VOLUME
45,
NUMBER I PHYSICAL
REVIEW
LETTERS
7
JULY
1980
50,
1564
(1976).
S. Nakahara, R.
Schutz and
L. R. Testardi,
in
Pro-
ceedings
of the International Conference on Metallurgi-
cal
Coatings,
San
Diego,
1980
(to
be
published).
A. Segmuller
and
A.
E.
Blakeslee,
J.
Appl.
Cryst.
6,
19
(1973).
The films here will be
specified
by
the ideal
Ni-Cu
thicknesses.
However,
this
is not
meant
to
imply
that
sharp
boundaries were
achieved
in
the
films.
I. S. Jacobs
and
C.
P.
Bean,
in
Magnetism III,
edited
by
G.
Rado
and
H.
Suhl
(Academic,
New
York,
1963),
p.
276.
G.
P. Felcher,
J.W.
Cable,
J.
Q.
Zheng,
J.B.
Ketterson
and J.E.
Hilliard
(private
communication)
have
recently
discussed
clustering
in
these materials.
R.
M. White and
C.
Herring,
to
be
published.
U.
Gradman, J.
Magn. Magn.
Mater.
6,
173
(1977).
Theory
for the Dielectric Function of Granular
Composite
Media
Ping
Sheng
Theoretical
Sciences
Group,
Exxon
Research
and
Engineering
ComPany,
Corporate
Research Science
Laboratories,
Linden,
New
Jersey
0703&
(Received
28
March 1980)
A new
theory
for the dielectric
function of
composite
solids is formulated which
dis-
plays
both
the
optical
dielectric
anomaly
and
the
percolation threshold,
thereby
provid-
ing
a
basis for
a
unified understanding
of
the
optical
and
percolation transport
properties
of
granular
materials.
The
results
of
the
theory
are shown
to
be in
good
agreement
with
experimental
data.
PACS
numbers:
77.
90.
+k
Recent
interest in
the
optical
and
transport
properties
of
granular
composite
materials has
spurred
renewed
theoretical
investigation
in the
calculation of
the dielectric
function'
'
for
a
het-
erogeneous composite
medium. There
are
at
present
two
prevalent
theories
for
the dielectric
constant of
composite
materials. One
is
the
Max-
well-Garnett
theory'
(MGT),
which
is
usually
preferred
for the calculation of
optical proper-
ties,
because
it
predicts
the existence
of
the
op-
tical
dielectric
anomaly
observed in
granular
metal films.
However,
because of the
inherently
asymmetrical
treatment of the two
constituents
of the
composite,
MGT
predictions
grossly
dis-
agree
with
experimental
optical
and
transport
re-
sults in
cermets when
the volume fraction of the
dispersed phase (in
MGT)
becomes
comparable
to or greater
than
that of the matrix
phase.
'
In
particular,
the
theory
does not
produce
the
ob-
served percolation
threshold
in
granular
metals.
The effective
medium
theory
(EMT)
proposed
by
Bruggeman' is the
other
widely
used
approach
to
the calculation
of
the dielectric constant for
corn-
posite
materials. The
EMT does
give
a percola.
—
tion
threshold,
but,
unlike
the
MGT,
it
yields
no
dielectric
anomaly. Moreover,
the
predicted
val-
ue of the percolation
threshold is low
compared
with
the
experimental
result.
'
In this
Letter I
present
a.
new
theory
for
the
dielectric
constant
of
granular
composite (also
known
as
cermet
films).
The
theory
displays
both
the
optical
dielectric
anomaly
and the
per-
colation threshold,
thereby providing
a
basis for
the unified
understanding
of the
optical
and
per-
colation transport properties
of
granular
compo-
site media.
The
study
of
hopping
conductivity
in
grangular
metals'
'
has
indicated
that the microstructure
of composite
films
is
primarily
determined
by
the
grain
formation
process
through
surface
dif-
fusion
of sputtered
or
evaporated
molecules.
The resulting
composition
homogeneity
on
the
scale
of
the surface
diffusion
length
has been
shown
to be responsible
for the characteristic
temperature
and electric field
dependence'
of
granular
metal
hopping
conductivity.
A.
s
the
starting
point
of the
present
theory,
consider
a
spherical region
with
the dimension of
a
diffu-
sion
length
inside
the material. Within
such
a
region
a
fraction
p
of
the
volume
is
taken
by
the
molecules of
component
1
and the rest
by
compo-
nent
2.
Here
p
is the
macroscopic
volume
com-
position
of
component
1.
