Random deposition models for thin film epitaxial growth

Physics and Astronomy Publications

Physics and Astronomy

1989

Random-deposition models for thin-film epitaxial

growth

James W. Evans

Iowa State University

, evans@ameslab.gov

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Abstract

The simple on-top site random-deposition model for film growth on a simple-cubic (100) substrate is

extended to other adsorption-site geometries and crystal structures. Nontrivial statistical correlations within

the growing film typically result. However, the master equations can be solved exactly to elucidate, e.g., growth

kinetics, Bragg intensity (IBr) behavior for surface diffraction, and spatial correlations within the film. Other

techniques facilitate analysis of the percolative structure of intralayer islands. For random deposition at

fourfold hollow sites on a face-centered cubic (100) substrate, despite the absence of diffusion, there are

distinct IBr oscillations similar to those observed experimentally during deposition of Pt on Pd(100) at low

temperature.

Disciplines

Condensed Matter Physics | Physics

Comments

This article is published as Evans, J. W. "Random-deposition models for thin-film epitaxial growth."

Physical

Review B

39, no. 9 (1989): 5655. doi:

10.1103/PhysRevB.39.5655

. Posted with permission.

PHYSICAL REVIEW B VOLUME 39, NUMBER 9 15MARCH 1989-II

Random-deposition

models

for

thin-film

epitaxial

growth

J.

W.Evans

Ames Laboratory and Department

of

Physics, Iowa State Uniuersity, Ames, Iotoa 50011

(Received 2November 1988)

The simple on-top site random-deposition model for61mgrowth on a simple-cubic (100)substrate

isextended toother adsorption-site geometries and crystal structures. Nontrivial statistical correla-tions within the growing film typically result. However, the master equations can be solved exactly

to elucidate, e.g.,growth kinetics, Bragg intensity (IB,)behavior for surface diffraction, and spatial

correlations within the film. Other techniques facilitate analysis ofthe percolative structure of

in-tralayer islands. Forrandom deposition atfourfold hollow sites on a face-centered cubic (100)

sub-strate, despite the absence ofdiffusion, there are distinct IF„oscillations similar to those observed experimentally during deposition ofPtonPd(100)atlow temperature.

I.

INTRODUCTION

Epitaxial growth

of

thin films has become an area

of

much current research, because

of

both its fundamental

interest and also its importance in device fabrication. A

variety

of

statistical mechanical models have been

developed which describe such processes. These include

the following.

(i) The random deposition model for growth on the

(100) face

of

a simple-cubic crystal [sc(100)] or (10) face

of

a 2D (two dimensional) square lattice [sq(10)] Atoms

are randomly deposited at constant rate on top

of

previ-ously occupied sites. Clearly there are no overhangs,

columns grow independently, and their height statistics is

Poisson. Thus the surface width g

of

the film scales like

(

h

₎

'~,

as (h

)

~

oo, where

(h )

is the mean height (i.

e.

,

the total coverage

6).

(ii) Ballistic deposition models for sc(100)

or

sq(10)

growth: ' _{Here a "vertical}

_{rain" of}

_{atoms stick at the}

first empty site reached with a filled nearest neighbor

(NN). Overhangs and avoids result.

If

L

isthe substrate

linear dimension then

g-(h

)~

with

L

=

Do, and

g-L

with

(h

)

=

oo. The exponents

a, P

are nontrivial and

di-mension dependent. Allowing multiple restructuring

after hitting may change these exponents.

(iii) The Eden model assumes equal addition

probabili-ties for all empty sites neighboring filled sites. '

Overhangs and voids again result, and exponents

a,

P

as-sume values for ballistic deposition (with no

restructur-ing). This scaling reflects long-wavelength fluctuations.

(iv) Kinetic solid-on-solid (SOS) models ' _{assume no}

overhangs and simple interactions between atoms.

Equi-libriurn SOS models exhibit a roughening transition at

some temperature Ttt

)

0,above which the height h

(r)

at

lateral position

r

satisfies

(~h(r)

—

h(0)~

)

—

~r~ in 2D

and 1n~r~ in

3D.

The kinetic model introduces dynamics

through microscopic absorption, desorption, and

diffusion rnechanisrns.

It

describes the transition from

nucleated growth for

T~

T~ to continuous growth for

T~

Ttt.

For

the latter, steady states with constant (or

periodic) d

(h

)

/dt _{and g}

occur.

(v) Cluster statistics models describing structure only,

not dynamics, involve counting certain interface

configurations

of

N atoms. These relate to deposition

models as random animals relate

to

kinetic cluster

growth models.

(iv) Terrace statistics models, mostly for 1Dsubstrates,

describe structure only, not dynamics. Finite-level

mod-els may describe the initial stages

of

epitaxy

or

near

layer-by-layer growth. Infinite-level models may

de-scribe perfect or disordered vicinal surfaces. ' Ad hoc

specifications

of

terrace width distributions, and/or

prob-abilities for stepping up or down, allow straightforward

calculation

of

diffracted intensity profiles. Simple

gen-eralizations for

20

substrates have been proposed.

"

Often thin-film growth is monitored utilizing

surface-sensitive diffraction techniques. ' In akinematic

scatter-ing theory, the diffracted intensity profile is determined

by the pair-correlation function

of

the scatterers

or

"sur-face atoms.

"

Here we consider only the amplitude

of

the

Bragg peak

I~,

which is determined by the net fraction

of

scatterers

S

in the various layers

j

as'

IB,

=

g

S,

e

"~

(1)

J

Here P is the phase angle between scattering from

adja-cent layers.

_IB,

is most sensitive to interface structure

near the anti-Bragg condition _P

=

(2n

+

1 )m., where

e'J~=(

—

1)j.

The appearance

of

intensity

_(IB,

)

oscilla-tions during

"near"

layer-by-layer growth is now

exploit-ed routinely as an analytic tool. Specifically, reQection

high-energy electron diffraction is often used during

molecular beam epitaxy

to

monitor the total coverage in

near layer-by-layer epitaxial growth. '

If

one makes the

common assumption ' that each atom produces a net

shielding

of

one atom in the adlayer directly beneath,

then _SJ

=

—

BJ

—

BJ.+& where OJ. is the coverage

of

layer

j.

The same result follows from an alternative prescription

described below.

Here we extend the random-deposition model to other

absorption-site geometries and crystal structures.

Specifically we consider random deposition at bridge sites

for a 1Dsubstrate, and at the fourfold hollow sites

start-ing from the (100) face

of

aface-centered-cubic (fcc)

crys-tal. The fourfold hollow adsorption-site requirement is

certainly natural for isomorphic

fcc

crystal growth, since

it reAects crystal geometry. Analysis

of

this type

of

mod-el is motivated by recent experiments on the deposition

of

Pt

on Pd(100) at low temperatures. '

'

Our work

pro-vides a new class

of

epitaxial-growth models which are

exactly solvable, but not trivial as is the sc(100)

random-deposition model. In the new models, the geometry

of

the adsorption-site requirement produces correlations

within layers as the film grows. Resulting growth

kinet-ics and structure is nontrivial and will be analyzed in the

following sections together with the associated behavior

of

_IB,

.

