IMA, 11/19/04 Multiscale Modeling of Epitaxial Growth Processes: Level Sets and Atomistic Models...
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Transcript of IMA, 11/19/04 Multiscale Modeling of Epitaxial Growth Processes: Level Sets and Atomistic Models...
IMA, 11/19/04
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
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
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
IMA, 11/19/04
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
A
B
δEEb
IMA, 11/19/04
SOS Simulation for coverage=.2
Gyure and Ross, HRLGyure & Ross
IMA, 11/19/04
SOS Simulation for coverage=10.2
IMA, 11/19/04
SOS Simulation for coverage=30.2
IMA, 11/19/04
Validation of SOS Model:Comparison of Experiment and KMC Simulation
(Vvedensky & Smilauer)
Island size density Step Edge Density (RHEED)
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
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)
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
ρt=DΔ ρ +F
ρ = ρeq
v =D [∂ ρ/ ∂n]
• Line tension (Gibbs-Thomson) in BC and velocityD ∂ ρ/ ∂n = c(ρ – ρeq ) + c κ
v =D [∂ ρ/ ∂n] + c κss
– similar to surface diffusion, since κss ~ xssss
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
ρt - D∆ ρ = F - dN/dt• Island nucleation rate
dN/dt = ∫ D σ1 ρ 2 dx
σ1 = capture number for nucleation• Level set equation (motion of )
φ t + v |grad φ| = 0 v = normal velocity of boundary
F
D
v
IMA, 11/19/04
The Levelset Method
Level Set Function Surface Morphology
t
=0
=0
=0
=0=1
Level Contours after 40 layers
In the multilayer regime, the level set method produces resultsthat are qualitatively similar to KMC methods.
IMA, 11/19/04
LS = level set implementation of island dynamics UCLA/HRL/Imperial group, Chopp, Smereka
IMA, 11/19/04
Nucleation Rate:
Nucleation: Deterministic Time, Random Position
2),( tDdt
dNx
Random Seedingindependent of
Probabilistic Seedingweight by local 2
Deterministic Seedingseed at maximum 2
max
IMA, 11/19/04C. Ratsch et al., Phys. Rev. B (2000)
Effect of Seeding Style on Scaled Island Size Distribution
Random Seeding Probabilistic Seeding Deterministic Seeding
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
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
IMA, 11/19/04
Step Edge Components
•adatom density ρ•edge atom density φ•kink density (left, right) k•terraces (upper and lower)
IMA, 11/19/04
Unsteady Edge Model from Atomistic Kinetics
• Evolution equations for φ, ρ, k ∂t ρ - DT ∆ ρ = F on terrace∂t φ - DE ∂s
2 φ = 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+ + DT n·grad ρ = - f+ – v ρ+ + DT n·grad ρ = 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
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+ = DT ρ+ - DE φ – f- = DT ρ- - DE φ
• 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 DE φ + DT (2ρ+ + ρ-) – 5 DK
– g= 2 (DE φ + DT (2ρ+ + ρ-)) φ – 8 DK kr kℓ – h= (2DE φ + DT (3ρ+ + ρ-)) kr kℓ – 8 DS
IMA, 11/19/04
BCF Theory
• Equilibrium of step edge with terrace from kinetic theory is same as from BCF theory
• Gibbs distributionsρ = 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
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
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
IMA, 11/19/04
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
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 y y
– Y= O(ε-1/2) = wavelength of disurbances
• Results at leading order– v = (f+ + f- ) + DE φyy
– k = c3 v / φ
– c1 φ2 - c2 φ-1 v = (φ X y ) y
• Linearized formula for φ– φ = c3 (f+ + f- )2/3 – c4
curvature
edge diffusion
IMA, 11/19/04
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 DT n·grad ρ = DT ρ - DE φ ≡ 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
± DT n·grad ρ = DT (ρ - φ * ) + γ κ v = (f+ + f- ) + c (f+ + f- ) ss + γ κss
detachment
IMA, 11/19/04
Numerical Solutions for KineticStep Edge Equations
• David Shao (UCLA)• Single island• First order discretization• Asymmetric linear system
IMA, 11/19/04
Circular Island → Square:Initial and Final Shape
IMA, 11/19/04
Circular Island → Square
Angle=angle relative to nearest crystallographic direction
IMA, 11/19/04
Circular Island → Square:Kink Density
initial final
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Circular Island → Square:Normal Velocity
initial final
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Circular Island → Square:Adatom and Edge Atom Densities
Adatom density held constant in this computation for simplicity
IMA, 11/19/04
Star-Shaped Island
Island boundary
IMA, 11/19/04
Star-Shaped Island
Kink density
IMA, 11/19/04
Star-Shaped Island
Edge atom density
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 adatoms2
0 ( , , , )t xD F s x t d d
IMA, 11/19/04
Dynamics of Steps• Characteristic form of equations
20
0
1( ) (( ( ) ) )
2
( ) ( ) ( , )
t x x
n n
s D n s w s n w s
s n n s d d
•PDE for s
0
0
2
( ) ( ) ( , )
( ) ( )
( ( ) )
1
2
t n n
t
t x
t
s s n n s d d
x wn D n
n w
w
Cancellation of 2 edges
Motion due to attachment
Rotation due to differential attachment
Decrease in curvature due to expansion
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
IMA, 11/19/04
Motion of Steps
• Creation of islands at nucleation sites (a=atomic size)
• Geometric constraint on steps
( ( ) ) 0xn s
0
( ) 0
s
n x
1 20( ) ...ts a D
- Characteristic form (τ along step edge)
- PDE
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