A Level-Set Method for Modeling Epitaxial Growth and Self-Organization of Quantum Dots Christian Ratsch, UCLA, Department of Mathematics •Russel Caflisch •Xiabin Niu •Max Petersen •Raffaello Vardavas Collaborators $$$: NSF and DARPA Santa Barbara, Jan. 31, 2005 Outline • Introduction • The basic island dynamics model using the level set method • Include Reversibility Ostwald Ripening • Include spatially varying, anisotropic diffusion self-organization of islands
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A Level-Set Method for Modeling Epitaxial Growth and Self-Organization of Quantum Dots
Outline Introduction The basic island dynamics model using the level set method Include Reversibility Ostwald Ripening Include spatially varying, anisotropic diffusion self-organization of islands . - PowerPoint PPT Presentation
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A Level-Set Method for Modeling Epitaxial Growth and Self-Organization of Quantum Dots
Christian Ratsch, UCLA, Department of Mathematics
•Russel Caflisch
•Xiabin Niu
•Max Petersen
•Raffaello Vardavas
Collaborators
$$$: NSF and DARPA
Santa Barbara, Jan. 31, 2005
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
What is Epitaxial Growth?
Atomic Motion Time Scale ~ 10-13 seconds Length Scale: AngstromIsland Growth Time Scale ~ seconds Length Scale: Microns
o
9750-00-444
(a) (a)
(h)
(f) (e) (b)
(c)
(i)
(g)
(d)
– = “on” – “arrangement”
Why do we care about Modeling Epitaxial Growth?
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
• 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.
KMC Simulation of a Cubic, Solid-on-Solid Model
ES: Surface bond energyEN: Nearest neighbor bond energy0 : Prefactor [O(1013s-1)]
• Parameters that can be calculated from first principles (e.g., DFT)
• Completely stochastic approach
• But small computational timestep is required
D = 0 exp(-ES/kT) F
Ddet = D exp(-EN/kT)
Ddet,2 = D exp(-2EN/kT)
KMC Simulations: Effect of Nearest Neighbor Bond EN
Large EN:IrreversibleGrowth
Small EN:CompactIslands
Experimental Data
Au/Ru(100)
Ni/Ni(100)Hwang et al., PRL 67 (1991) Kopatzki et al., Surf.Sci. 284 (1993)
440°C0.083 Ml/s20 min anneal
380°C0.083 Ml/s60 min anneal
KMC Simulation for Equilibrium Structures of III/V SemiconductorsExperiment(Barvosa-Carter, Zinck)
KMC Simulation(Grosse, Gyure)
Problem:Detailed KMC simulations are extremely slow !
Similar work by
Kratzer and Scheffler
Itoh and Vvedensky
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 diffusionself-organization of islands
The Island Dynamics Model for Epitaxial Growth
9750-00-444
(a) (a)
(h)
(f) (e) (b)
(c)
(i)
(g)
(d)
Atomistic picture(i.e., kinetic Monte Carlo)
F
D
v
• 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)
Island dynamics
The Level Set Method: SchematicLevel Set Function Surface Morphology
t
=0
=0
=0
=0=1
• 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: 0||
nvt=0
dtdNDF
t22
• Diffusion equation for the adatom density (x,t):
)( nnDvn• Velocity:
0
2),( tDdtdN x• Nucleation Rate:
• Boundary condition:
C. Ratsch et al., Phys. Rev. B 65, 195403 (2002)
Typical Snapshots of Behavior of the Model
t=0.1
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
ii-1 i+1 i+2
Need 4 points to discretize with third order accuracy
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”)
i-3 i-2 i+3 i+4
Set 1 Set 2 Set 3
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)
Solution of Diffusion Equation dtdNDF
t22
• Standard Discretization: 2
11
111
1
)(2x
Dt
ki
ki
ki
ki
ki
• Leads to a symmetric system of equations:
• Use preconditional conjugate gradient method
bAρ 1k
Problem at boundary:
i-2 i-1 i i+1
x1
0f
xx
xxiiif
ixx
1
1
1
21
)(
Matrix not symmetric anymore
xxxiiig
ixx
1
)(
: Ghost value at i“ghost fluid method”
g
g; replace by:
Nucleation Rate:
Fluctuations need to be included in nucleation of islands
2),( tDdtdN x
Probabilistic Seedingweight by local 2
max
C. Ratsch et al., Phys. Rev. B 61, R10598 (2000)
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 diffusionself-organization of islands
• 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 re-attachment of atoms from islands Significant computational cost !
• In Levelset formalism: Simply modify velocity (via a modified boundary condition), but keep timestep fixed
•Stochastic break-up for small islands is important
Extension to Reversibility
)( nnDvnVelocity:
),( det xDeq
2),( tDdtdN xNucleation Rate:
• Boundary condition:
• 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)
Details of stochastic break-up
•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
3/1tR
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 diffusionself-organization of islands
Nucleation and Growth on Buried Defect Lines
Growth on Ge on relaxed SiGe buffer layer
Dislocation lines are buried underneath. • Lead to strain field• This can alter potential energy surface: