A LevelSet Method for Modeling Epitaxial Growth and SelfOrganization of Quantum Dots

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A Level-Set Method for Modeling Epitaxial Growth
and Self-Organization of Quantum Dots
Christian Ratsch, UCLA, Department of Mathematics

• Outline
• The level-set method for epitaxial growth
• Results for irreversible and reversible
  aggregation
• Spatially varying diffusion can be used for
  self-organization of islands (quantum dots)
• Coupling of level-set formalism with an elastic
  model.

• Collaborators
• Xiaobin Niu
• Raffaele Vardavas
• Russel Caflisch

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Modeling thin film growth

• Methods used
• (Atomistic) KMC simulations
• Completely stochastic method
• Rate parameters can be obtained from DFT
• Continuum equations (PDEs)
• essentially deterministic
• no microscopic details.
• parameters can be obtained from atomistic model
  (but difficult)

• New Method
• Level set method
• PDE - based, (almost) deterministic
• atomistic details can be included
• microscopic parameters can be obtained from DFT

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Idea of the level set appproach
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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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Numerical details

• Level set function
• 3rd order essentially non-oscillatory (ENO)
  scheme for spatial part of levelset function
• 3rd order Runge-Kutta for temporal part
• Diffusion equation
• Implicit scheme to solve diffusion equation
  (Backward Euler)
• Use ghost-fluid method to make matrix symmetric
• Use PCG Solver (Preconditioned Conjugate
  Gradient)

S. Chen, M. Kang, B. Merriman, R.E. Caflisch, C.
Ratsch, R. Fedkiw, M.F. Gyure, and S. Osher, JCP
(2001)
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A typical level set simulation
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Fluctuations need to be included in nucleation of
islands
C. Ratsch et al., Phys. Rev. B 61, R10598 (2000)
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Detachment of adatoms and breakup of islands

• For small islands, calculate probability of
  island break-up.
• This probability is related to Ddet, and local
  environment
• Pick random number to decide break-up
• If island is removed, atoms are distributed
  uniformly in an area that corresponds to the
  diffusion length

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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)
Petersen, Ratsch, Caflisch, Zangwill, Phys. Rev.
E 64, 061602 (2001).
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Computational efficiency

• Fast events can be included without decreasing
  the numerical timestep (due to mean-field
  treatment of adatoms)

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Modeling self-organization of quantum dots

• Ultimate goal Solve elastic equations at every
  timestep, and couple the strain field to the
  simulation parameters (i.e., D, Ddet).
• This is possible because the simulation timestep
  can be kept rather large.
• Needed Spatially varying, anisotropic diffusion
  and detachment rates.
• Modifications to the code will be discussed!
• So far We assume simple variation of potential
  energy surface.
• Next (with some preliminary results) couple
  with elastic code of Caflisch, Connell, Luo, Lee

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Vertical alignment of stacked quantum dots
Stacked InAs quantum dots on GaAs(001)

• Islands nucleate on top of lower islands
• Size and separation becomes more uniform
• Interpretation buried islands lead to strain
  (there is a 7 misfit)
• Spatially varying potential energy
  surface
• Spatially varying nucleation
  probability

B. Lita et al. (Goldman group), APL 74, 2824
(1999)
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Aligned islands due to buried dislocation lines
Ge on relaxed SiGe buffer layers

• Islands align along lines
• Dislocation lines are buried underneath
• Interpretation buried dislocation lines lead to
  strain
• Spatially varying potential energy
  surface
• Spatially varying nucleation
  probability

Level Set formalism is ideally suited to
incorporate anisotropic, spatially varying
diffusion and thus nucleation without extra
computational cost
H. J. Kim, Z. M. Zhao, Y. H. Xie, PRB 68, 205312
(2003).
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Modifications to the level set formalism for
non-constant diffusion
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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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Comparison with experimental results
Results of Xie et al. (UCLA, Materials Science
Dept.)
Simulations
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Isotropic diffusion with sinusoidal variation in
x- and y-direction
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Anisotropic diffusion with variation of
adsorption energy
What is the effect of thermodynamic drift ?
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Transition from thermodynamically to kinetically
controlled diffusion
Constant adsorption energy (no drift)
Constant transition energy (thermodynamic drift)
In all cases, diffusion constant D has the same
form

• No drift (right) nucleation dominated by fast
  diffusion
• Large Drift (left) nucleation dominated by drift

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Time evolution in the kinetic limit

• A properly modified PES (in the kinetic limit)
  leads to very regular, 1-D structures
• Can this approach used to produce quantum wires?

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Combination of island dynamics model with elastic
code

• In contrast to an atomistic (KMC) simulation, the
  timestep is rather large, even when we have a
  large detachment rate (high temperature).
• A typical timestep in our simulation is O(10-2
  s) compare to typical atomistic simulation,
  where it is O(10-6 s).
• This allows us to do an expensive calculation
  at every timestep.
• For example, we can solve the elastic equations
  at every timestep, and couple the local value of
  the strain to the microscopic parameters.
• This work is currently in progress .. but here
  are some initial results.

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Our Elastic model

• Write down an atomistic energy density, that
  includes the following terms (lattice statics)
  (this is work by Caflisch, Connell, Luo, Lee, et
  al.)
• Nearest neighbor springs
• Diagonal springs
• Bond bending terms

• This can be related to (and interpreted as)
  continuum energy density

• Minimize energy with respect to all
  displacements ?u E u 0

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Numerical Method

• PCG using Algebraic MultiGrid (poster by
  Young-Ju Lee)
• Artificial boundary conditions at top of
  substrate (poster by Young-Ju Lee)
• Additional physics, such as more realistic
  potential or geometry easily included

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Couple elastic code to island dynamics model

• Example
• Epilayer is 4 bigger than substrate (I.e., Ge
  on Si)
• Choose elastic constants representative for Ge,
  Si
• Deposit 0.2 monolayers

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Modification of diffusion field
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Results with strain-dependent detachment rate
Constant diffusion
Change diffusion as a function of strain at every
timestep

• It is not clear whether there is an effect on
  ordering
• More quantitative analysis needed

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Modification of detachment rates

• The detachment rate has only physical meaning at
  the island edge (where it changes the boundary
  condition req)
• The model shown here indicates that it is more
  likely to detach from a bigger (more strained
  island) than from a smaller one.
• Previous (KMC) work suggests that this leads to
  more uniform island size distribution.

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Conclusions

• We have developed a numerically stable and
  accurate level set method to describe epitaxial
  growth.
• Only the relevant microscopic fluctuations are
  included.
• Fast events can be included without changing the
  timestep of the simulations.
• This framework is ideally suited to include
  anisotropic, spatially varying diffusion.
• A properly modified potential energy surface can
  be exploited to obtain a high regularity in the
  arrangement of islands.
• We have combined this model with a strain model,
  to modify the microscopic parameters of the model
  according to the local value of the strain.

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Essentially-Non-Oscillatory (ENO) Schemes
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Solution of Diffusion Equation