Correlated Istand Nucleation in Layer-by-Layer Growth

HAL Id: jpa-00247192

https://hal.archives-ouvertes.fr/jpa-00247192

Submitted on 1 Jan 1996

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Correlated Istand Nucleation in Layer-by-Layer Growth

E. Somfai, D. Wolf, J. Kertész

To cite this version:

E. Somfai, D. Wolf, J. Kertész. Correlated Istand Nucleation in Layer-by-Layer Growth. Journal de

Physique I, EDP Sciences, 1996, 6 (3), pp.393-401. �10.1051/jp1:1996164�. �jpa-00247192�

Correlated Island Nucleation in Layer-by-Layer Growth

E. Somfai (*),

D.E.

Wolf (**) ^{and J. Kertész} (***)

Theoretische

Physik,

FB 10, Gerhard Mercator Universitàt, 47048

Duisburg, Germany

(Received

20 October 1§95, received in final form and

accepted

9 November

1§95)

PACS.68.55.Bd Molecular and atomic beam

epitaxy

PACS.05.40.+j

Fluctuation

phenomena,

random processes, and Brownian motion PACS.05.70.Ln

Nonequihbrium thermodynamics,

irreversible _processes

Abstract.

Temporal

correlations of nucleation events in

layer-by-layer growth

_are studied

by simulating

_{a one} dimensional model. In order to focus _on fluctuations due to the nucle- ation process,

deposition

and diffusion _are treated deterministically. The correlations

decay algebraically

with effective exponents

depending

_on the order of the nucleations in

a

layer.

The autocorrelation function of ail nucleation events

decays

_as ^t~~'~~ with time t. In spite ^{of these}

correlations,

the

corresponding

randomness _in the

growth

_process does not lead to _a

rough-

emng of the surface. The _reason is that stochastic nucleation

imphes only

conserved current fluctuations.

1. Introduction

Recently

there has been

increasing

interest in nanostructured thin films

produced

_e-g-

by

molecular beam

epitaxy (MBE) [1-4].

In this context a better theoretical

understanding

of

layer-by-layer growth

is

highly

desirable. This

type

^of

growth proceeds by

nucleation of _two- dimensional

islands,

their

growth

and coalescence

leading

to the

completion

of _{a new} atomic

layer

of the film. Whereas

spatial

correlations among these islands within _a

layer

_are

essentially

understood

(see

Ref. _{[5] and} references

therein),

_we address here the

question

to what extent the nudeation of

subsequent layers

remains correlated with the first _one. This allows to estimate how many

layers

_one can grow without

washing

out trie initial nucleation

pattem.

In MBE atoms _are

deposited

onto _a substrate. In

layer-by-layer growth they

diffuse until

they

either stick to the

edge

of _a stable twodimensional island _or meet with other adatoms to form _{a new} nudeus of such _an island. The existence of

temporal

correlations _{is very}

plausible

for

high _symmetry

surfaces: the first nucleation event of _{a new}

layer

is

hkely

to

happen

close to the _{one m} the

previous layer,

because the

corresponding

island _grows ahead of the other

ones and

provides

the

largest

_area without

traps

for adatoms in the next

layer.

The aim of this _paper is to

study

the character of these correlations.

(*) Department

of Atomic

Physics,

Eôtvôs

University,

Puskin _u 5-7, 1088

Budapest, Hungary

Present address: Randall Lab. of

Physics,

Univ. of

Michigan,

Ann

Arbor,

MI 48109-1120, USA

(**)

Author for

correspondence: HLRZ,

KFA

Jülich,

52425 Jühch,

Germany;

e-mail:

d.wolf©kfa-juelich.de

(***) Institute of

Physics,

Technical

University

of

Budapest,

Budafoki ùt 8, H-llll

Budapest, Hungary

Q

Les

Éditions

_de

Physique

1996

394 JOURNAL DE PHYSIQUE I N°3

.----~----~----~-~--,

A A B

Fig.

l. Cells A and B _are sinks for

adatoms,

_{I.e. on} trie time scale /ht ail adatoms within them will condense to the Island

edges

which exist somewhere in them. Here

m has the meanmg of condensed materai

(shaded)

_on top ^of

completed

layers, and the adatom

density

ih

= 0. In ail other cells

m = flà.

Cells A _were invaded

by

the

edges

of _a

growing Island,

in B _a nucleation event has

happened.

Ail cells, mdicated

by

the dashed fines, _are of width /h~ and

height

1

(in

units of the atomic lattice

constant).

It is an

experimental

fact that

loyer-by-loyer growth

_con be sustained for hundreds of

loyers,

if the

growth

conditions _are

carefully

chosen

[6].

^In

particular

a Schwoebel

instability

_[7] should

not occur. We show that randomness associated with the nudeation _process does not

destroy

layer-by-layer growth,

_as

long

_as shot noise and diffusion _{noise can} be

neglected.

Whereas the latter two _are well understood

[8],

very little is known about the _nature of fluctuations due _to the nudeation _process. In this paper we focus _on these

properties by introducing

_a

model where all other _sources of noise _are

intentionally

eliminated in order to _see the bare contribution of trie nudeation events. This is achieved

by simulating deposition

and diffusion

on a coarse scale in _a deterministic continuum

approximation.

2. The Model

We _assume that

desorption

as well _as defects in the film _con be

neglected

and that the adatoms

are bound

irreversibly

^when

they

meet a step or another adatom. Then the

growth

is controlled

by

the ratio of trie surface diffusion _constant D _to the

deposition

rate F. For

simplicity

_we

consider the _one dimensional _case. Then the

typical

distance between the island _{centers is}

m~

(D /F)~R _[9].

We divide the substrate into cells of A~ lattice constants. This

coarsening length

must of

course be chosen smaller thon the

typical

distance between nudeation events, otherwise _one

could not resolve them. Each cell contains _an

integer

number

h(~, t)

of

completed loyers

and

a

partial filling

0 <

m(~, t)

< A~ _on

top

^{of them}

(Fig. 1).

In order _{to suppress} shot noise and fluctuations in the

hopping

of

adatoms, m(x, t)

has to vary

co~lt1~luously

rather than atom

by

atom. For

example, during

_one simulation

update corresponding

to a time increment At it

increases

by

FA~At due to

deposition.

It also

changes according

to the diffusion

equation

m(~,

_{t +}

At)

=

m(~, t)

+

DAt(A~)~~

_x

ji(~

₊

à~, t) 21(~, t) +1(~ à~, t)), (i)

where

ni(x, t) /Ax

_is the adatom

density.

^{For cells in} which there is _no island

edge,

ni coincides with _m.

Cells _in which there is _an island

edge

oct _as sinks for

adatoms,

ni

=

0, thereby filling

up

irreversibly

until _m

ix, t)

exceeds AZ. Then _{a new}

loyer

is

completed

in that cell: The island

edge

invades _a

neighboring

cell _or annihilates with another

edge (coalescence), h(x, t)

_is

incremented

by 1,

and

m(x, t)

is reduced

by

^AZ.

2000

o oo ~oio

~ o~

°° o .

' ~ °woo _°

o°oo.

"

,

~~

°.°,

°° .-.~ o

~

o

~° _{' Îw}

, w

~

)

~~°°

oo

Ob~

_Q'à°

Î

_{~ d} ~''

.

~

°,.

:~~j

~~

~~

flàw

~~

^~

°.

o

~ ~~

~

1000 o° OOh

o Î..

*°é'*

"ao ~ . °

o ;ooo~ao, i

. . »

~ oo_o 8* #o O (o ~

.o: o ~

°° ~$

.ào^~ ~'k°

&

~% -

~ ;: .o÷oo. .. . °°

500 .~ .~~

%~.~,

^° o

o , .~o . . .

.~. Oc oo ^{" . *}

~,

O.O. o

~ , ~_ " %a

°',"

oa o

~o

o ° " ~~_ (

o .oO ai

. °6 '" " O.°"

. ooo- -°

...oo _8'O ° " . +

0 200 400 600 800 1000

h

Fig.

2. Positions of first

(.)

and second

(o)

nucleation events for

layers

^of

heights

< h < looo

(D/F =10~)

New islands nudeate

only

_in cells without _an island

edge.

The nudeation rate has two terms

R~u~(z, t)

=

1(~, t) ifà~

₊

cD(à~)-2 _(1(~

₊

à~, t)

+

1(~ àa,, t) )j. (2) They

descnbe the two main processes

leading

to _a nudeation event: if

already

_one adatom is

in the cell at ~

(probability ni),

_a second _one

may1)

be

deposited

into the _same cell _{or may}

ii)

diffuse into it from its

neighborhood. Dunng

tiine At nucleation

happens

with _a

probability

1-

exp(-RnucAt).

