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

2. Simulation
NATURE
Experiment
Theory

3. NATURE Simulation (Monte Carlo) Experiment Theory (MBE, LEED, RHEED)
(Growth eqns.)

4. Why MBE? - the promise of designer materials e.g. multilayers
________________
quantum wires
(vicinal surfaces)
Theoretical questions:
binding energies - Quant Mech
large scale structures - Stat Mech

5. Theoretical Background
 non-equilibrium (equilibrium  roughening transition)
define:
height above L L substrate
mean height
surface width
structure factor 
local order parameter

6. Comprehensive growth equation:
random noise 
h = deviation of surface height from the mean

7. Edwards-Wilkinson growth equation: (sedimentation)
random noise 
h = deviation of surface height from the mean

8. 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 . . .

9. Surface Width Dynamic Finite Size Scaling
Define: z =/ = dynamic exponent

10. 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

11. 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

12. Atomistic Edwards-Wilkinson Model
BUT, what if more than one neighboring site has the same depth? 
constant flux

13. Atomistic Edwards-Wilkinson Model
BUT, what if more than one neighboring site has the same depth?
Generate a random number to decide! 
constant flux

15. Monte Carlo is Serious Science!

16. 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!

17. MBE Model Growth (KMC) UDP Model RHEED intensity-growth of GaAs
(Neave et al, 1985)

18. MBE Model Growth (KMC) What happens at “long times”?
Dynamic finite size scaling shows z=1.65
But the Edwards- Wilkinson eqn. yields z=2.0

19. Atomistic 2+1 dim EW Model
Interfacial width: What happens at “long times”?
(Note: For large systems >1010 random numbers are needed per run)

20. Atomistic 2+1 dim EW Model
Interfacial width: Dynamic Finite Size Scaling
…for the EW equation  = 0 , so

21. Atomistic 2+1 dim EW Model
Interfacial width: Dynamic Finite Size Scaling

22. Atomistic 2+1 dim EW Model
Interfacial width: Dynamic Finite Size Scaling

23. Atomistic 2+1 dim EW Model
Structure Factor: Dynamic Finite Size Scaling

24. Atomistic 2+1 dim EW Model
Structure Factor: Dynamic Finite Size Scaling

25. Atomistic 2+1 dim EW Model
Structure Factor: Dynamic Finite Size Scaling
Data do NOT scale for z=2.0 !

26. Atomistic 2+1 dim REW Model
Restricted Edwards-Wilkinson Model: When two or more neighboring sites have equal depth, the particle does not diffuse!

27. Atomistic 2+1 dim REW Model
Interfacial width: What happens at “long times”?

28. Atomistic 2+1 dim REW Model
Interfacial width: Dynamic finite size scaling
Data scale (for long times) with z=2.0 !

29. Atomistic 2+1 dim REW Model
Structure factor: Dynamic Finite Size Scaling

30. Atomistic 2+1 dim REW Model
Structure factor: Dynamic Finite Size Scaling

31. Growth equation: h = deviation of surface height from the mean
surface stiffness
Measure surface properties numerically:
Average quantities over b b blocks of sites 

32. Growth equation: h = deviation of surface height from the mean
“generalized noise”
Measure surface properties numerically:
Average quantities over b b blocks of sites 

33. Atomistic 2+1 dim EW Model
Surface stiffness
Stiffness decays to a constant value

34. Atomistic 2+1 dim EW Model
Non-equilibrium contribution to the interface velocity*
* i.e. “generalized noise”

35. Atomistic 2+1 dim EW Model
Non-equilibrium contribution to the interface velocity

36. Atomistic 2+1 dim EW and REW Models
Non-equilibrium contribution to the interface velocity
For random noise, U(b,t) should decay to 0 !

37. Time-displaced Correlation Function
To study temporal correlations in U(b,t), define
blocking factor

38. Time-displaced Correlation Function
EW model (finite size effects)
Note: C(b,) is independent of L

39. Time-displaced Correlation Function
EW model (time dependence)
C(b,) decays non-exponentially !

40. Time-displaced Correlation Function
REW model
Correlations decay exponentially fast to 0 !

41. Time-displaced Correlation Function
Dynamic scaling:
where

42. Time-displaced Correlation Function
EW model z

43. Time-displaced Correlation Function
EW model (define: P(b,)=C(b,)/  (b)-1 )
For b > 15, get scaling with z = 1.65

44. Time-displaced Correlation Function
EW model
No scaling for z = 2.0 !

45. 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?