In

Sec.

II

we briefly describe a rate-equation analysis

of

the sc(100)random-deposition model and discuss

associ-ated

Iz,

behavior. The rate-equation approach is adapted

toprovide an exact analysis

of

the kinetics and correlated

statistics

of

random deposition at bridge sites on a 1D

substrate in

Sec.

III

and at the fourfold hollow sites on a

fcc(100)substrate in Sec.

IV.

(These rate equations are

just the hierarchical form

of

the master equations. )

Specifically we focus on elucidating the behavior

of

the

6,

the short-range intralayer correlations, and the

asso-ciated

Ir„.

In

Sec.

Vwe characterize the (nonlocal)

struc-ture

of

the islands or clusters

of

filled sites within each

layer exploiting concepts and techniques

of

percolation

theory. Surface structure is also discussed brieAy. Some

simple generalizations

of

these models which still permit

exact analysis are described in

Sec.

VI.

There we also

in-dicate the diSculties inherent in rate-equation analysis

of

other thin-film growth models. Finally in

Sec.

VII

we

re-view our findings.

III.

RANDOM DEPOSITION ATBRIDGE

SITES

Consider random deposition, with impingement rate k,

at bridge sites, starting from a perfect 1D substrate at

t

=0.

Figure 1 portrays the resulting film geometry.

Note that for a site in layer

j

to be filled, a triangular

"cone"

of

K(j)=

gj,

i

=j(j+1)/2

atoms must be

filled. Clearly,

if

the cones associated with two particular

sites in the film intersect, then the occupancies

of

those

sites will be correlated. We shall provide amore detailed

discussion

of

these correlations below, but here we only

wish to emphasize that they introduce a nontrivial

structural complexity to the model which was absent

from that

of

Sec.

II.

Despite this fact, it is possible to recursively solve

ex-actly the hierarchy

of

rate equations

of

the

6

and the

various subconfiguration probabilities towhich these

cou-ple. Clearly the statistics

of

the first layer is random, and

d6,

/dt

=k(1

—

6,

). The requirement

of

two adjacent

first-layer atoms forsecond-layer deposition is

rejected

in

the equation d

82ldt

=

k

_{(8, —}

8z).

Determination

of

63, 64, 65, 66, . . .

requires 2,4, 10,26,

.

.

.

equations,

re-spectively, in addition tothose determining the preceding

6

.

These numbers increase dramatically with

j

reAecting the greatly increasing number

of

configurations

that must be considered for higher layers. Some

of

these

equations are shown schematically in

Fig. 2.

Results

from numerical integration

of

these are presented below.

Figure 3 shows the time dependence

of

the

6,

for

j

«6,

and

of

the total deposition rate

d6/dt.

The latter

decreases initially like

k(1

—

6)-ke

to an apparent

asymptotic limit

of

-0.

5k. A different representation

of

6

behavior is provided through

f

defined by

61+&

=f~(6/).

Fr—

om Fig. 4, we see that

f

~

f

_+,

indi-.

cating that growth

of

successive layers is increasingly less

layer-by-layer-like. This is a consequence

of

the

in-creased clustering, described quantitatively below, in

suc-cessively higher layers. Figure 5 shows the

correspond-ing

6

dependence

of

_Iz,

calculated from the

6

assuming

that each atom in layer

j

shields (a net of) one atom in

layer

j

—

1.

Equivalent results are obtained

if

one

as-sumes that an atom supporting (and thus obscured by)

0,

1, 2 atoms it the layer above has a scattering factor

of

1,

—,

',

0, respectively (see the Appendix). One finds very

weak oscillations in

_IB,

here reflecting the fact that the

bridge site-adsorption requirement makes growth

some-what more layer-by-layer-like than for the model

of Sec.

II.

THEsc(100) RANDOM DEPOSITION MODEL

Let k denote the rate

of

random deposition on top

of

previously occupied sites, starting from a sc(100)

sub-strate at time t

=0.

Here and in the following sections,

we label the top substrate layer by

j

=0

(so

60=1)

and

higher layers by'

j

)

_0.

_{Since the total} number

of

adsorp-tion sites is constant in this model, the total coverage

8=

g~"

i

61

equals

kt.

The rate

of

change

of

6.

is

determined by the fraction

of

columns

S,

=6.

_,

—

6.

of

height

j

—

1through

—_4e

Br

dldt

6j=kS

The solution (2) satisfies'

_6J

=

QI", Sl, with

Sk

=

6

"e

/k!.

Note that if

f

is defined via

6~.

+&=fi(6J),

then from Stirling's formula,

x

f

(x)—

=

₀₍

_j

'~

) uniformly in

x,

as

j

—

+

~

. From the above

results for

S.

, one can immediately calculate the Bragg

intensity using

(1).

At the anti-Bragg condition, one finds

that

exhibits no oscillations since this process is so far from

layer-by-layer growth. Finally we remark that since this

model describes the growth

of

independent columns, the

above results hold forany substrate dimension.

FIG.

1. Film geometry for random deposition at bridge sites. The cones associated with the third-layer atom A, fourth-layer atom B,and fourth-layer pair C, are indicated. Occupancies of

5657

39 RANDOM-DEPOSITION MODELS FORTHIN-FILM EPITAXIAL.

.

.

d/dt

d/dt 2

I

~3I

d/dt ₀₂

ra

t 2 2

d/dt Q2Q2

=

2

1 1 1

,'4t

d/dt tI

=

_{3 3}

—

3 3 r

',3 3

'I"

_OXO

=

2

d/dt

0

—

0

(

2. Q1 x Q2 —Qz02

—Qd

—

₀₀

=O-O

+

_EKE

+222

1.0

IX

0.6—

0.2—

f1

——-_~2

08

_—

_--f4

—

—f5

d/dt

=

EQ

—

0

I ! I !

0.6 0.7 0.8 0.9 1.0

0.0 0.1 0.2 0.& 0.4 0 5 0-6

LAYER & COVERAGE

FIG.

2. Rate equations for random deposition atbridge sites

with k

=1.

Probabilities ofconfigurations are represented by

the configurations themselves. Solid (dashed) circles represent

filled (empty) sites inthe layers indicated.

II.

Here minima in

_IB,

occuroccur at zeros

of

1

—

2e,

+

2e2

—

(and

I

ti,), and maxima where

Furtherh insig' ' ht in' too

e

and

f

behavior can be gleaned

from the following.

umber

of

(i)Short time anal-ysis Let K.

(o)

denote the number o

atoms in theh cone associaociatede with the filled configuration

o.,

i.e.

, the number

of

atoms in o. plus the minimum

num-b

ero

f

additionali deposited atoms required to support

h '

o.

Let

N(o)

denote the number

of

ways (p yh si-

'-those in o.

.

e o.

m anal sis

cal orders

of

filling)

of

creating this cone. From ana y '

o

f

the structure

of

the rate equations, ' one can

isfies

P

(o

)

show that the probability

of

o

satisfies

FIG.

4.

.

_y,~

e

-)

=

8

for random deposition atbridge sites.

—

N

o)/K(o)!](kt)

' ', as

t~0

Thus

.

one finds that

e

N'&'

(kt)

-

t-0,

K(j)!

where

K(

j

')=

=j

'(

j

'+1)/2,

as previously, and N 1

=1,

i

of

(4) is

N(2)=2

N(3)=16,

etc. Of

course, the utility o

~ ~ ~

greatly reduced with increasing

j.

(j+2)/j

Equation (4)implies that

f

j(x)

scales like

x

1 _J,and

J"~(

)=f

(x)~&+2~/'&, for small

x.

In fact, Table

I

shows that

_fj(x)/fj(x)

is roughly independent

of

g

forall

x

and that

f,

(x)=a(x)f

(x)"+"

where a

(x)

varies monotonically from aa

0

0

=

=0.

4 to

a(1)=1.

However, since the surface width diverges as

1..0

ca

o.6 O

o.

o-2 4

/ /

/

/

/ / / /

/ /

/

/

I

/

I I I I I

I

/

/ / / /

/ / / / / /

10

/5,

'6

I

12

0.8

0.6

0.&

0.0 0.5 1.0 1.5 (1)

(2) (3) (0)

0.936(1)

0.$43(2)

0.232 (3)

0,032 (4)

0.001(5)

2.0

2.0

time

FICx. 3. Time dependence (in units~ ~ of kt)

t) of

e

(in

mono-layers) and de//d(kt) (dotted line) for random deposition at

bridge sites.

TOTAL

COVERAGE

FIG.

5. Coverage dependence of IB,for ranfor random depositionp

nd minima are

at bridge sites. Layer coverages at maxima and

1

8

TABLE

I.

Values of

_f,

(xl/f,

(xlfor the bridge-site deposition model.

J=2

J

=3

j=4

j=5

0.624

0.551

0.532

0.528

0.699

0.646

0.635

0.636

0.760

0.721

0.715

0.721

0.813

0.790

0.785

0.791

0.863

0.846

0.848

0.855

0.911 0.903

0.907

0.914

0.958

0.957

0.962

0.966

j ~

~,

one expects that

_BJ+,

-BJ.

, and so

f/(x)-x,

as

j ~

oo. This behavior is not incorporated in (5).

(ii) Long time -analysis E.xact integration

of

the rate

equations reveals that probabilities

P

(cr)

of

filled

configurations o. are given by linear combinations

of

terms t~e q _{with integer pand q. One canreadily}

identi-fy the form

of

the dominant term in 1

P(o)—

as

t~

. ce.

In particular, one finds that

(2t)'

6

—

1

—

e ' as

t~~

(j

—

1)!

and

if

o. has n-filled sites in its uppermost layer

j

then

P(cr)-6"

as

t~~.

From (6), it follows immediately

that

f

(x)-1+(2/j)(1

—

x)ln(1

—

x)

as

x~1

.

Infact one can show that

f,

(x)

=

—

2T2[(1

—

x)ln(1

—

x)],

where

_{T [g]}

removes terms

x ",

with n &m, from the

Taylor expansion

of

_g

(x).

Finally we provide a detailed analysis

of

spatial

corre-lations within the growing film. Clearly

if

the cones

asso-ciated with two sites within the film intersect, then the

pair probability for both to be filled will exceed the

prod-uct

of

the corresponding pair

of

layer coverages. (Less

"supporting" atoms are required for these sites to be

filled than

if

they were far separated. ) These positive

correlations correspond to clustering induced by the

geometric aspects

of

the kinetic model rather than by

at-tractive interactions, which are absent here. Specifically,

consider the intralayer pair correlations, C

(i)

P,

(j

j)

P(j)—

, whe-re

—

P,

(j

j)

is the proba-bility

of

a pair

of

layer-j atoms separated by I lattice vectors and

P(j)—

=

6

.

These are nonzero only for the finite range

I

~

j

—

1 (see

Fig.

6), which demonstrates the

non-Markovian nature

of

the film statistics for

j

~2

(see

below). Figure 7 shows the

6

dependence

of

the NN

correlations C

(1).

Divergence

of

the surface width as

j

~

Oo, together with the geometric bound on the surface

slope

of

~/4,

suggest the creation

of

increasingly longer

stretches

of

layer-j atoms for fixed

0

as

j

~

Oo. This

im-plies divergence

of

the intralayer correlation length, so

C

(l)

6.

(1

—

6.

)forall Ias

j

In general, two-cluster correlations will be positive

if

the cones associated with the constituent clusters

of

sites

intersect (Fig. 1) and will be zero otherwise. The latter

result, which follows from the rate equations, allows

fac-torization

of

corresponding configuration probabilities

and has been exploited to reduce the number

of

equations

required for determination

of

the

0-'s.

Note, however,

that the statistics

of

nonfactorizing clusters

of

sites is

quite nontrivial. Let

P(j)

=B,

,

P(jj

),

P(jjj

),

. . .

denote

the probability

of

finding one, two, three,

. . .

consecutive

filled sites in layer

j.

One finds that

P

(jjj

)

P(jj

—

)

/P

_(j

)

are nonzero but small for

j

&_{2, with maxima}

_of

2.

2X

10,

1.

2X

10,

2.

2X10 at

6.

=0.

55,

0.

63,

0.

63,

for

j

=2,

3, 4, respectively. Thus the statistics

of

higher

(j

~

2)layers do not satisfy an exact Markov

prop-erty, as often assumed in terrace statistics models.

We also calculate the Ursell-Mayer three-point

correla-tions'

C(jjj

)

=p(jjjj)

2p(jj

)p—

(j

)

p(j

j—

)p(j

)+—

2p(j

)

These achieve maxima

of

4.

6X

10,

6.

5X 10

9.

2X10

at

6

=0.

44,

0.

31,

0.

28 for

j

=2,

3, 4, respec-tively.

IV. RANDOM DEPOSITION ATFOURFOLD HOLLOW

SITES

Consider random deposition, at impingement rate k, at

fourfold hollow sites starting from an fcc(100)substrate

at t

=0.

Figure 8portrays resulting film geometry.

For

a

site in layer

j

to be filled, a pyramidal cone

of

K(j)=

pi=,

i

=j

(j

+1)(2j+1)/6

atoms is required.

If

cones associated with two sites intersect, then their

occu-pancies will be correlated. Thus nontrivial structural

0.25

0,20

0, 10

FIG.

6. Intralayer pair correlations for half-filled layers

39 RANDOM-DEPOSITION MODELS FORTHIN-FILM

EPITAXIAL.

.

.

5659

0.10

~ ~

0=''.

.

:=

&

—

0

0.08—

z

O

0.06—

O o04

z

0.02

/

/

/

/

/

/

/

/ /

/

/

d/dt 2

d/dt

cI/dt 2 2 2 2

2 2 2

(}/dt

+

2 2 2 2 2 2',

1 r1

=

~o*B+0

~

'-

'oB

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

LAYER COVERAGE

FIG.

7. Dependence ofC,

(1)

on

ej,

for2~

j

~

5.

2

4/'4~ _(ZZ2)

=

2. _1,2 2

=

+

Qi XQ2

—

Q222I

L-1

3

0

.

',

'0

.

0

0

2 2 l . a j

complexity results in this model as for that

of Sec.

III.

Again one can recursively solve the rate equation for the

8

and various subcon6guratioa probabilities to which

these couple. The statistics

of

the 6rst layer is random,

and d6& /dt

=

k(1

—

6&

).

The fourfold hollow

adsorption-site requirement is refiected in the equation

d62/dt

=k(6, —

62).

Determination

of

63, 64,

.

. .

re-quires S,89,

. .

.

equations, respectively, in addition

to

those determining the preceding

6

(see Fig.

9).

These

numbers increase more quickly than in

Sec.

III

rejecting

the greater number

of

con6gurations in the higher

dimen-sions. Results from numerical integration

of

these

equa-tions arepresented below.

Figure 10 shows the time dependence

of

e.

for

j

~4

and

of

d

B/dt.

The latter decreases initially like k(1

—

6)—

ke

"'

to an apparent asymptotic limit

of

-0.

4k. Set

6

_+,

—

=

f

(6.

), as previously. Then

Fig.

11

shows that

_f1

~

f

_+,

indicating less layer-by-layer growth

in successive layers, just as in

Sec.

III.

Figure 12 shows

the

8

dependence

of IB„calcu1ated

from the assumption

that each atom in layer i shields a net

of

one in layer

i

—

1,

or

equivalently, that each atom supporting katoms

in the layer above has a scattering factor

of (4

—

k)/4

for

FIG.

9. Rate equations for random deposition at fourfold hollow sites with k

=

1(notation asinFig. 2).

0.8

O.6

O

P2

/'4

/ /

/ /

/

/ /

/

0

~

k

~

4 (see the Appendix}. Oscillations in

_Iz,

are now

quite noticeable since the fourfold hollow adsorption-site

requirement means that 61Jn growth is initially much

more layer-by-layer-like than in the models

of

Secs.

II

and

III.

Minima occur where 1

—

26,

+26&

—

.

(and

Iz,

}

=0,

and maxima where 6'&

—

_6&+63

—

.

=0.

This

I~,

behavior matches quite closely the low-energy

electron-diffraction Bragg' intensity variations observed

O.o

1 2 3 4 5

I

9 10

E.

r

i

Ai&

i

P

FICx. 8. Film geometry for random deposition at fourfold hollow sites. Layer-1 and -2 atoms are indicated.

time

FICx. 10. Time dependence (in units ofkt) ofe~ (in

mono-layers) and de/d(kl;) (dotted line) for random deposition at

t.p

f)

---

f3

f)

(x)

J +7J+6)I()oj2+5)

«r

small

~.

Table

at

ws

f(x),

J(x)=a

(x)f

( )

J

i(x)

values i

varies from

a(0

east for small

of

thee surface width a

=

.

However diver

J

~.

On

'

again implies that .

.

gence

also show that

fj'(x)

x

(4r)i J

t

(J

—

1)) as

r~ao

and that

p(

)

(7)

sites in it

j

a filled confi

e".

for

s uppermost la e

.

g«ation

~

with

J.

In thjs model on fi

fj

(x)

1+(4/

)(

nds that

x)ln(1

—

x

0.8

0

06

+

~_m 0 4

0.2

O.p ———I

0.] p2 0.3 p4 0.6 07

LAYER

i

C

~

(e

—

=

~+& for random de osi

'

,

(e,

(

)—

=

e

eposition at fourfold

experimentally durjn

'gg

r o P'

e can,

e.

g., accur

6=0.

(e

=

d

e=182 (e =

5) atthe second.

again provide further i ht i to

e

c

, q. &4' ' _v

and

q. ( ) 11

ld

J

+

1)/6

and

( ) '

vai

but n

=24,

N(3)=25 159680,

etc

s i e

x

J+"

'J', and thus like

2 and

2.

2

~ ~

~2' '

1.0

and that

)

asx

x~1

4T4~(1

—

x)ln(1—

Here, 'us

m as in Sec.

~ust as in

Sec.

III,

th

e positive

jf

cones,

the two-cluster

co

ters inter

s associated with

orrelations

ersect and are z

t

e const1tue

ero otherwise. S '

uent

clus-P Ircorrelations

C.(l

f

Pecifically, t

Parated by p

=

(I

l

i for aPair

of

la

dence pf ll

~

—J

—

1.

Fjgure 13sh

0 pnly

nonzero C g

ows the

e

cong .

y intergratin e

(Jl) values

g«ation

probab'1 q

at1O» for 182d 1&erent

pp y, suggesting d

guments

of Sec.

«elation len th

'vergence

vanishin o

s

_J

00 The b

intralayer

~ . g

tw p-cluste

ov{

c

pnd1

t.

1mpllcatlon

f

~ correlations h

two clust

s or confi . .

as the fpl

On fOr

ers Pffilled site

g rat1pnS e second la

PW1ng

NN bonds h

sa e

t

11nked b

Again th1s

f

gu1.ation r

secpnd

t

en the

c

.

yfirstor

eature js u

PrP ability fac

t1OnS requ1red to .

e e numbe

sed tp reduc th

tpr1zes.

Ilpn

f

termine the

e,

ber pfequa

Il actorjzjn

c

e .'s. The statistics

_of

p

ties

n - ayer configurationsions have equal probabili-ha

0.8

M

z

_0.6

z

0.4

lX

0.2

(1)

(2) (3)

0.973(1) 0.698(2)

0.147(3) 0'002(4)

0.992(1)

0.875(2) 0.417(3)"

0.036(4)

~2 2

2~ and ~2

~,

22

~

22

2.

2 2~ ~

222

and

222

.

Here thee sites indicated b' ' e y 2'

o ers are uns eci

1

0.0

020

.4

06 08

1.0

].

2 1.4 1.6 1

TABLE

II.

Values of (x

d o 'tio oc1

e.

1.

e ourfold hollow-site

TOTAL

.

_{Coverage de p}

o ollow sites. La e

&,for random de osi

'

n c. F1g.5).

a maxima and mini-

J=2

J

=3

0.27

0.19

0.36

0.29

0.44

0.38

0.53

0.47

0.62

0.57

0.72

0.69

0.84

39 RANDOM-DEPOSITION MODELS FORTHIN-FILM

EPITAXIAL.

.

.

