A Level-Set Method for Modeling Epitaxial Growth and Self-Organization 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
Santa Barbara, Jan. 31, 2005
Collaborators

• Russel Caflisch
• Xiabin Niu
• Max Petersen
• Raffaello Vardavas

NSF and DARPA
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What is Epitaxial Growth?
epi taxis on arrangement
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Why do we care about Modeling Epitaxial Growth?

• Many devices for opto-electronic application are
  multilayer structures grown by epitaxial growth.
• Interface morphology is critical for performance
• Theoretical understanding of epitaxial growth
  will help improve performance, and produce new
  structures.

• Methods used for modeling epitaxial growth
• KMC simulations Completely stochastic method
• Continuum Models PDE for film height, but only
  valid for thick layers
• New Approach Island dynamics model using level
  sets

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KMC Simulation of a Cubic, Solid-on-Solid Model
D G0 exp(-ES/kT)
F
Ddet D exp(-EN/kT)
Ddet,2 D exp(-2EN/kT)
ES Surface bond energy EN Nearest neighbor bond
energy G0 Prefactor O(1013s-1)

• Parameters that can be calculated from first
  principles (e.g., DFT)
• Completely stochastic approach
• But small computational timestep is required

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KMC Simulations Effect of Nearest Neighbor Bond
EN
Large EN Irreversible Growth
Small EN Compact Islands
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KMC Simulation for Equilibrium Structures of
III/V Semiconductors
Experiment (Barvosa-Carter, Zinck)
KMC Simulation (Grosse, Gyure)
Similar work by Kratzer and Scheffler Itoh and
Vvedensky
380C 0.083 Ml/s 60 min anneal
440C 0.083 Ml/s 20 min anneal
Problem Detailed KMC simulations are extremely
slow !
F. Grosse et al., Phys. Rev. B66, 075320 (2002)
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Outline

• Introduction
• The basic island dynamics model using the level
  set method
• Include Reversibility Ostwald Ripening
• Include spatially varying, anisotropic diffusion
• self-organization of islands

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The Island Dynamics Model for Epitaxial Growth
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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 Basic Level Set Formalism for Irreversible
Aggregation

• Governing Equation

• Boundary condition

C. Ratsch et al., Phys. Rev. B 65, 195403 (2002)
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Typical Snapshots of Behavior of the Model
t0.1
j
r
t0.5
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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)

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Essentially-Non-Oscillatory (ENO) Schemes
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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)

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Solution of Diffusion Equation
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Fluctuations need to be included in nucleation of
islands
Nucleation Rate
rmax
r
C. Ratsch et al., Phys. Rev. B 61, R10598 (2000)
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A Typical Level Set Simulation
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Outline

• Introduction
• The basic island dynamics model using the level
  set method
• Include Reversibility Ostwald Ripening
• Include spatially varying, anisotropic diffusion
• self-organization of islands

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Extension to Reversibility

• So far, all results were for irreversible
  aggregation but at higher temperatures, atoms
  can also detach from the island boundary
• Dilemma in Atomistic Models Frequent detachment
  and subsequent re-attachment of atoms from
  islands Significant computational cost !
• In Levelset formalism Simply modify velocity
  (via a modified boundary condition), but keep
  timestep fixed
• Stochastic break-up for small islands is important

• Boundary condition

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Details of stochastic break-up

• For islands larger than a critical size,
  detachment is accounted for via the (non-zero)
  boundary condition
• For islands smaller than this critical size,
  detachment is done stochastically, and we use an
  irreversible boundary condition (to avoid
  over-counting)

• calculate probability to shrink by 1, 2, 3, ..
  atoms this probability is related to detachment
  rate.
• shrink the island by this many atoms
• atoms are distributed in a zone that corresponds
  to diffusion area

• Note our critical size is not what is typical
  called critical island size. It is a numerical
  parameter, that has to be chosen and tested. If
  chosen properly, results are independent of it.

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Sharpening of Island Size Distribution with
Increasing Detachment Rate
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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Ostwald Ripening
M. Petersen, A. Zangwill, and C. Ratsch, Surf.
Science 536, 55 (2003).
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Outline

• Introduction
• The basic island dynamics model using the level
  set method
• Include Reversibility Ostwald Ripening
• Include spatially varying, anisotropic diffusion
• self-organization of islands

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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 for
non-constant Diffusion
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What we have done so far
Assume a simple form of the variation of the
potential energy surface (i.e., sinusoidal) For
simplicity, we look at extreme cases only
variation of adsorption energy, or only variation
of transition energy (real case typically
in-between)
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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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Anisotropic Diffusion with Sinusoidal Variation
in x-Direction
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Isotropic Diffusion with Sinusoidal Variation in
x- and y-Direction
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(No Transcript)
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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)
But In all cases, diffusion constant D has the
same form
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What is next with spatially varying diffusion?

• So far, we have assumed that the potential energy
  surface is modified externally (I.e., buried
  defects), and is independent of growing film
• Next, we want to couple this model with an
  elastic model (Caflisch et al., in progress)
• Solve elastic equations after every timestep
• Modify potential energy surface (I.e., diffusion,
  detachment) accordingly
• This can be done at every timestep, because the
  timestep is significantly larger than in an
  atomistic simulation

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Conclusions

• We have developed a numerically stable and
  accurate level set method to describe epitaxial
  growth.
• The model is very efficient when processes with
  vastly different rates need to be considered
• This framework is ideally suited to include
  anisotropic, spatially varying diffusion (that
  might be a result of strain)
• 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/cratsch