T. P. Schulze Department of Mathematics University of ... · 5/10/2004 · Minimal search KMC...

Some new tools for simulating

nano-scale crystal growth

T. P. Schulze

Department of Mathematics

University of Tennessee

Thanks to NSF (grant 0103825) for supporting thisresearch.

Outline

1. Kinetic Monte Carlo (KMC) & KMC algorithms

2. Burton-Cabrera-Frank (BCF) model & hybrid scheme

3. Off lattice KMC

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Kinetic Monte-Carlo

• Early work:

– F.F. Abraham and G.W. White (1970)

– G.H. Gilmer and P. Bennema (1972)

• Stochastic adatom deposition.

• Hopping rate k(∆φ, T) = k0 exp (−∆φ/kBT)

• Barriers can be measured, calculated or modeled...

φ

S=0 S>0

• Example:

∆φ = ES + mEN + max[(mi − mf),0]EB

(Smilauer & Vvdensky 1995)

Poisson Processes

The Poisson process assumes discrete events with ex-ponentially distributed waiting times:

p(t) = r exp (rt), t > 0,

where r is the rate at which the process occurs, i.e.the average frequency per unit time.

The expected waiting time is

〈tp(t)〉 =

∫ ∞

0

tp(t)dt =1

r.

The CDF, probability of waiting less than time t, is

P({t < T}) =

∫ T

0

p(t)dt = 1 − exp (−rT).

Multiple Processes

For two independent processes occurring with rates r1and r2, the probability of at least one event occurringbefore time T is

P({t1 < T or t2 < T}) = P1(T) + P2(T) − P1(T)P2(T)

= 1 − exp [−(r1 + r2)T ]

The corresponding distribution of waiting times re-mains exponential with a rate that is the sum of therates of the individual processes:

p(t) = (r1 + r2) exp [−(r1 + r2)t].

This generalizes to an arbitrary number of processes.

Kinetic Monte-Carlo algorithms

Principal algorithm: Bortz, Kalos &. Lebowitz (1975)

1. Randomly select the time it takes for the nextevent to occur

2. Decide which event it is using relative rates

This simulates M independent Poisson processes withrates rm that sum to give an overall rate R.

Performance of KMC algorithms:

• Rejection algorithm: usually not good for KMC

• Vanilla BKL with linear search: O(M)

• Binning technique: O(M1/2) (Mayksum 1988)

• Binary search: O(log(M))(Blue, Beichl & Sullivan 1995)

• Maintaining inverse list can eliminate this cost:(Schulze 2002)

Enm : (n, m) → (i, j, k)

E−1i,j,k : (i, j, k) → (m, n)

Minimal search KMC algorithm:

1. Compute the overall rate R = SN =∑N

n=1 rnCn;retain the partial sums Sn

2. Select a random number r ∈ [0, R).

3. Search through the list of partial sums Sn untilSn > r

4. Select an event from the set of events that occurat this rate by computing

m = Int

(

(Sn − r)

rn

)

+ 1.

5. Execute that event and update the configuration{hij}.

6. For the (local) events that have their rates changedfrom rni to rnf:

(a) Move them to the end of column nf ; add one

to Cnf ; update E−1i,j,k.

(b) Move the event listed as EniCni into the vacatedposition in column ni; reduce Cni by one; up-date E−1i,j,k.

Performance

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p

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Computation time for 105 events of a 2p × 2p

simulation for binary search (upper curve) and

minimal-search KMC (lower curve).

• If necessary, event lists can be packed into

a flat array of length M

• For arbitrary rates, this can be combined

with the rejection algorithm.

Other facets of film-growth modelling

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��

Terrace/Step models

Reference: Burton, Cabrera and Frank (1954)

x x31

x2

ρt = Dρxx − τ−1ρ + F ; x ∈ (xi, xi+1)

±Dρx|± = α±(ρ|± − ρe), x = xi

νdxidt

=

[

Dρx +dxidt

ρ

]+

−, x = xi

Hybrid Scheme

Collaborators: P. Smereka (Mich.), W. E (Princeton)

• Diffusion equation on terraces:

ρt = Dρxx + F ∈ ΩBCF

• KMC simulation near step edges:

Ω = ΩBCF ∪ ΩKMC

Two-grid system

• N × N sites partitioned into Nc × Nc cells

• Cell-width M × M (N = MNC)

• Cell is ∈ ΩKMC if

– it contains an edge

– its neighbors contain an edge

KMC region

• Adatoms are defined as sites where hij

– is one greater than at lateral nearest neighbors,

– all of which have the same height.

• For Hybrid simulation, we add events correspond-ing to:

– hops out of the continuum with rate ∼ Dρ

– nucleation ∼ ρ2?

– other processes as desired...

