A Level-Set Method for Modeling Epitaxial Growth and Self-Organization of Quantum Dots
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Transcript of A Level-Set Method for Modeling Epitaxial Growth and Self-Organization of Quantum Dots
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
9750-00-444
(a) (a)
(h)
(f) (e) (b)
(c)
(i)
(g)
(d)
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
9750-00-444
(a) (a)
(h)
(f) (e) (b)
(c)
(i)
(g)
(d)
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: