Russel Caflisch
Mathematics Department, UCLA

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

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

Growth and Analysis Facility at HRL
MBE Chamber
ABES
• substrate temperature
STM Chamber
RHEED
• surface structure
• morphology
• monolayer thickness
REMS
• desorbed and scattered flux
PEO
• morphology
• monolayer thickness
Effusion Cells
• In, Ga, Al evaporators
• Valved As, Sb crackers

STM Image of InAs
HRL whole-wafer STM surface quenched from 450 °C, “low As”
250nmx250nm
1.8 V, Filled States
Barvosa-Carter, Owen, Zinck (HRL)
20nmx20nm

AlSb Growth by MBE
Barvosa-Carter and Whitman, NRL

RHEED signatures
Growth t max t min
Recovery
I=I
0 e (-t/ t
)
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

• 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

Basic Processes in Epitaxial Growth
(a) (a)
(i)
9750-00-444
(c)
(g)
(h)
(f)
(e)
(b)
(d)
(a) deposition
(b) diffusion
(c) nucleation
(d) attachment
(e) detachment
(f) edge diffusion
(g) diffusion down step
(h) nucleation on top of islands
(i) dimer diffusion

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)

Atomistic Description of Epitaxial Growth

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 = r
0 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

SOS Simulation for coverage=.2
Gyure and Ross, HRL

SOS Simulation for coverage=10.2

SOS Simulation for coverage=30.2

Continuum Description of Epitaxial Growth

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 R=D/F (for a=lattice constant=1)
– R varies between 10 6 and 10 10 , for MBE
– F=Deposition flux, D=Diffusion coefficient

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

Boundary Conditions and Boundary Velocity
• Boundary condition at island boundaries
 
0

 n
 

 f   
(irreversible aggregation)
(equilibrium: BCF)
(mixed type)
•Normal velocity of boundary 
: v = D [
 n

] (irreversible aggregation) v = D [
 n

] -v detach
(attachment/detachment - Petersen) v = D [
 n

] + c
 ss
(edge diffusion)

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: dN

D
 
1
2 dt
• where N(t) is the total number of islands nucleated up to time t and < > 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

• Finite difference equation for 
– explicit method has severe timestep restriction  t < c
 x 2 /D
– implicit method required
• Resulting system has form
A x n = b
– A and b depend on surface geometry,i.e. on island boundaries
– difficult to make A symmetric

Computing the adatom density
• Away from boundaries, use standard spatial discretization
 xx




1
 x
 i

1

 x
 i

 i

 x
 i

1



•
Near boundaries, use subcell discretization based on ghost fluid method (Fedkiw)
 xx




1
 x
 a
  f
*
 x i 
 i

 x
 i

1



•
Spatially first order accurate
•
Resulting matrix system is symmetric.
• Solve using (cholesky) preconditioned conjugate gradient method

Update of the level set function
•
Since n

 
 
 
 t
 v 

and v n
 v n
0 simplifies to
 
 t
 v n

0.
•
The partial differential equation above is an example of a
Hamilton-Jacobi equa tion.
•
High order accurate, essentially non-oscillatory numerical methods
(ENO schemes) were developed by Osher et. al., to solve general
Hamilton-Jacobi equations.
•
Using a method of lines approach, we use a third order ENO scheme



Level Contours after 40 layers
In the multilayer regime, the level set method produces results that are qualitatively similar to KMC methods.

Simulation of Epitaxial Growth
The Island Dynamics/Level Set Method
S. Chen, M. Kang

Validity and Qualitative Features
• Conservation of Mass
• Dependence on nucleation site selection rule
• Comparison to KMC

Island Merger by Level Set Approach
•
Efficient and accurate numerical method
•
Merging of boundaries is automatically handled
• Method conserves mass time = .1
time = .9

Dependence on Nucleation Style
Location distribution must be correctly represented random: 1; probabilistic:

2 ; deterministic: max


Scaling of the Island Size Distribution
(Stroscio et al
PRB, 1994)

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

Comparison of Level Set Method and Alternatives
Level set
Topological changes Easy
New physics Easy
BCs required
Asymptotic analysis needed
Stiff numerics
Yes
No
No
Phase field Front tracking
Easy
Difficult
Special logic required
Easy
No
Yes
Yes
No
Yes Maybe

Kinetic Theory for Step Edge Dynamics and Adatom Boundary Conditions with Weinan E

Step Edge Components
•adatom density 
•edge adatom density 
•kink density (left, right) k
•terraces (upper and lower) 

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

Kinetic Theory for Step Edge
Dynamics

Equilibrium Solution
•Solution for F=0 (no growth)
•Same as BCF theory
•D
T
, D
E
, D
K are diffusion coefficients (hopping rates) on
Terrace, Edge, Kink in SOS model
Comparison of results from theory(-) and KMC/SOS (  )

Kinetic Steady State
•Solution for F>0
•k >> k eq
•P edge
=F edge
/D edge
“edge Peclet #”
Comparison of scaled results from theory(-) and KMC/SOS (  ) for L=25,50,100

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

Constants in BCs

• 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