1. Metropolis-type evolution rules for surface growth models
with the global constraints
on one and two dimensional substrates
Yup Kim, H. B. Heo, S. Y. Yoon
KHU

2. 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, (2002)
Dimer-type surface growth
J. D. Noh, H. Park, D. Kim and M. den Nijs, PRE 64, (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, (R) (2002)

3. nh : the number of columns with height h2
Partition function,
nh : the number of columns with height h
Steady state or Saturation regime ,
1. Normal RSOS (z =1)
Normal Random Walk (1D)
(EW)
2. Two-particle correlated (dimer-type) growth (z = -1)
nh=even number,
Even-Visiting Random Walk (1D)

4. ? ? 3. Self-flattening surface growth (z = 0)
Self-attracting random walk (1D)
Phase diagram (1D)
z = 0
z = 1
z =-1
Normal
Random Walk
Even-Visiting ?
Self-attracting z
Phase diagram (2D)
z = 0
z = 1
z =-1
Normal
Random Walk
Even-Visiting ?
Self-attracting z

5. 2. Generalized Model Evaluate the weight4
2. Generalized Model
( nh : the number of sites which have the same height h )
Evaluate the weight
in a given height configuration
Choose a column randomly.
Decide the deposition (the evaporation) attempt with probability p (1-p)
Calculate for the new configuration from the decided deposition
(evaporation) process
Acceptance parameter P is defined by

6. If P  1 , then new configuration is accepted unconditionally. 5
If P  1 , then new configuration is accepted unconditionally.
If P < 1 , then new configuration is accepted only when P  R.
where R is generated random number 0< R < 1 (Metropolis algorithm)
Any new configuration is rejected if it would result in violating the RSOS contraint
( a primitive lattice vector in the i – th direction )
P  R
hmax
hmin
p =1/2
L = 10
z = 0.5
n+2 = 1
n+1 = 3
n 0 = 2
n-1 = 2
n-2 = 2 w
n´ +2 = 2
n´ +1 = 2
n´ 0 = 2
n´ -1 = 2
n´ -2 = 2 w´

7.  Equilibrium model (1D, p=1/2)6
3. Simulation Results
 Equilibrium model (1D, p=1/2)

8. 7
0.22
0.33
1.1
0.9
 0.19
 0.22
1/4 
 0.33
 0.34
1/2
 (L) -1
-0.5
0.5 1
1.5 z

9.  Equilibrium model (2D, p=1/2)7
 Equilibrium model (2D, p=1/2) z -1
-0.5
0.5 1
1.5 a
0.176
0.175
0.179

10. Scaling Collapse to in 2D equilibrium state.7
Scaling Collapse to in 2D equilibrium state.
, Z = 2.5

11. Phase diagram in equilibrium (1D)7
Phase diagram in equilibrium (1D)
z =-1
-1/2
z = 0
1/2
z = 1
3/2
z = 0.9
z = 1.1
2-particle corr.
growth
Self-flattening
surface growth
Normal
RSOS
Phase diagram in equilibrium (2D) z
z =-1
z = -0.5
z = 0
z = 0.5
z = 1
z = 1.5
Even-Visiting
Random Walk
Self-attracting
Random Walk
Normal
Random Walk

12. Growing (eroding) phase (1D, p=1(0) )9
Growing (eroding) phase (1D, p=1(0) )
z  0
: Normal RSOS model (Kardar-Parisi-Zhang universality class) p
 (L)
1.5
 0.52
0.5
 0.51
 0.49

13. 10
z  0
Normal RSOS Model
(KPZ)

14. 11
z  0
z=-0.5 p=1 L=128

15.  Equilibrium model (1D, p=1/2)12
4. Conclusion
 Equilibrium model (1D, p=1/2) z -1
-1/2
1/2
0.9 1
1.1
3/2
2-particle corr. growth
(EVRW)
Self-flattening
surface growth
(SATW)
Normal
RSOS
(Normal RW)
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 (?)

16. Scaling Collapse to in 1D equilibrium state. ( = 1/3 , z = 1.5)
12-1
Scaling Collapse to in 1D equilibrium state. ( = 1/3 , z = 1.5)

17. Monomer & Extremal & Dimer & 2-site7
Monomer & Extremal & Dimer & 2-site
0.175
0.162
Dimer
Monomer
Slope a
Model
Extremal
0.174
2-site