When
a grain
is formed
inside
this
region
by
diffusion and coalescence.
there are
two
possible
outcomes:
Component
1
forms
the
grain
and
component
2
the
coating,
which
we
denote
as
a type-1
unit;
or
component
2
may
form
the
grain
and
component
1
the
coating,
which is denoted
as
a
type-2 unit. The
relative
probability
of occurrence
for the two
cases can
be estimated
by
counting
the number
of
equally
possible
final
configurations
(corresponding
to
60 1980 The
American
Fhysical
Society
VOLUME
45,
NUMBER
1
PHYSICAL REVIEW
LETTERS
7
Jvr.
v
1980
d
ifferent
positions
of the
grain
inside the
region).
By
assuming
the
grain
to
be
spherical,
it
is
clear
that,
in the case
of
type
1,
the number
of config-
urations
is
proportional
to
v,
=
(1
-p'/2)',
the
free
volume
(up
to a constant
mulI:iplicative
factor)
ac-
cessible
by
the
grain
center
of
mass inside the
spherical region.
By
the
same
reasoning
the
num-
ber of
configurations
for
the
alternative
case
is
proportional
to
v,
=
[1
—
(1-p)'
']'.
It
follows
that,
at
any
value
of
P,
the
relative
probability
of
oc-
currence
for
type
1 unit is
f
=v,
/(v,
+v,
),
and that
for
type
2
unit is
1
—
f.
This
simplified
picture,
though
it
neglects
the
interaction
between
neigh-
boring regions
in
the
grain-forming
process,
nevertheless does
capture
the
essential
structur-
al variation
in
granular
composites
as
a
function
of
composition
p.
For
0-p
(0.
35,
the
value of
f
decreases
slowly
from 1
atp
=0
to
-0.
92
atp
=
0.35.
The
preponderance of
type-1
unit in
this
range
is
in
accord
with the
observed
structure
of
grains
of
component
1
embedded
in the matrix
of
component
2.
As
p
is
increased
from
0.
35
to 0.
65,
f
drops
sharply,
reaching
-0.
08
atp
=0.
65.
This
composition
region,
with
the
two
kinds
of
struc-
tural units
competing
for
dominance,
clearly
cor-
responds
to the
structural
transition
regime
seen
in
electron
micrographs'
at similar
values of
p.
Beyond
p
=
0.
65,
the matrix
inversion
is
basically
complete,
and we
have
grains
of
component
2
dis-
persed
in the
matrix of
component
1.
To calculate
the
dielectric constant
of
the
com-
posite,
let us
consider the
embedding
of
a type-1
or
a type-2
unit in
a
uniform effective medium of
dielectric constant
&.
Upon
application
of
uni-
form
electric
field,
the inclusion
will
produce
a
dipole
moment
D,
,
(i),
where
the
subscript
corre-
sponds
to the
type
of unit
and
i
denotes the
con-
figuration.
If
we
approximate
D,
,
(i)
with the
di-
pole
moment of the
concentric
configuration
Dy
the
effective-medium condition
of
zero
average
dipole
moment'
results
in
the
equation
fD,
+
(1
—
f
)D2
=
0.
It
may
be
noted
that
Eq.
(1)
reduces
to the MGT
if
f
is
set
equal
to 1.
Although the
discussion to
this
point
has
been confined to
the
case
of
spheri-
cal
units,
Eq.
(1)
is
generally
applicable
to
parti-
cles of
spheroidal
shapes
as
mell,
since
the
pre-
ceding
arguments
remain
unchanged
if one
con-
siders
a
spheroidal particle
enclosed
in
a
simi-
Lar-shaped
region.
In
that
general
case,
how-
ever
Dy
2
stands for
the
orientRtionRlly
averaged
dipole
moment
of
coated
confocal
spheroidal par-
ticles
embedded in an effective
medium
~:
D,
=-',
D(e,
e„e»p,
A(a,
u),
B(a))+
2D(q,
q»q»p13
—
2/l(a1u),
3
—
2B(a)},
p
~+(i
~v)~B(~))+
2D(
t~2t~1
1-p1
3
—
2/i(P,
v)t
3
—
2B(P)),
(2a)
(2b)
where
a
is
the
ratio between
the
minor
(major)
and
major
(minor)
axes of the
elliptic
cross
section for
the
type-1
oblate
(prolate)
spheroidal
unit,
p
is
the
similar
quantity
for the
type-2 unit u
=(p/a)'
'
and
v
=
[(1-p)/0]'
'.
D,
&,
and
B
have the
following
functional
forms:
)
[&a+(3-A)y](y
—
x)I/.