The constant c in

(2)

^{is of} order 1. We did simulations for _c

= 2.5

(results below)

and for 0.6. The exponents

reported

below did not

depend

_on this.

3. Simulation Results

For the simulations _we chose At

=

0.2(Ax)~ ID.

This shows that _coarse

graining

makes the

algorithm

_very eficient [10]: the number of

updates

needed for

growing

_a film of

given

thickness

is reduced

by

_a factor

(Ax)~.

As each

update

involves

L/Ax cells, computation

time is reduced

by

another factor AZ. In most of the simulations _we used D

IF

₌

10~,

for which _m 12 _was found. Coarse graining over AZ

= 3 did not

change

the results but reduced the

computation

time

by

_a factor _m

(Az)~

= 27

compared

to a simulation without _coarse graining.

We recorded all

positions zn(h)

of the n-th nudeation events

in

=

1,2,3,.. N(h))

in the

layer

at

height

h.

Figure

2 shows the

positions

_xi

là)

and

x2(h)

of the first two nudeation

events

dunng

the

growth

of1000

loyers. Strong temporal

correlations

along

the h-direction

are obvious. In all _our simulations _{no more} than two

incomplete layers

_were observed at any

time,

so that the nudeation events at

height

h

happen

at _a time

(in

units of

monolayers)

h -1 _< t < h +1.

We evaluated the

spatial

correlation function _among all nucleation events,

Gir)

₌

lvlx, h)vlz

_{+ r,}

h)1 lvl~, (3)

the autocorrelation function _among all nucleation events,

A(T)

₌

lv(z,hlv(x,h

+

TII lvl~, (41

396 JOURNAL DE

PHYSIQUE

I N°3

lO~ _Q

~~3

A io'

O 10~

O O

10~

~~ ¹⁰⁰

1

Fig.

3. Correlation between ail nucleation sites _m two layers _~ersus their

height

separation. ^Fit has

slope

-1.35.

and the autocorrelation functions _among the n-th nucleation events,

~ni~i

"

i~n(~>hi~"(~>h

+

~)i i~"i~ i~i

where

Unix, h)

is 1 _or 0

depending

_on whether _or not the n-th nucleation event at

height

h

happened

in cell _x, and

v(x,h)

=

£~ Unix, h).

The

averaging (...)

was done _{over x,} h

(discarding

the 100 initial

layers)

and _over 20 _runs.

iv)

is the average number of nucleation events in _a

layer

divided

by

the number of

cells, lui

₌

Axli. Similarly, (uni

₌ AZ

IL _apart

for

n m

Lli,

where _an n-th nudeation event does not _{occur in} every

layer.

The

spatial

coi-relation function

Girl

shows the

expected

anticorrelation at short distances:

within _a distance of _a nucleation site further nucleation events _are

suppressed.

On

larger

distances _no correlation _can be observed among the nucleation events.

Figure

3 shows that the

temporal

correlation among all the nudeation events

decays

with _a power law

Air)

m~

T~°,

with _a

= 1.35 + o-1-

(6)

Although

10000

layers

_{were grown,}

A(T)

coula

only

be evaluated _over two decades _{in T} because of the bad statistics for

large height

differences

T.

The correlation among the first nudeation events in _every

layer decays

much _more

slowly, Figure 4,

Ai(T)

m~

aiT~°i,

with _ai

" 0.48 + 0.08.

(7)

Similarly,

the correlations _among the n-th nucleation events _were fitted with _{a power} law between _T

= 5 and

T =

1000, giving

^effective exponents an

(Fig. 5). They

_vary between low values

m~ 0.4 for the first and the last nudeation events to values between o-1and

0.8, indicating

that the intermediate nudeation events decorrelate faster.

Apart

from _a decrease

by

about _one decade for the first 5 nudeation _events the

prefactors

an of the _power laws _remain

essentially

constant

(Fig. 6).

The

Sharp drop

of the

amplitudes

for _{n >} 22 is due to the fact that the number of nudeation events in _one

layer

_was about 22 for _our

system

^{size and exceeded this value}

only

_very

seldomly.

0bviously,

there must be cancellation of the

An

terms

by

the _cross correlations

(vnvi)

(vn)(vm)

with _n

#

_{m since}

A(T) decays

faster than the

An(T)-s.

