Crossover from phonon- to photon-mediated charge transport observed
in metal-cluster compounds
J. A. Reedijk, L. J. Adriaanse, H. B. Brom, and L. J. de Jongh
Kamerlingh Onnes Laboratory, Leiden University, P.O. Box 9506, 2300 RA Leiden, The Netherlands
G. Schmid
Institut fu
¨
r Anorganische Chemie, Universita
¨
t GH Essen, Universita
¨
tsstrasse 5-7, D-45117 Essen, Germany
~Received 12 January 1998!
We have studied the frequency-dependent conductivity of randomly packed aggregates of ligand-stabilized,
monodisperse metal clusters from dc to the infrared over 12 decades in frequency. At low frequencies the ac
conductivity of these compounds is dominated by phonon-assisted hopping between localized states, and
shows scaling behavior as a function of temperature and frequency. Due to the unique structure of these cluster
aggregates, the transition from phonon- to photon-mediated charge transport is clearly observed. The data are
found to be in excellent quantitative agreement with the two-site tunneling model. @S0163-1829~98!50324-4#
Charge transport in disordered systems, where conduction
proceeds by hopping between localized states, has been a
widely studied subject over the last decades. A powerful
method to study hopping transport is to perform conductivity
experiments where both temperature and applied frequency
are varied. In this way, not only the various theoretical hop-
ping models can be tested,
1
but also insight in the structural
properties of the material at mesoscopic length scales may be
gained.
2,3
Hopping conduction between localized states i and j can
take place when the overlap between the wave functions
f
at
sites i and j, i.e., I
ij
5
*
dr
f
*
(r2r
i
)H
f
(r2r
j
) with H the
Hamiltonian of the system, is nonzero. This overlap, together
with the site energy difference E
j
2 E
i
, determines the en-
ergy needed to make a hop. When the field strength E and
the applied frequency
v
are small, this hop energy has to be
supplied by lattice vibrations, and the transport proceeds by
phonon-assisted hopping. At high frequencies, however, the
photon energy \
v
may become sufficient to induce a reso-
nant tunneling transition. This phononless tunneling process
will start to contribute to the conductivity when \
v
becomes
comparable to the overlap energy I
ij
.
4
A class of materials that is often used for studying charge
transport in disordered systems are random composites of
metal particles in an insulating host. When the conducting
filler fraction p is lower than the percolation threshold p
c
,
the electron wave functions are localized on the metal par-
ticles, and transport occurs via tunneling through the insulat-
ing barriers. Discontinuous films and granular metals with
relatively high tunneling probabilities due to a sufficient
dense packing of the metal particles, have been frequently
investigated in dc conductivity experiments,
5–7
while the
low-frequency ac conductivity of these compounds has been
relatively little studied.
8,9
Diluted insulator/metal composites
(p!p
c
) have often been used in far-infrared ~i.e., around 1
THz! absorption spectroscopy,
10–12
with the idea that experi-
ments on diluted systems should be more sensitive to the
properties of the individual metal particles.
In this work, we report on the frequency-dependent con-
ductivity of several samples consisting of small metal nano-
particles stabilized by ligand shells. These unique materials
enable us to present a dielectric study of small metal particle
systems which runs almost continuously from the low-
frequency region into the infrared. The data clearly show a
gradual transition from the phonon-assisted hopping charac-
teristics at low frequencies to the photon-induced resonant
tunneling processes predicted at far-infrared frequencies.
EXPERIMENTAL
All samples studied are well-defined stoichiometric metal
cluster compounds, in which monodisperse metal cluster
cores are surrounded by stabilizing ligand shells.
13,14
The
ligands are organic molecules like phenantroline or
chinchonidine.
13
The metal cores consist of a central atom
surrounded by a number of shells of fcc packed atoms. The
core diameters are 1.4 nm for the two shell Au cluster ~ab-
breviated as Au-2!, 2.0 nm for the four shell Pt ~Pt-4!, and
3.5 nm for a 50/50 mixture of seven and eight shell Pd ~Pd-
7/8! clusters. The metal volume fractions are p'20 vol %
for Pt-4 and two Au-2 samples, and p'40 vol % for Pd-
7/8. Diffraction experiments indicate that the densely packed
clusters do not form a regular lattice, but are randomly
stacked.
