Transcript No Slide Title

A Level-Set Method for Modeling Epitaxial Growth and
Self-Organization of Quantum Dots
Christian Ratsch, UCLA, Department of Mathematics
Santa Barbara, Jan. 31, 2005
Collaborators
Outline
• Introduction
•Russel Caflisch
•Xiabin Niu
• The basic island dynamics model using the
level set method
•Max Petersen
• Include Reversibility
•Raffaello Vardavas
• Include spatially varying, anisotropic diffusion
self-organization of islands
$$$: NSF and DARPA
Ostwald Ripening
What is Epitaxial Growth?
epi – taxis = “on” – “arrangement”
(a)
(a)
9750-00-444
(i)
(c)
(g)
(h)
(f)
Atomic Motion
Island Growth
(d)
(b)
(e)
-13
o
Time Scale ~ 10 seconds Length Scale: Angstrom
Time Scale ~ seconds
Length Scale: Microns
Why do we care about Modeling Epitaxial Growth?
• Many devices for opto-electronic application are
multilayer structures grown by epitaxial growth.
• Interface morphology is critical for performance
• Theoretical understanding of epitaxial growth
will help improve performance, and produce new
structures.
Methods used for modeling epitaxial growth:
• KMC simulations: Completely stochastic method
• Continuum Models: PDE for film height, but only valid for thick layers
• New Approach: Island dynamics model using level sets
KMC Simulation of a Cubic, Solid-on-Solid Model
F
D = G0 exp(-ES/kT)
Ddet = D exp(-EN/kT)
Ddet,2 = D exp(-2EN/kT)
ES: Surface bond energy
EN: Nearest neighbor bond energy
G0 : Prefactor [O(1013s-1)]
• Parameters that can be calculated from first principles (e.g., DFT)
• Completely stochastic approach
• But small computational timestep is required
KMC Simulations: Effect of Nearest Neighbor Bond EN
Large EN:
Irreversible
Growth
Small EN:
Compact
Islands
Experimental Data
Au/Ru(100)
Ni/Ni(100)
Hwang et al., PRL 67 (1991)
Kopatzki et al., Surf.Sci. 284 (1993)
KMC Simulation for Equilibrium Structures of III/V Semiconductors
Experiment
KMC Simulation
(Barvosa-Carter, Zinck)
(Grosse, Gyure)
Similar work by
380°C
0.083 Ml/s
60 min
anneal
Kratzer and Scheffler
Itoh and Vvedensky
440°C
0.083 Ml/s
20 min
anneal
Problem:
Detailed KMC
simulations are
extremely slow !
F. Grosse et al., Phys. Rev. B66, 075320 (2002)
Outline
• Introduction
• The basic island dynamics model using the level set method
• Include Reversibility
Ostwald Ripening
• Include spatially varying, anisotropic diffusion
self-organization of islands
The Island Dynamics Model for Epitaxial Growth
Atomistic picture
(i.e., kinetic Monte Carlo)
Island dynamics
F
(a)
(a)
9750-00-444
(i)
v
(c)
(g)
(h)
(f)
(e)
(b)
(d)
D
• Treat Islands as continuum in the plane
• Resolve individual atomic layer
• Evolve island boundaries with levelset method
• Treat adatoms as a mean-field quantity (and solve diffusion equation)
The Level Set Method: Schematic
Level Set Function j
Surface Morphology
j=0
j=0
t
j=0
j=1
j=0
• Continuous level set function is resolved on a discrete numerical grid
• Method is continuous in plane (but atomic resolution is possible !), but has
discrete height resolution
The Basic Level Set Formalism for Irreversible Aggregation
• Governing Equation:
j
 vn | j | 0
t
j=0
• Diffusion equation for the adatom density (x,t):
• Boundary condition:
 0
• Velocity:
vn  D(n     n    )
dN
 D  ( x, t ) 2
• Nucleation Rate:
dt
C. Ratsch et al., Phys. Rev. B 65, 195403 (2002)