The anticorrelations _are

o o

o

10'

Ai o

o

~~s

O

o

10~

Fig.

4. Correlation between first nucleation events in two

layers

_~ersus their

height

separation. Fit has

slope

-0.48.

08

o

07

o °

O O

06 O

on

o o

05

~

o

~ ^o

~ O

0.4 o O O

O ~

~ ^O

~ ~

~~

0 5 10 15 25 30

n

Fig.

5.

Temporal

correlations _among n'th nucleation events

decay

with exponents on.

plausible

_as it is very

unlikely

that two nudeation events with

in mi

m

L/2i

_appear in the _same cell. Dur results show that there is _an

approximate

cancellation _in the

leading

_T-

dependence

of the

positive

and

negative

terms in

A(T)

and the fast

decay (large exponent)

is due to the

non-vanishing

corrections. We cannot be _sure, of _course, that _{we see} the

asymptotic

behavior: If the cancellation _is not

perfect, finally

the smallest

exponent

should overtake.

However,

we have _not detected any crossover of this kind.

4. A

Simplifled Analytical Approach

In order to understand the results _on the correlations

A, An

let _us discuss _a somewhat

simplified

picture:

we assume that the total number of nudeation events in _a

layer

is _a

constant,

N. As indicated

by

the

rapid decay

of the

spatial correlations,

there is _a charactenstic "exclusion zone" of size around each nudeation event. Therefore _{we imagine} the whole

system

divided

398 JOURNAL DE PHYSIQUE I N°3

io~

O

o O

O~

10'

