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

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

Phase diagram (2D)

z = 0 z = 1z =-1Normal

Random WalkEven-Visiting Random Walk

?

Self-attracting Random Walk

z

LK

WG

ln2

12

Choose a column randomly.x

2. Generalized Model

4

,1)()( xhxh

1)()( xhxh

Acceptance parameter P is defined by

)})(({

)})('({

rhw

rhwP

Decide the deposition (the evaporation) attempt with probability p (1-p)

Calculate for the new configuration from the decided deposition(evaporation) process

)})('({ rhw

)}('{ rh

)1(2

1)})(({

max

min

hnh

hh

zrhw

( nh : the number of sites which have the same height h ) r

Evaluate the weight in a given height configuration )}('{ rh

( a primitive lattice vector in the i – th direction ) ie

Any new configuration is rejected if it would result in violating the RSOS contraint

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

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)

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

2.5 3.0 3.5 4.0 4.5 5.00.4

0.5

0.6

0.7

0.8

0.9

1.0

z = -1

z = -0.5

z = 0

z = 0.5

z = 1

z = 1.5

W2

ln L

7

WzG L

tLg

KtLW ln

2

1),(2

z

G

z

G

LtLK

LttK

,ln2

1

,ln2

1

Equilibrium model (2D, p=1/2)

z -1 -0.5 0 0.5 1 1.5

a 0.176 0.176 0.176 0.175 0.176 0.179

176.02

1a

KG

7

Scaling Collapse to in 2D equilibrium state.

WzG L

tLg

KtLW ln

2

1),(2

176.02

1a

KG , Z = 2.5

Phase diagram in equilibrium (1D)

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

7

z = 0 z = 1z =-1Normal

Random WalkEven-Visiting Random Walk

Self-attracting Random Walk

z

Phase diagram in equilibrium (2D)

LK

WG

ln2

12

176.0

2

1a

KG

z = -0.5 z = 0.5 z = 1.5

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

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

7

0.175

0.162Dimer

0.175Monomer

Slope aModel

Extremal 0.174

2-site

Monomer & Extremal & Dimer & 2-site