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 Semiconductors

Experiment(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 diffusion

self-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: Schematic

Level 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

dt

dNDF

t22

• Diffusion equation for the adatom density (x,t):

)( nnDvn• Velocity:

0

2),( tDdt

dNx• 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 dt

dNDF

t22

• Standard Discretization: 2

11

111

1

)(

2

xD

t

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

x

xxiiig

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),( tDdt

dNx

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 diffusion

self-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),( tDdt

dNxNucleation 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 diffusion

self-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:

• Anisotropic diffusion• Spatially varying diffusion

Hypothesis:Nucleation occurs in regions of fast diffusion

Results of Xie et al.(UCLA, Materials Science Dept.)

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

)()( DnDnnv• Velocity:

2),(2

)()(t

DD

dt

dN yyxx xxx

• Nucleation Rate:

)(0

0)()(

x

xxDD

yy

xx

D

D• Replace diffusion constant by matrix:

Diffusion in x-direction Diffusion in y-direction

drift2)(

dt

dNF

t

D• Diffusion equation:

adad~drift EDED yyyxxx

drift

no drift

Possible potential energy surfaces

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)

Isotropic Diffusion with Sinusoidal Variation in x-Direction

)sin(~ axDD yyxx

fast diffusion slow diffusion

• Islands nucleate in regions of fast diffusion

• Little subsequent nucleation in regions of slow diffusion

Only variation of transition energy, and constant adsorption energy

Comparison with Experimental Results

Results of Xie et al.(UCLA, Materials Science Dept.)

Simulations

Anisotropic Diffusion with Sinusoidal Variation in x-Direction

)sin(~ axDxx .constDyy )sin(~ axDyy .constDxx

• 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

)sin()sin(~ ayaxDD yyxx

Comparison with Experimental Results

Results of Xie et al.(UCLA, Materials Science Dept.)

Simulations

Anisotropic Diffusion with Variation of Adsorption Energy

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

What is the effect of thermodynamic drift ?

Etran

Ead

Transition from thermodynamically to kinetically controlled diffusion

But: In all cases, diffusion constant D has the same form:

D

x

Constant adsorption energy(no drift)

Constant transition energy (thermodynamic drift)

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 atwww.math.ucla.edu/~cratsch