Multi-Site-Correlated Surface Growths with Restricted Solid-on-Solid Condition Yup Kim, T. S. Kim(Kyung Hee University) and Hyunggyu Park(Inha University)

1  Abstract Provided that the heights of randomly chosen k columns are all equal in a surface growth model, then the simultaneous deposition processes are attempted with a probability p and the simultaneous evaporation processes are attempted with the probability q=1-p. The whole growth processes are discarded if any process violates the restricted solid-on-solid (RSOS) condition. If the heights of the chosen k columns are not all equal, then the chosen columns are given up and a new selection of k columns is taken. The recently suggested dissociative k -mer growth is in a sense a special case of the present model. In the k-mer growth the choice of k columns is constrained to the case of the consecutive k columns. The dynamical scaling properties of the models are investigated by simulations and compared to those of the k-mer growth models. We also discuss the ergodicty problems when we consider the relation of present models and k-mer growth models to the random walks with the global constraints.

2.Model  The growth rule for the -site correlated growth  P : probability of deposition  q = 1-p : probability of evaporation  The growth rule for the -site correlated growth <1> Select columns { } (  2) randomly. <2-a> If then for =1,2..., with a probability p. for =1,2..., with q =1-p. With restricted solid-on-solid(RSOS)condition, <2-b> then new selection of columns is taken.  The dissociative -mer growth ▶ A special case of the -site correlated growth. Select consecutive columns

 An arbitrary combination of (2, 3, 4) sites of the same height Model (k-site)  The models with extended ergodicity  An arbitrary combination of (2, 3, 4) sites of the same height q p p q p q  Nonlocal topological constraint : All height levels must be occupied by an (2,3,4)-multiple number of sites. Mod (2,3,4) conservation of site number at each height level. Dynamical Scaling Law for Kinetic Surface Roughening

 Physical Backgrounds for This Study 4  Physical Backgrounds for This Study Steady state or Saturation regime, 1. Simple RSOS with Normal Random Walk(1d)  =-1, nh=even number, Even-Visiting Random Walk (1d)

- mer 3. Simulation results - site (1) 1-dimension ; eff ( ) 5 3. Simulation results (1) 1-dimension ; eff ( ) (i) p = q = 1/2 - site a  1 / 3 ( L→ ∞ ) - mer a  1 / 3 ( L→ ∞ )

- mer - site (ii) P =0.6 (p > q) & P =0.1 (p < q) ▶ p (growing phase), q (eroding phase) a  1 ( L→ ∞ ) - site a  1 ( L→ ∞ ) - mer

Surface Morphology of k-site growth model 7 Surface Morphology of k-site growth model P = 0.6 (p > q) & P = 0.1 (p < q) Groove formation (relatively Yup-Kim and Jin Min Kim, PRE. (1997))

Surface Morphology of k-mer growth model 8 Surface Morphology of k-mer growth model P = 0.6 (p > q) & P = 0.1 (p < q) Facet structure (J. D. Noh, H. Park, Doochul Kim and M. den Nijs, PRE. (2001))

- mer 0.097 4-site 0.14 3-site 0.193 2-site model 0.098 4-mer 0.10 (i) p = q =1/2 eff ( ) - site 0.097 4-site 0.14 3-site 0.193 2-site model  - mer 0.098 4-mer 0.10 Trimer 0.108 Dimer model 

- mer 0.214 4-site 0.325 3-site 0.46 2-site model - site  10 (ii) P = 0.6 (p > q) - site 0.214 4-site 0.325 3-site 0.46 2-site model  Groove phase - mer Sharp facet

Slope a  a values 2-dimension Model N. RSOS 0.174 p = 0.5 0.174 11 2-dimension ( )  a values Model Slope a N. RSOS 0.174 p = 0.5 0.174 Dimer growth 0.16 Two-site growth 0.174

4. Conclusion L   , Ergodicity problem eff eff 12 4. Conclusion L   , 1. p = q = 1/2  1/3 ( k-site, 3,4-mer) ? 1/3 (Dimer growth model) Ergodicity problem 2. p ≠ q  1 k-mer (faceted) (J. D. Noh, H. Park, Doochul Kim and M. den Nijs, PRE. (2001))  1 k-site (groove formation )  Saturation Regime  Conserved RSOS model(?) (Yup-Kim and Jin Min Kim, PRE. (1997)) (D. E. Wolf and J. Villain Europhys. Lett. 13, 389 (1990)) eff eff eff eff