• This requires a list of boundary segments

Continuum region

ρn+1j = ρnj + D

∆t

∆x2

(

ρnj−1 − 2ρnj + ρ

nj+1

)

+ F∆t

• ρnj is average “adatom” density in cell j

• Discrete time step ∆t,

• Coarse grid ∆x = aM ,

• There are many KMC events per ∆t

t∆ {t

Interface

• If adatom ΩKMC → ΩBCF

– remove the adatom from list

– set ρn+νj = ρnj + 1/M

– mass is immediately spread over cell

• If adatom ΩBCF → ΩKMC

– add adatom to list

– set ρn+νj = ρnj − 1/M

– rate for this event type is Dρj

– event not added to list if ρj < 1/M

• ν indicates fractional timestep (one event)

Moving boundary

• If a cell changes type, a local conversion processtakes place:

– To change KMC-cell → BCF-cell

∗ Locate adatoms using inverse list

∗ Remove hopping events from list

∗ Update cell-list

– To change BCF-cell → KMC-cell

∗ Randomly position Int(Mρj) adatoms in cell

∗ Add hopping events to list

∗ Update cell-list

Nucleation, etc.

• To nucleate islands within the BCF region

– Add nucleation events with rate ∼ (Mρj)2?

– If event is chosen, convert cell and its neighbors

• This approach has proven useful in level-set simu-lations of BCF model

• A similar procedure applies to vacancy formation,chemical reactions, etc.

• Deposition and evaporation can be handled at thecontinuum level.

1D Hybrid Simulations

Adatoms per cell

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• Continuum regions tend toward parabolic adatomdensity profile

• Step velocities, step widths and number of adatomson surface fluctuate

Average density and fluctuations

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σe

D/F

++++++++

+

+

3333333

33

3

++++++++

+

+

++++++++

+

+

0

0.005

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0.015

0.02

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σρ

D/F

++++++++++

33333333

3

3

++++++++++ ++++++++

++

Computational cost (1D)

• For 1D KMC:

– N net attachment events per layer

– Each adatom performs O(L2) hops

– Total cost is O(NL2) (or O(N logN L2))

• For 1D hybrid scheme:

– Total cost = KMC-cost + BCF-cost

– KMC-cost reduced to ⇒ O(NM2)

– BCF-cost is: O(

(NM2)( LNMD

)( 1∆t

)(NM

))

(events/layer)(time/event)(calls/time)(operations/call)

– CFL condition requires ∆t < M2

2D

– Total cost is O(NM2) + O(NLM2

)

Accuracy

• Errors inherent to BCF model:

– Errors resulting from interactions

– Errors from neglected events

• Errors relative to “adatom” KMC:

– Discretization error ∼ M2 ≡ (∆x)2

– Variance reduction

– Coupling error

cell scale

ρ

Step edge instability

Experiment: Behm, et. al. Phys. Rev. Lett.(1994)

Theory: Bales & Zangwill, Phys. Rev. B

(1990)

Simulation: Rost, Smilauer & Krug, Surf. Sci-

ence (1996).

KMC and Hybrid surface

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• KMC (left) took about 12 days to simiulate.

• Hybrid (right) took about 56 hours with cell width10

KMC and Hybrid surface

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• KMC (left) took about 12 days to simiulate.

• Hybrid (right) took about 70 hours with cell width15

Hybrid and Adaptive KMC surface

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The simulaion on the right uses a crude, coarse-grainedKMC instead of solving a discretized diffusion equation.

It was slightly faster.

Off-lattice KMC

Collaborators:

Weidong Guo (post-doc, UTK)Weinan E (Princeton)

• In all KMC models, transition state theory predictsrates of the form R ∼ exp[−∆Φ]

• In lattice-based models, the energy landscape Φ isprescribed.

• In off-lattice KMC, we wish to allow an arbitraryset of atom positions {ri}.

• The potential energy of such a collection is oftenmodeled as a sum of pair-wise interactions, e.g.the Lennard-Jones potential:

Φ =∑

ij

Φij =∑

ij

[

(

σ

rij

)12

−

(

σ

rij

)6]

Off-lattice KMC Algorithm

1. Identify the set of likely transitions xik (i.e. newlocal minima) for each particle

xik = minx∈Sik

Φ(x, {rj 6=i})

2. Identify the barrier ∆Φ = Φ∗ −Φ0 for each transi-tion, where

Φ∗ = minri(s)∈P

maxs∈[0,1]

Φ(ri, {rj 6=i})

3. Use the BKL algorithm to randomly select andexecute an event

4. Locally relax new configuration, with occasionalglobal relaxation

Off-lattice KMC

• This is rather slow.

• Applications might include nano-tubes, wires, pipeflow,gears, levers...

Summary

• Cross-indexed lists

– provides a minimal-search KMC algorithm

– combine with rejection for arbitrary rate sets

• Hybrid scheme

– significantly faster than KMC

– many complications with respect to accuracy

• Work in progress

– Off-lattice KMC

– Adaptive/multi-grid KMC

For further details...

T.P. Schulze,“A Hybrid Method for Simulating Epitax-ial Growth,” Journal of Crystal Growth 263 (2004)605-615.

T.P. Schulze, P. Smereka and Weinan E, “CouplingKinetic Monte-Carlo and Continuum Models with Ap-plication to Epitaxial Growth,” J. Comp. Phys. 189(2003) 197-211.

T.P. Schulze, “Kinetic Monte-Carlo with Minimal Search-ing,” Phys. Rev. E 65 (2002) 036704.

http://www.math.utk.edu/ schulze/publications.htm

T. P. Schulze Department of Mathematics University of ... · 5/10/2004 · Minimal search KMC...

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