Multiscale Modeling of Epitaxial Growth Processes: Level Sets and Atomistic Models

1. Multiscale Modeling of Epitaxial Growth Processes: Level Sets and Atomistic Models Russel Caflisch1,Mark Gyure2, Bo Li4, Stan Osher 1, Christian Ratsch1,2, David Shao1 and Dimitri Vvedensky3 1UCLA, 2HRL Laboratories 3Imperial College, 4U Maryland www.math.ucla.edu/~material IMA, 11/19/04

2. Outline • Epitaxial Growth • molecular beam epitaxy (MBE) • Step edges and islands • Mathematical models for epitaxial growth • atomistic: Solid-on-Solid using kinetic Monte Carlo • continuum: Villain equation • island dynamics: BCF theory • Kinetic model for step edge • edge diffusion and line tension (Gibbs-Thomson) boundary conditions • Numerical simulations • Coarse graining for epitaxial surface • Conclusions IMA, 11/19/04

3. Solid-on-Solid Model • Interacting particle system • Stack of particles above each lattice point • Particles hop to neighboring points • random hopping times • hopping rate D= D0exp(-E/T), • E = energy barrier, depends on nearest neighbors • Deposition of new particles • random position • arrival frequency from deposition rate • Simulation using kinetic Monte Carlo method • Gilmer & Weeks (1979), Smilauer & Vvedensky, … IMA, 11/19/04

4. IMA, 11/19/04

5. Kinetic Monte Carlo • Random hopping from site A→ B • hopping rate D0exp(-E/T), • E = Eb = energy barrier between sites • not δE = energy difference between sites B Eb δE A IMA, 11/19/04

6. SOS Simulation for coverage=.2 Gyure & Ross Gyure and Ross, HRL IMA, 11/19/04

7. SOS Simulation for coverage=10.2 IMA, 11/19/04

8. SOS Simulation for coverage=30.2 IMA, 11/19/04

9. Validation of SOS Model:Comparison of Experiment and KMC Simulation(Vvedensky & Smilauer) Step Edge Density (RHEED) Island size density IMA, 11/19/04

10. Difficulties with SOS/KMC • Difficult to analyze • Computationally slow • adatom hopping rate must be resolved • difficult to include additional physics, e.g. strain • Rates are empirical • idealized geometry of cubic SOS • cf. “high resolution” KMC IMA, 11/19/04

11. High Resolution KMC Simulations • InAs • zinc-blende lattice, dimers • rates from ab initio computations • computationally intensive • many processes • describes dynamical info (cf. STM) • similar work • Vvedensky (Imperial) • Kratzer (FHI) High resolution KMC (left); STM images (right) Gyure, Barvosa-Carter (HRL), Grosse (UCLA,HRL) IMA, 11/19/04

12. Island Dynamics • Burton, Cabrera, Frank (1951) • Epitaxial surface • adatom density ρ • continuum in lateral direction, atomistic in growth direction • Adatom diffusion equation, equilibrium BC, step edge velocity ρt=DΔ ρ +F ρ = ρeq v =D [∂ ρ/ ∂n] • Line tension (Gibbs-Thomson) in BC and velocity D ∂ ρ/ ∂n= c(ρ – ρeq ) + c κ v =D [∂ ρ/ ∂n] + c κss • similar to surface diffusion, since κss ~ xssss IMA, 11/19/04

13. F v D Island Dynamics/Level Set Equations • Variables • N=number density of islands • k = island boundaries of height k represented by “level set function”  k (t) = { x : (x,t)=k} • adatom density (x,y,t) • Adatom diffusion equation ρt - D∆ρ = F - dN/dt • Island nucleation rate dN/dt = ∫ D σ1ρ2 dx σ1 = capture number for nucleation • Level set equation (motion of  ) φt + v |grad φ|= 0 v = normal velocity of boundary  Papanicolaou Fest 1/25/03

14. The Levelset Method Level Set Function j Surface Morphology j=0 j=0 t j=0 j=1 j=0 IMA, 11/19/04

15. Level Contours after 40 layers In the multilayer regime, the level set method produces results that are qualitatively similar to KMC methods.

16. LS = level set implementation of island dynamics UCLA/HRL/Imperial group, Chopp, Smereka IMA, 11/19/04

17. Random Seeding independent of r Probabilistic Seeding weight by local r2 Deterministic Seeding seed at maximum r2 Nucleation: Deterministic Time, Random Position Nucleation Rate: rmax r IMA, 11/19/04

18. Effect of Seeding Style on Scaled Island Size Distribution Probabilistic Seeding Deterministic Seeding Random Seeding IMA, 11/19/04 C. Ratsch et al., Phys. Rev. B (2000)

19. Island size distributions Experimental Data for Fe/Fe(001), Stroscio and Pierce, Phys. Rev. B 49 (1994) Stochastic nucleation and breakup of islands IMA, 11/19/04

20. Kinetic Theory for Step Edge Dynamicsand Adatom Boundary Conditions • Theory for structure and evolution of a step edge • Mean-field assumption for edge atoms and kinks • Dynamics of corners are neglected • Validation based on equilibrium and steady state solutions • Asymptotics for large diffusion IMA, 11/19/04

21. Step Edge Components • adatom density ρ • edge atom density φ • kink density (left, right) k • terraces (upper and lower)  IMA, 11/19/04

