A Level-Set Method for Modeling Epitaxial Growth and Self-Organization of Quantum Dots

Christian Ratsch, UCLA, Department of Mathematics

Collaborators:

•Xiaobin Niu

•Raffaele Vardavas

•Russel Caflisch

Outline:

•The level-set method for epitaxial growth

•Results for irreversible and reversible aggregation

•Spatially varying diffusion can be used for self-organization of islands (quantum dots)

• Coupling of level-set formalism with an elastic model.

Modeling thin film growth

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Methods used

(Atomistic) KMC simulations:

• Completely stochastic method• Rate parameters can be obtained from DFT

Continuum equations (PDEs):

• essentially deterministic• no microscopic details.• parameters can be obtained from atomistic model (but difficult)

Time scale Length scale

Atomic motion

~ 1013

secondsÅngstroms

Islands/ devices

Seconds - hours

Microns and larger

New Method

Level set method:

• PDE - based, (almost) deterministic• atomistic details can be included• microscopic parameters can be obtained from DFT

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Atomistic picture(i.e., kinetic Monte Carlo)

Idea of the level set appproach

F

D

v

Island dynamics

•Describe motion of island boundaries by a level-set function

•Adatoms are described in a mean-field approach with a diffusion equation

The level set method: schematic

Level set function Surface morphology

t

=0

=0

=0

=0=1

• Level set function is continuous in plane, but has discrete height resolution

• Adatoms are treated in a mean field picture

The level set method: the basic formalism

)( nnDvn• Velocity:

• Governing Equation: 0||

nvt

),( det xDeq

=0

Seeding position chosen stochastically(weighted with local value of 2)

2),( tDdt

dNx• Nucleation Rate:

dt

dNDF

t22

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

• Boundary condition:

• Stochastic break-up of islands (depends on: , size, local environment)detD

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)

S. Chen, M. Kang, B. Merriman, R.E. Caflisch, C. Ratsch, R. Fedkiw, M.F. Gyure, and S. Osher, JCP (2001)

A typical level set simulation

Fluctuations need to be included in nucleation of islands

Nucleation rate: 2),( tDdt

dNx

Probabilistic seedingweight by local 2

max

C. Ratsch et al., Phys. Rev. B 61, R10598 (2000)

Validation:

Scaling of island densities

Nucleation Theory:N ~ (D/F)-1/3

Scaled island size distribution

Detachment of adatoms and breakup of islands

• Detachment of atoms (from boundary) is accounted for by boundary condition:

• The numerical timestep remains unchanged. Thus, no increase in CPU time!

• Stochastic element is needed for breakup of islands

),( det xDeq

• For “small” islands, calculate probability of island break-up.• This probability is related to Ddet, and local environment• Pick random number to decide break-up• If island is removed, atoms are distributed uniformly in an area that corresponds to the diffusion length

Validation: Scaling and sharpening of island size distribution

Experimental Data for Fe/Fe(001),Stroscio and Pierce, Phys. Rev. B 49 (1994)

Petersen, Ratsch, Caflisch, Zangwill, Phys. Rev. E 64, 061602 (2001).

Computational efficiency

• Fast events can be included without decreasing the numerical timestep (due to mean-field treatment of adatoms)

Modeling self-organization of quantum dots

• Ultimate goal: Solve elastic equations at every timestep, and couple the strain field to the simulation parameters (i.e., D, Ddet).

• This is possible because the simulation timestep can be kept rather large.

• Needed: Spatially varying, anisotropic diffusion and detachment rates.Modifications to the code will be discussed!

• So far: We assume simple variation of potential energy surface.

