Dynamic Scaling of Surface Growth in Simple Lattice Models

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Dynamic Scaling of Surface Growth in Simple
Lattice Models
Croucher ASI on Frontiers in Computational
Methods and Their Applications in Physical
Sciences Dec. 6 - 13, 2005 The Chinese
University of Hong Kong
D. P. L. S. Pal K. Binder ?
Background ? Models and Simulation Method ?
Results Surface properties Temporal
correlations ? Summary and Conclusions

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Simulation

NATURE
Experiment
Theory
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Simulation (Monte Carlo)

NATURE
Experiment (MBE, LEED, RHEED)
Theory (Growth eqns.)
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Why MBE?
- the promise of designer materials e.g. mu
ltilayers ________________
quantum wires (vicinal surfaces)
Theoretical questions binding energies - Quant
Mech large scale structures - Stat Mech
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Theoretical Background
? non-equilibrium (equilibrium ? roughening
transition) define height above L? L
substrate mean height surface
width structure factor
? local order parameter
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Comprehensive growth equation
random noise ?
h deviation of surface height from the
mean
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Edwards-Wilkinson growth equation(sedimentation)
random noise ?
h deviation of surface height from the
mean
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Question What happens when t ? ? ? simple
model studies EW sedimentation model (Edwards
Wilkinson, 1982) KPZ equation (Kardar, Parisi
Zhang, 1986) random deposition (Family,
1986) restricted SOS model (Kim Kosterlitz,
1989) growth-diffusion model (Wolf Villain,
1990) MBE models (Pal and Landau, 1993) and
many more . . .
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Surface Width Dynamic Finite Size Scaling
Define z ?/? dynamic exponent
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Computational Study of Film Growth
Surface Science ? Statistical
Mechanics Multiple processes deposition
diffusion Methods ? Ab initio Molecular
Dynamics ? Classical Molecular Dynamics
(phenomenological potentials) O ? O O
OOO OO OO ? surface ? Discrete stochastic SOS
models ? surface
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Atomistic Edwards-Wilkinson Model
? L? L square lattice substrate (p. b. c.) ?
Growing film held at constant temperature T ?
Particles fall randomly on the surface, then
diffuse to the neighboring site with the
greatest depth
? constant flux
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Atomistic Edwards-Wilkinson Model

• BUT, what if more than one neighboring site has
  the same depth?

? constant flux
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Atomistic Edwards-Wilkinson Model

• BUT, what if more than one neighboring site has
  the same depth?
• Generate a random number to decide!

? constant flux
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(No Transcript)
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Monte Carlo is Serious Science!
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Simulations of MBE Monte Carlo (MC) versus
Kinetic Monte Carlo (KMC)
Deposition diffusion ?
O OOOOOOOOOO MC KMC
In KMC we must consider more than just the final
particle state!
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MBE Model Growth (KMC)
UDP Model
RHEED intensity-growth of GaAs
(Neave et al, 1985)
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MBE Model Growth (KMC)
What happens at long times?
Dynamic finite size scaling shows z1.65 But the
Edwards- Wilkinson eqn. yields z2.0
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Atomistic 21 dim EW Model
Interfacial width What happens at long times?
(Note For large systems gt1010 random numbers are
needed per run)
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Atomistic 21 dim EW Model
Interfacial width Dynamic Finite Size
Scaling for the EW equation ? 0 , so
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Atomistic 21 dim EW Model
Interfacial width Dynamic Finite Size Scaling
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Atomistic 21 dim EW Model
Interfacial width Dynamic Finite Size Scaling
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Atomistic 21 dim EW Model
Structure Factor Dynamic Finite Size Scaling
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Atomistic 21 dim EW Model
Structure Factor Dynamic Finite Size Scaling
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Atomistic 21 dim EW Model
Structure Factor Dynamic Finite Size Scaling

• Data do NOT scale for z2.0 !

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Atomistic 21 dim REW Model
Restricted Edwards-Wilkinson Model When two or
more neighboring sites have equal depth, the
particle does not diffuse!
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Atomistic 21 dim REW Model
Interfacial width What happens at long times?
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Atomistic 21 dim REW Model
Interfacial width Dynamic finite size scaling

• Data scale (for long times) with z2.0 !

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Atomistic 21 dim REW Model
Structure factor Dynamic Finite Size Scaling
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Atomistic 21 dim REW Model
Structure factor Dynamic Finite Size Scaling
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Growth equation
h deviation of surface height from the mean
surface stiffness Measure surface properties
numerically
Average quantities over b? b blocks of sites ?
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Growth equation
h deviation of surface height from the mean
generalized noise Measure
surface properties numerically
Average quantities over b? b blocks of sites ?
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Atomistic 21 dim EW Model
Surface stiffness
Stiffness decays to a constant value
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Atomistic 21 dim EW Model
Non-equilibrium contribution to the interface
velocity
i.e. generalized noise
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Atomistic 21 dim EW Model
Non-equilibrium contribution to the interface
velocity
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Atomistic 21 dim EW and REW Models
Non-equilibrium contribution to the interface
velocity
For random noise, U(b,t) should decay to 0 !
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Time-displaced Correlation Function
To study temporal correlations in U(b,t), define
blocking factor
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Time-displaced Correlation Function
EW model (finite size effects)
Note C(b,?) is independent of L
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Time-displaced Correlation Function
EW model (time dependence)
C(b,?) decays non-exponentially !
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Time-displaced Correlation Function
REW model
Correlations decay exponentially fast to 0 !
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Time-displaced Correlation Function
Dynamic scaling
where
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Time-displaced Correlation Function
EW model
z
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Time-displaced Correlation Function
EW model (define P(b,?)C(b,?)/? ? (b)-1 )
For b gt 15, get scaling with z 1.65
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Time-displaced Correlation Function
EW model
No scaling for z 2.0 !
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Summary and Conclusions

• Simple lattice models for surface growth have
  behavior that depends on the noise
• The atomistic EW model does not have the same
• behavior as the EW equation!
• but the REW model ? the EW equation.
• Time-displaced correlations are non-exponential
  for the atomistic EW model ? the use of a random
  number to choose one of the degenerate neighbor
  sites creates a 2nd source of (correlated) noise.
• Challenge for the future
• - Study other models by simulation to extract
  the noise ? Universality classes?