22. Unsteady Edge Model from Atomistic Kinetics • Evolution equations for φ, ρ, k ∂t ρ - DT∆ρ = F on terrace ∂t φ - DE∂s2φ = f+ +f- -f0 on edge ∂t k - ∂s (w ( kr - k ℓ))= 2 ( g - h ) on edge • Boundary conditions for ρ on edge from left (+) and right (-) • v ρ+ + DTn·gradρ = - f+ • v ρ+ + DTn·gradρ = f- • Variables • ρ = adatom density on terrace • φ = edge atom density • k = kink density • Parameters • DT, DE, DK, DS = diffusion coefficients for terrace, edge, kink, solid • Interaction terms • v,w = velocity of kink, step edge • F, f+ , f- , f0 = flux to surface, to edge, to kinks • g,h = creation, annihilation of kinks IMA, 11/19/04

23. Constitutive relations • Geometric conditions for kink density • kr + kℓ= k • kr - k ℓ= - tan θ • Velocity of step • v = w k cos θ • Flux from terrace to edge, • f+ =DTρ+ - DEφ • f- = DTρ- - DEφ • Flux from edge to kinks • f0 = v(φ κ + 1) • Microscopic equations for velocity w, creation rate g and annihilation rate h for kinks • w= 2 DEφ + DT (2ρ+ + ρ-) – 5 DK • g= 2 (DEφ + DT (2ρ+ + ρ-)) φ – 8 DKkr kℓ • h= (2DEφ + DT (3ρ+ + ρ-)) kr kℓ – 8 DS IMA, 11/19/04

24. BCF Theory • Equilibrium of step edge with terrace from kinetic theory is same as from BCF theory • Gibbs distributions ρ = e-2E/T φ = e-E/T k = 2e-E/2T • Derivation from detailed balance • BCF includes kinks of multi-heights IMA, 11/19/04

25. Equilibrium Solution • Solution for F=0 (no growth) • Same as BCF theory • DT, DE, DK are diffusion coefficients (hopping rates) on Terrace, Edge, Kink in SOS model Comparison of results from theory(-) and KMC/SOS () IMA, 11/19/04

26. Kinetic Steady State • Deposition flux F • Vicinal surface with terrace width L • No detachment from kinks or step edges, on growth time scale • detailed balance not possible • Advance of steps is due to attachment at kinks • equals flux to step f = L F F f L IMA, 11/19/04

27. Kinetic Steady State • Solution for F>0 • k >> keq • Pedge=Fedge/DE“edge Peclet #” • = F L / DE Comparison of scaled results from steady state (-), BCF(- - -), and KMC/SOS (∆) for L=25,50,100, with F=1, DT=1012 IMA, 11/19/04

28. Asymptotics for Large D/F • Assume slowly varying kinetic steady state along island boundaries • expansion for small “Peclet number”f / DE = ε3 • f is flux to edge from terrace • Distinguished scaling limit • k = O(ε) • φ = O(ε2) • κ = O(ε2) = curvature of island boundary = X yy • Y= O(ε-1/2) = wavelength of disurbances • Results at leading order • v = (f+ + f- ) + DEφyy • k = c3 v / φ • c1 φ2 - c2φ-1 v = (φ X y ) y • Linearized formula for φ • φ = c3(f+ + f- )2/3 – c4 edge diffusion curvature IMA, 11/19/04

29. Macroscopic Boundary Conditions • Island dynamics model • ρt – DT∆ρ = F adatom diffusion between step edges • X t = v velocity of step edges • Microscopic BCs for ρ DTn·grad ρ = DTρ - DEφ≡ f • From asymptotics • φ*= reference density = (DE / DT) c1((f+ + f-)/ DE)2/3 • γ = line tension = c4 DE • BCs for ρ on edge from left (+) and right (-), step edge velocity detachment • ± DTn·grad ρ = DT (ρ - φ*) + γκ • v = (f++ f- ) + c (f++ f- ) ss + γκss IMA, 11/19/04

30. Numerical Solutions for KineticStep Edge Equations • David Shao (UCLA) • Single island • First order discretization • Asymmetric linear system IMA, 11/19/04

31. Circular Island → Square:Initial and Final Shape IMA, 11/19/04

32. Circular Island → Square Angle=angle relative to nearest crystallographic direction IMA, 11/19/04

33. Circular Island → Square:Kink Density initial final IMA, 11/19/04

34. Circular Island → Square:Normal Velocity initial final IMA, 11/19/04

35. Circular Island → Square:Adatom and Edge Atom Densities Adatom density held constant in this computation for simplicity IMA, 11/19/04

36. Star-Shaped Island Island boundary IMA, 11/19/04

37. Star-Shaped Island Kink density IMA, 11/19/04

38. Star-Shaped Island Edge atom density IMA, 11/19/04

39. Coarse-Grained Description of an Epitaxial Surface • Extend the previous description to surface • Surface features • Adatom density ρ(x,t) • Step edge density s(x,t,θ, κ) for steps with normal angle θ, curvature κ • Diffusion of adatoms IMA, 11/19/04

40. Dynamics of Steps • Characteristic form of equations Cancellation of 2 edges Motion due to attachment Rotation due to differential attachment Decrease in curvature due to expansion • PDE for s IMA, 11/19/04

41. Dynamics of Steps • Cancellation of 2 edges • Rotation due to differential attachment • Decrease in curvature due to expansion • Motion due to attachment IMA, 11/19/04

42. Motion of Steps • Geometric constraint on steps • - Characteristic form (τ along step edge) • Creation of islands at nucleation sites (a=atomic size) • - PDE IMA, 11/19/04

43. Conclusions • Level set method • Coarse-graining of KMC • Stochastic nucleation • Kinetic model for step edge • kinetic steady state ≠ BCF equilibrium • validated by comparison to SOS/KMC • Numerical simulation do not show problems with edges • Atomistic derivation of Gibbs-Thomson • includes effects of edge diffusion, curvature, detachment • previous derivations from thermodynamic driving force • Coarse-grained description of epitaxial surface • Neglects correlations between step edges IMA, 11/19/04