QUASI-EQUILIBRIUM MONTE-CARLO: OFF-LATTICE KINETIC MONTE CARLO - - PowerPoint PPT Presentation

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

Cells Unit

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 elastic

effects are negligible

• surface forces

dominate

• minimize

surface area Stranski-Krastanov (SK)

• expected when elastic

effects are significant.

• commonly observed

in experiments

• results from

an interplay between elastic and surface forces Volmer-Weber (VM)

• expected when elastic

effects are

• verwhelming
• not commonly
• bserved in

experiments

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Kinetic Monte Carlo - Basic Idea

Current State Transistion State Final State Energy 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 neighbor

bonds with strengths: γSS, γSG, γGG

• Nearest and next to nearest neighbor

springs with contants: kL and kD

• System evolves by letting the surface

atoms 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

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• ω is a prefactor, kBT is the thermal energy
• rij is the distance between atoms i and j
• ǫij = √ǫiǫj and σij = σi+σj

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• 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

0.01 0.02 0.03 0.04 −0.02 0.02 0.04 0.06 0.08 0.1

∆ U − ∆ Uappx µ = −0.04

0.02 0.04 0.06 −0.05 0.05 0.1 0.15

µ = −0.05

0.05 0.1 −0.05 0.05 0.1 0.15 0.2 0.25

∆ Uloc − ∆ Uideal

loc

µ = −0.06

0.05 0.1 −0.1 0.1 0.2 0.3 0.4

∆ Uloc − ∆ Uideal

loc

∆ U − ∆ Uappx µ = −0.07

0.05 0.1 0.15 0.2 −0.1 0.1 0.2 0.3 0.4 0.5

∆ Uloc − ∆ Uideal

loc

µ = −0.08

−10 −5 1.5 2 2.5 3 3.5 4

µ ⋅ 102 slope

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 10

0.03511

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0.03512

10

0.03513

10

0.03514

Substrate Depth (Monolayers) Strain Energy 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 = C µhf

h2

s , C > 0

−400 −300 −200 −100 100 200 300 400 −50 50 Tensile Strain −400 −300 −200 −100 100 200 300 400 50 100 150 Compressive Strain

Figure 4: Curvature (κ) –9 ML of substrate and 3ML of film. µ = ±0.04

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−400 −300 −200 −100 100 200 300 400 50 100 Tensile Strain −400 −300 −200 −100 100 200 300 400 50 100 150 Compressive 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 50 40 60 80 100 120

Figure 6: µ = −0.02, deposition flux (F) = 1ML/s

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−100 −50 50 100 50 100 150 5.4 5.45 5.5 5.55 5.6 Figure 7: µ = −0.02, ¯ rij 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 11

Figure 10: , ¯ rij to nearest neighbors, F = 0.1ML/s

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4 % misfit 4.5 % misfit 5 % misfit

9 10 11 9 10 11 9 10 11

Figure 11: , ¯ rij to nearest neighbors, F = 0.1ML/s

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−300 −200 −100 100 200 300 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 50 100 150 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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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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