Multi-Site-Correlated Surface Growths with Restricted Solid-on-Solid Condition

Yup Kim, T. S. Kim(Kyung Hee University) and Hyunggyu Park(Inha University)

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.

1

2

2.Model

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> If then new selection of columns is taken.

The dissociative -mer growth ▶ A special case of the -site correlated growth. Select consecutive columns

kxxx k,...,, 21

)(...)()( 21 kxhxhxh

1)()( ii xhxh i k

k

k

1)()( ii xhxh i k

.1,0)1()( xhxh

)(...)()( 21 kxhxhxh

k

k

k

k

).1(,...,2,1 11312 kxxxxxx k

Model (k-site) The models with extended ergodicity

An arbitrary combination of (2, 3, 4) sites of the same height

p q

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.

zL

tfLW

)(

)(z

z

LtL

Ltt

Dynamical Scaling Law for Kinetic Surface Roughening

3

Physical Backgrounds for This Study Steady state or Saturation regime,

1. Simple RSOS with 2

1

RSOSrh

RSOSr ZZ

hP}{

1,1

)}({Normal Random Walk(1d)

2

11

RWz

)1(,)1(

)}({}{

21

21

max

min

max

min h

RSOSr

h

n

h

h

h

h

h

n

RSOSr ZZ

hP

=-1, nh=even number,

Even-Visiting Random Walk (1d)

0)}({ RSOSrhP

3

1

zLt

LL

tLWtLWLeff ln)2ln(

),(ln),2(ln)(

)10/ln(ln

)10/(ln)(ln)(

tt

tWtWteff

4

5

3. Simulation results

(i) p = q = 1/2

L→ ∞ )

L→ ∞ )

(1) 1-dimension ; eff ( )zLt

k - site

- merk

6

(ii) P =0.6 (p > q) & P =0.1 (p < q)▶ p (growing phase), q (eroding phase)

L→ ∞ )

L→ ∞ )

k - site

- merk

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))

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))

9

(i) p = q =1/2

0 20000 40000 60000 80000 1000000.09

0.10

0.11

0.12

0.13

0.14

0.15

0.16

0.17

t

eff

4-mer

Trimer

Dimer

eff

t

0 20000 40000 60000 80000 1000000.09

0.10

0.11

0.12

0.13

0.14

L = 10000

0 20000 40000 60000 80000 1000000.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

eff

eff

tt

2-site

eff

t

0 10000000.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

3-site

0 50000000.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

L = 10000, 1024, 1024 (2,3,4-site)

4-site

zLt eff ( )

0.0974-site

0.143-site

0.1932-site

model

0.0984-mer

0.10Trimer

0.108Dimer

model

- merk

k - site

10

(ii) P = 0.6 (p > q)

4 5 6 7 8 9 10 11 12

0

40

80

120

160

200

L = 10000

Dimer Trimer 4-mer

W 2

ln t

0 20000 40000 60000 80000 100000

0.46

0.47

0.48

0.49

0.50

0.51

0.52

eff

eff

tt6 X 10

74 X 10

72 X 10704 X 10

72 X 10

70

2-site

eff

t

0.26

0.28

0.30

0.32

0.34

0.36

0.38

3-site

0.20

0.22

0.24

0.26

0.28

0.30

0.32

L = 10000, 1024, 1024 (2,3,4-site)

4-site

0.2144-site

0.3253-site

0.462-site

model

k - site

- merk

Groove phase

tw 2

tw ln2 Sharp facet

2-dimension

a values

11

0.16Dimer growth

0.174Two-site growth

0.174p = 0.5

0.174N. RSOS

Slope aModel

( )

zLt

12

1. p = q = 1/2 1/3 ( k-site, 3,4-mer) ? 1/3 (Dimer growth model) Ergodicity problem 2. p ≠ qk-mer (faceted) (J. D. Noh, H. Park, Doochul Kim and M. den Nijs, PRE. (2001)) 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))

4. Conclusion

zLt

eff

eff

L ,

eff

eff