VOLUME
31,
NUMBER
1
PHYSICAL
REVIEW
LETTERS
2
Jvz,
v
1973
and Professor
R.
J.
Elliott and
Dr.
Richard
For-
man for several discussions
on
the
properties
of
cuprous
oxide.
We
also wish
to
thank
Dr.
For-
man,
W.
S.
Brower,
Jr.
,
and
H.
S.
Parker
for
the
generous
loan
of
the
crystal
used
in
our
ex-
periments.
)Work
supported
by
the National
Science Foundation.
*Alfred
P. Sloan
Research Fellow.
E.
F.
Gross,
Vsp.
Fiz. Nauk
76,
433
(1962),
and
103,
431
(1971)
[Sov.
Phys. Usp.
5,
195
(1962)
and
14,
104
{1971))
.
A.
Compaan
and H.
Z.
Cuxnmins,
Phys.
Rev. B
6,
4753
(1972).
Y. Petroff,
P. Y.
Yu,
and Y. B.
Shen, Phys.
Bev.
Lett.
29,
1558
(1972).
P. Y.
Yu,
Y. R.
Shen,
Y.
Petroff,
and
L. M.
Falicov,
Phys.
Rev. Lett.
30,
283
(1973).
E.
F.
Gross,
A. G.
Zhilich,
B.
P.
Zakharchenya,
and
A. V.
Varfalomeev,
Fiz.
Tverd. Tela
3,
1445
(1961)
[Sov.
Phys.
Solid
State
3,
1048
(1961)
j.
S. Nikitine,
J.B.
Grun,
and M.
Certier,
Phys.
Kon-
dens.
Mater.
1,
214
(1963).
W.
S.
Brower,
Jr.
,
and H. S.
Parker,
J.
Cryst.
Growth
8,
227
(1971).
A.
Compaan
and
H.
Z.
Cummins,
Bull. Amer.
Phys.
Soc.
18,
75
(1973).
Air
Products and
Chemicals,
Inc.
,
Allentown,
Pa.
The
data
points
in
Fig.
2
indicate
the
observed
Ra-
man
intensity
divided
by
the incident laser
power.
Ab-
sorption
corrections,
which are
less
than
4'
in this
fre-
quency
range,
have
not
been included.
B.
Loudon,
Advan.
Phys.
13,
423
(1964);
J. L.
Bir-
man,
"Theory
of
Crystal Space Groups
and
Infra-Bed
and
Raman
Processes in
Insulating
Crystals"
in
Hand-
buch
der
Physik,
edited
by
S.
Flugge
(Springer,
New
York,
1973, to be
published),
Vol.
25,
Part 2b.
'
See,
for
example,
M. Born and
M.
Bradburn,
Proc.
Roy.
Soc.
,
Ser.
A 188,
161
(1947);
L. N.
Ovander,
Opt.
Spectrosk.
9,
571
{1960)
[Opt.
Spectrosc.
9,
302
(1960)]
.
'3J.
L.
Birman,
to be
published.
P.
F. Williams and S. P.
S.
Porto,
Bull.
Amer.
Phys.
Soc.
18,
411
(1973);
P.
F.
Williams,
Ph. D.
thesis,
Uni-
versity
of
Southern
California,
1973
{unpublished)
.
P.
Y.
Yu,
Y. R.
Shen,
and
Y.
Petroff,
Bull. Amer.
Phys.
Soc. 18,
411
(1973).
R. M.
Martin,
Phys.
Rev. B
4,
3676
(1971).
Hopping
Conductivity in
Granular Metals
Ping Sheng
Institute
fo~
Advanced
Study,
Princeton, ¹se
Jersey
08540
B. Abeles
and Y. Arie
RCA
Laboxatomes,
Princeton,
Nezo
Jersey
08540
{Beceived
23
April 1973)
We
present
evidence
that in
granular
metals the
observed
temperature
dependence
of
the
low-field
conductivity,
exp(-
b
j&
)
with
o
=
a,
can
be
ascribed to a
relationship
sE~
=const
between
s,
the
separation
of
neighboring
metal
grains,
and
E~,
the electrostatic
energy
required
to create
a
positive-negative
charged
pair
of
grains.
This
relationship
results from
simple
considerations of the
structure
of
granular
metals.
The
predictions
of
the
theory,
for both
the high-
and the
low-field
electrical
conductivity, are
in
excel-
lent
accord with
experimental results in
granular Ni-Si02
films.
The
low-fieM
electrical
conductivity
a~
of
many
disordered
materials
has a
temperature
dependence
that
can
be
expressed in
the
form
o~-exp(
—
b/T
).
The value n
=
4,
found
in
many
of the
amorphous
semiconductors
and
semicon-
ducting
glasses,
has
been
predicted
by
Mott'
us-
ing
a
model
of
hopping
conductivity
between
local-
ized states.
There
are, however,
disordered
materials
such
as
granular metals,
'
and some
disordered
semiconductors,
'
which
exhibit
an
n
=
&
behavior for
which no definitive
theory
has
been
proposed.
In this Letter
we
present
evi-
dence
that in
granular metals,
consisting
of
fine
metallic
particles
dispersed in
a
dielectric
ma-
trix,
the n
=
&
behavior
can
be
explained
by
a
structural effect.
Furthermore,
the
theory
we
propose
predicts
the
temperature and electric
field
dependences
of
conductivity in the
high-field.
regime
and relates the
high-
and
low-field
be-
haviors
through a
single
structural
parameter.
We
present
measurements of
the
temperature
and field
dependences of the
resistivities
in
gran-
ular
Ni-Si02
films
and find
excellent
agreement
with
the
predictions
of the
theory.
VOLUME
31,
NUMBER
I
PHYSICAL
REVIEW LETTERS
2
JULY
1973
10
IO
II
10'
E
IO
I
8
@10
10"
10
10
F
o
lo
~~
IO
lo
10
IO
5
I
I I
0
I.O
1.5
2.0
IO/P
(cm-V
I)
5
FIG. 1.
The
high-field
resistivity
I/o's
as a
function
of
the reciprocal
electric field
I/8
for several
temper-
atures
of
a
Ni-Si02
film,
measured with the
electric
field normal to the film.
The
composition is
indicated
in
the
figure.
The
solid lines were calculated
from
Eq.
(2)
with
the
values
of the
parameters
8,
and
2(CO/k)~/2
given
in columns 2
and
3
of Table
I.
I
2' 5 3.0
The Ni-SiO,
films,
about
1000
A
thick,
were
prepared"
by
cosputtering
Ni
and
silica.
Elec-
tron
microscopy"
and
x-ray
analysis'
show
that
the
films
consist of small
crystalline
nickel
grains
embedded in
amorphous
SiO,
.
The
resis-
tivities,
high-field
and
low-field,
are
presented
in
Figs.
l.
and
2.
Electrical conduction in
granular
metals
re-
sults from the
transport
of electrons
and
holes
by
tunneling
from
charged
metal
grains
to
neu-
tral
grains.
"
In order to
generate a
charge
car-
rier,
an
electron
has
to
be removed from
a
neu-
tral
grain
and
placed
on a
neighboring
neutral
grain.
Such a
process requires
an
energy
E,
=
(e'/d)F(s/d),
where
e
is
the electronic
charge,
d
is
the
grain
size,
s
is
the
separation
between
the
grains,
and
+
is
a
function
whose form
de-
pends
on
shape
and
arrangement
of
the
grains.
"
In the
low-field
regime
the
charge
carriers are
thermally
activated. In the high-field
regime
where the
voltage
drop,
4V,
between neighbor-
ing
grains
is
much
larger
than
kT/e,
where
k
is
the
Boltzmann
constant,
the
majority
of
charge
carriers are
created
by
field-induced
tunneling
between neutral
grains.
'
From
the
above
discus-
sion,
it is
clear that the
charging
energy
E,
plays
a
central
role
in
the granular-metal
con-
duction
process, and,
therefore,
a
closer look
at
the structural
origin
of
F-,
is
necessary.
We
now
propose,
using
a
simple
plausibility
argu-
ment,
a
structural
relationship between
E,
and
s. It
is known
from
electron
rnicrographs
that
2
0.
20
0.
05
O.I5
-
I/2
(oK
)-I/2
FIG.
2.
The
low-field
resistivity
1/oL
as a
function
of
T
for
three different
compositions of
Ni-SiO&
films. The
compositions
and the
corresponding
mean
sizes
of
the nickel
grains
are
indicated
in
the
figure.
The full
lines
represent the relation
logo@,
=-
2(CO/
kT)~/2;
the
values
of
2(CO/k)~
~
are
given
in
column
4
of Table
I.
0.25
O,
I
in
granular
metals there
is
a
distribution
of
grain
sizes.
