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Capacitance variation of an assembly of clusters in the Coulomb blockade regime

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We fabricated tunnel junctions containing clusters embedded in a thin insulating layer. Low-temperature dc transport measurements reveal typical Coulomb blockade features. We measured the differential capacitance at different dc voltages and frequencies of these samples in the Coulomb blockade regime. The differential capacitance is constant at low dc bias and increases by 35% above a threshold voltage. This behavior can be well understood using a simple model which takes into account the Coulomb blockade effect. Such measurements could be used as a probe of Coulomb blockade properties of clusters assemblies. This effect is also promising for the fabrication of capacitors whose capacitance can be changed by varying the applied voltage.
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Capacitance variation of an assembly of clusters in the Coulomb
blockade regime
J. Carrey, P. Seneor, N. Lidgi, H. Jaffre
`
s, F. Nguyen Van Dau, A. Fert, A. Friederich,
F. Montaigne, and A. Vaure
`
s
Unite
´
Mixte de Physique CNRS/THALES, Domaine de Corbeville, 91404 Orsay Cedex, France,
and Universite
´
Paris-Sud, 91405 Orsay Cedex, France
Received 8 April 2003; accepted 10 November 2003
We fabricated tunnel junctions containing clusters embedded in a thin insulating layer.
Low-temperature dc transport measurements reveal typical Coulomb blockade features. We
measured the differential capacitance at different dc voltages and frequencies of these samples in the
Coulomb blockade regime. The differential capacitance is constant at low dc bias and increases by
35% above a threshold voltage. This behavior can be well understood using a simple model which
takes into account the Coulomb blockade effect. Such measurements could be used as a probe of
Coulomb blockade properties of clusters assemblies. This effect is also promising for the fabrication
of capacitors whose capacitance can be changed by varying the applied voltage. © 2004 American
Institute of Physics. DOI: 10.1063/1.1638619
I. INTRODUCTION
The Coulomb blockade effect has been the subject of
intensive theoretical
1,2
and experimental
3–5
research during
the two past decades. Applications are promising in metrol-
ogy and microelectronics.
6
Single-electron capacitance spec-
troscopy has proven to be successful for studying the quan-
tized energy levels of a micrometer-size GaAs quantum
well.
7–9
In this article, we show that a similar method can be
applied to obtain Coulomb blockade properties for a set of
nanometric metallic clusters. We present a theoretical model
for the expected capacitance variation and experimental re-
sults for Co clusters embedded in an alumina tunnel barrier.
II. THEORY
Let us consider the structure of Fig. 1, where a small
island is separated from two electrodes by tunnel junctions of
capacitance C
L
and C
R
. When the charging energy of the
island E
c
e
2
/2(C
L
C
R
) is larger than the thermal energy
k
B
T, a single electron cannot tunnel from an electrode into
the island unless the applied voltage exceeds a threshold
value. This results in a ‘Coulomb gap’ in the current-
voltage
I(V)
curves and is known as the Coulomb block-
ade effect.
Let us now calculate the variation of differential capaci-
tance DCAassociated with this effect. In the following, we
will consider the case presented in Fig. 1, with C
R
C
L
.If
we call V the dc voltage applied between the two external
electrodes and Q the charge contained in the central island,
the resulting voltage drops V
L
and V
R
on each junction are
V
L
(C
R
V Q)/(C
L
C
R
) and V
R
(C
L
V Q)/(C
L
C
R
).
In the Coulomb blockade regime, Q is zero below the thresh-
old voltage V
T
e/2C
R
and increases by steps when V in-
creases above V
T
. If we take Q⫽⫺Ne, where N is the
number of electrons on the island and e is the absolute
charge of the electron, the total charge Q
tot
stored in the two
junctions is
Q
tot
C
L
C
R
C
L
C
R
V
C
R
C
L
C
R
Ne.
The DCA is thus
C
V
dQ
tot
dV
C
L
C
R
C
L
C
R
C
R
e
C
L
C
R
dN
dV
. 1
The first term is the ‘normal’ capacitance C
eq
C
L
C
R
/(C
L
C
R
), given by C
L
and C
R
in series. It does
not depend on the voltage V. The second term (C) is re-
lated to the charging and discharging of the island. The DCA
is relevant when an alternating voltage of amplitude V is
added to a constant dc voltage V, so that the total applied
voltage oscillates between V V/2. When VV
T
V/2,
the number of charges on the island is always zero so that
C C
eq
. When V is swept between V
T
V/2 and V
T
V/2, the island is charged and discharged during every
oscillation of the voltage. According to Eq. 1, this increases
the capacitance above C
eq
by C, with
C
C
eq
e
C
L
dN
dV
. 2
For example, if the capacitor of Fig. 1 contains a single
cluster with an area of 1 nm
2
, embedded between two 25 Å
thick alumina layers, when V 1 mV the DCA should in-
crease by a factor of 5000.
The amplitude of the effect is of course expected to be
lower for an island assembly with some distribution of size,
since all the islands do not have the same threshold value.
We present in Fig. 1 a theoretical calculation of the DCA
variation with voltage
C(V)
expected for an island assem-
bly having a Gaussian distribution of areas. The following
assumptions were made: i the mutual capacitance between
an island and one of the electrodes is C S/d, where d is
the insulating barrier thickness, S the island area, and the
dielectric constant of the barrier; ii only the first threshold
JOURNAL OF APPLIED PHYSICS VOLUME 95, NUMBER 3 1 FEBRUARY 2004
12650021-8979/2004/95(3)/1265/4/$22.00 © 2004 American Institute of Physics
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voltage (V
T
) of an island contributes significantly to the ef-
fect which is strictly true if the junctions are weakly
asymmetric
2
.
Below the threshold of the highest-capacitance island in
the assembly, no island can be charged or discharged, and the
DCA stays constant. Then increasing the voltage increases
the DCA as more and more islands can be charged and dis-
charged during each cycle. This increase reflects the distri-
bution in the voltage threshold values of the islands. Thus, as
there is a direct correspondence between the island surfaces
and their threshold voltage values, the maxima of the C(V)
curves correspond to the maxima of the surface distribution.
At sufficiently high voltage, all islands stay charged during a
cycle and the DCA decreases to its initial value.
III. RESULTS
A. Sample preparation
Using an Alcatel 610 sputtering system we grew samples
with the structure Ni
80
Fe
20
150 Å/Al
2
O
3
–25Å/
Co-t
Co
/Al
2
O
3
25 Å/Co150 Å//Si, where t
Co
is the nomi-
nal thickness of deposited Co. In this paper we present re-
sults on two samples with t
Co
4 and 10 Å. Both Al
2
O
3
layers were fabricated by the oxidation of a 15 Å thick Al
layer in an Ar/O
2
plasma as described in Ref. 10. The valid-
ity of this method of fabricating high-quality double-tunnel
junctions has been previously demonstrated.
10
Depositing 4
and10ÅofCoonAl
2
O
3
produces clusters whose growth
mode has already been studied.
11,12
The samples were litho-
graphically processed to produce junctions with surfaces
ranging from 300 300 to 10001000
m
2
, which were
then measured.
B. dc transport measurements
Both samples present a behavior typical of Coulomb
blockade.
11
In Fig. 2, we present the resistance as a function
of temperature at various voltages, and I(V) curves from the
samples with t
Co
10 and 4 Å. In the former, the resistance
increases by a factor of 20 between room temperature and 4
K when the bias voltage is 2 mV, whereas this increase is
only 60% when the bias is 100 mV. For the sample with
t
Co
4 Å, the factor is 150 at 2 mV and 3 at 100 mV. At 4 K,
as shown in the insets of Fig. 2, both samples present a gap
in their I(V) curves; this corresponds to the Coulomb block-
ade. For t
Co
4 and 10 Å, the gaps are about 30 and 10 mV,
respectively. Then, with increase of temperature, k
B
T over-
comes the charging energy of most clusters so that at room
temperature the I(V) curves for both samples show a classi-
cal double-tunnel junction behavior see Fig. 2. Coulomb
blockade effects disappear around 100 K for the t
Co
sample and around 70 K for the t
Co
10 Å sample.
C. ac transport measurements
Dynamic properties of the samples were measured using
a multifrequency LCR meter HP 4274A. The impedance Z
and the phase difference between current and voltage were
measured at different voltages and frequencies f in two-probe
mode. Data were analyzed assuming that the samples can be
modeled with the equivalent circuit shown in Fig. 3a. The
capacitance values are then extracted from Z, , and the
resistance of the leads R
S
using the equation
C
Z sin
2
f
R
S
2
Z
2
2R
S
Z cos
3
where
/2,0
. Z and
are functions of voltage and
frequency.
In samples without a cluster between the two alumina
barriers classic tunnel junctions, we do not find a signifi-
cant 1%variation of the DCA with voltage. On the other
hand, we present in Fig. 4a a C(V) curve obtained at 10 K
FIG. 1. Theoretical calculation of the DCA expected for a cluster assembly
with a Gaussian distribution of areas. The mean island area and the standard
deviation of the Gaussian distribution are 10 nm
2
. The density of islands is
2.26 10
16
m
2
. The islands are embedded between a 25 and a 50 Å thick
alumina layer, which has a dielectric constant 8. The inset represents a
single cluster separated from electrodes by two capacitances C
L
and C
R
.
FIG. 2. Resistance of the samples with t
Co
10 Å a and4Åb as a
function of temperature for different applied voltages. The resistance is nor-
malized by the resistance at 280 K. In the insets: I(V) curves at 300 and 4
K. a From lower to upper curves, the voltages are 100, 50, 25, 20, 15, 10,
5, and 2 mV. b From lower to upper curves, the voltages are 100, 50, 20,
10, 5, and 2 mV.
1266 J. Appl. Phys., Vol. 95, No. 3, 1 February 2004 Carrey
et al.
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for the sample containing4ÅofCo.Aclear gap of about 80
mV in the C(V) curve and a variation of the DCA with
voltage up to 17% are observed. As expected, the effect de-
creases for high applied voltages.
13
In Fig. 4b, we present the evolution of this C(V) curve
with temperature. As the temperature increases, the number
of clusters in the Coulomb blockade regime decreases, so
that the effect described above vanishes. Instead, a supple-
mentary spurious effect due to clusters which are not in the
Coulomb blockade regime and inherent to double-tunnel
junctions takes place. This effect tends to decrease the DCA
at increasing voltage. At intermediate temperatures from 30
to 50 K, the two effects compete so that the DCA does not
vary monotonically. For temperatures above 100 K, when no
more clusters are in the Coulomb blockade regime, one ob-
serves only a decrease of the DCA with voltage.
This effect can be understood if we model an island
outside the Coulomb blockade regime as in Fig. 3b. In such
a case, the total charge stored in the system is Q
tot
C
L
R
L
/(R
L
R
R
)
V assuming that C
R
R
R
C
L
R
L
). Since
the resistances R
L
and R
R
are not Ohmic resistances but
tunnel junctions, their values depend upon bias voltage. This
causes a dependence of the DCA on applied voltage:
C
V
dQ
tot
dV
R
L
C
L
R
L
R
R
C
L
V
d
dV
R
L
R
L
R
R
. 4
This equation can be put in a simpler form if we assume
that R
L
R
R
:
C
V
C
L
1
R
R
R
L
V
d
dV
R
R
R
L
. 5
As R
L
R
R
, the main potential drop occurs on the left
junction so V
R
is always small. At low voltages, tunnel junc-
tions have a nearly Ohmic resistance so we can reasonably
assume that R
R
does not vary with V. The previous equation
becomes
C
V
C
L
1
R
R
R
L
V
R
R
R
L
2
dR
L
dV
. 6
As R
L
is a decreasing function of voltage the two right-
hand terms in the parentheses induce a decrease of the DCA
with voltage. This effect can thus be the origin of the high-
temperature behavior of the DCA observed in Fig. 4b.
In Fig. 5, we present the C(V) curves obtained at4Kfor
the t
Co
10 Å sample with measurement frequency ranging
from 1 to 100 kHz. A maximum variation of 35% is observed
at 1 kHz. The decrease of the effect with frequency might be
related to the discharging characteristic time constant of the
islands. The effect decreases when the time constant is
greater than the oscillation half period. The discharging
FIG. 3. a Equivalent circuit representation of a sample: C and R
p
are the
double-junction capacitance and resistance, and R
s
is the serial resistance of
the leads and is about 20 . b Equivalent circuit representation of an
island outside the Coulomb blockade regime. C
L
(C
R
)andR
L
(R
R
) are the
capacitance and the resistance of the left right tunnel junction.
FIG. 4. C(V) curve measurement for the sample with t
Co
4 Å. The fre-
quency of measurement is 2 kHz, and the amplitude of the voltage oscilla-
tion is V 1 mV. a Curve at 10 K. b Evolution of the curves with
temperature. From lower to upper curves, the temperature are 4, 10, 30, 50,
100, and 300 K.
FIG. 5. a C(V) measurement at 4 K of the sample with t
Co
10 Å at
various frequencies of measurement, with an oscillating voltage V
1 mV. From upper to lower curves, the frequencies are 1, 2, 4, 10, 20, 40,
and 100 kHz. Please note that the curves at 1, 2, and 4 kHz are almost
indistinguishable. bA zoom at low applied voltage of the same curves.
1267J. Appl. Phys., Vol. 95, No. 3, 1 February 2004 Carrey
et al.
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time of an island
can be roughly estimated using the for-
mula
RC
island
R
S
/d, where R
S
15 cm
2
is the resis-
tance area product of our alumina layer, d 25 Å its thick-
ness, and its dielectric constant. This allows us to estimate
the average discharging time of an island as 40
s. The
effect should thus decrease strongly above a cutoff frequency
of 12 kHz. The experimental curves are in good agreement
with this simple calculation.
We also remark that, at low voltage, the DCA drops
down until the increase takes place. This decrease can also
be attributed to islands that are outside the Coulomb block-
ade regime. This reflects that, even at 4 K, a fraction of the
islands are not in the blocking regime in this sample. We
note that the decrease of this peak with frequency is faster
than the decrease of the effect due to Coulomb-blocked clus-
ters. At 20 kHz, the peak totally disappears so that a Cou-
lomb gap of about 10 mV can be observed see the inset in
Fig. 5. This probably originates from the fact that the non-
blocked islands are the ones with the largest capacitances,
and consequently with the largest charging times. Smaller
islands are in the Coulomb blockade regime and have higher
cutoff frequencies.
IV. CONCLUSION
In summary, we have shown that DCA measurements are
very sensitive to the Coulomb blockade effects of a cluster
assembly embedded in an insulator. When recording I(V)
curves on a cluster assembly, it is very difficult to extract a
distribution of threshold values from experimental data. On
the contrary, using DCA measurements, one could determine
the number of island which can be charged and discharged
for each dc voltage. Moreover, varying the frequency of
measurement allows us to get information about the charging
time of clusters. The DCA measurements could thus be used
as a probe for Coulomb blockade properties of cluster assem-
blies. This effect is also promising for the fabrication of ca-
pacitors whose capacitance can be changed by varying the
applied voltage.
ACKNOWLEDGMENTS
Many thanks to M. Bowen and E. Price for their careful
reading of the manuscript.
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One has to note that the dielectric breakdown of our junctions does not
allow us to apply voltages greater than about 1.8 V.
1268 J. Appl. Phys., Vol. 95, No. 3, 1 February 2004 Carrey
et al.
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We report on aggregation of Co on a smooth amorphous alumina surface, during sputter deposition of Co atoms at room temperature. The atomic structure of the Co grains appears to be size dependent: hcp in the larger bean-shaped clusters (greater than 3 nm), and often fcc in the smaller, almost spherical clusters. The deposits show relatively narrow distributions of the cluster sizes and of the intercluster distances, which results in a noticeable local order and explains how charge blocking (the so-called Coulomb blockade) could be observed in large sets of such particles by Schelp et al. (1997, Phys. Rev. B, 56, R5747). Electrical percolation occurs at a nominal thickness of about 2 nm.
Article
We present a theoretical study of two ultrasmall-capacitance normal tunnel junctions connected in series and driven by a dc voltage source, using the semiclassical junction approach. We show that the two junctions in series display a voltage offset similar to that of a single junction that is driven by a current source. We also show that two junctions in series can, for the right set of parameters, produce an I-V curve with distinct steps. A parallel array of such series units shows similar behavior. We suggest specific experimental realizations of two junctions in series and the parallel array using a scanning tunneling microscope and a granular array of small metal drops. We also compare our predictions with experiments in the literature.
Article
We present a theoretical analysis for a system composed of two mesoscopic tunnel junctions coupled in series. We show that the current-voltage characteristic for this system can be obtained analytically. The usefulness of the model is demonstrated through the fit of experimental data acquired with a cryogenic (4.2 K) scanning tunneling microscope. A simple extension of the model predicts additional structure in the system characteristics when discrete middle electrode states are present.
Article
Unusual structure and large electric-field\char21{}induced oscillations have been observed in the current-voltage curves of small-area tunnel junctions arranged in a low-capacitance ($\lesssim${}1 fF) multiple-junction configuration. This behavior arises from the tunneling of individual electrons charging and discharging the capacitance. The observations are in accord with what would be expected from a simple model of the charging energies and voltage fluctuations of e/C associated with such effects.
Article
We have studied electron tunneling in ultralow-capacitance silver particles sandwiched between artificial tunnel barriers. We have observed steps in the current-voltage characteristics of these systems with voltage widths of e/C (C being the capacitance of the particles), and associated current rises of e/RC. This represents the first observationof the so-called Coulomb staircase, expected for the charging of particles with extremely small capacitance by a discrete number of electrons.
Article
The capacitance signal resulting from single electrons tunneling into discrete quantum levels is observed. The electrons tunnel between a metallic layer and confined states of a single disk in a microscopic capacitor fabricate in GaAs. Charge transfer occurs only for bias voltages at which a quantum level resonates with the Fermi energy of the metallic layer. This creates a sequence of distinct capacitance peaks whose bias positions directly reflect the electronic spectrum of the confined structure. From the magnetic field evolution of the spectrum, the nature of the bound states is deduced.