13
The frequency-dependent complex conductivity was stud-
ied in the frequency range 5 Hz – 3 THz. To cover this huge
range in frequency, several experimental techniques were
combined.
3
For the capacitive method employed below 1
GHz, the sample was pressed between two electrodes. Be-
tween 200 MHz and 18 GHz, we used an open ended coaxial
probe. The frequency range 30–100 GHz was covered by
means of a filled waveguide technique, and between 0.2 and
0.6 THz by a quasioptical setup allowing the measurement of
both attenuation and phase of the transmitted signal without
the application of contacts
15
. Between 0.3 and 3 THz, we
used a Fourier transform infrared spectrometer.
Measurements at frequencies from dc up to 13 MHz were
done between room temperature and 4 K. Four-point dc mea-
surements indicated that, using pressed brass contacts, con-
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PHYSICAL REVIEW B 15 JUNE 1998-IIVOLUME 57, NUMBER 24
57
0163-1829/98/57~24!/15116~4!/$15.00 R15 116 © 1998 The American Physical Society
tact resistances were small (<5% of the sample resistance!.
Above 13 MHz, all experiments were performed at ambient
temperature.
RESULTS AND DISCUSSION
In an earlier study, the T dependence of the dc conduc-
tivity of these compounds was found to be well described by
variable-range hopping.
16
Figure 1 shows the frequency-
dependent conductivity ~the inset gives the related imaginary
part of the dielectric constant! at room temperature of two
Au-2 samples with different ligands over the entire fre-
quency range 5 Hz–3 THz. The sample with slightly thicker
ligands has lower conductivity at low frequencies. Dashed
and drawn lines are fits based upon, respectively, two site
and multiple hopping models ~see below!.
In Fig. 2, the frequency-dependent conductivity of the
cluster compound Pd-7/8 is plotted, again fitted with a mul-
tiple hopping model discussed below. Data on this sample
~which is well suited to study low-frequency scaling, since
its high dc conductivity allows experiments and scaling
down to helium temperatures! were taken up to 13 MHz at
temperatures between 4 K and 300 K.
17
In this temperature
range,
s
dc
varies over seven orders of magnitude. Using
scaled conductivity
s
8
(
v
,T)/
s
dc
(T) and scaled frequency
v
/
s
dc
(T), data sets of all temperatures fall on top of each
other. The same scaling behavior is seen in the T-dependent
low-frequency data on all other compounds studied ~includ-
ing Au-2! and the scaling curves are very similar for all
cluster compounds.
16
Many hopping models predict a scaling of
s
8
(
v
,T).
18–21
Often the frequency is scaled by a characteristic frequency
v
*
, which is proportional to T
s
dc
(T).
18
However, the
T-independent scaling
v
*
}
s
dc
(T) observed here is fre-
quently found experimentally and, e.g., expected in a diffu-
sion model when the carrier density n}T,
3
which is the case
for hopping transport near the Fermi level.
22
Bryksin
23
has calculated the ac conductivity for spatially
random systems with nearest-neighbor tunneling in an effec-
tive medium approximation. In this model, a wide distribu-
tion of hopping times due to complete spatial randomness is
assumed. This theory has later been extended
21
to systems
where conduction proceeds by means of variable-range hop-
ping. Both models predict the conductivity in the low and
intermediate frequency regime, where multiple hopping is
dominant, to obey the relation ~where
s
˜
5
s
/
s
dc
is the nor-
malized complex conductivity and
v
˜
is the scaled frequency
v
/
v
*
)
s
˜
ln
s
˜
5 i
v
˜
. ~1!
Solving this equation numerically,
s
8
(
v
˜
) can be com-
puted. Together with the room-temperature data on the Au-2
samples, the theoretical curve for
s
8
(
v
) is plotted in Fig. 1
~solid line!. In Fig. 2, the same theoretical curve is plotted
together with the scaled
s
8
(
v
) data for Pd-7/8. In both com-
pounds, a clear deviation between data and prediction is
present at frequencies around the ‘‘onset’’ frequency
v
0
,
where
s
8
(
v
) starts to deviate from the dc value. At higher
frequencies data and theory agree well. In all cluster com-
pounds studied, the discrepancy around
v
0
is found.
24
Simi-
lar deviations have also been observed in other
experiments
25,26
and computer simulations.
