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

1

ELASTIC ENERGY

CellsUnit

Due to misfit the bottom configuration has less elastic energy than

the top one.

2

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

3

Kinetic Monte Carlo - Basic Idea

Current State Transistion State Final StateE

nerg

y

Reaction Coordinate

∆Ε

energy

KMC is based on transition state theory

4

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

5

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)

6

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

7

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.

8

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.

9

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.

12

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

13

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

15

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

16

−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

17

Growth Modes

18

Ge

Si

−50 0 50

40

60

80

100

120

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

19

−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

11

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

24

−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

25

−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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