dN
 F  D 2   2
t
dt
Typical Snapshots of Behavior of the Model
t=0.1
j

t=0.5
Numerical Details
Level Set Function
• 3rd order essentially non-oscillatory (ENO) scheme for spatial part of
levelset function
• 3rd order Runge-Kutta for temporal part
Diffusion Equation
• Implicit scheme to solve diffusion equation (Backward Euler)
• Use ghost-fluid method to make matrix symmetric
• Use PCG Solver (Preconditioned Conjugate Gradient)
Essentially-Non-Oscillatory (ENO) Schemes
Need 4 points to discretize  j with third order accuracy
i-3
i-2
i-1
i
i+1
i+2
i+3
i+4
Set 1 Set 2 Set 3
This often leads to oscillations at the interface
Fix: pick the best four points out of a larger set of grid points to get rid of
oscillations (“essentially-non-oscillatory”)
Numerical Details
Level Set Function
• 3rd order essentially non-oscillatory (ENO) scheme for spatial part of
levelset function
• 3rd order Runge-Kutta for temporal part
Diffusion Equation
• Implicit scheme to solve diffusion equation (Backward Euler)
• Use ghost-fluid method to make matrix symmetric
• Use PCG Solver (Preconditioned Conjugate Gradient)

dN
 F  D 2   2
t
dt
Solution of Diffusion Equation
• Standard Discretization:
ik 1  ik
t
D
ik11  2ik 1  ik11
(x) 2
Aρk 1  b
• Leads to a symmetric system of equations:
• Use preconditional conjugate gradient method
f  0
Problem at boundary:
  f  i   i  i 1 

  



x

x



(  xx )i   1
1
1x  x 
2
i-2
1x
g
Matrix not symmetric anymore ; replace by:
  g  i   i  i 1 

  


x

x



(  xx )i  
x
i
i-1
g
: Ghost value at i
“ghost fluid method”
i+1
Fluctuations need to be included in nucleation of islands
Nucleation Rate:
dN
 D  ( x, t ) 2
dt
Probabilistic Seeding
weight by local 2
C. Ratsch et al., Phys. Rev. B 61, R10598 (2000)
max

A Typical Level Set Simulation
Outline
• Introduction
• The basic island dynamics model using the level set method
• Include Reversibility
Ostwald Ripening
• Include spatially varying, anisotropic diffusion
self-organization of islands
Extension to Reversibility
• So far, all results were for irreversible aggregation; but at higher
temperatures, atoms can also detach from the island boundary
• Dilemma in Atomistic Models: Frequent detachment and subsequent reattachment of atoms from islands
Significant computational cost !
• In Levelset formalism: Simply modify velocity (via a modified boundary
condition), but keep timestep fixed
• Boundary condition:

  eq ( Ddet , x)
Velocity:
vn  D(n     n    )
dN
 D  ( x, t ) 2
Nucleation Rate:
dt
•Stochastic break-up for small islands is important
Details of stochastic break-up
• For islands larger than a “critical size”, detachment is accounted for via the
(non-zero) boundary condition
• For islands smaller than this “critical size”, detachment is done stochastically,
and we use an irreversible boundary condition    (to avoid over-counting)
•calculate probability to shrink by 1, 2, 3, ….. atoms; this probability is
related to detachment rate.
•shrink the island by this many atoms
•atoms are distributed in a zone that corresponds to diffusion area
• Note: our “critical size” is not what is typical called “critical island size”. It is
a numerical parameter, that has to be chosen and tested. If chosen properly,
results are independent of it.
Sharpening of Island Size Distribution with Increasing Detachment Rate
Experimental Data for
Fe/Fe(001),
Stroscio and Pierce,
Phys. Rev. B 49 (1994)
Petersen, Ratsch, Caflisch, Zangwill, Phys. Rev. E 64, 061602 (2001).
Scaling of Computational Time
Almost no increase in computational time due to mean-field treatment of fast events
Ostwald Ripening
Verify Scaling Law
R  t 1/ 3
Slope of 1/3
M. Petersen, A. Zangwill, and C. Ratsch, Surf. Science 536, 55 (2003).
Outline
• Introduction
• The basic island dynamics model using the level set method
• Include Reversibility
Ostwald Ripening
• Include spatially varying, anisotropic diffusion
self-organization of islands
Nucleation and Growth on Buried Defect Lines
Results of Xie et al.
(UCLA, Materials Science Dept.) Growth on Ge on relaxed SiGe buffer layer
Dislocation lines are buried underneath.
•
•
Lead to strain field
This can alter potential energy surface:
• Anisotropic diffusion
• Spatially varying diffusion
Hypothesis:
Nucleation occurs in regions of fast diffusion
Level Set formalism is ideally suited to
incorporate anisotropic, spatially varying
diffusion without extra computational cost
Modifications to the Level Set Formalism for non-constant Diffusion
0 
 Dxx (x)