+[Bx+(3-B)y](e
y)
A(3-A)(e
—
y)(y
—
x)p+[Bx+
(3-B)y]
[gy
+(3
/I)
g]'
(3a)
1
1,
(1
—
y')'/'w
2
(1
y2)w2
(1
y2)1/2
(s2+y2w2)1/2
B(y)
A(y,
1/y'
')
evaluated
at
s
=0, where
s
is the
.
solution of
the
equation
(s'+w2)'(s2+yI
2)
=1,
and
tan
'gz-~tanh
'z
where y&1.
Equations
(1)-(3)
constitute
a complete
set
of
conditions
for
the
determination of
6
from the
in-
put
parameters
~„e„p,
a,
and
p.
To
compare
the
theory
mith
experimental
results,
we
shorn
in
Fig.
1
the
variation
of
normalized
de
conductivity
o
as a
function of
p
for
W-A1202 composite
films.
'
The
solid lines are calculated with
6y
i,
&,
=0,
and the
p
values
indicated
in
the
figure.
a
is
as-
sumed to be
1
for all
systems
in this
paper
be-
cause
electron mierographs'
have shown
the
met-
—
w(s'+
y'w'),
!
RLLi.
c
particles
to be
roughLy
round.
Tmo
features
of
Fig.
1
should
be
especially
noted.
First,
it
is
seen that
the annealed
sample
and the as-pre-
pared sample
exhibit
opposite
curvatures
in the
0'(p)
variation. This
characteristic
trait
is
re-
markably reproduced
by
the
theory
with
two
dif-
ferent values
of
P.
The
good agreement between
theory
and
experiment therefore
suggests
that
an-
nealing
modifies the
geometry
of the
insulator
particles
from
platelike before
annealing to more
spherical
after
annealing.
The
resulting increase
in
o(p)
upon
annealing
can
then
be
simply
explained
61
VOLUME
45,
NUMBER
1
PHYSICAL REVIEW LETTERS
7
JULY
1980
1.
0
I
I
I
0.
8—
~
ANN
~
AS
15%
1200A
-1—
0.
6—
29% 1400A
10%
0.
4
0.
2—
I I I /
0.
2
0.
4
0.
6
0.
8
1.0
0
54%
1700A
P
25%
FIG.
1.
Normalized
conductivity
c'
plotted
as
a
func-
tion of metal volume fraction
p
for
samples
of
W-A1203
cermets. The
data
are
from
Ref.
7.
Solid lines
are
calculated from the
theory.
Dashed line denotes
EMT
result.
6
0
50%
by
the fact
that,
for the
same amount of
insulator,
platelets
are
expected
to
be more
effective
than
spheres
in
impeding
current flow. The second
feature of the
figure
is the
generally
high
value
of
percolation
threshold
compared
with predic-
tions of EMT and
continuous
random-percolation
model.
"
The
result of the
present
theory
shows
that
the structural constraint
imposed
by
the
con-
dition
of local
composition
homogeneity
is
suffi-
cient to raise the
percolation threshold to
the
ob-
served
range
of
P,
=0.
35-0.
5.
Optical
transmission data of
Au-SiO,
composite
films'
are
plotted
in
Fig.
2
for five different Au
compositions.
"
The
absorption
peaks
seen
in
these curves
are
the
well-known
dielectric
anom-
alies
peculiar
to the
optics
of
composite media.
To obtain
the theoretical
curves,
I
have used
for
the
experi
mentally
known
constants"
of
Au,
modified
to
take into account the
decrease of
con-
duction electron relaxation time
7
due to
small-
particle
microstructure.
"
The
value
of
w
used
in
the
calculation
is
2.
5&&10
"
sec,
corresponding
to a particle
size of
50
A.
Also,
e,
=2.
2
and
P=2
were
employed,
where the
value
of
p
is
chosen
by
cursory
survey
of insulator
particle
shapes
from
electron
micrographs.
'
Since the
absolute
accuracy
of
experimental composition
determina-
tion
is
-+
51,
the theoretical
value
of
P
(labeled
to the
right
of each
curve)
is
treated as
a
slightly
adjustable
parameter
in
the
calculation.
Once
e
is
known,
the
transmission
through
the film
is
65%
1500
t
I
I
I
60%
80%-
-10
I
1.
6 1.
4
1.
2
1.
0
0.8
0.
6
0.
4
0.
2
obtained
by
the
usual
electromagnetic
wave
for-
malism,
'
with the
experimentally determined
film thickness
values marked
above
each curve.
From
Fig.
2
it is noticed
that the
present
theory
reproduces all the
characteristic
features
of the
FIG. 2.
Optical
transmission
as
a
function of
light
wavelength
for
a
series
of
Au-SiO&
composites.
