Modeling of SelfOrganization of Quantum Dots using the Level Set Method

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Modeling of Self-Organization of Quantum Dots
using the Level Set Method
Christian Ratsch, UCLA, Department of Mathematics
Collaborators Russel Caflisch, Raffaele
Vardavas, Xioabin Niu
Idea of the level-set approach
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The Level Set Method Schematic

• Level set function is continuous in plane, but
  has discrete height resolution
• Adatoms are treated in a mean field picture

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The Level Set Method The Basic Formalism

• Governing Equation

• Boundary condition

Seeding position chosen stochastically (weighted
with local value of r2)
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A Typical Level Set Simulation
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Validation Scaling and Sharpening of Island Size
Distribution
Experimental Data for Fe/Fe(001), Stroscio and
Pierce, Phys. Rev. B 49 (1994)
So far constant diffusion
Petersen, Ratsch, Caflisch, Zangwill, Phys. Rev.
E 64, 061602 (2001).
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Motivation for Spatially Varying, Anisotropic
Diffusion Nucleation and Growth on Buried Defect
Lines
Results of Xie et al. (UCLA, Materials Science
Dept.)

• Growth on Ge on relaxed SiGe buffer layer
• Dislocation lines are buried underneath.
• Lead to strain field
• This can alter potential energy surface
• Anisotropic diffusion
• Spatially varying diffusion
• Hypothesis
• Nucleation occurs in regions of fast diffusion

Level Set formalism is ideally suited to
incorporate anisotropic, spatially varying
diffusion without extra computational cost
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Modifications to the Level Set Formalism
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Isotropic Diffusion with Sinusoidal Variation in
x-Direction
Only variation of transition energy, and constant
adsorption energy

• Islands nucleate in regions of fast diffusion
• Little subsequent nucleation in regions of slow
  diffusion

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Anisotropic Diffusion with Sinusoidal Variation
in x-Direction

• In both cases, islands mostly nucleate in
  regions of fast diffusion.
• Shape orientation is different

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Isotropic Diffusion with Sinusoidal Variation in
x- and y-Direction
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Recall Experimental Results
Results of Xie et al. (UCLA, Materials Science
Dept.)
Simulations
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Anisotropic Diffusion with Sinusoidal Variation
in x-Direction with Drift
Now we have varying adsorption energy, i.e.,
thermodynamic drift
Spatially constant adsorption and transition
energies, i.e., no dirft
Most nucleation does not occur in region of fast
diffusion, but is dominated by drift
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Conclusions

• We have developed a numerically stable and
  accurate level set method to describe epitaxial
  growth
• This framework is ideally suited to include
  anisotropic, spatially varying diffusion.
• Islands nucleate preferentially in regions of
  fast diffusion (when the adsorption energy is
  constant)
• However, a strong drift term can dominate over
  fast diffusion
• A properly modified potential energy surface can
  be exploited to obtain a high regularity in the
  arrangement of islands.

More details and transparencies of this talk can
be found at www.math.ucla.edu/material