Multiscale Modeling of Epitaxial Growth:

1
Multiscale Modeling of Epitaxial Growth from Ab
Initio to Level-Sets
Christian Ratsch, UCLA, Department of Mathematics
Collaborators

• At UCLA
• Russel Caflisch
• Max Petersen
• Jennifer Garcia
• Frank Grosse

• At HRL Laboratories
• Mark Gyure
• Jennifer Zinck
• Bill Barvosa Carter

NSF and DARPA
2
Physical Processes During Thin Film Growth

• Islands nucleate
• Questions
• Where ?
• When ?
• Boundaries Move
• Key Quantity
• Velocity

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

• Stable surface reconstructions for InAs(001) via
  density-functional theory
• Microscopic diffusion parameters via
  density-functional theory
• Diffusion barrier for Ag/Ag(111), Ag/Pt(111)
• Growth Modeling Atomistic or continuum-type
  modeling ?
• The Level Set method to model thin film growth

5
Surface Reconstruction for InAs(001)
Low As pressure
High As pressure

• Different (2x4) reconstructions are observed
• Which ones ?

Barvosa-Carter, Ross, Ratsch, Grosse, Owen,
Zinck, Surf. Sci. 499, L129 (2002)
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Possible Structures
a(2x4)
a2(2x4)
a3(2x4)
b2(2x4)
b3(2x4)
b(2x4)
We also considered the corresponding (4x2)
structures (which are rotated by 90o, and In and
As atoms are interchanged)
7
Density-Functional Theory (DFT) Computational
Details

• Norm-conserving pseudopotentials
• Plane-wave basis set
• Ecut 12 Ryd for InAs
• Ecut 40-50 Ryd for Ag, Pt
• k-summation with special k-points according to
  Monckhorst and Pack
• Supercell with surface on one side,
  pseudo-hydrogen on the other side
• Damped Newton dynamics to optimize atomic
  structure
• All results shown here are with the
  local-density approximation (LDA), but main
  results have been checked with gradient
  corrections (GGA)
• Computer code used fhi98md

8
Phase Diagram for InAs(001) calculated with DFT
C. Ratsch et al., Phys. Rev. B 62, R7719 (2000).
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Density-Functional Theory to Calculate Surface
Diffusion Parameters
Once we know the surface reconstruction, we need
to study how atoms move on the surface
Transition state theory (Vineyard, 1957)
Model System Ag/Ag(111), Ag/Pt(111)
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Layer Dependent Island Densities
Ag/Pt(111) (Brune et al, Phys. Rev. B 52, 14380
(1995))
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Results and Comparison for Diffusion Parameters
Diffusion Barrier
Prefactor
Discrepancy for prefactor has been explained with
DFT calculations, because long range effects
become important for systems with small barrier
and standard nucleation theory does not apply any
longer (Fichthorn and Scheffler, PRL, 2000
Bogicevic et al., PRL, 2000)
Ratsch et al., Phys. Rev. B 55, 6750 (1997)
Phys. Rev. B 58, 13163 (1998).
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Modeling of Thin Film Growth

• What type of model atomistic or continuum ?
• Do we need to keep track of every atom ?
• How much detail do we need ?

• Atomistic KMC simulations Completely stochastic
  method
• Rate Equation Coupled ODEs. completely
  deterministic method
• Level-Set Method PDE - based, (almost)
  deterministic

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KMC Simulation for Equilibrium Structures at
Different Temperatures
Experiment (Barvosa-Carter, Zinck)
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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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

15
The Level Set Method Formalism

• PDE - based method, almost deterministic !

16
Scaling of the Island Size Distribution
Compare results to a simple cubic KMC simulation,
that includes the same physical processes
(diffusion, irreversible aggregation, fast edge
diffusion)
(Stroscio et al PRB, 1994)
C. Ratsch et al., Phys. Rev. B 61, R10598 (2000).
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A Typical Level Set Simulation
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Extension of Level Set method to III/V
Semiconductor Growth

• Determine rate limiting process for growth and
  calculate rate (via DFT)
• Velocity of island boundary is determined by
  this process
• Solve diffusion equation for each species

• Boundary conditions reflect orientation and
  local structure at step edge

• Level Set function j describes the surface
  morphology (island boundaries)

• Level Set function y describes boundary between
  reconstruction domains
• Level Set functions interact many modeling and
  numerical challenges !

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Conclusions

• DFT calculations predict equilibrium phase
  diagram for surface reconstructions
• DFT can be used to calculate microscopic
  parameters for diffusion
• Detailed high resolution KMC model keeps track
  of every atom (dimer) on the surface
  (reconstructions), but is too slow to describe
  nanoscale systems
• Level Set method
• - keeps track of island boundaries (for
  islands in each layer)
• - reproduces island density and island size
  distribution for simple, cubic model.
• - more work is needed to include microscopic
  details (surface reconstructions).

• Atomistic information from microscopic models
  (such as DFT) needs to be combined with level set
  method !!

Transparencies of this talk can be found at
www.math.ucla.edu/material
20
Surface Energy

• Surface energy g is the energy cost to introduce
  a surface
• For two-species system, the stoichiometry is
  important

g Esurf - mAsNAs - mInNIn
with mi lt mi(bulk)
Equilibrium condition mIn mAs mInAs
mInAs(bulk) -mIn(bulk) lt mAs lt mAs(bulk)