The Level Set Method to Model Epitaxial Growth

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The Level Set Method to Model Epitaxial Growth
Christian Ratsch UCLA, Department of Mathematics
Collaborators Russel Caflisch Max
Petersen Jennifer Garcia Mark Gyure Dimitri
Vvedensky

• Outline
• What is epitaxial growth?
• How is level set method used to model thin film
  growth ?
• Result
• Inclusion of microscopic effects

NSF and DARPA
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The Island Dynamics Model for Epitaxial Growth

• Conventional Approaches
• Atomistic Kinetic Monte Carlo simulation
  (stochastic, expensive)
• Continuum Models (deterministic, lacks detail)

• Treat Islands as continuum in the plane
• Resolve individual atomic layers
• Velocity of island boundaries ?
• How do islands nucleate ? Where ?
• Evolve island boundaries with levelset method
• Treat atoms as a mean field quantity

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The Level Set Method Schematic

• Continuous level set function is resolved on a
  discrete numerical grid
• Method is continuous in plane (but atomic
  resolution is possible !), but has discrete
  height resolution

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The Level Set Method Formalism

• PDE - based method, almost deterministic !

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Typical Snapshots of Behavior of the Model
t0.1
j
r
t0.5
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Validation Scaling of the Island Size
Distribution
Q Coverage sav Average island size ns Density
of islands of size s
C. Ratsch et al., Phys. Rev. B 61, R10598 (2000)
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A Typical Level Set Simulation
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Decay of Step Edge Oscillations
Layer-by-layer growth persists longer with no
(slow) edge diffusion
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Roughness Evolution
Surface roughens faster with fast edge diffusion
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Adatom Concentration
Fast edge diffusion Compact Islands
Slow edge diffusion Fractal Islands
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Effect of Edge Diffusion on Surface Roughness
Fast edge diffusion
No edge diffusion
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Extension to Reversibility

• Remark Stochastic element in the break-up of
  islands is needed !!
• No frequent detachment/re-attachment needed !

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Extension to Reversibility Sharpening of Island
Size Distribution
Experimental Data for Fe/Fe(001), Stroscio and
Pierce, Phys. Rev. B 49 (1994)
Petersen, Ratsch, Caflisch, Zangwill, Phys. Rev.
E 64, 061602 (2001)
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Scaling of Computational Time
Almost no increase in computational time due to
mean-field treatment of fast events
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Extensions and Applications of the Reversible
Aggregation Model
Ostwald Ripening (without flux)

• Strain dependent detachment
• Simple approach
• Simply make the detachment rate (i.e.shrink
  velocity) size dependent
• More sophisticated approach
• Solve elastic equations
• Couple with level set code
• Applications
• model formation and self-organization of quantum
  dots
• segregation at interfaces

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Sharpening due to Island Size Dependent
Detachment (Strain)
Linear Increase
Linear Decrease
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Atomic Size Effects
D/F106
D/F107
D/F108

• Island densities are too high
• Results of finite size of atoms (at the
  boundary)
• Idea Set r0 in a region of width a (atomic
  lattice constant)

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Estimate Atomic Size Effect
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Implementation of a Boundary Region
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Results for the Island Densities
D/F106
D/F107
D/F108
C. Ratsch et al., Phys. Rev. E 64, 020601 (2001)
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Conclusions

• We have developed a numerically stable and
  accurate levelset model to study epitaxial growth
• Spatial fluctuations are needed in the seeding of
  new islands
• Microscopic details such as detachment and edge
  diffusion can be included

• Future Directions
• The levelset method provides a natural framework
  to couple other external fields such as strain or
  hydrodynamics to the growth modeling
• Multiple species can be described with multiple
  (coupled) diffusion equations
• Include surface chemistry (surface
  reconstrcutions) Talk tomorrow, 430pm

Transparencies of this talk can be found at
www.math.ucla.edu/material