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 1

ELASTIC ENERGY Unit Cells Due to misfit the bottom configuration has less elastic energy than the top one. 2

GROWTH MODES Wetting Layer Layer-by-Layer Stranski-Krastanov Volmer-Weber (FM) (SK) (VM) -observed when elastic -expected when elastic -expected when elastic effects are negligible effects are significant. effects are overwhelming -surface forces -commonly observed dominate in experiments -not commonly observed in -minimize -results from experiments surface area an interplay between elastic and surface forces 3

Kinetic Monte Carlo - Basic Idea Current State Transistion State Final State Energy ∆Ε energy Reaction Coordinate 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/k B T ) • ∆ E = E (current state) − E (transition state) • ω is the attempt frequency, k B T 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 neighbor bonds with strengths: γ SS , γ SG , γ GG • Nearest and next to nearest neighbor springs with contants: k L and k D • 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) 6

The Model • Hopping Rate R k = ω exp [( U − U k ) /k B T ] • U = total energy, U p =total energy without atom p N �� � 6 � � 12 � σ ij σ ij � • U = φ ( r ij ) where φ ( r ij ) = 4 ǫ ij − r ij r ij i>j • ω is a prefactor, k B T is the thermal energy • r ij is the distance between atoms i and j • ǫ ij = √ ǫ i ǫ j and σ ij = σ i + σ j 2 • ǫ s = 0 . 4, ǫ f = 0 . 3387, σ s = 2 . 7153 • µ = σ s − σ f : misfit , σ f = (1 − µ ) σ s σ s • Periodic in the x -direction • Semi-infinite in the y -direction 7

KMC Computational Bottlenecks • In principle we need to compute R k = ω e (∆ U k /k B T ) 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 × ( N Si + N Ge ), N Si = 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 R p using local distortion around atom p . 97% accuracy. 9

Rates Approximated by a local distortion Figure 1: Configuration for generating best fit line 10

Rates Approximated by a local distortion µ = −0.04 µ = −0.05 µ = −0.06 0.1 0.15 0.25 0.08 ∆ U − ∆ U appx 0.1 0.2 0.06 0.15 0.04 0.05 0.1 0.02 0.05 0 0 0 −0.02 −0.05 −0.05 0 0.01 0.02 0.03 0.04 0 0.02 0.04 0.06 0 0.05 0.1 ∆ U loc − ∆ U ideal loc µ = −0.07 µ = −0.08 0.4 4 0.5 ∆ U − ∆ U appx 0.3 3.5 0.4 slope 0.2 3 0.3 0.2 0.1 2.5 0.1 0 2 0 −0.1 −0.1 1.5 0 0.05 0.1 0 0.05 0.1 0.15 0.2 −10 −5 0 ∆ U loc − ∆ U ideal loc ∆ U loc − ∆ U ideal loc µ ⋅ 10 2 Figure 2: Best Fit Lines for several misfits 11

The Algorithm 0. Detect the surface atoms and estimate the rates . j N � � 1. Compute partial sums p j = R k and R tot = R d + R k . k =1 k =1 2. Generate r ∈ (0 , R tot) 3. Hop first surface atom j for which p j > r . If r > p N , deposit. 4. Perform steepest descent on adatom or deposited atom. 5. Relax atom and neighbors in local region (4 σ s ). – 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

0.03514 10 Strain Energy per atom (eV) 0.03513 10 0.03512 10 0.03511 10 5 10 15 20 25 30 35 40 Substrate Depth (Monolayers) Figure 3: Strain Energy per atom vs Substrate Depth. µ = − 0 . 04 . 14

Curvature 15

κ = C µh f s , C > 0 h 2 Tensile Strain 50 0 −50 −400 −300 −200 −100 0 100 200 300 400 Compressive Strain 150 100 50 0 −400 −300 −200 −100 0 100 200 300 400 Figure 4: Curvature ( κ ) –9 ML of substrate and 3ML of film. µ = ± 0 . 04 16

Tensile Strain 100 50 0 −400 −300 −200 −100 0 100 200 300 400 Compressive Strain 150 100 50 0 −400 −300 −200 −100 0 100 200 300 400 Figure 5: Curvature ( κ ) –15 ML of substrate and 3ML of film. µ = ± 0 . 04 17

Growth Modes 18

120 Ge 100 80 60 Si 40 −50 0 50 Figure 6: µ = − 0 . 02 , deposition flux (F) = 1ML/s 19

150 5.6 5.55 100 5.5 50 5.45 5.4 0 −100 −50 0 50 100 Figure 7: µ = − 0 . 02 , ¯ r ij to nearest neighbors 20

3 % misfit 3.5 % misfit 4 % misfit Figure 8: FM, SK and VW growth, F = 0 . 1 ML/s 21

4 % misfit 4.5 % misfit 5 % misfit Figure 9: SK and VW growth, F = 0 . 1 ML/s 22

3 % misfit 9.6 9.4 9.2 3.5 % misfit 10.5 10 9.5 9 4 % misfit 11 10.5 10 9.5 9 Figure 10: , ¯ r ij to nearest neighbors, F = 0.1ML/s 23

4 % misfit 11 10 9 4.5 % misfit 11 10 9 5 % misfit 11 10 9 Figure 11: , ¯ r ij to nearest neighbors, F = 0 . 1 ML/s 24

2.5 200 2 150 100 1.5 50 1 0 −300 −200 −100 0 100 200 300 Figure 12: Volmer Weber growth showing energy of each atom. µ = − 0 . 1 , F = 1 . 0 ML/s 25

150 2.5 100 2 1.5 50 1 0 −150 −100 −50 0 50 100 150 Figure 13: VM growth showing energy of each atom. µ = − 0 . 1 , F = 0 . 25 ML/s 26

Figure 14: Edge dislocations in nature. By Peter J. Goodhew, Dept. of Engineering, University of Liverpool released under CC BY 2.0 license 27

Edge dislocations 150 120 140 100 130 80 60 120 40 110 20 100 0 Figure 15: Edge dislocations 28

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 29

Future Work Extend the new model to three dimensions 3D KMC - [Schulze and Smereka] 30