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

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Multiscale Modeling of Epitaxial Growth
Processes Level Sets and Atomistic Models
Russel Caflisch1, Mark Gyure2 , Bo Li4, Stan
Osher 1, Christian Ratsch1,2, 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
• 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 dE energy difference between sites

B
Eb
dE
A
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SOS Simulation for coverage.2
Gyure Ross
Gyure and Ross, HRL
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SOS Simulation for coverage10.2
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SOS Simulation for coverage30.2
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Validation of SOS ModelComparison of Experiment
and KMC Simulation(Vvedensky Smilauer)
Step Edge Density (RHEED)
Island size density
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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 ?
• continuum in lateral direction, atomistic in
  growth direction
• Adatom diffusion equation, equilibrium BC, step
  edge velocity
• ?tD? ? F
• ? ?eq
• v D ? ?/ ?n
• Line tension (Gibbs-Thomson) in BC and velocity
• D ? ?/ ?n c(? ?eq ) c ?
• v D ? ?/ ?n c ?ss
• similar to surface diffusion, since ?ss xssss

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Island Dynamics/Level Set Equations

• Variables
• Nnumber 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
• ?t - D? ? F - dN/dt
• Island nucleation rate
• dN/dt ? D s1 ? 2 dx
• s1 capture number for nucleation
• Level set equation (motion of ? )
• f t v grad f 0
• v normal velocity of boundary ?

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The Levelset Method
Level Set Function j
Surface Morphology
j0
j0
t
j0
j1
j0
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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.
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LS level set implementation of island dynamics
UCLA/HRL/Imperial group, Chopp, Smereka
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Nucleation Deterministic Time, Random Position
Nucleation Rate
rmax
r
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Effect of Seeding Style on Scaled Island Size
Distribution
Probabilistic Seeding
Deterministic Seeding
Random Seeding
C. Ratsch et al., Phys. Rev. B (2000)
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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 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

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Step Edge Components

• adatom density ?
• edge atom density f
• kink density (left, right) k
• terraces (upper and lower) ?

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Unsteady Edge Model from Atomistic Kinetics

• Evolution equations for f, ?, k
• ?t ? - DT ? ? F on terrace
• ?t f - DE ?s2 f f f- - f0 on edge
• ?t k - ?s (w ( kr - k l)) 2 ( g - h ) on edge
• Boundary conditions for ? on edge from left ()
  and right (-)
• v ? DT ngrad ? - f
• v ? DT ngrad ? f-
• Variables
• ? adatom density on terrace
• f 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?, 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 ?
• Velocity of step
• v w k cos ?
• Flux from terrace to edge,
• f DT ? - DE f
• f- DT ?- - DE f
• Flux from edge to kinks
• f0 v(f ? 1)
• Microscopic equations for velocity w, creation
  rate g and annihilation rate h for kinks
• w 2 DE f DT (2? ?-) 5 DK
• g 2 (DE f DT (2? ?-)) f 8 DK kr kl
• h (2DE f DT (3? ?-)) 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
• ? e-2E/T
• f 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 F0 (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

F
f
L
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Kinetic Steady State

• Solution for F0
• k keq
• PedgeFedge/DE edge Peclet
• F L / DE

Comparison of scaled results from steady state
(-), BCF(- - -), and KMC/SOS (???) for
L25,50,100, with F1, DT1012
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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 e3
• f is flux to edge from terrace
• Distinguished scaling limit
• k O(e)
• f O(e2)
• ? O(e2) curvature of island boundary X y y
• Y O(e-1/2) wavelength of disurbances
• Results at leading order
• v (f f- ) DE fyy
• k c3 v / f
• c1 f2 - c2 f-1 v (f X y ) y
• Linearized formula for f
• f c3 (f f- )2/3 c4 ?

edge diffusion
curvature
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Macroscopic Boundary Conditions

• Island dynamics model
• ?t DT ? ? F adatom diffusion between step
  edges
• X t v velocity of step edges
• Microscopic BCs for ?
• DT ngrad ? DT ? - DE f
  f
• From asymptotics
• ? reference density (DE / DT) c1((f f- )/
  DE)2/3
• ? line tension c4 DE
• BCs for ? on edge from left () and right (-),
  step edge velocity

detachment
DT ngrad ? DT (? - ? ) ? ?
v (f f- ) c (f f- ) ss ? ?ss
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Coarse-Grained Description of an Epitaxial
Surface

• Extend the previous description to surface
• Surface features
• Adatom density ?(x,t)
• Step edge density s(x,t,?, ?) for steps with
  normal angle ?, curvature ?
• Diffusion of adatoms

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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

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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 (t along step edge)
- PDE

• Creation of islands at nucleation sites (aatomic
  size)

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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
• 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