27
The failure of effective medium models
28–30
at frequen-
cies
v
'
v
0
can be understood by realizing that low-
frequency transport is dominated by multiple hopping over a
large number of connected sites, known as the ‘‘infinite clus-
ter’’ in percolation theory.
31
The complex, inhomogeneous
structure of this percolating network is not incorporated in
the effective medium theory of Ref. 21. Using arguments
FIG. 1.
s
8
(
v
) at 300 K between 5 Hz and 3 THz for two Au-2
samples with different ligands; the sample with the thicker ligands
has lower
s
8
at low
v
. The drawn lines represent fits with the
effective medium model (
v
/2
p
, 100 MHz!; a fit to the two-site
model, including phonon- and photon-assisted tunneling, is indi-
cated by the dashed line (
v
/2
p
. 100 MHz!. In the inset, the loss
function
e
9
(
v
)5
s
8
(
v
)/
e
0
v
is shown for the two Au-2 samples,
together with the fit to the two-site model ~dashed line!.
FIG. 2. Scaled conductivity
s
8
(
v
)/
s
dc
vs scaled frequency
v
/
s
dc
of the Pd-7/8 sample. Data taken at 5 Hz ,
v
/2
p
, 13 MHz
and temperatures 4 K, T, 300 K collapse onto a single curve. The
solid line represents a fit to the effective medium prediction.
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57 R15 117CROSSOVER FROM PHONON- TO PHOTON-MEDIATED . . .
from percolation theory, Hunt
32
showed that relaxation rates
of assemblies of connected sites can be orders of magnitude
slower than individual pair relaxation rates, and hence effec-
tively increase the conductivity for low frequencies
v
<
v
0
.
At higher frequencies
v
.
v
0
, where transport occurs only
on small groups of connected sites and hence the slow relax-
ation rates are irrelevant, the close agreement between the
data and the effective medium model ~see Fig. 2! indicates
that in the material no structure is present at mesoscopic
length scales, since this would result in a specific frequency
dependence of the conductivity at
v
.
v
0
.
2,3
s
8
„
v
… AT HIGH FREQUENCIES
In order to display more explicitly the features of the high
frequency conductivity data, the loss function
e
9
(
v
)
5
s
8
(
v
)/
e
0
v
is plotted in the inset of Fig. 1 for the two
Au-2 samples in the high frequency range ~100 MHz–3
THz!. In this frequency range, the two data sets fall on top of
each other, indicating that ligand effects are negligible at
high frequencies. With increasing
v
,
e
9
first decreases, and
then starts to increase around
v
/2
p
5 50 GHz. The same
minimum in the high-frequency loss function is found in the
four-shell Pt compound ~not shown!. Below we shall argue
that this minimum in
e
9
indicates the crossover from
phonon- to photon-induced tunneling transport.
In a system of isolated small metal particles, classical
electromagnetic theories predict
s
8
(
v
)}
v
2
at far-infrared
frequencies, due to the low-frequency tail of a surface plas-
mon resonance of the conduction electrons.
10
An approxi-
mate
s
8
}
v
2
behavior at for far-infrared ~FIR!, frequencies
has indeed been found many times in samples of metal par-
ticles diluted in a nonabsorbing host.
10–12
However, the ab-
solute value of the absorption was always orders of magni-
tude larger than the predictions for intraparticle relaxation.
Similar features are seen in the high frequency data
(
v
/2
p
. 10
11
Hz! of the Au-2 and Pt-4 compounds. Al-
though the data can be approximaly fitted with
s
8
}
v
2
, i.e.
e
9
}
v
, the absolute value of the measured conductivity (
s
8
'10 S/m at 1 THz! is almost four orders of magnitude larger
than expected from intracluster dipole absorption.
33
To explain the earlier observed strong FIR absorption in
metal particle systems, several models were proposed in a
paper by Curtin and Ashcroft.
34
One of the possibilities sug-
gested was an enhancement of the conductivity due to
photon-induced tunneling between metal particles. In this
model, the FIR conductivity for an assembly of metal par-
ticles, weakly coupled by tunnel junctions, is given by
s
8
}
v
2
at low frequencies, with a prefactor comparable to the
experimentally found values. In this model, relatively high
interparticle tunneling probabilities are assumed. Such high
tunneling probabilities can only occur in diluted systems
when, due to strong inhomogeneities, regions of densely
packed metal particles are present. Indeed, clustering of par-
ticles into dense regions has been observed many times in
experimental diluted small metal particle systems.