°OOO~

_O

QO°°~

O CC _° CCC

O 0ll

O

ios

o

o

~~~

0 5 10 15 20 25 30

n

Fig.

6. Amplitudes _an of

temporal

correlation function for n'th nucleation events.

into N cells of

length

1.

The cells in each

layer

_{are now} numbered in the order of the nucleation events

(one

_per

cell)

from 1 to N. From

layer

to

layer

_one will find

only

small

changes

in the _sequence: most

likely

the senal number attributed to _a cell will not

change.

As mentioned in the

beginning,

the first nucleation event in _a

layer happens

close to the _{one in} the previous

layer ii-e-

_in the _same

cell).

The _{reason is} that adatom sinks

(island edges)

vamsh there

first,

when the

layer

fills _{up, so} that

the adatom

density

will be

higher

there than elsewhere _on the surface. Hence the nucleation

probability

is

largest

there.

Similarly

the second nudeation _event in the _new

layer

will most

likely happen

in the _same cell _{as in} the _previous

layer:

after the first nudeation created a first sink for adatoms in the _new

layer,

_one has to look for the

region

of maximal adatom

density

outside its

depletion

_zone. This will be the region which has been devoid of island

edges

for the

next-to-longest period. By

induction _one expects that also the n-th nudeation event has _a

higher probability

to

happen,

where it took

place before,

^than

anywhere

else.

Nevertheless,

due to the stochastic nature of the nucleation _process, the first and the second nucleation events may be

interchanged occasionally,

for

example.

More

complicated

_permuta-

tions like

il

_-

2,

2 _- 3~3 _-

1)

_or

(1

-

3,

3 _-

1)

^will be less

probable

and the

probability

will

rapidly decay

with the range of the

jump

_in ^the _space of serial numbers. Thus _any fixed cell

performs

_a random walk in this space. The correlation function

An(T) corresponds

then to the

probability

^that a cell starts with the serial number _n and returns to it after _T

steps.

^{This is}

a

plausible

model for the observed

temporal

correlations in the absence of

spatial

_{ones, since} the mechanism

leading

to the power law time

decay

_is the

wandenng

of nucleation events not in real _space but in the _space of their serial numbers. It

predicts

_a

decay

like

An(T)

m~

T~~/~

with exponents _on

=1/2

which lits _very well to _n

= 1 but not to medium values of _n.

In this

simplified

model

A(T)

vanishes

identically:

there _{is a} nudeation event in every cell

in every

loyer.

In this _case the cancellation of the

positively

and

negatively

correlated terms

is exact. This _{gives some} confidence that indeed there will be _an almost

perfect

cancellation

in our

simulation,

_too. The

remaining

weak correlation may be due to the fluctuation of the nudeation site within _a cell of size _or due to the fluctuation of

N,

both of which have been

ignored

in the

simplified

model.

The measured values of _{an vary} between 0.3 and 0.8 with _error bars of at least o-1- Pre-

sumably

this variation _can be accounted for

by making

the random walk of the cells _over the

serial numbers _more realistic. For

example

the nudeation events with serial numbers close _to

N/2

will

exchange

their

positions

much easier than the first _or last _ones. From _one

layer

to the next several cells _may

change

their serial numbers

simultaneously,

and

larger step

sizes will be allowed. This makes the combinatonal

problem

_more dificult and

probably

decorrelates the

nudeation _{events on} diiferent

layers.

5. Continuum

Description

of Stochastic Nucleation

The simulations showed _no indication for kinetic

roughening [11]

even for values of

D/F

_as

small _as

10, system

^sizes up to L

= 10000 and _{up to} 10000

deposited layers

_{no more} than two

layers

_were

incomplete

at any time. This _agrees with the

finding

for another MBE-model

without shot- and diffusion noise

[12].

^{The randomness of the nudeation} events does not lead _to

significant

surface

roughness,

in

spite

of its

temporal

correlations. In the

following

_{we propose}

a continuum

description

of the

growth

_we simulated and

explain why

stochastic nucleation did not lead to the

roughening

^{of the}

surface,

while random diffusion and _{even more so} the shot

noise will make the surface

rough, ultimately.

Dur

starting point

^is a coarse

grained description

of the

growth,

ôth=F-Vj+i~, (8)

where

j

is the surface _current and J/ is a noise term. In _our simulations the

only

contribution to J/ is nudeation noise and

j

_{is a} functional of the adatom

density

that is determined

by

the

distribution of islaiids and the values of h. Two

questions

bave to be answered: what is the constitutive Iaw

specifying

the surface current _{as a} functional of

h,

and what is the

appropriate description

of the noise due to stochastic nucleation?

The first

question

is

easily

answered

by observing

that the

present

^model does not allow tilt induced currents. No matter what the

global

tilt of the substrate _is, the _average current

vanishes due to the

penodic boundary

conditions. Hence the _genenc form of the constitutive law is

j

=

V(KV~h

₊

_{À(Vh)~), (9)}

where the linear term

only

matters if

= 0

[7,13,14].

The fluctuations of the surface current _are

govemed by

those of the adatom

density, p(x,t),

ôj

₌

Vp. (10)

The

only

_source of

density

fluctuations _on the surface _are nucleation

events,

_since random

hopping

