equilibrium restricted solid-on-solid models with constraints on the distribution of surface heights...
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Equilibrium Restricted Solid-on-Solid Models with Constraints on the Distribution of Surface Heights
허희범 , 윤수연 , 김엽
경희대학교
Physical Backgrounds for This Study
Steady state or Saturation regime,
1. Simple RSOS2
1
2. Two-site correlated growth (Yup Kim, T. Kim and H. Park, 2002) Dimer growth (J. D. Noh, H. Park, D. Kim and M. den Nijs, PRE, 2001,
J. D. Noh, H. Park and M. den Nijs, PRL, 2000)
RSOSrh
RSOSr ZZ
hP}{
1,1
)}({
Normal Random Walk(1d)
2
11
RWz
)1(}{
21
max
min
h
RSOSr
n
h
h
h
Z ,zLt
1
3
1
=-1, nh=even number,
Even-Visiting Random Walk (1d)
0)}({ RSOSrhP
ZhP
h
h
n
RSOSr
h
max
min
)1(
)}({21
=1
3. Extremal Growth
2
RSOSr
hzZ
}{)0(
0;)01(max
min
21 h
h
hn
0;1max
min
h
h
hn
)(
}{21 cH
h RSOSr
)1)(( minmax hhcH
3
1
= 0
= 1 =-1Normal
Random WalkEven-Visiting Random Walk
2
13
1
= 1/2 = -1/2
3
1
??
4. Generalized Model for 1,1,0
Phase diagram
= 0
(1d)
Model
1. 크기가 L 인 1 차원 기판의 height distribution 중 , 최대높이 hmax 와 최소높이 hmin 를 찾는다 .
2. 기판의 한 site 를 임의로 선택한다 .
)}({ ixh
rx
3. 확률 (1-) 로 deposition (evaporation) 을 결정한다 .
4.
라 하고 , 만일 에서 deposition (evaporation) 이 일어났다고 가정했을 때 , 새로운 configuration 에 대하여 을 구한다 . 이때 , 이 과정을 허용할 확률을 다음과 같이 정의한다 .
3
,1)()( rr xhxh
1)()( rr xhxh
)1(2
1max
min
h
h
nh
hn
w
rx
'w
w
wP
'
)1,1,0(
5. 만약 확률 , P 가 일 경우 , deposition (evaporation) 의 과정을 허용한다 . 반대로 이면 , 임의의 random number Pr 을 발생시켜 Pr<P 일 경우에만 이 과정을 허용한다 .
•모든과정은 restricted solid-on-solid constraint 를 만족하여야 한다 .
( 여기서 는 d-dimensional hypercubic lattice 에서의 nearest-neighbor bond vectors 중의 한 site 를 말한다 .)
1)()( axhxh rr
a
4
1P1P
0)( ixh
n+2 =1n+1 =3n 0 =L-8n-1 =2n-2 =2
n´ +2 =2
n´ +1 =2
n´ 0 =L-8
n´ -1 =2
n´ -2 =2
w w´hmin
hmax
9259.0)5.01()5.01(
)5.01()5.01('
213
21
2212
21
w
wP
=1/2L= 10= 0.5
Pr<P
Simulation Results
1d , =1/2
LL
tLWtLWLeff ln)2ln(
),(ln),2(ln)(
zL
tfLW
z
z
Ltt
LtL
,
,
5
eff
0.5 0.33
-0.5 0.34
0.0 0.34
L = 16, 32, 64, 128, 256, 512, 1024
0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.0350.30
0.32
0.34
0.36
0.38
0.40
0.42
0.44 =0.5
=0
=-0.5
eff
1/L
zLt
L = 1024 = 0.22
5.1
z
6
0 1 2 3 4 5 6-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
=0.22
=0.5
=0
=-0.5
ln W
ln t
Scaling Collapse to in 1d. ( = 1/3 , z = 1.5)
zL
tfLW
7
1d , >0 , = 1 (growing phase) Normal RSOS Model
8
3.5 4.0 4.5 5.0 5.50.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
=0.5
=0.5ln W
ln L
0 1 2 3 4 5-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
=0.3
L=256
=0.5
ln W
ln t
zL
tfLW
z
z
Ltt
LtL
,
,
,5.0 ,3.0 67.1z
Normal RSOS Model
1d , <0 , = 1 (growing phase) (???)
9
0 1 2 3 4 5 6
-0.5
0.0
0.5
1.0
1.5
2.0
2.5 =-0.5
=0.86
ln W
ln t
3.5 4.0 4.5 5.0 5.50.5
1.0
1.5
2.0
2.5
3.0
=-0.5
=0.97
ln W
ln L
L=32,64,128,256
1d , <0 , = 0.6 (growing phase) (???)
10
1 2 3 4 5 6-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
=-0.5
=0.24
ln W
ln t
L=32,64,128,256
3.5 4.0 4.5 5.0 5.50.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0=-0.5
=0.84
ln W
ln L
Conclusion
1d , =1/2
11
= 0
= 1 =-1Normal
Random WalkEven-Visiting Random Walk
2
13
1
= 1/2 = -1/2
3
1
1d , >0 ,=1 Normal RSOS Model
1d , <0 ,=1 ?
1d , <0 ,=0.6 ?
growing(>1/2) or eroding (<1/2) phase Phase transition at =0(?)