Since the
grains
are formed
by
sur-
face diffusion
of the
sputtered
nickel
atoms and
SiO,
molecules,
the
composition
averaged
over
a
volume
larger
than
the
surface
diffusion
length,
which
is
of the
order of
a
few
grains,
is
expect-
ed
to be
the same
throughout
the volume
of the
sample
and
equal
to that of the
bulk. The
condi-
tion of
constant
composition
requires
s
and
d
to
be related.
Consider the
granular
metal
to be
divided
into
regions
each
of which
contains
grains
of
roughly
the
same size. In
order
for
the
com-
position
of each of these
regions
to be
equal
to
the
bulk
composition,
it is
necessary
for the
ra-
tio
s/d
to have
the same value for
each
region
even
though
d
is allowed to
vary
from
region
to
region.
It follows that
sE,
is
a constant
whose
value
depends
only
on the
composition
of the
sam-
ple
and
the
dielectric
constant of
the insulator.
We now
derive,
on the
basis
of
the rule
sE,
=
const,
the high-field
and
the
low-field behav-
iors of the
conductivity in
granular
metals.
"
For the
purpose
of
calculation we
use
for
the
distribution
function
D(s)
of
s
the
form
D(s)
=(s/
s,
')
exp(-
s/s,
),
where
s,
is
the
most
probable
value of s.
This form of
distribution
function
is
an
approximation
to the
one
determined from
electron
micrographs.
(a)
High field regime.
In
t-his
limit
t—
he
cur-
rent
density
j
is
proportional
to the
generation
rate of
charge
carriers
through
field-induced
tunneling.
"
To
simplify
the
calculation we
as-
sume that the
applied
electric field
8
gives
rise
to
equal voltage
drops
AV
=
h b
l
between
layers
of
neighboring
grains.
The
layers
are
separated
by
4l,
and on the
average
are
perpendicular
to
45
VOLUME
31,
NUMBER
I
PHYSICAL RKVIKW
LKTTKRS
2
Jvt.
v
1975
j
=
const
J
dsD(s)
f
e
2"'f(E)[1
—
f(E+e&V
—
E,
)]
The
current
term
due
to electrons tunneling
counte
for
the
case
ehV»kT, has
been
dropped
in
Eq.
(1).
is
significantly
larger
than
eh
1/',
we
have
neglected
i
shape,
and
have
regarded
y
as
a
constant'
=
(2m'/5
tion with
respect
to
E
yields
j/S
=
cr„=
o,
exp(
—
S,
/S)
g /g
Here
Cc=2ysE,
[1+
I/(2)jsc)],
and
SO=Cc/Ale.
At
T
=0
Eq.
(2)
reduces to
o
=o'cexp(-So/S).
This
behavior
can
be
qualitatively
understood
as
fol-
lows. When the
voltage
difference between
the
neighboring
layers
of
grains
is
41/',
the electron
can
only
tunnel between
those
pairs
of
neighbor-
ing
grains
with
E,
~
erat/'.
Since
sE,
=
const,
the
rate
of tunneling
is
—
exp(-
2ys)
=
exp(
—
const/
E,
)
=
exp(- const/eb
V).
With
S,
determined
by
the zero-temperature
behavior,
Eq.
(2)
repre-
sents
a
one-parameter
(C,)
description of
the
temperature and
field
dependences of high-field
conductivity.
Figure
1
shows the
experimental
data and the theoretical
curves
calculated from
Eq.
(2)
with
Co
as
the
fitting parameter.
Be-
cause
of
lack of
space only
the
data for one of
the
granular
Ni-SiO,
compositions
are shown.
The
fits are
equally
good
for other
compositions.
(b)
Low
field
regim-e.
—
In this limit the
charge
carriers are
thermally
activated. We assume
that
the
generated
positive-negative
charged
pair
of
grains
are
roughly
of same size
so that
each contributes
about half of the
charging
ener-
gy
E,
.
Let
us consider all those
charge
carriers
with
charging
energy
F-,
'.
Their
number
density
is
—
exp(-E,
'/kT).
When
a
weak
electric
field
(ehV«kT) is
applied,
the
charge
will drift
along
a
path
of
largest mobility
towards the electrodes.
The
charge
is
inhibited
in
tunneling
to
a much
smaller
grain
with
&,
»E,
'
because
the
electron
has
insufficient
energy.
The
charge, will,
there-
fore,
tunnel
to
grains
with
E,
~
E,
'.
