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.EvansAmes 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.
INTRODUCTIONEpitaxial growth
of
thin films has become an areaof
much current research, because
of
both its fundamentalinterest and also its importance in device fabrication. A
variety
of
statistical mechanical models have beendeveloped 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) faceof
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 thefirst empty site reached with a filled nearest neighbor
(NN). Overhangs and avoids result.
If
L
isthe substratelinear dimension then
g-(h
)~
withL
=
Do, andg-L
with
(h
)
=
oo. The exponentsa, P
are nontrivial anddi-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)
atlateral position
r
satisfies(~h(r)
—
h(0)~)
—
~r~ in 2Dand 1n~r~ in
3D.
The kinetic model introduces dynamicsthrough microscopic absorption, desorption, and
diffusion rnechanisrns.
It
describes the transition fromnucleated growth for
T~
T~ to continuous growth forT~
Ttt.For
the latter, steady states with constant (orperiodic) d
(h
)
/dt and goccur.
(v) Cluster statistics models describing structure only,
not dynamics, involve counting certain interface
configurations
of
N atoms. These relate to depositionmodels as random animals relate
to
kinetic clustergrowth models.
(iv) Terrace statistics models, mostly for 1Dsubstrates,
describe structure only, not dynamics. Finite-level
mod-els may describe the initial stages
of
epitaxyor
nearlayer-by-layer growth. Infinite-level models may
de-scribe perfect or disordered vicinal surfaces. ' Ad hoc
specifications
of
terrace width distributions, and/orprob-abilities for stepping up or down, allow straightforward
calculation
of
diffracted intensity profiles. Simplegen-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 scatterersor
"sur-face atoms.
"
Here we consider only the amplitudeof
theBragg peak
I~,
which is determined by the net fractionof
scatterers
S
in the various layersj
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 structurenear the anti-Bragg condition P
=
(2n+
1 )m., wheree'J~=(
—
1)j.
The appearanceof
intensity(IB,
)oscilla-tions during
"near"
layer-by-layer growth is nowexploit-ed routinely as an analytic tool. Specifically, reQection
high-energy electron diffraction is often used during
molecular beam epitaxy
to
monitor the total coverage innear layer-by-layer epitaxial growth. '
If
one makes thecommon assumption ' that each atom produces a net
shielding
of
one atom in the adlayer directly beneath,then SJ
=
—
BJ
—
BJ.+& where OJ. is the coverageof
layerj.
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, sinceit reAects crystal geometry. Analysis
of
this typeof
mod-el is motivated by recent experiments on the deposition
of
Pt
on Pd(100) at low temperatures. ''
Our workpro-vides a new class
of
epitaxial-growth models which areexactly 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 analysisof
the sc(100)random-deposition model and discuss
associ-ated
Iz,
behavior. The rate-equation approach is adaptedtoprovide an exact analysis
of
the kinetics and correlatedstatistics
of
random deposition at bridge sites on a 1Dsubstrate in
Sec.
III
and at the fourfold hollow sites on afcc(100)substrate in Sec.
IV.
(These rate equations arejust the hierarchical form
of
the master equations. )Specifically we focus on elucidating the behavior
of
the6,
the short-range intralayer correlations, and theasso-ciated
Ir„.
InSec.
Vwe characterize the (nonlocal)struc-ture
of
the islands or clustersof
filled sites within eachlayer exploiting concepts and techniques
of
percolationtheory. Surface structure is also discussed brieAy. Some
simple generalizations
of
these models which still permitexact analysis are described in
Sec.
VI.
There we alsoin-dicate the diSculties inherent in rate-equation analysis
of
other thin-film growth models. Finally in
Sec.
VII
were-view our findings.
III.
RANDOM DEPOSITION ATBRIDGESITES
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 befilled. Clearly,
if
the cones associated with two particularsites in the film intersect, then the occupancies
of
thosesites will be correlated. We shall provide amore detailed
discussion
of
these correlations below, but here we onlywish 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 equationsof
the6
and thevarious subconfiguration probabilities towhich these
cou-ple. Clearly the statistics
of
the first layer is random, andd6,
/dt=k(1
—
6,
). The requirementof
two adjacentfirst-layer atoms forsecond-layer deposition is
rejected
inthe equation d
82ldt
=
k(8, —
8z).
Determinationof
63, 64, 65, 66, . . .
requires 2,4, 10,26,.
.
.
equations,re-spectively, in addition tothose determining the preceding
6
.
These numbers increase dramatically withj
reAecting the greatly increasing number
of
configurationsthat must be considered for higher layers. Some
of
theseequations are shown schematically in
Fig. 2.
Resultsfrom numerical integration
of
these are presented below.Figure 3 shows the time dependence
of
the6,
forj
«6,
andof
the total deposition rated6/dt.
The latterdecreases initially like
k(1
—
6)-ke
to an apparentasymptotic limit
of
-0.
5k. A different representationof
6
behavior is provided throughf
defined by61+&
=f~(6/).
Fr—
om Fig. 4, we see thatf
~
f
+,
indi-.
cating that growth
of
successive layers is increasingly lesslayer-by-layer-like. This is a consequence
of
thein-creased clustering, described quantitatively below, in
suc-cessively higher layers. Figure 5 shows the
correspond-ing
6
dependenceof
Iz,
calculated from the6
assumingthat each atom in layer
j
shields (a net of) one atom inlayer
j
—
1.
Equivalent results are obtainedif
oneas-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 veryweak oscillations in
IB,
here reflecting the fact that thebridge site-adsorption requirement makes growth
some-what more layer-by-layer-like than for the model
of Sec.
II.
THEsc(100) RANDOM DEPOSITION MODELLet k denote the rate
of
random deposition on topof
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
(so60=1)
andhigher layers by'
j
)
0.
Since the total numberof
adsorp-tion sites is constant in this model, the total coverage
8=
g~"
i61
equalskt.
The rateof
changeof
6.
isdetermined by the fraction
of
columnsS,
=6.
,—
6.
of
heightj
—
1through—4e
Br
dldt
6j=kS
The solution (2) satisfies'
6J
=
QI", Sl, withSk
=
6
"e
/k!.
Note that iff
is defined via6~.
+&=fi(6J),
then from Stirling's formula,x
f
(x)—
=
0(
j
'~) uniformly in
x,
asj
—
+~
. From the aboveresults for
S.
, one can immediately calculate the Braggintensity using
(1).
At the anti-Bragg condition, one findsthat
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, theabove 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 of5657
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=
2d/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—
—f5d/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 siteswith k
=1.
Probabilities ofconfigurations are represented bythe configurations themselves. Solid (dashed) circles represent
filled (empty) sites inthe layers indicated.
II.
Here minima inIB,
occuroccur at zerosof
1
—
2e,
+
2e2
—
(andI
ti,), and maxima whereFurtherh insig' ' ht in' too
e
andf
behavior can be gleanedfrom the following.
umber
of
(i)Short time anal-ysis Let K.
(o)
denote the number oatoms in theh cone associaociatede with the filled configuration
o.,
i.e.
, the numberof
atoms in o. plus the minimumnum-b
ero
f
additionali deposited atoms required to supporth '
o.
LetN(o)
denote the numberof
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 structureof
the rate equations, ' one canisfies
P
(o
)show that the probability
of
o
satisfiesFIG.
4..
y,~e
-)=
8
for random deposition atbridge sites.—
No)/K(o)!](kt)
' ', ast~0
Thus.
one finds thate
N'&'(kt)
-
t-0,
K(j)!
where
K(
j
')=
=j
'(
j
'+1)/2,
as previously, and N 1=1,
i
of
(4) isN(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 likex
1 J,andJ"~(
)=f
(x)~&+2~/'&, for smallx.
In fact, TableI
shows thatfj(x)/fj(x)
is roughly independentof
gforall
x
and thatf,
(x)=a(x)f
(x)"+"
where a
(x)
varies monotonically from aa0
0
=
=0.
4 toa(1)=1.
However, since the surface width diverges as1..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
(inmono-layers) and de//d(kt) (dotted line) for random deposition at
bridge sites.
TOTAL
COVERAGE
FIG.
5. Coverage dependence of IB,for ranfor random depositionpnd minima are
at bridge sites. Layer coverages at maxima and
1
8
TABLE
I.
Values off,
(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 thatBJ+,
-BJ.
, and sof/(x)-x,
asj ~
oo. This behavior is not incorporated in (5).(ii) Long time -analysis E.xact integration
of
the rateequations reveals that probabilities
P
(cr)of
filledconfigurations 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 1P(o)—
ast~
. ce.In particular, one finds that
(2t)'
6
—
1—
e ' ast~~
(j
—
1)!
and
if
o. has n-filled sites in its uppermost layerj
thenP(cr)-6"
ast~~.
From (6), it follows immediatelythat
f
(x)-1+(2/j)(1
—
x)ln(1
—
x)
asx~1
.Infact one can show that
f,
(x)
=
—
2T2[(1
—
x)ln(1—
x)],
where
T [g]
removes termsx ",
with n &m, from theTaylor expansion
of
g(x).
Finally we provide a detailed analysis
of
spatialcorre-lations within the growing film. Clearly
if
the conesasso-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 pairof
layer coverages. (Less"supporting" atoms are required for these sites to be
filled than
if
they were far separated. ) These positivecorrelations correspond to clustering induced by the
geometric aspects
of
the kinetic model rather than byat-tractive interactions, which are absent here. Specifically,
consider the intralayer pair correlations, C
(i)
P,
(jj)
P(j)—
, whe-re—
P,
(jj)
is the proba-bilityof
a pairof
layer-j atoms separated by I lattice vectors andP(j)—
=
6
.
These are nonzero only for the finite rangeI
~
j
—
1 (seeFig.
6), which demonstrates thenon-Markovian nature
of
the film statistics forj
~2
(seebelow). Figure 7 shows the
6
dependenceof
the NNcorrelations C
(1).
Divergenceof
the surface width asj
~
Oo, together with the geometric bound on the surfaceslope
of
~/4,
suggest the creationof
increasingly longerstretches
of
layer-j atoms for fixed0
asj
~
Oo. Thisim-plies divergence
of
the intralayer correlation length, soC
(l)
6.
(1
—
6.
)forall Iasj
In general, two-cluster correlations will be positive
if
the cones associated with the constituent clusters
of
sitesintersect (Fig. 1) and will be zero otherwise. The latter
result, which follows from the rate equations, allows
fac-torization
of
corresponding configuration probabilitiesand has been exploited to reduce the number
of
equationsrequired for determination
of
the0-'s.
Note, however,that the statistics
of
nonfactorizing clustersof
sites isquite nontrivial. Let
P(j)
=B,
,P(jj
),P(jjj
),. . .
denotethe probability
of
finding one, two, three,. . .
consecutivefilled sites in layer
j.
One finds thatP
(jjj
)P(jj
—
)/P
(j
)are nonzero but small for
j
&2, with maximaof
2.
2X
10,
1.
2X
10,
2.
2X10 at6.
=0.
55,0.
63,0.
63,
forj
=2,
3, 4, respectively. Thus the statisticsof
higher
(j
~
2)layers do not satisfy an exact Markovprop-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.
6X10,
6.
5X 109.
2X10
at6
=0.
44,0.
31,
0.
28 forj
=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
asite in layer
j
to be filled, a pyramidal coneof
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 layers39 RANDOM-DEPOSITION MODELS FORTHIN-FILM
EPITAXIAL.
.
.
56590.10
~ ~
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)
onej,
for2~j
~
5.2
4/'4~ (ZZ2)
=
2. 1,2 2=
+
Qi XQ2—
Q222IL-1
30
.
',
'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 whichthese couple. The statistics
of
the 6rst layer is random,and d6& /dt
=
k(1—
6&).
The fourfold hollowadsorption-site requirement is refiected in the equation
d62/dt
=k(6, —
62).
Determinationof
63, 64,
.
. .
re-quires S,89,
. .
.
equations, respectively, in additionto
those determining the preceding
6
(see Fig.9).
Thesenumbers increase more quickly than in
Sec.
III
rejecting
the greater number
of
con6gurations in the higherdimen-sions. Results from numerical integration
of
theseequa-tions arepresented below.
Figure 10 shows the time dependence
of
e.
forj
~4
and
of
dB/dt.
The latter decreases initially like k(1—
6)—
ke"'
to an apparent asymptotic limitof
-0.
4k. Set6
+,
—
=
f
(6.
), as previously. ThenFig.
11shows that
f1
~
f
+,
indicating less layer-by-layer growthin successive layers, just as in
Sec.
III.
Figure 12 showsthe
8
dependenceof IB„calcu1ated
from the assumptionthat each atom in layer i shields a net
of
one in layeri
—
1,or
equivalently, that each atom supporting katomsin the layer above has a scattering factor
of (4
—
k)/4
forFIG.
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 inIz,
are nowquite 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&
—
.
(andIz,
}=0,
and maxima where 6'&—
6&+63
—
.
=0.
This
I~,
behavior matches quite closely the low-energyelectron-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)
---
f3f)
(x)
J +7J+6)I()oj2+5)«r
small~.
Tableat
ws
f(x),
J(x)=a
(x)f
( )J
i(x)
values ivaries from
a(0
east for small
of
thee surface width a=
.
However diverJ
~.
On'
again implies that ..
gence
also show that
fj'(x)
x
(4r)i J
t
(J
—
1)) asr~ao
and that
p(
)(7)
sites in it
j
a filled confi
e".
fors uppermost la e
.
g«ation
~
withJ.
In thjs model on fifj
(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 fourfoldexperimentally durjn
'gg
r o P'e can,
e.
g., accur6=0.
(e
=
d
e=182 (e =
5) atthe second.
again provide further i ht i to
e
c
, q. &4' ' vand
q. ( ) 11
ld
J
+
1)/6
and( ) '
vai
but n=24,
N(3)=25 159680,
etcs i e
x
J+"
'J', and thus like2 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,
the positive
jf
cones,
the two-cluster
co
ters inter
s associated with
orrelations
ersect and are z
t
e const1tueero otherwise. S '
uent
clus-P Ircorrelations
C.(l
f
Pecifically, t
Parated by p
=
(I
li for aPair
of
ladence pf ll
~
—J
—
1.
Fjgure 13sh0 pnly
nonzero C g
ows the
e
cong .
y intergratin e
(Jl) values
g«ation
probab'1 qat1O» for 182d 1&erent
pp y, suggesting d
guments
of Sec.
«elation len th
'vergence
vanishin o
s
J
00 The bintralayer
~ . g
tw p-cluste
ov{
c
pnd1t.
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 bAgain th1s
f
gu1.ation r
secpnd
t
en thec
.
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
0.6z
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
and222
.
Here thee sites indicated b' ' e y 2'
o ers are uns eci
1
0.0
020
.406 08
1.0].
2 1.4 1.6 1TABLE
II.
Values of (xd o 'tio oc1
e.
1.e ourfold hollow-site
TOTAL
.
Coverage de po 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.
.
.
5661V. PERCOLATIVE INTRALAYER ISLAND
STRUCTURE AND SURFACE STRUCTURE
The structure
of
filled clusters or islandsof
sites withineach layer
j
isoftenof
interest in nearly or exactlylayer-by-layer epitaxial growth. Clearly as
6
increases, acriti-cal average
6',
is reached at which these filled regionsfirst link up sufficiently
to
span the substrate (thepercola-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 modelof
Sec. IV, andsome-what arbitrarily define clusters as maximal sets
of
filledsites (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 characterizationof
thedivergence
of
average cluster size, radiusof
gyration,etc.
,at
6;,
andof
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 aconsequence, the fractal dimension
of
the percolationcluster at
6'
should equal the random percolation valueof
4,'.
However we do expect clustering to lower theper-colation threshold and the cluster ramification (cluster
perimeter length tosize ratios). '
Small-cell real-space renormalization-group
(RSRG)
techniques provide simple estimates
of
the6'.
Thecor-responding
56'
=6'
—
6'.
values may providesemiquan-titative estimates
of
deviations in the exact6'
from theexact random-percolation value
of
0.
593.
(Hopefullysys-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 withour expectations
of
lower threshold values. Morereli-able, and essentially exact determination
of
the6J
can beextracted from finite-size scaling (FSS) analysis in
con-junction with computer simulation. ' ' ' We obtain the
FSS
values56,
= —
0.049+0.
001,
56&= —
0.070+0.
003,
56„'=
—
0.089+0.
005,
in surprisingly (perhapsacciden-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)
onej
forj
=2
(solid lines) andJ
=3.
tally) good agreement with
RSRG
results.FSS
analysisalso allows us
to
check the critical exponents. We alwaysfind random-percolation values confirming that a
per-colating cluster in layer
j
will have therandom-percolation fractal dimension
of
—,",
at6
.
Figure 14shows typical film geometries near the percolation
thresh-olds
of
the first and fourth layers. Note that fourth-layerthreshold clusters are
"coarser"
than the first-1ayerran-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 thestatistics
of
these is distinct from the filled regions.Characterization
of
the surface structureof
the grow-ing film isalsoof
interest. Figure 15 depicts the evolvingsurface profile during random deposition at fourfold
hol-low sites. One can clearly see the inAuence
of
thegeometry
of
the adsorption site on the local orshort-range structure. However, growth
of
regionsof
the filmseparated laterally by greater than
O((h
)
) latticevec-TABLE
III.
K: maximum errors for Kirkwood factorization ofprobabilities ofvarious configurations in terms ofprobabilities ofall constituent pair probabilities (except for
+,
indicating that only NN pairs were used). UM: maximum amplitudes ofthree-point (and four-pointfor',
~j) Ursell-Mayer correlations.6,
values corresponding to maxima are given in parentheses.[
—
m] meansX 10—m
K
J=2
K
J
=3
UMJ=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
.
I0 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 00 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 10 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 0I 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 10 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 35 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-jatoms 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 alsosuggested by the analysis
of
Kardar et a/. We shal1con-sider these questions infuture work.