• Replace diffusion constant by matrix: D  D(x) 
 0
Dyy (x) 

Diffusion in x-direction
• Diffusion equation:

dN
 F    D( )  2
 drift
t
dt
drift ~ Dxx x Ead  Dyy y Ead
• Velocity:
vn  n  D( )  n  D(  )
• Nucleation Rate:
Dxx (x)  Dyy (x)
dN

 (x, t ) 2
dt
2
Diffusion in y-direction
Possible potential
energy surfaces
no drift
drift
What we have done so far
Assume a simple form of the variation of the potential energy surface
(i.e., sinusoidal)
For simplicity, we look at extreme cases: only variation of adsorption
energy, or only variation of transition energy (real case typically inbetween)
Isotropic Diffusion with Sinusoidal Variation in x-Direction
Dxx  Dyy ~ sin(ax)
Only variation of transition
energy, and constant adsorption
energy
• Islands nucleate in regions
of fast diffusion
• Little subsequent nucleation
in regions of slow diffusion
fast diffusion
slow diffusion
Comparison with Experimental Results
Results of Xie et al.
(UCLA, Materials Science Dept.)
Simulations
Anisotropic Diffusion with Sinusoidal Variation in x-Direction
Dxx ~ sin(ax) Dyy  const.
Dyy ~ sin(ax) Dxx  const.
• In both cases, islands mostly nucleate in regions of fast diffusion.
• Shape orientation is different
Isotropic Diffusion with Sinusoidal Variation in x- and y-Direction
Dxx  Dyy ~ sin(ax) sin(ay)
Comparison with Experimental Results
Results of Xie et al.
(UCLA, Materials Science Dept.)
Simulations
Anisotropic Diffusion with Variation of Adsorption Energy
What is the effect of thermodynamic drift ?
Etran
Ead
Spatially constant
adsorption and transition
energies, i.e., no drift
small amplitude
large amplitude
Regions of fast surface diffusion
Most nucleation does not occur in region of fast diffusion, but is dominated by drift
Transition from thermodynamically to kinetically controlled diffusion
Constant transition
energy
(thermodynamic drift)
Constant adsorption
energy
(no drift)
D
But: In all cases, diffusion
constant D has the same form:
x
What is next with spatially varying diffusion?
•So far, we have assumed that the potential energy surface is modified
externally (I.e., buried defects), and is independent of growing film
•Next, we want to couple this model with an elastic model (Caflisch et
al., in progress);
•Solve elastic equations after every timestep
•Modify potential energy surface (I.e., diffusion, detachment)
accordingly
•This can be done at every timestep, because the timestep is significantly
larger than in an atomistic simulation
Conclusions
• We have developed a numerically stable and accurate level set method to
describe epitaxial growth.
• The model is very efficient when processes with vastly different rates need
to be considered
• This framework is ideally suited to include anisotropic, spatially varying
diffusion (that might be a result of strain):
• Islands nucleate preferentially in regions of fast diffusion (when the
adsorption energy is constant)
• However, a strong drift term can dominate over fast diffusion
• A properly modified potential energy surface can be exploited to obtain
a high regularity in the arrangement of islands.
More details and transparencies of this talk can be found at
www.math.ucla.edu/~cratsch