5661

V. PERCOLATIVE INTRALAYER ISLAND

STRUCTURE AND SURFACE STRUCTURE

The structure

of

filled clusters or islands

of

sites within

each layer

j

isoften

of

interest in nearly or exactly

layer-by-layer epitaxial growth. Clearly as

6

increases, a

criti-cal average

6',

is reached at which these filled regions

first link up sufficiently

to

span the substrate (the

percola-tion threshold). At that point we shall argue (below) that

the filled clusters have aramified, fractal structure. Henceforth, to be specific, we shall consider only the

fcc(100)

random-deposition model

of

Sec. IV, and

some-what arbitrarily define clusters as maximal sets

of

filled

sites (within a given layer

j)

connected by NN bonds.

Since the first layer

(j

=1)

has random statistics,

stan-dard results from random-percolation theory apply to

give

6;

=0.

593and to provide a characterization

of

the

divergence

of

average cluster size, radius

of

gyration,

etc.

,

at

6;,

and

of

the ramified, fractal cluster structure, '

'

We expect that the intralayer correlations present in

higher layers will not change the critical exponents

characterizing the divergence

of

the average cluster size,

etc.

, since the correlation length is always finite. '

'

As a

consequence, the fractal dimension

of

the percolation

cluster at

6'

should equal the random percolation value

of

_4,

'.

However we do expect clustering to lower the

per-colation threshold and the cluster ramification (cluster

perimeter length tosize ratios). '

Small-cell real-space renormalization-group

(RSRG)

techniques provide simple estimates

of

the

6'.

The

cor-responding

56'

=6'

—

6'.

values may provide

semiquan-titative estimates

of

deviations in the exact

6'

from the

exact random-percolation value

of

0.

593.

(Hopefully

sys-tematic errors cancel.) From the exact horizontal

span-ning probabilities for 2X2 cells, we obtain

RSRG

esti-mates

of

56&=

—

0.

051,

563=

—

0.

074, compatible with

our expectations

of

lower threshold values. More

reli-able, and essentially exact determination

of

the

_6J

can be

extracted from finite-size scaling (FSS) analysis in

con-junction with computer simulation. ' ' ' _{We obtain the}

FSS

values

56,

= —

0.049+0.

001,

56&

= —

0.070+0.

003,

56„'=

—

0.089+0.

005,

in surprisingly (perhaps

acciden-0.08

0.07

~J3

Zi ₀p6

0.

05—

0.

04—

O.O3

00)

0.01

0.00

0.1 0.2 0.3 0.4- 0.5 0.6 0.7 0.8 0.9

LAYER COVERAGE

FICx. 13. Dependence ofall nonzero

Cj(l)

on

ej

for

j

=2

(solid lines) and

J

=3.

tally) good agreement with

RSRG

results.

FSS

analysis

also allows us

to

check the critical exponents. We always

find random-percolation values confirming that a

per-colating cluster in layer

j

will have the

random-percolation fractal dimension

of

—,

",

at

6

.

Figure 14

shows typical film geometries near the percolation

thresh-olds

of

the first and fourth layers. Note that fourth-layer

threshold clusters are

"coarser"

than the first-1ayer

ran-dom percolation clusters, reflecting the development

of

positive correlations in the fourth layer. As an aside, we

note that percolation thresholds forempty regions in lay-ers

j

&1 must be determined independently, since the

statistics

of

these is distinct from the filled regions.

Characterization

of

the surface structure

of

the grow-ing film isalso

of

interest. Figure 15 depicts the evolving

surface profile during random deposition at fourfold

hol-low sites. One can clearly see the inAuence

of

the

geometry

of

the adsorption site on the local or

short-range structure. However, growth

of

regions

of

the film

separated laterally by greater than

O((h

)

) lattice

vec-TABLE

III.

K: maximum errors for Kirkwood factorization ofprobabilities ofvarious configurations in terms ofprobabilities of

all constituent pair probabilities (except for

+,

indicating that only NN pairs were used). UM: maximum amplitudes ofthree-point (and four-point

for',

~j) Ursell-Mayer correlations.

6,

values corresponding to maxima are given in parentheses.

[

—

m] means

X 10—m

K

J=2

K

J

=3

UM

J=2

UM

J

=3

2.5[

—

4]

(0.53)

1.1[

—

3]

(0.62)

5.3[

—

3]

(0.44)

1.2[

—

2]

(0.43)

1,1[

—

4]

(0.55)

2.5[

—

3]

(0.44)

J

J

J

5.1[

—

5]

(0.57)

1.2[

—

3]

(0.44)

—

1.6[

—

2]

(0.60)

—

_3.1[

—

_2]

(0.56)

—

_9.5[

—

_3]

(0.73)

—

_1.6[

—

_2]

(0.75)

2.6[

—

3]*

(0.70)

4.9[

—

3]

{0.67)

—

3.8[

—

2]

(0.65)

—

_8.2[

—

_2]

(0.59)

6.0[

—

3]

(0.79)

8.6[

—

3]

(0.&3)

—

_8.5[

—

3]*

(0.64)

—

_1.8[

—

_2]

0 1 1 0 I 1 0 1 I 0 I 0 I 0 0 I 0 'I. il I I 0 I 1 0 I 1 0 1 0 1 0 I 0 I 0 I 0 ~I 0 0 1 0 I 0 0 I 0 0 I 0 0 0 I 0 I I 0 I 0 0 0 0

.

I

0 il 0

1 1

0 0 0 1 1

0 2 0

1 1 0 0 1 1 , L 1 0 0 0 1 0 0 1 1 0 ~00 0 0 0

1

0 0 5

0 0 )

1

0 0 0

1

0 0 0 1

0 0 '5

1 1

O 0 0

1

0 0 0

0 1 1 0 0 I 1 0 1 1 0 I i~i ol 0 'i [)

0 0 I 0 0 1, 0 1 0 0 1 1 0 0 ' _'LJ ) 0 0 1 0 0 I 0 1 0 1 0 ') 0 0 0 1 ') 0 1 '5 5 ) 5 1 I 1 0 0 1 1 o _Jo 0 0 I 0 0 I 0 0 I I 0 _]0 0 0 0 0 0 0 0

.

I.

1 I

0 0 0

1 1

0 0 0

I I 0 0 0

0 0

.

