QUASI-EQUILIBRIUM MONTE-CARLO: OFF-LATTICE KINETIC MONTE CARLO SIMULATION OF HETEROEPITAXY WITHOUT SADDLE POINTS

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1 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

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

3 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 overwhelming -not commonly observed in experiments 3

4 Kinetic Monte Carlo - Basic Idea Current State Transistion State Final State Energy Ε energy Reaction Coordinate KMC is based on transition state theory 4

5 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

6 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

7 The Model Hopping Rate R k = ω exp [(U U k )/k B T] U = total energy, U = N i>j U p =total energy without atom p φ(r ij ) where φ(r ij ) = 4ǫ ij [ ( σij r ij ) 12 ( σij r ij ) 6 ] 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 = , σ s = µ = σ s σ f σ s : misfit, σ f = (1 µ)σ s Periodic in the x-direction Semi-infinite in the y-direction 7

8 KMC Computational Bottlenecks In principle we need to compute R k = ωe ( 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 = Thus full on computations of heteroepitaxy are very time consuming and memory intensive. 8

9 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

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

11 Rates Approximated by a local distortion 0.1 µ = µ = 0.05 µ = 0.06 U U appx µ = µ = loc 4 U loc U ideal U U appx slope loc U loc U ideal loc U loc U ideal µ 10 2 Figure 2: Best Fit Lines for several misfits 11

12 The Algorithm 0. Detect the surface atoms and estimate the rates. j 1. Compute partial sums p j = R k and R tot = R d + 2. Generate r (0, R tot ) k=1 N R k. k=1 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 (4σ 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 Return to Step 1. 12

13 Previous Work Using The Lennard-Jones Potential F. Much, M. Ahr, M. Biehl, and W. Kinzel, Europhys. Lett, 56 (2001) F. Much, M. Ahr, M. Biehl, and W. Kinzel, Comput. Phys. Commun, 147 (2002) M. Biehl, M. Ahr, W. Kinzel, and F. Much Thin Solid Films, 428 (2003) F. Much, and M. Biehl Europhys. Lett, 63 (2003) They compute saddle points but we do not. They use a substrate depth of 6 monolayers which is not ideal 13

14 Strain Energy per atom (ev) Substrate Depth (Monolayers) Figure 3: Strain Energy per atom vs Substrate Depth. µ =

15 Curvature 15

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

17 100 Tensile Strain Compressive Strain Figure 5: Curvature (κ) 15 ML of substrate and 3ML of film. µ = ±

18 Growth Modes 18

19 Ge Si Figure 6: µ = 0.02, deposition flux (F) = 1ML/s 19

20 Figure 7: µ = 0.02, r ij to nearest neighbors

21 3 % misfit 3.5 % misfit 4 % misfit Figure 8: FM, SK and VW growth, F = 0.1ML/s 21

22 4 % misfit 4.5 % misfit 5 % misfit Figure 9: SK and VW growth, F = 0.1ML/s 22

23 3 % misfit % misfit % misfit Figure 10:, r ij to nearest neighbors, F = 0.1ML/s 23

24 4 % misfit % misfit % misfit Figure 11:, r ij to nearest neighbors, F = 0.1ML/s 24

25 Figure 12: Volmer Weber growth showing energy of each atom. µ = 0.1, F = 1.0ML/s 25

26 Figure 13: VM growth showing energy of each atom. µ = 0.1, F = 0.25ML/s 26

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

28 Edge dislocations Figure 15: Edge dislocations 28

29 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

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