11
In a more general treatment of hopping conduction be-
tween localized states in disordered systems, Bo
¨
ttger and
Bryksin calculated the conductivity in a two-site
approximation,
19
valid for high frequencies where the mul-
tiple hopping process is negligible. Two processes were
found to contribute to the conductivity: ‘‘classical’’ phonon-
assisted tunneling, dominant at lower frequencies, and reso-
nant phononless tunneling, dominating at high
v
. The
phonon-assisted contribution was found to follow:
s
phonon
8
~
v
!
5
p
4
384
e
2
v
k
B
T
a
2 5
n
0
2
ln
4
~
n
ph
/
v
!
, ~2!
where n
0
is the density of states at the Fermi level,
a
is the
inverse electron localization length, and
n
ph
is the phonon
‘‘attempt’’ frequency.
The probability of a photon-induced tunneling transition
between states i and j depends strongly on the overlap en-
ergy I
ij
5
*
dr
f
*
(r2r
i
)H
f
(r2r
j
), which can be conve-
niently written as I
ij
5I
0
e
2
a
u
r
i
2r
j
u
. Integrating over all pairs,
Bo
¨
ttger and Bryksin
19
derive for the phononless contribution
to the conductivity
s
photon
8
~
v
!
5
p
2
6
e
2
\
v
2
a
2 5
n
0
2
ln
4
~
2I
0
/\
v
!
. ~3!
Apart from a small numerical factor, this coincides with an
earlier result of Mott, derived for zero temperature.
4
The
conductivity at high frequencies is now given by the sum of
the two contributions, i.e.,
s
8
(
v
)5
s
phonon
8
1
s
photon
8
. The
data on the cluster compounds can be adequately fitted with
this equation, as is shown in Fig. 1 for the Au-2 samples. For
this fit, three parameters (
n
ph
, I
0
, and
a
2 5
n
0
2
) were varied.
From the fit for Au-2, we find for the attempt frequency
n
ph
5 83 10
12
Hz, which lies in the range of usually sug-
gested values of 10
12
to 10
14
Hz. For the overlap prefactor
we find I
0
5 1.2 eV. The corresponding rate I
0
/\ should be
interpreted as the attempt frequency for photon-assisted hop-
ping, i.e., as the electronic collision rate within the clusters.
This collision rate can be estimated by
v
F
/d with d5 1.4 nm
for Au-2, giving
v
F
5 23 10
6
m/s, close to the Fermi velocity
of bulk gold.
Comparing the absolute value of the measured conductiv-
ity with the predictions for the two-site model, we find that
they coincide when n
0
2
a
2 5
5 10
4
eV
22
cm
21
. With reason-
able esimates n
0
5 63 10
20
eV
21
cm
23
and
a
2 1
5 3Å,
35
full
quantitative agreement with the two-site model is found for
frequencies 100 MHz,
v
/2
p
, 3 THz.
We note that in the two-site approximation, the distribu-
tion of hopping lengths p(R) follows from a completely ran-
dom site distribution to be p(R)}R
2
. In the cluster com-
pounds, the hopping length distribution will be different
especially at small R, since the ligand shells prevent the clus-
ters to be separated by less than 2d, where d is the ligand
shell thickness, i.e., p(R)5 0 for R, 2d. We numerically
found that the changed hopping length distribution has neg-
ligible influence on the conductivity as long as the ‘‘charac-
teristic pair separation’’ R(
v
) @given by (2
a
)
2 1
ln(
n
ph
/
v
)
for the phonon contribution, and
a
2 1
ln(2I
0
/\
v
) for the pho-
ton contribution
19
# exceeds the minimum pair separation R
5 2d. For the Au-2 samples, d'3.5 Å, i.e., Eq. ~2! may be
used for
v
/2
p
, 12 GHz, and Eq. ~3! is valid for
v
/2
p
, 50 THz.
In summary, the frequency-dependent conductivity data
of the metal cluster compounds studied clearly show a cross-
RAPID COMMUNICATIONS
R15 118 57REEDIJK, ADRIAANSE, BROM, de JONGH, AND SCHMID
over from phonon-assisted hopping at low frequencies to
photon-induced transport around THz frequencies. The data
taken at low frequencies, where multiple hopping dominates,
can be fitted adequately with an effective medium model.
