surface structures of growth models with constraints from the surface-height distribution
DESCRIPTION
Surface Structures of Growth Models with Constraints From the Surface-Height Distribution. 허희범 , 윤수연 , 김 엽 경희대학교. 1. 1. Motivation. In equilibrium state, Normal restricted solid-on-solid model : Edward-Wilkinson universality class Two-particle correlated surface growth - PowerPoint PPT PresentationTRANSCRIPT
Surface Structures of Growth Models with ConstraintsFrom the Surface-Height Distribution
허희범 , 윤수연 , 김 엽
경희대학교
1. Motivation
In equilibrium state,
Normal restricted solid-on-solid model: Edward-Wilkinson universality class
Two-particle correlated surface growth- Yup Kim, T. S. Kim and H. Park, PRE 66, 046123 (2002)
Dimer-type surface growth- J. D. Noh, H. Park, D. Kim and M. den Nijs, PRE 64, 046131 (2001) - J. D. Noh, H. Park and M. den Nijs, PRL 84, 3891 (2000)
Self-flattening surface growth-Yup Kim, S. Y. Yoon and H. Park, PRE(RC) 66, 040602(R) (2002)
1
1. Normal RSOS (z =1)
2. Two-particle correlated (dimer-type) growth (z = -1)
)1(}{
21
max
min
h
RSOS
n
h
h
hh
zZ
2
3
1
nh=even number,
Even-Visiting Random Walk (1D)
0)}({ RSOShP
Z
z
hP
h
hh
n
RSOS
h
max
min
)1(
)}({21
RSOSh
RSOS ZZ
hP}{
1,1
)}({Normal Random Walk (1D)
4
1,
2
11
RWz
Partition function,
(EW)
)( zLt Steady state or Saturation regime ,
nh : the number of columns with height h
3. Self-flattening surface growth (z = 0)
3
)(
}{ 2
1cH
h RSOSr
Z
)1)(( minmax hhcH 3
1
Self-attracting random walk (1D)
Phase diagram (1D)
z = 0 z = 1z =-1Normal
Random WalkEven-Visiting Random Walk
2
1
3
1
3
1 ??
Self-attracting Random Walk
?
z
Height distribution 중 , 주어진 높이 h 에 있는 column 의 개수 nh 를 계산한다 .
x
크기가 L 인 1 차원 substrate 의 한 column 를 임의로 선택한다 .
2. Generalized Model
4
,1)()( xhxh
1)()( xhxh
)1(2
1)})(({
max
min
hnh
hh
zrhw
로 정의하고 , 만일 에서 deposition (evaporation) 이 일어났다고 가정했을
때 , 새로운 configuration 에 대하여 weight 을 구한다 . 이때 , 이 과정을 허용할 확률을 다음과 같이 정의한다 .
x
)})('({ rhw
)})(({
)})('({
rhw
rhwP
)}({ rh
확률 p (1-p) 로 deposition (evaporation) 을 결정한다 .
Weight 를
( nh : 높이 h 를 갖는 column 의 개수 )
( 여기서 는 d-dimensional hypercubic lattice 에서의 nearest-neighbor bond vectors 중 ,
한 site 를 말한다 .) ie
모든 과정은 restricted solid-on-solid constraint 를 만족하여야 한다 .
1)ˆ()( ierhrh
5
n+2 = 1n+1 = 3n 0 = 2n-1 = 2n-2 = 2
wn´ +2 = 2
n´ +1 = 2
n´ 0 = 2
n´ -1 = 2
n´ -2 = 2
w´
hmax
0)( ixh
hmin
p =1/2L = 10z = 0.5
P R
만약 확률 P 가 P1 일 경우 , deposition (evaporation) 의 과정을 무조건 허용한다 . 반대로 P<1 이면 , 임의의 random number R 을 발생시켜 P R 일 경우에만 이 과정을 허용한다 . (Metropolis algorithm)
9259.0)5.01()5.01(
)5.01()5.01(
213
21
2212
21
)})(({
)})('({
rhw
rhwP
0.00 0.01 0.02 0.030.2
0.3
0.4
0.5
z = - 1
z = 1.5 z = 0.5 z = 0
z = - 0.5
eff
1/L
3. Simulation Results
6
Equilibrium model (1D, p=1/2)
zL
tfLW
z
z
Ltt
LtL
,
,
LL
tLWtLWLeff ln)2ln(
),(ln),2(ln)(
z
0.00 0.01 0.02 0.030.2
0.3
0.4
0.5
1/3
z=0.9
z=1.1
eff
1/L
0 1 2 3 4 5
-0.5
0.0
0.5
1.0
=0.22
z=-0.5
z=0
z=0.5
z=0.9
z=1.1
z=1.5
ln W
ln t
7
0.22
0.33
1.1
0.22
0.33
0.9
0.19 0.22 0.22 0.22 1/4 0.22
0.33 0.34 0.33 0.331/2
0.33 (L)
-1-0.500.511.5z
8
h h
Lh
h
n
rh
nLandhhSwhere
zzZ h
1
]...1[)2/1(
minmax
)}({
2max
min
max
min
max
min
]1[)}({
h
hh
nsfsf
h
hh
n
rh
S hh zZZzeZ
)}({)}({
)()2/1(rh
Ssfsf
L
rh
SL eZZzzZ
z
0z
8-1
Phase diagram in equilibrium (z =1/2)
z = 0
z = 1z =-1
Normal RSOS
2-particle corr. growth
2
1
3
1
3
1
Self-flattening surface growth
3
1
3
1
3
1
-1/2 1/2 3/2
z = 0.9 z = 1.1
3
1
3
1
9
Growing (eroding) phase (1D, p=1(0) )
p (L)
1.5 0.52
0.5 0.51
0 0.49
zL
tfLW
z
z
Ltt
LtL
,
,
z 0
)1,31,2
1( zz
: Normal RSOS model (Kardar-Parisi-Zhang universality class)
,5.0 ,33.0 5.1z
Normal RSOS Model (KPZ)
10
z 0
11
z 0
z=-0.5 p=1 L=1280.00 0.01 0.02 0.03
0.5
1.0
1.5
2.0
z = -0.1 z = -0.5
z = -1
eff
1/L
11-1
z 0 morphology
z = 0.5 , L=256
11-2
z < 0 morphology
z = - 0.5 , L=256
12
4. Conclusion
Equilibrium model (1D, p=1/2)
0 1-1
Normal RSOS
(Normal RW)
2-particle corr. growth
(EVRW)
2
1
3
1
3
1
Self-flattening surface growth
(SATW)
3
1
3
1
3
1
3/21/2-1/2
Growing (eroding) phase (1D, p = 1(0) )
1. z 0 : Normal RSOS model (KPZ universality class)
2. z 0 : Groove phase ( = 1)
Phase transition at z=0 (?)
z0.9 1.1
3
1
3
1
Scaling Collapse to in 1D equilibrium state. ( = 1/3 , z = 1.5)
zL
tfLW
12-1