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