The superlinear rise of
s
8
(
v
) at high frequencies is ex-
plained with a photon-assisted tunneling model. The param-
eters found from the phonon- and photon-mediated conduc-
tivities in the two site regime, which at room temperature
starts above 100 MHz, are fully consistent.
ACKNOWLEDGMENTS
This work was part of the research program of the ‘‘Stich-
ting voor Fundamenteel Onderzoek der Materie’’ ~FOM!
which is partially supported by the ‘‘Nederlandse Organi-
satie voor Wetenschappelijk Onderzoek’’ ~NWO!. The dis-
cussions with H.C.F. Martens, and the assistance of G.A. van
Albada with the FIR experiments are gratefully acknowl-
edged, as is the support of the European Community under
the HCM program.
1
See, e.g., M.P.J. van Staveren, H.B. Brom, and L.J. de Jongh,
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2
M. Ben-Chorin, F. Mo
¨
ller, F. Koch, W. Schirmacher, and M.
Eberhard, Phys. Rev. B 51, 2199 ~1995!.
3
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1755 ~1997!.
4
N.F. Mott and E.A. Davies, Electronic Processes in Non-
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B. Abeles, Ping Sheng, M.D. Coutts, and Y. Arie, Adv. Phys. 24,
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8
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10
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12
Y.H. Kim and D.B. Tanner, Phys. Rev. B 39, 3585 ~1989!.
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Clusters and Colloids, edited by G. Schmid ~VCH, Weinheim,
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14
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15
AB Millimetre MVNA S/N 21 User’s Manual, Paris, France.
16
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17
Due to the lack of sufficient sample, experiments on this com-
pound could not be extended into the THz region.
18
S. Summerfield, Philos. Mag. B 52,9~1985!.
19
H. Bo
¨
ttger and V.V. Bryksin, Hopping Conduction in Solids
~VCH Akademie-Verlag, Berlin, 1985!.
20
W. Schirmacher, Ber. Bunsenges. Phys. Chem. 95, 368 ~1991!.
21
O. Bleibaum, H. Bo
¨
ttger, and V.V. Bryksin, Phys. Rev. B 54,
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22
V. Ambegaokar, B.I. Halperin, and J.S. Langer, Phys. Rev. B 4,
2612 ~1971!.
23
V.V. Bryksin, Fiz. Tverd. Tela ~Leningrad! 22, 2441 ~1980!@Sov.
Phys. Solid State 22, 1421 ~1980!#.
24
When fitting the data with other effective-medium approximation
predictions for systems with less wide jump frequency distribu-
tions, the discrepancy around
v
0
remains.
25
K.K. Bardhan and R.K. Chakrabarty, Phys. Rev. Lett. 72, 1068
~1994!.
26
A.R. Long, J. McMillan, N. Balkan, and S. Summerfield, Philos.
Mag. B 58, 153 ~1988!.
27
J.C. Dyre, Phys. Rev. B 49,11709~1994!.
28
Also other theoretical considerations ~Refs. 29 and 30! indicate
that the low-frequency hopping conductivity does not follow Eq.
~1!.
29
V.V. Bryksin, Fiz. Tverd. Tela 26, 1362 ~1984!@Sov. Phys. Solid
State 26, 827 ~1984!#.
30
J.C. Dyre and T.B. Schro”der, Phys. Rev. B 54,14884~1996!.
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D. Stauffer and A. Aharony, Introduction to Percolation Theory
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32
A. Hunt, Philos. Mag. B 64, 579 ~1991!; J. Non-Cryst. Solids
183, 109 ~1995!.
33
Using Eq. ~8! of Ref. 10 with a51.4 nm, f5 0.2, and
s
0
5 10
7
S/m, we find
s
8
'2310
2 3
S/m at 1 THz.
34
W.A. Curtin and N.W. Ashcroft, Phys. Rev. B 31, 3287 ~1985!.
35
The density of states n
0
in Au-2 is estimated assuming a uniform
band of width W'0.3 eV ~which is approximately equal to the
charging energy of a cluster!, in which all clusters contribute
one state; this gives n
0
5 63 10
20
cm
2 3
eV
2 1
. The localization
radius
a
2 1
is related to the binding energy to the cores via E
b
5 \
2
a
2
/2m; when
a
2 1
5 3 Å, this binding energy E
b
5 1 eV.
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