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 (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
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 = 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 = 0.3387, σ s = 2.7153 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
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 = 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 0.1 µ = 0.04 0.15 µ = 0.05 µ = 0.06 U U appx 0.08 0.06 0.04 0.02 0 0.1 0.05 0 0.25 0.2 0.15 0.1 0.05 0 0.02 0 0.01 0.02 0.03 0.04 0.4 µ = 0.07 0.05 0 0.02 0.04 0.06 µ = 0.08 0.05 0 0.05 0.1 loc 4 U loc U ideal U U appx 0.3 0.2 0.1 0 0.5 0.4 0.3 0.2 0.1 0 slope 3.5 3 2.5 2 0.1 0 0.05 0.1 loc U loc U ideal 0.1 0 0.05 0.1 0.15 0.2 loc U loc U ideal 1.5 10 5 0 µ 10 2 Figure 2: Best Fit Lines for several misfits 11
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 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
Strain Energy per atom (ev) 10 0.03514 10 0.03513 10 0.03512 10 0.03511 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 h 2 s 50, C > 0 Tensile Strain 0 50 400 300 200 100 0 100 200 300 400 150 Compressive Strain 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
100 Tensile Strain 50 0 400 300 200 100 0 100 200 300 400 150 Compressive Strain 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
120 100 Ge 80 60 Si 40 50 0 50 Figure 6: µ = 0.02, deposition flux (F) = 1ML/s 19
150 5.6 100 5.55 5.5 50 5.45 0 100 50 0 50 100 Figure 7: µ = 0.02, r ij to nearest neighbors 5.4 20
3 % misfit 3.5 % misfit 4 % misfit Figure 8: FM, SK and VW growth, F = 0.1ML/s 21
4 % misfit 4.5 % misfit 5 % misfit Figure 9: SK and VW growth, F = 0.1ML/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 = 0.1ML/s 24
200 150 100 50 0 300 200 100 0 100 200 300 2.5 2 1.5 1 Figure 12: Volmer Weber growth showing energy of each atom. µ = 0.1, F = 1.0ML/s 25
150 2.5 100 2 50 1.5 1 0 150 100 50 0 50 100 150 Figure 13: VM growth showing energy of each atom. µ = 0.1, F = 0.25ML/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 150 140 130 80 100 120 120 60 110 1000 20 40 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