has been eliminated. These

density

fluctuations _are correlated

only

_on short scales of order 1. On _{a coarse}

grained

scale which is

implied by

the continuum

description

these

density

fluctuations _can be viewed _as

spatially

uncorrelated

noise, (p(x, t)p(x', t))

m~

à(x x').

On the other

hand,

the

temporal

correlation _among all the nudeation events

directly

trans- lates _{into a}

temporal

correlation _among the adatom

density fluctuations, (p(x,t)p(x,t'))

rw

(t t'(~°,

where _a is the _same exponent as in

(6).

Hence _we find that nucleation events lead to

conserved, temporally

correlated current fluc- tuations

lô)(X> t)à)lX~ t')1

_~

v~ô(X Z')

_t ~'

l~~ (ii)

The fluctuations _J/ of the

growth velocity

ⁱⁿ

(8)

_are then the

divergence

of the current fluctu-

ations,

1-e-

l~(X>t)~lX'>t'))

_~

v~ôlX X')

t

t' l~~ (12)

400 JOURNAL DE

PHYSIQUE

I N°3

Now _we show that the

spatial

anticorrelation of the _current fluctuations _on short scales

ii-e-

on scale _{1) as}

expressed by

the

regularized V~ô(z z')

counteracts the

temporal

correlation and renders it ineffective for kinetic

roughening.

As

usual, roughening

is characterized

by dynamical scaling

of the

height-height

correlation

function, ((h(z, t) h(z

_{+ r,}

t))~)

m~

r~~g(r/t~/~)

with the

roughness exponent (

^and the

dynamical

exponent z

[11].

Comparing

the

scaling

dimensions of the deterministic _terms in

(8) yields

the exponent identities [7,

8,

_14]

z ₌ 4 for

=

0,

z ₌ 4

(

for

#

0.

(13)

Comparing

the

scaling

dimension of the noise term with that of the left hand side of

(8)

_one

obtains the

scaling

relation

[8,13,14]

2( z(2 a)

= -4

d', ₍₁₄₎

where d' is the dimension of the

surface,

d'

= 1 in the present case.

Together

with

(13)

this fixes the exponents. ^We

only

need to

specify

the linear _case (À ₌

0),

(o

"

(4 ^d' 4a) /2, (15)

as the

roughness exponent

^for the nonhnear

equation

is

given by (

₌

2(o/(4-a). Equation (15) implies

that for

suficiently

fast

decaying temporal correlations,

_a >

1-d'/4,

random nudeation

does not

roughen

the surface. In the

present

_case, ^for the times

simulated,

the effective value of _a is

larger

thon

3/4,

which

explains

the absence of

roughening.

6.

Summary

In summary we have shown in _a one-dimensional model that

long

time correlations _in nudeation

events occur

during layer-by-layer growth.

These _are not

accompanied by spatial

correlations

since the mechanism

leading

to the _power law time

decay

_is the

wandering

of nudeation events

not in real space but in the space of their serial numbers. In

spite

^of the correlations the nudeation noise does not

roughen

the surface _since the conservation of current fluctuations

leading

to short _range anticorrelations

compensates

the slow

decay.

There _are several

interesting questions

related to the

problems investigated

here. First, it is clear that in real

systems

^{nucleation noise} coexists with the shot _noise and the diffusion _noise.

It should be

important

to understand how the latter affect the

temporal

correlations described above. Another open

problem

_is the behavior _in

higher

dimensions. Dur

simphfied

random walk

argument

^is

independent

of the

dimensionality

and therefore _we

expect

^slow

decay

of

correlations there _as well.

However,

the random walk

picture

_{uses a}

coarsening

scale which

is

equivalent

to the island _size. For smaller scales the

migration

of nudeation events _on the island

(see

the

wigghng

of connected

patches

in

Fig. 1)

^becomes important and the different

dimensionality

_may

play

a role and could lead to _a faster

decay

of correlations. Thus _~ve do not expect ^that

roughening

due to nucleation _noise would _{occur since a} >

1/2 already

leads to smooth surfaces for 2+1 dimensions.

Acknowledgments

We hke _to thank Harald Kallabis for _a careful

reading

of the manuscript. ^{E-S- and J-K- thank} the

University

of

Duisburg,

D.E.W. and J-K- thank the Isaac Newton

Institute, University

of

Cambridge

^for

hospitahty.

This work _was

supported by

DFG

(SFB 166)

and OTKA.

References

iii

Kinetics of

0rdering

and Growth _at

Surfaces,

^M-G.

Lagally

Ed.

(Plenum Press,

New

York, 1990).

[2] Surface

Disordering: Growth, Roughening

and Phase

Transitions,

R.

Jullien,

J.

Kertész,

P. Meakin and D.E- Wolf Eds.

(Nova Science, Commack, 1992).

[3] Villain J. and

Pimpinelli A., Physique

de la Croissance

Cristalline,

Collection Aléa

Saclay (Eyrolles, Paris, 1995).

[4] Barabàsi A.-L. and

Stanley H-E-,

Fractal

Concepts

in Surface Growth

(Cambridge

Uni-

versity Press, 1995).

[5] Schroeder M. and ivolf

D.E., Phgs-

Reu. Lett. 74

(1995)

2062.

[6] Kunkel R. et

ai-, Phgs.

Reu. Lett. 65

(1990)

733.

[7] Villain

J.,

J.

Phgs.

1France1

(1991)

19.

[8]

Tang

L.H. and Nattermann

T., Phgs.

Reu. Lett. 66

(1991)

2899.

[9]

Pimpinelh

A. et

ai., Phys.

Reu- Lett. 69

(1992)

985.

[10] ^{Wolf D.E. in Scale}

Invariance, Interfaces,

and

Non-Equilibrium Dynamics,

M. Droz et ai.

Eds.

(Plenum,

New

York, 1995).

[Il] Family

F- and Vicsek

T.,

J.

Phys.

A 18

(1985) L75; Dynamics

of Fractal

Surfaces,

F-

Family

and T. Vicsek Eds.

(World Scientific, Singapore, 1991).

[12]

^{Elkinani I. and Villain}

J.,

J.

Phys.

I France 4

(1994)

949.

[13]

^{Wolf D.E- and Villain}

J., Europhys.

Lett. 13

(1990)

389.

[14]

^{Lai Z-W- and Das Sarma}

S., Phys.

Reu. Lett. 66

(1991)

2348.