However,
since smaller
E,
is associated
with
a
larger
tun-
neling
barrier
by
the rule
sE,
=
const,
the
opti-
mal
path
follows the
regions
with
the
least
de-
viations of
E,
from
E,
'.
The
corresponding
mo-
bility
is
proportional
to
exp(
—
2ys'),
where
s'
=
const/E,
',
and the total
conductivity,
being
the
sum
of
products
of
mobility, charge,
and
num-
ber
density
of
charge
carriers over
all
charging
the
applied
electric
field,
which is
negligible
e
in
our
system
the barrier
height
(3.
6
eV)
.
(1)
the effect of
applied
field
on
the barrier
A
change
of
variables in
Eq.
(1)
and
integra-
r to
Sine
n
Eq
)1I2
Z
C„S
exp(
—
Z)Z
dZ
1
—
exp—
(2)
g~-
f
ds'D(s')
exp[
—
2ys'
—
(E,
'/kT)].
By
performing
the
integration
by
the method of
steepest
descent,
the dominant
temperature
de-
pendence
of
the
low-field
conductivity is
given
by
cz
—
exp[
—
2(CO/kT)'~~].
The pre-exponential
factor of the
conductivity
is not
given
by
the
above
argument
because its exact form
depends
on more
complicated considerations of
statisti-
cal fluctuations
and
the
counting
of
percolation
paths. However,
the
temperature
dependence
of
the
pre-exponential
factor
is
expected
to be weak.
In
Table
I
we
give
the
values of
So
and
compare
the
values of
2(C,
/k)"'
obtained
from the
slopes
of
lno1
versus I/vT
plots
(Fig. 2)
with those
ob-
tained from
the high-field
data. The
agreement
between
the
two sets of
values
of
2(C,
/k)'
'
is
truly
remarkable
in view of the
fact that
two
dif-
ferent
physical
mechanisms,
field
generation
in
the high-field
regime
and
temperature activation
with
percolation
mobility
in the
low-field
regime,
are
operative in the two
limits.
Direct
calcula-
tions of
C,
from the
expression
2ysE,
(1+
I/2ys,
)
were found
to
be in
agreement
with those
in
Ta-
ble
I
within the
uncertainties
of
the
parameters,
e.
g.
,
for the
sample
with
24% volume
Ni,
if
we
take
y
=
1
A
(Ref.
7),
s
=
5 A
(estimated from
TABLE
I.
Parameters
of the
granular
metals.
2{g
/k)1/2
S
(
oKi/2)
%
vol Ni
(10
V/czn)
S
dependence
&
dependence
160
106
56
280
122
69
8,
50
1,
04
0.
28
8
24
44
the direction
of
the
macroscopic field.
The
rate
of
tunneling
for
an
electron with
energy
E
from
a
grain
on one
layer
to
a
neighboring
grain
on
the next
layer
is
—
exp(-2),
s)f
(E)
[1
—
f
(E+
eb
V
-E
)],
where
f
(E)
=
1/[1
+
exp(E/kT)]
is
the Fermi
function,
and
X
=
[2m
(p
-
E)/k
]'"
with
p
the effective
bar-
rier
height.
The current
density
j
can
be written
as
VOLUME
31,
NUMBER
I
PHYSICAL
REVIEW
LETTERS
2
JUr.
v
1973
electron
microscopy),
and
E,
=80
meV
(estimat-
ed from
a
plot
of
ines
versus
l/T),
we
obtain
2[(CO/k)'~2]
=
120
('K)'~'.
The
self-consistency
of
the
model receives
further
confirmation
in that
the
separation between the
layers
of
grains
cal-
culated from
the
relation
So=CO/&le
agrees
with
the
mean value
of
s+d
determined
from the
elec-
tron
microscopy.
It
is
instructive
to
point
out the
similarities
and
the
differences
between
Mott's
model
and
ours.
In
Mott's
model the
density
of
charge
car-
riers
is assumed to be
temperature
independent,
and the
percolation
paths
for the
charge
carriers
are
determined
by
optimizing
the
mobility.
In
our
model the
charge
carriers are
thermally
ac-
tivated,
tunneling
occurs between nearest neigh-
bors
only,
and
the
optimization is
applied
to the
product
of
mobility
and number
density
of
charge
carriers.
The differences
between
charging
en-
ergies
in our
model
are
analogous
to
the
relative
displacements
of
the
energy
levels for the
local-
ized states in
Mott's
model.