I'~

I 0 0 I 0 0 0 0 0 0 0 0

0 0 0 0 0 le I 0 I 1 0 2 1 1 0 0 1 0 0 1 0 I 0 ) 0 I 0 I 1 2

0 0 0

I'

0

PJ

0 O 0

0 0 1I 0 0 I I 1

0 0 0 0 I I I

0

1

0 0 I

L

0 0 0 0 0 0 0 0 0

1 1 I

0 0 0

1 1 1

QD 0 0 1 I 0 0 1 I 0 0 1 I 0 0 0 0 0 0 0 0

1 1 I

:L'I:

0 0 0 0 0 0 0

1 I

'I*

' 0 I 0 I 0 I 0 0 0 0 0 0 I I 1

0 0 0 I I

0 0 0 I I

0 0 0 0 0 0 0 0 0 0 0 I 2 0 I 0 0 0 0 0

Li

'

0 0 0 0 II 1

0 0

1

l o o 0 0 0

1 I 1 1

olo olo

0 0 0 0

1

0 O O 0 I 0 0 0

1 0 0 'LI' 0 0 I I 0 0 I 0 0 1 0 0 I 1 0 0 1 I 0 0 I 0 0 I

~

0 I I 0 0 1 0 0 I 0 0 I 1 0 0 I 0 0 I 0 0 0 0 I 0 0 I 1 2 0 I I 0 0 1 I 0 0 0 0 Ql 0 0 0 0 0

I 0 0 0

1 1

0 0 0

1 I

0 2 0 I I I 1 I 0 0 0 I 1

0 0 0 0

1

0 0 0 2

1 I 1 1

o ol 2

Io 0 0 0

, FI.

0 0

I

0 I I

0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

n) 0 1 II 0 0 Ql 0 I 1

0 0 0 I 0 0 0

I 0 0 0

I 0 0 0

1

0 0 0

1

0 0 2 1

o~

ll ~0 1

0 0 0

1

0 0 0 I 0 0 0

1 1

0 1 0 0 0 0

0

i! il

0 0 0 0 0 O 0

2 14

3 3

2 2

)~4 2 2

3/3 3 1

3 3 3 3

2

gA 2

2

3 3

4

3 3 3 3

2 +4 2

3 3 3 3

Q4 2 2 2

3 3 1 3

2 2 2

3 3 3 3

5 3 3 3

1 3 3 3

1 3 l3 3

2 4 4

3 3 3 3

2 2 Q4 2

3 3 3 3

2 Q4 2 2

3 3 3 3

2 2 2 2

I 3 3 3

2 2 2 2

3 3 3

5 3 3 3

3 2 1 2 ) 3 3

I4 2 2 2

2 2

i

'L '

2 2

2 .' 2

3 3 3 3

2 2

1 3 3 'S

4

3 13 3

3 3I 4) 3 3 3 ~35 3 3 2 3 3 2 1 2 3 3 3

4 ' 4 2

3 3 3 2

1 3

2 2

3 3 3

41 2 2

3 3 3

2

3 5 5 3

2 1 2 3 3 3 2 3 2 3 I ' 2 2 3) 5 3 2 3 3 2 3 2 3 2 3 2 1 2 1 1 2 5 3

3 3 3

5 S

5 S 3

i

'I '

3 3

I

5 5 )

3 l,

L

3 3 1 2 ) ) )

5 S 5

3 2 3 2 3 14 3 2 I 2 3 2 3 2 3 2 3 5 4 3 3 3 '3 2 3 3 S ' 1 'I 3 3

L',

2 2 2 1 ' 3 Q) 3 3 3 4 5 5 4 5 3 4 5 3 3

5 3 S

2 4 2 2 2 )4 4 41 2 2

3 3 2 2 3 1 2 2 3 2 I 2 3 2 3 3 2 2 3 1 2 2 1 'i 3

5 3 l3

3~

3 2 2 1 0 1 2 2 3 5 3 3 5 4 3 3 3 2 3 1 2 1 2 3 3 5 3_~3 I 3 1 3 2 3 4~4 2 3 3 3~3 3I 5 3 2 3 2 3 2 1 2 3 2 3 4 3 4

3 3 3

2 2

3 3 1

2 2

3 1 1

2 2

3 3 3

3 3 3

3 1 I

0

3 1 I

2 2

3 3 3

3 3 3

2 2

1 1 3

0 2

1 1 3

2 2

3 3 3

5 3 3

5 3 3

3 3~ 3

2 4

J4

3 3 3

2 2

3 3 3

4

5 5 3

5 3 3 ZI 3 2 2 3 2 2 3 2 3 )l 3 2 3 3 3 3 4J 3 2 2 1 5 3 i 3 2 1 5 5 i ) 2 2 3 3 3 2 3 2 3 2 3 2 3 2 3 2 3 Z 3 2 2 3 2 2 1 2 2 3 2 2 Lq 2 2 1 2 2 ) 3 2 2 1 2 2 3 3 2 3 2 3 2 3 3 4 3 ) 2 1 2 1 2 I 2 3 Q4 3 2 3 2 3 2 3 I 3 2 2 I 3 2 2 3 3 2 2 3 1 2 2 5 3 'I 3 3 L

3 3 3 3 3

4 3 1 3 1 3

3_~3 3

3 3 )

2 4 4 4 2

2 4 2

3 5 5 5 3

4 4 2

) 3 3, 5 3 2

3 3 3 3 3

2 2

1 I 3 3 (3

2 2

3 I 3 3 )

) 3 3 3 )

2 2

3 3~5 3

2 4 ' 2

3 I 3 3 5

2 2 2 14

3 3 1 3 3

2 2 2

3 3 3

4I 2 4 2 2

I3 3 3 1

4 4 4 2 2

3 3 3 3

5 5 5 f 3

4 4 4 4 2

3j' 3 5 5

4

3 3,'

5 3 3

2 4 4I 2 2

1 3 5 3 I

2 4 4 2 2

) 3~3 3 3

2 2 4 2

3 3 3 I) 3

4 4I 2 2

3 3 3 3 3

2 2 2 2

3 3 3 3 3

2 2 2 l4

1 3 3 3 3

3 3 3 2 3 2 1 2 3 1 3 2 1 1 2 I 2 3 \ 2 3 2 3 2 3 2 3 4 3 2 3

FIG.

14. Simulated film geometry near the percolation threshold for the first (fourth) layer is shown on the left (right). Layer-j

atoms are indicated by

j,

and the lines indicate perimeters oflayer-one (four) clusters.

tors is independent, so the Poisson result for surface

width

g-(h

)'~

may be expected to apply. This is also

suggested by the analysis

of

Kardar et a/. We shal1

con-sider these questions infuture work.

VI. OTHER SOLVABLEDEPOSITION MODELS

dB, /dt=k(l

—

8,

),

dB,

/dt

=k(8",

8,

),

(8)

—

independent

of

the configuration

of

the X-atom

adsorp-tion ensemble. The

dB,

._{/dt for i}

~3

_{do depend on this}

configuration. Clearly as N increases, the process

be-comes initially more layer-by-layer-like. Consider the

point where

dB,

/dt

=d

82/dt

which corresponds closely

to

the first maximum

of

Ia,

=(l

—

28&+28&)

for

N~

4.

Here one finds that

8,

(and

Bz)

equals

0.

770(and

0.

120)

for N

=4,

0.

789 (and

0.

010)

for N

=5,

0.

809 (and

0.

089) for N

=6,

0.

838 (and

0.

075) for N

=

8,

0.

858 (and

0.

065)

for N

=10,

0.