VI. OTHER SOLVABLEDEPOSITION MODELS
dB, /dt=k(l
—
8,
),dB,
/dt=k(8",
8,
),
(8)—
independent
of
the configurationof
the X-atomadsorp-tion ensemble. The
dB,
./dt for i~3
do depend on thisconfiguration. 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 closelyto
the first maximumof
Ia,
=(l
—
28&+28&)
forN~
4.
Here one finds that
8,
(andBz)
equals0.
770(and0.
120)for N
=4,
0.
789 (and0.
010)
for N=5,
0.
809 (and0.
089) for N=6,
0.
838 (and0.
075) for N=
8,0.
858 (and0.
065)for N
=10,
0.
868 (and0.
052) for N=12.
Obviously theheight
of
the first maximum increases with N.Mode}s involving random deposition at threefold
hol-low sites (N
=3)
of
anfcc(ill)
substrate have been usedto
describe poor wetting and disorder at certaincrystal-melt interfaces. However, in these mode1s, there is
nat-ural steric constraint that occupation
of
one substratedeposition 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 (andThere is clear1y considerable Aexibility in the choice
of
adsorption-site requirement in random-deposition
mod-els.
For
example, we could modify the modelsof
Secs.III
and IV by demanding a larger"raft"
of
layer-i atomssurround 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 modelsdeveloped 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 therate
of
deposition onto the substrate differs slightly from1 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.
.
.
5663that for higher layers. Clearly increasing the former
makes the process initially more layer-by-layer-like.
A key feature
of
the exactly solvable deposition modelsdescribed above is that the state
of
a given layer isin-dependent
of
the stateof
all higher layers. As aconse-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 stateof
layeri
+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.
CONCLUSIONSIn 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 theadsorption-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 sitegeometry seems essential to an understanding
of
theBragg intensity oscillations observed recently in
experi-mental growth
of Pt
films on Pd(100) at cryogenictem-peratures. '
'"
This observation motivated our analysisof
the coverage dependenceof
the Bragg intensity inpre-vious sections. Clearly a full-diffracted intensity-profile
analysis is also
of
interest. In future work on the fourfoldhollow-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
ysi
Q~J J
FIG.
16. Diagrammatic proof of (A1) for the bridge-site modelX
=2.
ACKNOWLEDGMENTS
Ames Laboratory is operated for the
U.S.
Departmentof
Energy by Iowa State University under Contract No.W-740S-ENG-82. This work was supported by the
Division
of
Engineering, Mathematics, and Geoscienceswith Budget Code No.
KC-04-01-03.
The author wouldalso like to acknowledge
P.
A.
Thiel foruseful discussionsand 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 generalizationis obvious. Note that for
j
=0
and random first-layerstatistics, (A1)follows from the identity
N
)k iv—k k=0
For
the bridge-site deposition model, each atom cansupport (and thus be obscured by) up to N
=2
atoms inthe layer above. In the fourfold hollow deposition model,
N=4.
In either case, suppose that an atom supportingexactly 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
layerj
atoms obscured by exactly katoms in layer
j+1.
Then the net fractionof
scatterersin layer
j
equalsF.
Reif, Statistical and Therma/ Physics (McGraw-Hill, NewYork, 1965), p. 42.
J.
D.
Weeks, G.H.Gilmer, andK.
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 Modelingof
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 andG.
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,
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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.
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Appl. Phys. 57, 1121 (1985).J.
M.Pimbley andT.
-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 andT.
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Appl. Phys. 57, 4583J.
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 andRef?ection Energy Imaging
of
Surfaces, edited by P.K.
Larsen(Plenum, New York, in press); S.
T.
Purcell,B.
Heinrich, andA.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 andR.
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 forall
j
~1.
J.
W.Evans and D.E.
Sanders,J.
Vac. Sci. Technol. A 6,726(1988);
D.
E.
Sanders andJ.
W.Evans, Phys. Rev. A38, 4186(1988).
H.
E.
Stanley,R.
J.
Reynolds, S.Redner, andF.
Family, inReal-Space Renormalization, Vol. 30ofCurrent Physics, edit-ed by
T.
W.Burkhardt andJ.
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. Let6,
denote associated percolation thresholds, andset
56,
*=6,
*—
6&.
We obtain2X2
cell RSRG estimates of66&
=
0.003, 56&=
—
0.010, and FSS estimates of66p
=
0.002+0.001,56
= —
0.004+0.002,66*
=
0.003+0.
002. The exact6,
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 estimateof37.
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&