Transcript Multiscale Modeling of Epitaxial Growth Processes: Level Sets and Atomistic Models Russel Caflisch1, Mark Gyure2 , Bo Li4, Stan Osher 1, Christian Ratsch1,2, David.

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
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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
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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, …
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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
A
Eb
δE
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SOS Simulation for coverage=.2
GyureHRL
& Ross
Gyure and Ross,
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SOS Simulation for coverage=10.2
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SOS Simulation for coverage=30.2
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Validation of SOS Model:
Comparison of Experiment and KMC Simulation
(Vvedensky & Smilauer)
Island size density
Step Edge Density (RHEED)
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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
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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)
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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
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Island Dynamics/Level Set Equations
F
• 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 
Papanicolaou Fest 1/25/03
v
D
The Levelset Method
Level Set Function j
Surface Morphology
j=0
j=0
t
j=0
j=1
j=0
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Level Contours after 40 layers
In the multilayer regime, the level set method produces results
that are qualitatively similar to KMC methods.
LS = level set implementation of island dynamics
UCLA/HRL/Imperial group,
Chopp, Smereka
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Nucleation:
Deterministic Time, Random Position
Nucleation Rate:
max

dN
 D  ( x, t ) 2
dt
Random Seeding
independent of 
Probabilistic Seeding
weight by local 2
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Deterministic Seeding
seed at maximum 2
Effect of Seeding Style on Scaled Island Size Distribution
Random Seeding
C. Ratsch et al., Phys. Rev. B (2000)
Probabilistic Seeding
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Deterministic Seeding
Island size distributions
Experimental Data for
Fe/Fe(001),
Stroscio and Pierce,
Phys. Rev. B 49 (1994)
Stochastic
nucleation and
breakup of
islands
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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
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Step Edge Components
•adatom density ρ
•edge atom density φ
•kink density (left, right) k
•terraces (upper and lower) 
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Unsteady Edge Model
from Atomistic Kinetics
• Evolution equations for φ, ρ, k
∂t ρ - DT ∆ ρ = F
on terrace
∂t φ - DE ∂s2 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
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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
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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
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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 ()
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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
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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
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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
edge diffusion
– v = (f+ + f- ) + DE φyy
– k = c3 v / φ
– c1 φ2 - c2 φ-1 v = (φ X y ) y
curvature
• Linearized formula for φ
– φ = c3 (f+ + f- )2/3 – c4 j
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Macroscopic Boundary Conditions
• Island dynamics model
– ρt – DT ∆ ρ = F adatom diffusion between step edges
– Xt=v
velocity of step edges
detachment
• 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
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Numerical Solutions for Kinetic
Step Edge Equations
•
•
•
•
David Shao (UCLA)
Single island
First order discretization
Asymmetric linear system
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Circular Island → Square:
Initial and Final Shape
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Circular Island → Square
Angle=angle relative to nearest crystallographic direction
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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
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Star-Shaped Island
Island boundary
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Star-Shaped Island
Kink density
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Star-Shaped Island
Edge atom density
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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
t  D x   F   0   s ( x, t , ,  )d d
2
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Dynamics of Steps
• Characteristic form of equations
st   s 
n n  0
n( ) n( ) s( ,  )d d 
Cancellation of 2 edges
xt  wn( )   0 D  n( )
Motion due to attachment
t  (n( )   x   ) w
Rotation due to differential attachment
1
t  
w 2
2
Decrease in curvature due to expansion
•PDE for s
1
w 2 s  ((n( )   x   ) w) s
2
n( ) n( ) s( ,  )d d  
st   0 D  n( ) x s 
 s 
n n  0
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Dynamics of Steps
• Cancellation of 2 edges
• Rotation due to differential attachment
• Decrease in curvature due to expansion
• Motion due to attachment
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Motion of Steps
• Geometric constraint on steps
- Characteristic form (τ along step edge)
s  0
n( ) x  0
   
- PDE
(n( ) x   )s  0
• Creation of islands at nucleation sites (a=atomic size)
st (  a1 )   0 D 2  ...
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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
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