Modeling of Epitaxial Growth using LevelSets

1
Modeling of Epitaxial Growth using Level-Sets
Christian Ratsch, UCLA, Department of Mathematics
Collaborators
_at_ UCLA Russel Caflisch Max Petersen Myungjoo
Kang Susan Chen Chris Andersen Stan
Osher Jennifer Garcia Raffaello Vardavas
_at_ HRL Laboratories Mark Gyure
_at_ Georgia Tech Andy Zangwill Max Petersen
_at_ Imperial College Dimitri Vvedensky
_at_ IPAM - RIPS Lisa Feigenbaum Dan Shaevitz Chris
Tiee Mike Sheffler
NSF and DARPA
2
Physical Processes During Epitaxial Growth
epi taxis on arrangement
3
Hierarchy of Theoretical Approaches
Time (s)
Continuum Methods
103
Level Set
Device growth
1
Kinetic Monte Carlo
Formation of islands
10-3
Classical MD accelerated
10-6
Atomic motion
Classical MD
10-9
Ab-Initio MD
Atomic vibrations
10-12
DFT
size
of atoms (lateral)
1
109
103
106
1nm
length
1mm
1mm
1m
circuit
islands
device
waver
4
Outline

• Modeling of Epitaxial Growth using Kinetic Monte
  Carlo Simulations
• The Island Dynamics Model using the Level Set
  method
• Include Reversibility (Ostwald Ripening, Strain
  dependent detachment)
• Include Edge Diffusion
• Include Surface Chemistry such as Surface
  Reconstructions
• A model with coupled level set functions (IPAM -
  RIPS 2002)

• Atomistic size effect at island boundary

5
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

6
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 at
Different Temperatures
Experiment (Barvosa-Carter, Zinck)
KMC Simulation (Grosse, Gyure)
380C, 0.083 Ml/s, 60 min anneal
440C, 0.083 Ml/s, 20 min anneal
Problem Detailed KMC simulations are extremely
slow !
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Modeling Thin Film Growth

• Methods used
• KMC simulations Completely stochastic method
• Rate Equation Coupled ODEs. completely
  deterministic method

• New Method
• Level-Set Method PDE - based, (almost)
  deterministic

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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 Level Set Method The Basic Formalism

• Velocity
• Nucleation Rate

• 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)

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Essentially-Non-Oscillatory (ENO) Schemes
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Solution of Diffusion Equation
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Typical Snapshots of Behavior of the Model
t0.1
j
r
t0.5
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Validation Comparison to an Atomistic KMC
Simulation
Scaling of island densities Nucleation Theory
predicts N (D/F)-1/3
Scaling of island size distribution
Q Coverage sav Average island size
Slope -1/3
Ratsch, Gyure, Chen, Kang, Vvedensky, PRB 61,
R10598 (2000)
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A Typical Level Set Simulation
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Outline

• Modeling of Epitaxial Growth using Kinetic Monte
  Carlo Simulations
• The Island Dynamics Model using the Level Set
  method
• Include Reversibility (Ostwald Ripening, Strain
  dependent detachment)
• Include Edge Diffusion
• Include Surface Chemistry such as Surface
  Reconstructions
• A model with coupled level set functions (IPAM -
  RIPS 2002)

• Atomistic size effect at island boundary

19
Extension to Reversibility

• Remark Stochastic element in the break-up of
  islands is needed !!
• Microscopic parameters can be calculated from
  first principles (DFT)
• No frequent detachment/re-attachment needed !

• Alternative Approach Change Boundary Condition
  Work currently in progress

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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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Ostwald Ripening
Petersen, Zangwill, Ratsch, Surface Science, in
press
23
Island Size Dependent Detachment Rate (Strain)

• Simple approach make the detachment rate
  (i.e.shrink velocity) size dependent
• More sophisticated Solve elastic equations and
  couple with level set code (Current work of
  Caflisch, Connell, Luo)
• Possible because time step is large (elastic
  problem is expensive)
• Couple to strain dependent rates as obtained
  from DFT !

