martin.burger_at_jku.at