868 (and

0.

052) for N

=12.

Obviously the

height

of

the first maximum increases with N.

Mode}s involving random deposition at threefold

hol-low sites (N

=3)

of

an

fcc(ill)

substrate have been used

to

describe poor wetting and disorder at certain

crystal-melt interfaces. However, in these mode1s, there is

nat-ural steric constraint that occupation

of

one substrate

deposition site blocks occupation

of

the three NN sites.

Thus at most half these sites can fill, and for random

deposition only

37.

9%

(Ref. 24) are filled at saturation.

Clearly the statistics

of

the first layer is not random (and

There is clear1y considerable Aexibility in the choice

of

adsorption-site requirement in random-deposition

mod-els.

For

example, we could modify the models

of

Secs.

III

and IV by demanding a larger

"raft"

of

layer-i atoms

surround the layer-(i

+

1 )adsorption site.

Consider, in general, random-deposition processes

where the first-layer statistics israndom, and layer-(i

+1)

adsorption requires an ensemble

of

N-layer i atoms.

For

impingement rate k,one has that

not exactly solvable), in contrast to all

of

the models

developed above in this work. In fact, here the intralayer

correlations have infinite range but fast superexponentia1

decay. Also, for these models, deposition in higher

lay-ers is typically not epitaxial.

Another simple model modification involves specifying

different adsorption rates for the different layers. This

re-sults in no structural change in the hierarchical rate

equations, so exact solution is again possible. In the

growth

of

bimetallic thin films, it is conceivable that the

rate

of

deposition onto the substrate differs slightly from

1 222 22 2 2 2 2

I I II I I I I I I I I III I I I I I I II I1I II III I I III I I I I I I I I 0 0 0 0 00 00 00 00 0 0 0 0 0 0 0 0 0II

2 3 333 3 33 3 3 3 3 33 3

2 22 2 2222 2 22 2222 22 2 22 222 222222222 2 22222 22

III I I IIII I I I II I I I I I I I I I1

0 0 0 0

3

5

4~68 ~l 6 l l 6 44 l 4

3 33 3 3 3 3 333333333 3' 33 33333 3 333 333

222 2 2 222 22222 2222 22

I II II

66 5 55 5 5 5 5 5 5 55 5 5

4 ~ 64 4 4 I4~4l6 66 44 6~8~l 6l l~ l44 48 6~

33\ 33)3 33 33 3 3 33 33 33 33

2

55 5 5

~ 4~

33

e 4

777 77 7 7 ') 7 7 66 6 66 66 666 6666666 6 6 6 66 6 5 55 5 5 5 5 55 5

7 77 7 666 66 6 6 6 6 6 6 66 66 6 66

555 5 5 55 5 55 55

0

8 9 9 9 99 99

4 4 4 8 8d4 8as I 84 4 44

77 77~7 7 77 7 77 77 7 7 777

6li 66 6

0 0

9 9 999 9 99

I a ~ ~4 44 48 8844 7777 7 7 7 7 777

6

2

I I I I 1 1

gp o doooo

I I ')

86

I I II I I 0 0 I0 0 00 0 0 0 0 9 9 9 999 9 9 99

88d I8

2

I I I I1II I

0 0 0 0 0 0 0 0 0 0 00

9 99 9 9 9 9

48 ~4

6 6 6 6 6 6 6

55555 55 5 5 5 5 55 5 55 55 5 5 55 5 5 555 55

444 4 ~ 6 ~ 4~~ ~ l~ ~8 4~l

333 )333 3 33 3 3

222 22

0 0 0 0 0 9 9 9 9 9 9 999 9 9

88 888 8~I4 ~4 8 44 ~

7 77 7 7 7 77 7 7 7 7 7 7 7 7 7 7 6 6 66

0 0

I 99 99 9 9

~ ~ ~I ~ ~I ~44444

7 7 7 77

7

5 5 555 5 5 5 5 5

24 ~ ~l4 444 l ~ 4l6 6 6~6~64 ~84 4 l~

3 3 ) ''.33333333 3 3 3 3 333 3 3 33 Il33 3

2 22 22 22

2 I

I!)1I I I I II II I II II I I I I I

OOO0OOO OOOO I O 0I0I0 0 0 0 0I 0 0 0 0 99 9 9 9

8I I 4I4

77 7 0 0

3P 9 9 9 9 9

II 44

I I 00 0 0 00 0I

9 9 9 9I9 9 9 9 9

~~44 I8 84 I 4~ 4 8I4 ~ ~8

'77 7 7 7 777

6666

,I

0 0 9 9 444

7 6 6 6 5 5 5 I 0 9 4 7 6 5 I 00

9 9 9 9 9 9I0

38 I48

FIG.

15. Simulated thin-film surface profiles in the (1,1)

39 RANDOM-DEPOSITION MODELS FORTHIN-FILM EPITAXIAL.

.

.

5663

that for higher layers. Clearly increasing the former

makes the process initially more layer-by-layer-like.

A key feature

of

the exactly solvable deposition models

described above is that the state

of

a given layer is

in-dependent

of

the state

of

all higher layers. As a

conse-quence, the hierarchical rate equations couple

"down-wards" only, making an exact solution viable. .Ballistic

deposition, Eden, and kinetic SOS models do not satisfy

this property,

i.e.

,

de;/dt

depends on the state

of

layer

i

+1

and sometimes higher layers. One could, however,

remove this dependence in the sc(100) Eden model by

im-posing the additional constraint that attachment only

occurs at empty sites with a filled NN beneath or in the

same layer. One still has the complication that the

hierarchical equations couple laterally to increasing1y

larger configurations (in contrast to our models).

Howev-er one can still obtain exact results for

8,

=1

—

e

(trivially) and for

62.

VII.

CONCLUSIONS

In this paper, we have extended the trivial sc(100)

random-deposition model to other adsorption-site

geometries and crystal structures. Although the resulting

models incorporate nontrivial statistical correlations, one

can still obtain exact results for layer coverages (and

oth-er quantities).

It

is clear that restrictions on the

adsorption-site geometry, which are almost certainly

op-erable in real physical systems, have a dramatic effect on

the extent to which the initial film growth is

layer-by-layer-like. Appropriate treatment

of

adsorption site

geometry seems essential to an understanding

of

the

Bragg intensity oscillations observed recently in

experi-mental growth

of Pt

films on Pd(100) at cryogenic

tem-peratures. '

'"

This observation motivated our analysis

of

the coverage dependence

of

the Bragg intensity in

pre-vious sections. Clearly a full-diffracted intensity-profile

analysis is also

of

interest. In future work on the fourfold

hollow-site random-deposition model, we shall show that

the diffuse intensity has nontrivial angular dependence even before the second layer has significant population.

This contrasts the standard sc(100)model, again

show-ing the important inAuence

of

geometry.

qi

y

si

_Q~

J J

FIG.

16. Diagrammatic proof of (A1) for the bridge-site model

X

=2.

