1. 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
The basic island dynamics model using the level set method
Include Reversibility Ostwald Ripening
Include spatially varying, anisotropic diffusion self-organization of islands
Russel Caflisch
Xiabin Niu
Max Petersen
Raffaello Vardavas
$$$: NSF and DARPA

2. What is Epitaxial Growth?
epi – taxis = “on” – “arrangement” o

3. 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

4. KMC Simulation of a Cubic, Solid-on-Solid Model
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

5. 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)

6. KMC Simulation for Equilibrium Structures of III/V Semiconductors
Experiment
(Barvosa-Carter, Zinck)
KMC Simulation
(Grosse, Gyure)
Similar work by
Kratzer and Scheffler
Itoh and Vvedensky
380°C
0.083 Ml/s
60 min anneal
440°C
0.083 Ml/s
20 min anneal
Problem:
Detailed KMC simulations are extremely slow !
F. Grosse et al., Phys. Rev. B66, (2002)

7. 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

8. The Island Dynamics Model for Epitaxial Growth
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

9. The Level Set Method: Schematic
Level Set Function j
Surface Morphology t
j=0
j=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

10. The Basic Level Set Formalism for Irreversible Aggregation
j=0
Governing Equation:
Diffusion equation for the adatom density r(x,t):
Boundary condition:
Velocity:
Nucleation Rate:
C. Ratsch et al., Phys. Rev. B 65, (2002)

11. Typical Snapshots of Behavior of the Modelj r
t=0.5

12. 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)

13. Essentially-Non-Oscillatory (ENO) Schemes
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

14. 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)

15. Solution of Diffusion Equation
Standard Discretization:
Leads to a symmetric system of equations:
Use preconditional conjugate gradient method
Problem at boundary:
i-2
i-1 i
i+1
Matrix not symmetric anymore
: Ghost value at i
“ghost fluid method”
; replace by:

16. Fluctuations need to be included in nucleation of islands
Nucleation Rate:
rmax r
Probabilistic Seeding
weight by local r2
C. Ratsch et al., Phys. Rev. B 61, R10598 (2000)

17. A Typical Level Set Simulation

18. 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

19. 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 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
Boundary condition:
Nucleation Rate:
Velocity:

20. 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.

21. 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, (2001).

22. Scaling of Computational Time
Almost no increase in computational time due to mean-field treatment of fast events

23. Ostwald Ripening Verify Scaling Law
Slope of 1/3
M. Petersen, A. Zangwill, and C. Ratsch, Surf. Science 536, 55 (2003).

24. 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

25. 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

26. Modifications to the Level Set Formalism for non-constant Diffusion
Replace diffusion constant by matrix:
Diffusion in x-direction
Diffusion in y-direction
drift
no drift
Possible potential energy surfaces
Diffusion equation:
Velocity:
Nucleation Rate:

27. 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 in-between)

28. Isotropic Diffusion with Sinusoidal Variation in x-Direction
fast diffusion
slow diffusion
Only variation of transition energy, and constant adsorption energy
Islands nucleate in regions of fast diffusion
Little subsequent nucleation in regions of slow diffusion

29. Comparison with Experimental Results
Results of Xie et al.
(UCLA, Materials Science Dept.)
Simulations

30. Anisotropic Diffusion with Sinusoidal Variation in x-Direction
In both cases, islands mostly nucleate in regions of fast diffusion.
Shape orientation is different

31. Isotropic Diffusion with Sinusoidal Variation in x- and y-Direction

32. Comparison with Experimental Results
Results of Xie et al.
(UCLA, Materials Science Dept.)
Simulations

33. 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

34. Transition from thermodynamically to kinetically controlled diffusion
Constant transition energy (thermodynamic drift)
Constant adsorption energy
(no drift)
From constant Transition energy to const Absorption energy.
In a certain position, D is always a constant. Et- Ea = ~ 1.111eV (D = 104 ~ 108) D x
But: In all cases, diffusion constant D has the same form:

35. 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

36. 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