However,
in
our
case not
only
the
differences
but also
the
magni-
tudes
of
E,
play
an
important role.
This
is
es-
pecially
obvious
in the
high-field
regime
where
the
governing
factor for field
generation
of
charge
carriers
is
the
value
of
E,
rather than the
dif-
ferences
in
E,
.
To
conclude,
we
wouM
like
to
make
the
follow-
ing
comments:
The field
dependence,
exp(-
go/
8),
of
o„
in the low-temperature
regime is
par-
tially
the result of
high
tunneling
barriers
in
Ni-SiO, . In materials
where the
tunneling
bar-
rier
is
low,
the
high-field
conductivity
might
well follow
some
other
form
of
behavior
(such
as
the Frenkel-Poole
effect).
However,
the
re-
lationship
sE,
=
const
should still
yield
the
n
=
&
behavior in
the
low-field
regime. Attemps
are
being
made to extend the
concept
of
structural
effects
to
other
disordered
materials.
It is
in-
teresting
to note
that the
inhomogeneous
trans-
port
regime
in disordered
materials,
recently
treated
by
Cohen
and
Jortner,
'~
bears
close
anal-
ogy
to
granular
metals.
~N.
F.
Mott,
Phil.
Mag.
19,
885
(1969); V.
Ambegao-
kar,
B.
I. Halperin,
and
J.S.
Langer, Phys.
Rev.
B
4,
2612
(1971).
~J.
I,
Gittleman,
Y. Goldstein,
and
S.
Bozowski,
Phys.
Rev.
B
5,
8609
{1972).
H. R.
Zeller,
Phys.
Rev. Lett.
28,
1452
(1972).
J,
J.
Hauser,
Phys.
Rev,
B
(to
be
published).
D.
Redfield,
Bull. Amer.
Phys.
Soc.
18,
361
(1978),
and
Phys.
Rev.
Lett.
80,
1319
(1978).
J.
J,
Hanak,
H.
W.
Lehmann,
and
R,
K.
Wehner,
J,
Appl.
Phys.
48,
1666
(1972}.
7P.
Sheng
and
B.Abeles,
Phys.
Rev. Lett.
28,
84
(1972).
BM.
S. Abrahams,
C, J.
Buiocchi,
M.
Rayl,
and
P.
J.
Wojtowicz, J.
Appl.
Phys.
48,
2537
(1972).
B.
Abeles,
P.
Sheng,
M. Coutts,
and
Y.
Arie,
to
be
published,
will contain
more
experimental
and
theoreti-
cal
details.
C.
A. Neugebauer
and
M.
B.
Webb,
J,
Appl.
Phys. 33,
74
(1962};
C.
A. Neugebauer,
Thin Solid Films
6,
448
(1970).
In Ref.
7,
for
example,
E(s/d}
=2(s/d}/x
[~2+
(s/d}],
where
I("
is
the
dielectric
constant.
The
present
treatment of high-field
conductivity
dif-
fers
from
that in
Ref.
7
in
that we no
longer
consider
E~
as constant.
M.
H.
Cohen and
J.Jortner,
Phys.
Rev. Lett.
80,
699
(1978};
R. Landauer,
J.
Appl.
Phys.
28,
779
(1952}.
Scaling
Theory
for Metastable
States
and Their Lifetimes
K.
Binder*
and
E.
Stoll
TBM
Zurich
Research
Laboratory,
8803
Ruschlikon,
Szvitzexland
(Received
6
April
1978)
The
response
of
the
magnetization
to
a sudden reversal
of
the
magnetic
field
is studied
in the kinetic
Ising
model
by
means
of
computer experiments
on
square
N»
lattices.
It
is found
that
the
nonequilibrium relaxation
function
fulfills
a
dynamic
scaling
hypothesis.
The
magnetization
of the metastable
state
agrees
with
predictions
of
the cluster model
and
also with an
analytic
continuation of
the
linear-model
equation
of
state.
Apart
from
systems
where
the
mean-field
ap-
proximation
is
valid,
'
no reliable
predictions
about the
properties
of
metastable states
exist.
Monte
Carlo calculations have
been
performed
on
the
N
&
W
square
kinetic
Ising
model„"
and we
obtained the magnetization
of the metastable
states,
their
lifetimes,
and the
detailed
non-
equilibrium
behavior.
'
Within
the
accuracy
of
the
numerical calculations
(roughly
1%)
the
re-
sults
agree
with
simple scaling
ideas.