Post on 28-Mar-2018
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QUASI-EQUILIBRIUM MONTE-CARLO: OFF-LATTICE
KINETIC MONTE CARLO SIMULATION OF HETEROEPITAXY
WITHOUT SADDLE POINTS
Henry A. Boateng
University of Michigan, Ann Arbor
Joint work with
Tim Schulze and Peter Smereka
Support from NSF FRG grant 0854870
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ELASTIC ENERGY
CellsUnit
Due to misfit the bottom configuration has less elastic energy than
the top one.
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GROWTH MODES
Wetting Layer
Layer-by-Layer(FM)
-observed when elasticeffects are negligible
-surface forcesdominate
-minimizesurface area
Stranski-Krastanov(SK)
-expected when elasticeffects are significant.
-commonly observedin experiments
-results froman interplaybetween elastic andsurface forces
Volmer-Weber(VM)
-expected when elasticeffects areoverwhelming
-not commonlyobserved inexperiments
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Kinetic Monte Carlo - Basic Idea
Current State Transistion State Final StateE
nerg
y
Reaction Coordinate
∆Ε
energy
KMC is based on transition state theory
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Kinetic Monte Carlo - Basic Idea
• Rates are based on transition state theory which gives
R = ω exp(−∆E/kBT )
• ∆E = E(current state) − E(transition state)
• ω is the attempt frequency, kBT is the thermal energy
• One needs to know or assume what are the important events
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Ball and Spring Model [Baskaran, Devita, Smereka, (2010)]
• Atoms are on a square lattice
• Semi-infinite in the y-direction
• Periodic in the x-direction
• Nearest and next to nearest neighborbonds with strengths: γSS , γSG, γGG
• Nearest and next to nearest neighborsprings with contants: kL and kD
• System evolves by letting the surfaceatoms hop: Surface Diffusion
• Atoms hop to discrete sites, hence does not capture dislocations.∗ without intermixing this model is due to:
Orr, Kessler, Snyder, and Sander (1992)Lam, Lee and Sander (2002)
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The Model
• Hopping Rate Rk = ω exp [(U − Uk)/kBT ]
• U = total energy, Up=total energy without atom p
• U =
N∑
i>j
φ(rij) where φ(rij) = 4ǫij
[
(
σij
rij
)12
−(
σij
rij
)6]
• ω is a prefactor, kBT is the thermal energy
• rij is the distance between atoms i and j
• ǫij =√
ǫiǫj and σij =σi+σj
2
• ǫs = 0.4, ǫf = 0.3387, σs = 2.7153
• µ =σs−σf
σs: misfit, σf = (1 − µ)σs
• Periodic in the x-direction
• Semi-infinite in the y-direction
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KMC Computational Bottlenecks
• In principle we need to compute Rk = ωe(∆Uk/kBT ) for all
atoms.
• This means removing each surface atom and relaxing the full
system with nonlinear conjugate gradient (NlCG)
• Relax the whole system after each hop or deposition
• NlCG involves a hessian matrix with dimension D × D where
D = 2 × (NSi + NGe), NSi = 256 × 40
• Thus full on computations of heteroepitaxy are very time
consuming and memory intensive.
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Mitigating Bottlenecks
• We perform local relaxation (local NlCG) in a small region
around hopped/deposited atom
• Local NlCG uses a cell-list
• Global relaxations periodically, also triggered by a flag
• Approximate Rp using local distortion around atom p. 97%
accuracy.
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Rates Approximated by a local distortion
Figure 1: Configuration for generating best fit line
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Rates Approximated by a local distortion
0 0.01 0.02 0.03 0.04−0.02
0
0.02
0.04
0.06
0.08
0.1
∆ U
− ∆
Uap
pxµ = −0.04
0 0.02 0.04 0.06−0.05
0
0.05
0.1
0.15µ = −0.05
0 0.05 0.1−0.05
0
0.05
0.1
0.15
0.2
0.25
∆ Uloc − ∆ Uidealloc
µ = −0.06
0 0.05 0.1−0.1
0
0.1
0.2
0.3
0.4
∆ Uloc − ∆ Uidealloc
∆ U
− ∆
Uap
px
µ = −0.07
0 0.05 0.1 0.15 0.2−0.1
0
0.1
0.2
0.3
0.4
0.5
∆ Uloc − ∆ Uidealloc
µ = −0.08
−10 −5 01.5
2
2.5
3
3.5
4
µ ⋅ 102
slop
e
Figure 2: Best Fit Lines for several misfits
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The Algorithm
0. Detect the surface atoms and estimate the rates .
1. Compute partial sums pj =
j∑
k=1
Rk and Rtot = Rd +N
∑
k=1
Rk.
