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, …

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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

A

B

δEEb

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SOS Simulation for coverage=.2

Gyure and Ross, HRLGyure & 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

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

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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

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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 ∂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

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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 distributionsρ = 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

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

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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

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Numerical Solutions for KineticStep 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 adatoms2

0 ( , , , )t xD F s x t d d

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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

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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

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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