• Next (with some preliminary results): couple with elastic code of Caflisch, Connell, Luo, Lee

Vertical alignment of stacked quantum dots

B. Lita et al. (Goldman group), APL 74, 2824 (1999)

•Islands nucleate “on top” of lower islands

•Size and separation becomes more uniform

•Interpretation: buried islands lead to strain (there is a 7% misfit)

Spatially varying potential energy surface

Spatially varying nucleation probability

Stacked InAs quantum dots on GaAs(001)

Aligned islands due to buried dislocation lines

Ge on relaxed SiGe buffer layers

Level Set formalism is ideally suited to incorporate anisotropic, spatially varying diffusion and thus nucleation without extra computational cost

• Islands align along lines

• Dislocation lines are buried underneath

• Interpretation: buried dislocation lines lead to strain

Spatially varying potential energy surface

Spatially varying nucleation probability

H. J. Kim, Z. M. Zhao, Y. H. Xie, PRB 68, 205312 (2003).

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 variations of potential energy surface

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

Isotropic diffusion with sinusoidal variation in x- and y-direction

)sin()sin(~ ayaxDD yyxx

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

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

Constant adsorption energy(no drift)

Constant transition energy (thermodynamic drift)

x

D

•No drift (right): nucleation dominated by fast diffusion

•Large Drift (left): nucleation dominated by drift

2),( tDdt

dNx

Time evolution in the kinetic limit

• A properly modified PES (in the “kinetic limit”) leads to very regular, 1-D structures• Can this approach used to produce quantum wires?

Combination of island dynamics model with elastic code

•In contrast to an atomistic (KMC) simulation, the timestep is rather large, even when we have a large detachment rate (high temperature).

•A typical timestep in our simulation is O(10-2 s); compare to typical atomistic simulation, where it is O(10-6 s).

•This allows us to do an “expensive calculation” at every timestep.

•For example, we can solve the elastic equations at every timestep, and couple the local value of the strain to the microscopic parameters.

•This work is currently in progress ….. but here are some initial results.

•Write down an atomistic energy density, that includes the following terms (lattice statics) (this is work by Caflisch, Connell, Luo, Lee, et al.):

Nearest neighbor springs

Diagonal springs

Bond bending terms

Our Elastic model

• Minimize energy with respect to all displacements: u E [u] = 0

•This can be related to (and interpreted as) continuum energy density

22 )2()2( yyxyxxdiagyyxyxxdiag SSSkSSSkE

)( 22yyxx SSkE

2xymSE

yyxxxyyyxx SSSSSE 222 )(

Numerical Method

• PCG using Algebraic MultiGrid (poster by Young-Ju Lee)

• Artificial boundary conditions at top of substrate (poster by Young-Ju Lee)

• Additional physics, such as more realistic potential or geometry easily included

Couple elastic code to island dynamics model

Sxx

Syy

Example:

• Epilayer is 4% bigger than substrate (I.e., Ge on Si)

• Choose elastic constants representative for Ge, Si

• Deposit 0.2 monolayers

Modification of diffusion field

The dependence of D on strain can be based on DFT results.

Example: Stain dependent diffusion for Ag/Ag(111)

C. Ratsch, A.P. Seitsonen, and M. SchefflerPhys. Rev. B 55, 6750-6753 (1997). )(0, yyxxdiffdiff SSconstEE

Constant diffusion Change diffusion as a function of strain at every timestep

•It is not clear whether there is an effect on ordering

•More quantitative analysis needed

Results with strain-dependent detachment rate

Modification of detachment rates

• The detachment rate has only physical meaning at the island edge (where it changes the boundary condition eq)

• The model shown here indicates that it is more likely to detach from a bigger (more strained island) than from a smaller one.

• Previous (KMC) work suggests that this leads to more uniform island size distribution.

220det,det yyxx SSconstEE

No change of Ddet Strain induced change of Ddet at every timestep

Results with strain-dependent detachment rate

•Maybe fewer islands are close together upon strain induced increase of Ddet (?)

•Obviously, a more quantitative analysis is needed!

Conclusions

• We have developed a numerically stable and accurate level set method to describe epitaxial growth.

• Only the relevant microscopic fluctuations are included.

• Fast events can be included without changing the timestep of the simulations.

• This framework is ideally suited to include anisotropic, spatially varying diffusion.

• A properly modified potential energy surface can be exploited to obtain a high regularity in the arrangement of islands.

• We have combined this model with a strain model, to modify the microscopic parameters of the model according to the local value of the strain.

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

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: