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 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

Simulation NATURE Experiment Theory

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

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

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

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

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

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 . . .

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

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

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

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

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

Monte Carlo is Serious Science!

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!

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 !

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

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

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

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

Time-displaced Correlation Function Dynamic scaling: where

Time-displaced Correlation Function EW model z

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

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

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?