Interface Dynamics in Epitaxial Growth

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Interface Dynamics in Epitaxial Growth
Russel Caflisch Mathematics Department, UCLA

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Collaborators

• UCLA Anderson, Connell, Fedkiw, Gibou, Kang,
  Merriman, Osher, Petersen (GaTech), Ratsch
• HRL Barvosa-Carter, Owen, Grosse, Gyure, Ross,
  Zinck
• Imperial Vvedensky
• Support from DARPA and NSF under the Virtual
  Integrated Prototyping (VIP) Initiative and from
  ARO
• www.math.ucla.edu/thinfilm

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Outline

• Epitaxial Growth
• molecular beam epitaxy (MBE)
• layer-by-layer growth
• Kinetic Monte Carlo
• atomistic description
• Arrhenius rates
• Continuum model
• island dynamics
• level set method
• boundary conditions
• Kinetic model for step edge
• density of edge adatoms and kinks on boundary
• obtain curvature diffusion
• Conclusions

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Growth and Analysis Facility at HRL
ABES

• substrate temperature

MBE Chamber
STM Chamber

• surface structure
• morphology
• monolayer thickness

Effusion Cells
PEO

• In, Ga, Al evaporators
• Valved As, Sb crackers

• desorbed and scattered flux

• morphology
• monolayer thickness

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STM Image of InAs
HRL whole-wafer STMsurface quenched from 450C,
low As
20nmx20nm
250nmx250nm 1.8 V, Filled States
Barvosa-Carter, Owen, Zinck (HRL)
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AlSb Growth by MBE
Barvosa-Carter and Whitman, NRL
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RHEED signatures
Zinck, Owen, Barvosa-Carter (HRL)

• RHEED reflective high energy electron
  diffraction
• intensity a - b?(step edge density)
• 1 oscillation per crystal layer
• amplitude and decay rate for oscillations is
  indicator of surface quality

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Epitaxial Growth

• Growth of thin film as single crystal
• crystal properties determined by substrate
• Layer-by-layer growth
• layer (nearly) complete before initiation of next
  layer
• Surface features in layer-by-layer growth
• adatoms
• islands
• step edges
• Data
• STM atomistic picture after growth
• RHEED diffraction intensity c - (step edge
  density)
• Nanoscale morphology can significantly affect
  device performance

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Basic Processes in Epitaxial Growth
(a) deposition (f) edge diffusion (b)
diffusion (g) diffusion down step (c)
nucleation (h) nucleation on top of islands (d)
attachment (i) dimer diffusion (e) detachment
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Hierarchy of Models

• Large range of length and time scales
• atomic scale 1 Å 10-10 m
• surface feature scale 10 nm 10-8 m
• device scale 1 ?m 10-6 m
• wafer scale 1mm 10-3 m
• Hierarchy of models and simulation methods
• ab initio (1 Å, 1 fs)
• molecular dynamics (1 Å, 1 fs)
• Kinetic Monte Carlo (KMC) (1 nm, 1 ?s)
• continuum (10 nm, 1 ms)
• bulk (1 ?m, 1 s)

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Atomistic Description of Epitaxial Growth

• The Kinetic Monte Carlo Method

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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 r depends on nearest neighbors
• r r0 e-E / kT
• E energy barrier between state before and after
  hop
• Deposition of new particles
• random position
• arrival frequency from deposition rate
• Simulation using kinetic Monte Carlo method

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SOS Simulation for coverage.2
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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Continuum Description of Epitaxial Growth

• The Island Dynamics/Level Set Method

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Continuum Equations Island Dynamics/Level Set
Model

• Description of epitaxial surface
• ?k island boundaries of height k represented
  by level set function ?
• ?k (t) x ?(x,t)k
• Normal velocity v of step edge or island boundary
  is essential quantity
• N Number of islands
• adatom density ?(x,y,t)
• Level of description
• continuum in lateral directions (x,y)
• discrete (at atomic level) in growth direction z
• Valid for growth of very thin layers
• application quantum well devices, layer
  thickness 20 Å
• coarse-grained eqtns (e.g KPZ or Villain) for
  thin film height h(x,y) not valid
• Diffusion dominant
• inverse Peclet number RD/F (for alattice
  constant1)
• R varies between 106 and 1010 , for MBE
• FDeposition flux, DDiffusion coefficient

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Island Dynamics/Level Set Equations