2. Generate r ∈ (0, Rtot)
3. Hop first surface atom j for which pj > r. If r > pN , deposit.
4. Perform steepest descent on adatom or deposited atom.
5. Relax atom and neighbors in local region (4σs).
– update the rates of the atoms that were relaxed
– If maximum norm of the force on a boundary > tolerance,
perform global relaxation and Step 0.
6. Return to Step 1.
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Previous Work Using The Lennard-Jones Potential
• F. Much, M. Ahr, M. Biehl, and W. Kinzel, Europhys. Lett, 56
(2001) 791-796
• F. Much, M. Ahr, M. Biehl, and W. Kinzel, Comput. Phys.
Commun, 147 (2002) 226-229
• M. Biehl, M. Ahr, W. Kinzel, and F. Much Thin Solid Films, 428
(2003) 52-55
• F. Much, and M. Biehl Europhys. Lett, 63 (2003) 14-20
• They compute saddle points but we do not.
• They use a substrate depth of 6 monolayers which is not ideal
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5 10 15 20 25 30 35 40
100.03511
100.03512
100.03513
100.03514
Substrate Depth (Monolayers)
Str
ain
Ene
rgy
per
atom
(eV
)
Figure 3: Strain Energy per atom vs Substrate Depth. µ = −0.04.
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Curvature
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κ = Cµhf
h2s
, C > 0
−400 −300 −200 −100 0 100 200 300 400
−50
0
50
Tensile Strain
−400 −300 −200 −100 0 100 200 300 4000
50
100
150Compressive Strain
Figure 4: Curvature (κ) –9 ML of substrate and 3ML of film.
µ = ±0.04
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−400 −300 −200 −100 0 100 200 300 400
0
50
100Tensile Strain
−400 −300 −200 −100 0 100 200 300 4000
50
100
150Compressive Strain
Figure 5: Curvature (κ)–15 ML of substrate and 3ML of film. µ =
±0.04
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Growth Modes
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Ge
Si
−50 0 50
40
60
80
100
120
Figure 6: µ = −0.02, deposition flux (F) = 1ML/s
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−100 −50 0 50 1000
50
100
150
5.4
5.45
5.5
5.55
5.6
Figure 7: µ = −0.02, r̄ij to nearest neighbors
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3 % misfit
3.5 % misfit
4 % misfit
Figure 8: FM, SK and VW growth, F = 0.1ML/s
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4 % misfit
4.5 % misfit
5 % misfit
Figure 9: SK and VW growth, F = 0.1ML/s
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3 % misfit
3.5 % misfit
4 % misfit
9.2
9.4
9.6
9
9.5
10
10.5
9
9.5
10
10.5
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Figure 10: , r̄ij to nearest neighbors, F = 0.1ML/s
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4 % misfit
4.5 % misfit
5 % misfit
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10
11
9
10
11
9
10
11
Figure 11: , r̄ij to nearest neighbors, F = 0.1ML/s
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−300 −200 −100 0 100 200 3000
50
100
150
200
1
1.5
2
2.5
Figure 12: Volmer Weber growth showing energy of each atom.
µ = −0.1, F = 1.0ML/s
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−150 −100 −50 0 50 100 150
0
50
100
150
1
1.5
2
2.5
Figure 13: VM growth showing energy of each atom.
µ = −0.1, F = 0.25ML/s
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Figure 14: Edge dislocations in nature.
By Peter J. Goodhew, Dept. of Engineering, University of Liverpool
released under CC BY 2.0 license
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0
20
40
60
80
100
120
100
110
120
130
140
150
Edge dislocations
Figure 15: Edge dislocations
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Summary
Our model
• Predicts the right curvature due to Stoney’s formula
• Clearly captures the effect of misfit strength on the growth
modes (FM, SK, VM)
• Captures dislocations and its physical effects
• Easily incorporates intermixing
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Future Work
Extend the new model to three dimensions
3D KMC - [Schulze and Smereka]
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