ACKNOWLEDGMENTS

Ames Laboratory is operated for the

U.S.

Department

of

Energy by Iowa State University under Contract No.

W-740S-ENG-82. This work was supported by the

Division

of

Engineering, Mathematics, and Geosciences

with Budget Code No.

KC-04-01-03.

The author would

also like to acknowledge

P.

A.

Thiel foruseful discussions

and comments on the manuscript.

APPENDIX: SURFACE ATOM SCATTERING FACTORS

S,

=

g

fke,

(k)=e, —

e,

+i

k=0

(Al)

The last identity exploits probability relationships. These

are illustrated in

Fig.

16for N

=2,

but the generalization

is obvious. Note that for

j

=0

and random first-layer

statistics, (A1)follows from the identity

N

)k iv—k k=0

For

the bridge-site deposition model, each atom can

support (and thus be obscured by) up to N

=2

atoms in

the layer above. In the fourfold hollow deposition model,

N=4.

In either case, suppose that an atom supporting

exactly k atoms in the layer above has a scattering factor

fk

=(N

k)/N.

Thus

—

the scattering factor iszero (unity)

for completely obscured (unobscured) atoms. Let

e

(k)

denote the fraction

of

layer

j

atoms obscured by exactly k

atoms in layer

j+1.

Then the net fraction

of

scatterers

in layer

j

equals

F.

Reif, Statistical and Therma/ Physics (McGraw-Hill, New

York, 1965), p. 42.

J.

D.

Weeks, G.H.Gilmer, and

K.

A. Jackson,

J.

Chem. Phys.

65, 712 (1976).

3P.Meakin, CRC Crit Rev.Solid State Mat.Sci.13,143(1988).

P.Meakin and

R.

Jullien, in Modeling

of

Optical Thin Films, edited by M.

R.

Jacobson [Proc. SPIE821, 46(1988)].

5M. Plischke and Z. Ratz, Phys. Rev.A32,3825(1985);

J.

Ker-tesz and D.

E.

Wolf,

J.

Phys. A21,747(1988); M. Kardar,

G.

Parisi, and Y.-C.Zhang, Phys. Rev.Lett. 56,889(1986).

J.

D.

Weeks and

G.

H. Gilmer, Adv. Chem. Phys. 40, 157

(1979).

7H. M. V. Temperley, Proc. Cambridge Philos. Soc.48, 683

(1952); V.Privman and N.M.Svrakic,

J.

Stat. Phys. 51,1091

(1988).

80n Growth and Form, edited by H.

E.

Stanley and N.

Ostrow-sky (Nijhoff, Dordrecht, 1986).

C. S.Lent and P.

I.

Cohen, Surf. Sci. 139, 121(1984);

J.

M.

Pimbley and

T.

-M.Lu,

J.

Appl. Phys. 57, 1121 (1985).

J.

M.Pimbley and

T.

-M.Lu,

J.

Appl. Phys. 55, 182(1984).

If

h(r) isthe height above some reference vicinal surface, then here random walk ideas imply that ₍~h(r)

—

_h(0)

~ )

—

~r~Iw,

where wisthe average terrace width.

T.

-M.Lu, G. C.Wang, and M.G.Lagally, Surf. Sci.108, 494

(1981);

J.

M.Pimbley and

T.

-M.Lu,

J.

Appl. Phys. 57, 4583

J.

M.Van Hove, C. S.Lent, P.

R.

Pukite, and P.

I.

Cohen,

J.

Vac.Sci.Technol. B

I,

741 (1983);

E.

Bauer, in RHEED and

Ref?ection Energy Imaging

of

Surfaces, edited by P.

K.

Larsen

(Plenum, New York, in press); S.

T.

Purcell,

B.

Heinrich, and

A.S.Arrott, Phys. Rev.B35,6458 {1987);

J.

Vac.Sci.

Tech-nol. B6, 794 (1988); M.

J.

Henzler and M. Horn,

J.

Cryst.

Growth 81,428(1987).

D.

K.

Flynn, W.Wang. S.-L.Chang, M.C.Tringides, and P.

A.Thiel, Langmuir 4, 1096 (1988).

~4D.

_K.

_Flynn,

_J.

_{W.Evans, and P.A.Thiel,}

_J.

_Vac.Sci.

Tech-nol. A (to be published);

J.

W.Evans, D.

K.

Flynn, and P.A.

Thiel, Ultramicrosc. (to be published).

I5The analogous result for submonolayer filling processes has been noted previously. See,e.g.,

J.

W.Evans and

R.

S.Nord, Phys. Rev.A31,3831 (1985).

6E.G. D.Cohen, Physica 28,1025(1962).

D.Stauffer, Phys. Rep. 54, 1(1979);

D.

Staufter, A. Coniglio, and M. Adam, in Advances in Polymer Science (Springer-Verlag, Berlin, 1982),Vol.44.

Note that all layers in the sc(100)model ofSec.

II

have ran-dom statistics. Thus random percolation theory applies for

all

j

~1.

J.

W.Evans and D.

E.

Sanders,

J.

Vac. Sci. Technol. A 6,726

(1988);

D.

E.

Sanders and

J.

W.Evans, Phys. Rev. A38, 4186

(1988).

H.

E.

Stanley,

R.

J.

Reynolds, S.Redner, and

F.

Family, in

Real-Space Renormalization, Vol. 30ofCurrent Physics, edit-ed by

T.

W.Burkhardt and

J.

M.

J.

van Leeuwen (Springer, Berlin, 1982).

H.Saleur and

B.

Derrida,

J.

Phys. {Paris)46,1043(1985).

Consider clusters of empty layer

j

sites connected by NN bonds. Let

_6,

denote associated percolation thresholds, and

set

56,

*=6,

*

—

6&

.

We obtain

2X2

cell RSRG estimates of

66&

=

0.003, 56&

=

—

0.010, and FSS estimates of

66p

=

0.002+0.001,

56

= —

0.004+0.002,

66*

=

0.003

+0.

002. The exact

6,

equals 1

—

6;

or0.407.

A.Bonissent in Crystals: Growth, Properties, and Applications,

edited by H. C. Freyhardt (Springer-Verlag, Berlin, 1983),

Vol.

9.

~D.

E.

Sanders (unpublished simulation estimate);

J.

W.Evans,

J.

Math. Phys. 25, 2527 (1984) provides an analytic estimate

of37.

5%.

J.

W. Evans,

D.

R.

Burgess, and D.

K.

Hoftman,

J.

Math. Phys. 25,3051(1984).

~_{LetP~ (P&)denote the probability offinding an empty layer-2}

site with an empty (filled) layer-1 site beneath it. Then

6&

=

1

—

P,

—

P,

',

and one can show that

P,

(kt)

=e

"'I

_l+

I"'ds

exp[

—

3s

—

2(e

'

—

l)]I

and P'z

=e

"'

_I

_dse'Pz(s).

J.

W.Evans (unpublished).