Application Formation and self-organization of
Quantum Dots !
24
Outline

• Modeling of Epitaxial Growth using Kinetic Monte
  Carlo Simulations
• The Island Dynamics Model using the Level Set
  method
• Include Reversibility (Ostwald Ripening, Strain
  dependent detachment)
• Include Edge Diffusion
• Include Surface Chemistry such as Surface
  Reconstructions
• A model with coupled level set functions (IPAM -
  RIPS 2002)

• Atomistic size effect at island boundary

25
Edge Diffusion

• Island boundaries in the level set approach are
  smooth (because adatoms (are treated as a
  continuum quantity).
• Fast edge diffusion is included in KMC
  simulations
• Edge diffusion makes islands compact
• But Edge diffusion has strong effect on
  roughness evolution
• Include edge diffusion with a curvature-dependent
  velocity

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Effect of Edge Diffusion on Surface Roughness
Fast edge diffusion
No edge diffusion
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Adatom Concentration
Fast edge diffusion Compact Islands
Slow edge diffusion Fractal Islands
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Roughness Evolution
C. Ratsch et al, Phys. Rev. B 65, 195403 (2002)
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Outline

• Modeling of Epitaxial Growth using Kinetic Monte
  Carlo Simulations
• The Island Dynamics Model using the Level Set
  method
• Include Reversibility (Ostwald Ripening, Strain
  dependent detachment)
• Include Edge Diffusion
• Include Surface Chemistry such as Surface
  Reconstructions
• A model with coupled level set functions (IPAM -
  RIPS 2002)

• Atomistic size effect at island boundary

30
Typical Surfaces Reconstruct Example InAs(001)
Recall KMC Simulation (Grosse, Gyure, part of VIP)
440C, 0.083 Ml/s
380C, 0.083 Ml/s
Dynamics (i.e., adatom mobility) on different
surfaces is different.
Problem Detailed KMC simulations are extremely
slow !
31
Extension of Level Set Model to III/V
Semiconductor Growth

• 2 coupled level set functions
• j describes the surface morphology (island
  boundaries)
• y describes boundary between reconstruction
  domains

• Velocity of island boundary depends on the
  underlying reconstruction
• Velocity of reconstruction boundary depends on
  chemistry (dimer desorption)
• Solve diffusion equation for each species

• Boundary conditions reflect orientation and
  local structure at step edge

32
A Toy Model for Coupled Level Sets (IPAM-RIPS
2002, Lisa Feigenbaum, Dan Shaevitz, Mike
Sheffler, Chris Tiee)

• Integer crossings of f represent island
  boundaries
• zero crossings of y represent a/b boundaries
• We dont solve diffusion equations instead

Sample Event a island spreads into b region ?
Level set function y
Level set function ?
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Fixed Boundary between Reconstruction Domains 1,
va vb
b region
a region
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Shape Evolution at Boundary between
Reconstruction Domains
Level set function f
Level set function y
slow velocity
fast velocity
Island seeded just below boundary
Island seeded just above boundary
Ratsch et al., Applied Math Letters, in press.
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Fixed Boundary between Reconstruction Domains 2
va vb
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Moving Boundary between Reconstruction Domains
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Outline

• Modeling of Epitaxial Growth using Kinetic Monte
  Carlo Simulations
• The Island Dynamics Model using the Level Set
  method
• Include Reversibility (Ostwald Ripening, Strain
  dependent detachment)
• Include Edge Diffusion
• Include Surface Chemistry such as Surface
  Reconstructions
• A model with coupled level set functions (IPAM -
  RIPS 2002)

• Atomistic size effect at island boundary

38
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 level set method to describe epitaxial
growth on macroscopic scale Many detailed
microscopic processes such as detachment, edge
diffusion can be included. The levelset method
provides a natural framework to couple other
external fields such as strain or hydrodynamics
to the growth modeling More details such as
surface chemistry (reconstructions) can be
included in principle, this is numerically very
efficient, but more work is needed. Atomic size
effects at island boundary can be included
More details and transparencies of this talk can
be found at www.math.ucla.edu/material