1. IMA, 11/19/04 Multiscale Modeling of Epitaxial Growth Processes: Level Sets and Atomistic Models Russel Caflisch 1, Mark Gyure 2, Bo Li 4, Stan Osher 1, Christian Ratsch 1,2, David Shao 1 and Dimitri Vvedensky 3 1 UCLA, 2 HRL Laboratories 3 Imperial College, 4 U Maryland www.math.ucla.edu/~material

2. IMA, 11/19/04 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

3. IMA, 11/19/04 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= D 0 exp(-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, …

4. IMA, 11/19/04

5. Kinetic Monte Carlo Random hopping from site A→ B hopping rate D 0 exp(-E/T), –E = E b = energy barrier between sites –not δE = energy difference between sites A B δEδE EbEb

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

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

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

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

10. IMA, 11/19/04 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

11. IMA, 11/19/04 High Resolution KMC Simulations High resolution KMC (left); STM images (right) Gyure, Barvosa-Carter (HRL), Grosse (UCLA,HRL) 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)

12. IMA, 11/19/04 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 ρ t =DΔ ρ +F ρ = ρ eq v =D [∂ ρ/ ∂n] Line tension (Gibbs-Thomson) in BC and velocity D ∂ ρ/ ∂n = c(ρ – ρ eq ) + c κ v =D [∂ ρ/ ∂n] + c κ ss –similar to surface diffusion, since κ ss ~ x ssss

13. Papanicolaou Fest 1/25/03 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 ρ t - D∆ ρ = F - dN/dt Island nucleation rate dN/dt = ∫ D σ 1 ρ 2 dx σ 1 = capture number for nucleation Level set equation (motion of  ) φ t + v |grad φ| = 0 v = normal velocity of boundary  F D v

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

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

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

17. IMA, 11/19/04 Nucleation Rate: Nucleation: Deterministic Time, Random Position Random Seeding independent of  Probabilistic Seeding weight by local  2 Deterministic Seeding seed at maximum  2   max

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

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

20. IMA, 11/19/04 Kinetic Theory for Step Edge Dynamics and 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

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

22. IMA, 11/19/04 Unsteady Edge Model from Atomistic Kinetics Evolution equations for φ, ρ, k ∂ t ρ - D T ∆ ρ = Fon terrace ∂ t φ - D E ∂ s 2 φ = f + + f - - f 0 on edge ∂ t k - ∂ s (w ( k r - k ℓ ))= 2 ( g - h ) on edge Boundary conditions for ρ on edge from left (+) and right (-) –v ρ + + D T n·grad ρ = - f + –v ρ + + D T n·grad ρ = f - Variables –ρ = adatom density on terrace –φ = edge atom density –k = kink density Parameters –D T, D E, D K, D S = diffusion coefficients for terrace, edge, kink, solid Interaction terms –v,w = velocity of kink, step edge –F, f +, f -, f 0 = flux to surface, to edge, to kinks –g,h = creation, annihilation of kinks

23. IMA, 11/19/04 Constitutive relations Geometric conditions for kink density –k r + k ℓ = k –k r - k ℓ = - tan θ Velocity of step –v = w k cos θ Flux from terrace to edge, –f + = D T ρ + - D E φ –f - = D T ρ - - D E φ Flux from edge to kinks –f 0 = v(φ κ + 1) Microscopic equations for velocity w, creation rate g and annihilation rate h for kinks –w= 2 D E φ + D T (2ρ + + ρ - ) – 5 D K –g= 2 (D E φ + D T (2ρ + + ρ - )) φ – 8 D K k r k ℓ –h= (2D E φ + D T (3ρ + + ρ - )) k r k ℓ – 8 D S

24. IMA, 11/19/04 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

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

26. IMA, 11/19/04 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 L F f

27. IMA, 11/19/04 Kinetic Steady State Solution for F>0 k >> k eq P edge =F edge /D E “edge Peclet #” = F L / D E Comparison of scaled results from steady state (-), BCF(- - -), and KMC/SOS (  ∆) for L=25,50,100, with F=1, D T =10 12

28. IMA, 11/19/04 Asymptotics for Large D/F Assume slowly varying kinetic steady state along island boundaries –expansion for small “Peclet number” f / D E = ε 3 –f is flux to edge from terrace Distinguished scaling limit –k = O(ε) –φ = O(ε 2 ) –κ = O(ε 2 ) = curvature of island boundary = X y y –Y= O(ε -1/2 ) = wavelength of disurbances Results at leading order –v = (f + + f - ) + D E φ yy –k = c 3 v / φ –c 1 φ 2 - c 2 φ -1 v = (φ X y ) y Linearized formula for φ –φ = c 3 (f + + f - ) 2/3 – c 4  curvature edge diffusion

29. IMA, 11/19/04 Macroscopic Boundary Conditions Island dynamics model –ρ t – D T ∆ ρ = F adatom diffusion between step edges –X t = vvelocity of step edges Microscopic BCs for ρ D T n·grad ρ = D T ρ - D E φ ≡ f From asymptotics –φ * = reference density = (D E / D T ) c 1 ((f + + f - )/ D E ) 2/3 –γ = line tension = c 4 D E BCs for ρ on edge from left (+) and right (-), step edge velocity ± D T n·grad ρ = D T (ρ - φ * ) + γ κ v = (f + + f - ) + c (f + + f - ) ss + γ κ ss detachment

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

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

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

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

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

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

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

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

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

39. IMA, 11/19/04 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

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

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

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

43. IMA, 11/19/04 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