Data
are
from
Ref. 3.
For
clarity,
the
curves
are
displaced
with
respect
to
one another.
The theoretical
curves
are normalized to
the
experimental
values at
0.
3
pm.
The theoretical values of
p
are labeled to
the
right
of
pairs
of
curves,
whereas
the
experimental values
of
p
and the
film
thickness
are
given
above
each
pair
of
curves.
62
VOLUME
45,
NUMBER
I
PHYSICAL
REVIEW LETTERS
7
JULY
1980
1.0
0.
8—
0.
6—
0.
4—
I
I
I
I
0
0
self-consistency of the
theory
and
demonstrates
the essential
underlying
role
of microstructure
in
both the
optical
and
electrical
properties
of
physical
composite systems.
The author
wishes
to thank
R. Cohen
for
helpful
discussions and B.
Abeles
and
R.
Stephens
for
pointing
out.
the
revision
in the
values of cermet
composition.
0.
2—
I
0.
2
/
/
/
/
/
/
l~
0.
W
0.
6
0.8
1.
0
FIG.
3.
Normalized
conductivity
o
as
a
function of
p
for
Au-SiO& cermets.
Data
from
Ref. 3. Dashed
line
denotes
EMT result.
data.
Compared
to
MGT,
the
agreement
of this
theory
with
experimental results
is
far more
sat-
isfactory
in
two
respects.
First,
whereas in
the
MGT the
dielectric
anomaly persists
up
to
p
=
l,
the new
theory
shows
a
rapid
disappearance
of
the dielectric
anomaly
above
p
=
0.
7,
in
accord
with observed
data.
Second,
the MGT
exhibits
infrared
behavior
typical
of the
matrix
compo-
nent
over the
entire
range
of
P.
Therefore,
if
one
(arbitrarily)
chooses
an insulator
matrix,
the
infrared
transmission
will
be
independent of
frequency
for
all
p.
However,
in
actual
cases the
infrared
transmission
undergoes
a
transition
from
being
constant
at
small
p
values
to
increas-
ing
with
frequency
at
p
&0.
6.
Such a transition
is
indeed
correctly
reproduced
by
the
present
theo-
ry.
The dc
electrical
transport
data'
of the
Au-
SiO,
composites are
displayed
in
Fig.
3. The
sol-
id
line
is
calculated
from
the
theory
with the
same
value of
P
as in
Fig.
2.
The
excellent
agree-
ment
with the
experimental results
confirms
the
'A
large
number of
articles
and
references
may
be
found in
Electrical
TxansPo& and
OPtical
MoPewties
of
Inhomogeneous Media-1977,
edited
by
J.
C. Garland
and
D.
B.
Tanner,
AIP
Conference
Proceedings
No.
40
(American
Institute of
Physics,
New
York,
1978).
D.
Stroud and
F. P.
Pan,
Phys.
Rev. B
17,
1602
(1978).
'R.
W.
Cohen,
G. D.
Cody,
M.
D.
Coutts,
and
B.
Abe-
les,
Phys.
Rev.
B
8,
3689
(1973).
D.
A.
G.
Bruggeman,
Ann.
Phys.
(Leipzig) 24,
636
(1935)
.
'B.
Abeles, H. L.
Pinch,
and
J.I.
Gittleman,
Phys.
Rev. Lett.
35,
286
(1975).
P.
Sherg,
B.
Abeles,
and
Y.
Arie,
Phys. Rev. Lett.
31,
44
(1973).
B.
Abeles,
P.
Sheng,
M.
D.
Coutts,
and
Y.
Arie,
Adv.
Phys.
24,
407
(1975);
B.
Abeles,
Appl.
Solid
State
Sci.
6,
1
(1976).
P.
Sheng
and B.
Abeles,
Phys.
Rev. Lett.
28,
34
(1972).
R.
Landauer,
J.
Appl. Phys.
23,
779
(1962).
'
H. Scher and
H,.
Zallen, J.
Chem.
Phys.
53,
3759
(1970)
.
~'The
author
has been
advised
by
B.
Abeles
and
R.
Stephens that there
is
a
systematic shift
in
the
values of
composition
published in
Ref. 4.
The
com-
positions
used
in the
present
paper
represent the
re-
vised
values.
P. B.
Johnson
and
R. W.
Christy,
Phys.
Rev. B
6,
437O
(1972).
'
See Ref. 3 for
detailed
prescription
and
justification
of
the
procedure.
'40.
S.
Heavens, OPtical
ProPe&ies
of
Thin
Solid
films
(Dover,
New
York,
1965),
p.
55.
63
