DriftDrift 와 결함이 있는 계의 표면 거칠기와 결함이 있는 계의 표면 거칠기

Sooyeon Yoon & Yup KimDepartment of Physics, Kyung Hee University

제 13 차 열 및 통계물리 워크샵

),(),( 2

2 txht

txhx

Surface roughness

222 ),( hhtLW

L : system sizeh : the height of surface

Family-Vicsek Scaling behavior

z

z

z

LtL

Ltt

L

tfLtLW

,

,

),(

2

2

22

EW universality class

zz 2,

4

1,

2

1

• G. Pruessner (PRL 92, 246101 (2004))

),(v),( 2

2 txhht

txhxx

v : drift velocity

with Fixed Boundary Condition (FBC)0),(),0( tLxhtxh

)v/(~

)v(~)(

21

21

212

LtL

txxW

Background of this studyBackground of this study

• Edward-Wilkinson Eq.

1,4

1,

4

1 z

Anomalous exponents

1. What is the simple stochastic discrete surface growth model to describe the EW equaiton with drift and FBC ?

• Numerical Integration

• Toy models : Family model, Equilibrium Restricted Solid-On-Solid (RSOS) model …

Stochastic analysis for the Langevin equation ( S.Y. Yoon & Yup Kim, JKPS 44, 538 (2004) )

2. Application The effect of the defect and drift for the surface growth ?

MotivationMotivation

),(),(),( )1( txtxhK

t

txh

• Continuum Langevin Equation :

NiihH 1**

• Fokker-Planck Equation :

),(2

1),(

),( )2(2

,

)1( tHPKhh

tHPKht

tHPij

jijiii

i

'

')1( )',()()(H

iii HHhhHK

)',()()()( '

'

')2( HHhhhhHK jjH

iiij HH ,' is the transition rate from H′ to H.

A stochastic analysis of continuum Langevin equation A stochastic analysis of continuum Langevin equation for surface growthsfor surface growths

,0)( ti )'(2)'()'()( )2( ttDttKtt ijijijji

If we consider the deposition(evaporation) of only one particle at the unit evolution step.

( a is the lattice constant. )

ahi

ahi

(deposition)

(evaporation)'ih

2

)1(

2

)(

a

D

a

HK iid

2

)1(

2

)(

a

D

a

HK iie

S.Y. Yoon & Yup Kim, JKPS 44, 538 (2004))

• Master Equation :

''

),()',(),'(),'(),(

HH

tHPHHtHPHHt

tHP

For the Edward-Wilkinson equation with drift,

][2

v2]v[ 11112

22

)1( iiiiiii hhhhhhhK

• Evolution rate on the site

ModelModel

),(v),( 2

2 txhht

txhxx

)deposition(1)()(':2

)(2

)1(

ththa

D

a

HKii

iid

)nevaporatio(1)()(':2

)(2

)1(

ththa

D

a

HKii

iie

d (e )

• Determine the evolution of the center point (x0=L/2) by the defect strength.

0pyprobabilit,02

0pyprobabilit,02

0

0

Lxh

Lxh

x0=L/2

p x d (e )

or

Simulation ResultsSimulation Results

Scaling Properties of the Surface Width

222 ),( hhtLW

zL

tfLtLW 22 ),(

2,4

1,

2

1 z (PBC, p=1)

4096L 145 2~2L

1,4

1,

4

1 z (FBC, p=0)

Analysis of the Interface Profile

xxC ~)(

2~)( xxG2

1

4

1 )()()(

~00 xhxxhxC

~

,20Lx ,1024L 1v,12

|)()(|)( 00 xhxxhxC

0 200 400 600 800 10000

5

10

15

y(x)=Bx1/2 p=0 p=0.1

C(x

)

x

y(x)=Ax1/4

200 |)()(|)( xhxxhxG

0 200 400 600 800 10000

100

200

y(x)=Ax1/2

y(x)=Bx

p=0 p=0.1

G(x

)

x

PBCinfunc.ncorrelatioheightheight:

)]()([2)()cf2/

1

2

L

x

rxhxhLrG

-200 0 200 400 600 800 1000 12000

100

200

G(x

)

x

Crossover (EWanomalous roughening) according to the defect strength

1v,12

4096L

125 2~2L

)1.0(2,4

1,

2

1 pz

222 ),( hhtLW

)0(1,4

1,

4

1 pz

zL

tfLtLW 22 ),(

3.0v,5.02

0 200 400 600 800 10000

5

10

15

y(x)=Bx1/4

y(x)=Bx1/2

C(x

)

x

p=0 p=0.1

y(x)=Bx1/2

• Phase transition of RSOS model with a defect site

: H.S.Song & J.M.Kim (Sae Mulli, 50, 221 (2005))

)),0(),((),( PhPrhPrC

r : the distance from the center pointP : defect strength

c

csat

PPBr

PPBrArPrC

:~

:~),(

87.0at32 cP

P=0, facet P=1, RSOSPc

BrrC ~)(BrArrC ~)(

p=0 p=1

41with

~)(

BxxC

EW

Application of the surface growth by the defect (Queuing problem)

||

||

1

1

axx

hhx

h

ii

iii

)( 00 iaxxxh iiii

1

~vi x

hi

( : particle density )

G slowly go out ! fast get in !

( a : lattice constant )iaxi 0

1i 2 3

....

..

Conclusion Conclusion

We studied the phase transition of the stochastic model which satisfies the Edward-Wilkinson equation with a drift and a defect on the 1-dimensional system.

),(v),( 2

2 txhht

txhxx

0),2/(with 0 tLxh

1. The scaling exponents are changed by the drift and the perfect defect.

3. Application to the queuing phenomena ( at p=0 : perfect defect ).

)0at(1,4

1,

4

1 pz

Anomalous exponents

2. Crossover

EW (p0) Anomalous roughening (p=0)