• Adatom diffusion equation
• ?t ? - D?2 ? F - dN/dt
• Island nucleation rate
• dN/dt ? D ?1 ? 2 dx
• ?1 capture number for
  nucleation
• Level set equation (motion of ? )
• ?t ? v ? ?? 0
• v normal velocity of boundary ?
• To be determined
• boundary conditions for ?
• boundary velocity v determined next
• nucleation site

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Boundary Conditions and Boundary Velocity

• Boundary condition at island boundaries
• (irreversible aggregation)
• (equilibrium BCF)
• (mixed type)

• Normal velocity of boundary ?
• v D ?n ? (irreversible
  aggregation)
• v D ?n ? -vdetach
  (attachment/detachment - Petersen)
• v D ?n ? c ?ss (edge diffusion)

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Seeding of new islands

• Islands nucleate by random binary collisions
  between adatoms.
• Assuming that nucleation takes place continuously
  in time, the rate at which new islands are seeded
  is given by
• where N(t) is the total number of islands
  nucleated up to time t and lt gt denotes a spatial
  average.
• Nucleation site chosen at random with spatial
  density ? 2
• Every time N(t) increases by 1, it is time to
  seed a new island.
• Initially ? is set to -.5 at every gridpoint. A
  new island is seeded by raising the value of ? by
  1 at the nucleation site and at a few neighboring
  gridpoints.
• Atomistic fluctuations
• in nucleation site are important
• in nucleation time are not important

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Computing the adatom density

• Finite difference equation for ?
• explicit method has severe timestep restriction
  ?t lt c ?x2/D
• implicit method required
• Resulting system has form
• A xn b
• A and b depend on surface geometry,i.e. on island
  boundaries
• difficult to make A symmetric

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Computing the adatom density

• Away from boundaries, use standard spatial
  discretization

• Near boundaries, use subcell discretization
  based on ghost fluid method (Fedkiw)

• Spatially first order accurate
• Resulting matrix system is symmetric.
• Solve using (cholesky) preconditioned conjugate
  gradient method

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Update of the level set function

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Evolution of and time 1.7
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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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Simulation of Epitaxial GrowthThe Island
Dynamics/Level Set Method

• S. Chen, M. Kang

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Validity and Qualitative Features

• Conservation of Mass
• Dependence on nucleation site selection rule
• Comparison to KMC

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Island Merger by Level Set Approach

• Efficient and accurate numerical method
• Merging of boundaries is automatically handled
• Method conserves mass

time .1 time .9
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Dependence on Nucleation StyleLocation
distribution must be correctly representedrandom
1 probabilistic ? 2 deterministic
max ?
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Scaling of the Island Size Distribution
(Stroscio et al PRB, 1994)
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Island Dynamics vs. KMC

• Island dynamics is faster than KMC in principle
• adatom hopping time
• KMC must resolve
• handled continuously by island dynamics
• No faster in practice (so far)
• nucleation requires atomistic grid, small times
• solution of diffusion equation is slow
• Some features easier to test
• variation in statistic of fluctuations
• capture zones of islands (Gibou)
• stability
• Some physics easier to add, some harder
• strain easier for island dynamics
• reconstruction easier for KMC

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Comparison of Level Set Methodand Alternatives
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Kinetic Theory for Step Edge Dynamicsand Adatom
Boundary Conditions

• with Weinan E

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Step Edge Components

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

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Adatom and Kink Dynamics on a Step Edge
Attachment at kinks ? kink velocity w
Kink pair creation ? kink creation rate g
Kink pair collision ? kink loss rate h
Reverse processes do not occur in typical MBE
growth ? no detailed balance ? nonequilibrium
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Kinetic Theory for Step Edge Dynamics
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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

• Solution for Fgt0
• k gtgt keq
• PedgeFedge/Dedge edge Peclet

Comparison of scaled results from theory(-) and
KMC/SOS (???) for L25,50,100
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Macroscopic Boundary Conditions from Step Edge
Model

• Assume slowly varying kinetic steady state along
  island boundaries
• Result is Gibbs-Thomson BC, but derived from
  atomistic theory rather than from thermodynamics
• Reference density ? from kinetic steady, not
  equilibrium
• ? is curvature of island boundary

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Constants in BCs
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Conclusions

• Island dynamics model
• appropriate for very thin films continuum in
  x,y discrete in z
• level set simulation method
• validated by comparison to SOS/KMC
• derivation of boundary conditions
• Additional physics
• attachment/detachment (Petersen)
• strain
• edge diffusion
• multiple species
• reconstruction effects