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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
´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
The Coulomb blockade effect has been the subject of
intensive theoretical1,2 and experimental3–5 research during
the two past decades. Applications are promising in metrol-
ogy and microelectronics.6Single-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.
Let us consider the structure of Fig. 1, where a small
island is separated from two electrodes by tunnel junctions of
capacitance CLand CR. When the charging energy of the
island Ece2/2(CLCR) is larger than the thermal energy
kBT, 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 CRCL.If
we call Vthe dc voltage applied between the two external
electrodes and Qthe charge contained in the central island,
the resulting voltage drops VLand VRon each junction are
In the Coulomb blockade regime, Qis zero below the thresh-
old voltage VTe/2CRand increases by steps when Vin-
creases above VT. If we take Q⫽⫺Ne, where Nis the
number of electrons on the island and eis the absolute
charge of the electron, the total charge Qtot stored in the two
junctions is
The DCA is thus
dV .1
The first term is the ‘‘normal’ capacitance Ceq
CLCR/(CLCR), given by CLand CRin 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 Vis
added to a constant dc voltage V, so that the total applied
voltage oscillates between VV/2. When VVTV/2,
the number of charges on the island is always zero so that
CCeq . When Vis swept between VTV/2 and VT
V/2, the island is charged and discharged during every
oscillation of the voltage. According to Eq. 1, this increases
the capacitance above Ceq by C, with
dV .2
For example, if the capacitor of Fig. 1 contains a single
cluster with an area of 1 nm2, embedded between two 25 Å
thick alumina layers, when V1 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: ithe mutual capacitance between
an island and one of the electrodes is CS/d, where dis
the insulating barrier thickness, Sthe island area, and the
dielectric constant of the barrier; iionly the first threshold
12650021-8979/2004/95(3)/1265/4/$22.00 © 2004 American Institute of Physics
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voltage (VT) of an island contributes significantly to the ef-
fect which is strictly true if the junctions are weakly
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.
A. Sample preparation
Using an Alcatel 610 sputtering system we grew samples
with the structure Ni80Fe20 150 Å/Al2O3–25Å/
Co-tCo /Al2O325 Å/Co150 Å//Si, where tCo is the nomi-
nal thickness of deposited Co. In this paper we present re-
sults on two samples with tCo4 and 10 Å. Both Al2O3
layers were fabricated by the oxidation of a 15 Å thick Al
layer in an Ar/O2plasma 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
2O3produces clusters whose growth
mode has already been studied.11,12 The samples were litho-
graphically processed to produce junctions with surfaces
ranging from 300300 to 10001000
m2, 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 tCo10 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
tCo4 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 tCo4 and 10 Å, the gaps are about 30 and 10 mV,
respectively. Then, with increase of temperature, kBTover-
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 tCo
sample and around 70 K for the tCo10 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 fin 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 RSusing the equation
are functions of voltage and
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. 4aaC(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 nm2. The density of islands is
2.261016 m2. 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 CLand CR.
FIG. 2. Resistance of the samples with tCo10 Å aand4Åbas 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. aFrom lower to upper curves, the voltages are 100, 50, 25, 20, 15, 10,
5, and 2 mV. bFrom 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 Qtot
CLRL/(RLRR)Vassuming that CRRRCLRL). Since
the resistances RLand RRare not Ohmic resistances but
tunnel junctions, their values depend upon bias voltage. This
causes a dependence of the DCA on applied voltage:
This equation can be put in a simpler form if we assume
that RLRR:
As RLRR, the main potential drop occurs on the left
junction so VRis always small. At low voltages, tunnel junc-
tions have a nearly Ohmic resistance so we can reasonably
assume that RRdoes not vary with V. The previous equation
As RLis 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 tCo10 Å sample with measurement frequency ranging
from 1 to 100 kHz. Amaximum 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. aEquivalent circuit representation of a sample: Cand Rpare the
double-junction capacitance and resistance, and Rsis the serial resistance of
the leads and is about 20 .bEquivalent circuit representation of an
island outside the Coulomb blockade regime. CL(CR)andRL(RR) are the
capacitance and the resistance of the left righttunnel junction.
FIG. 4. C(V) curve measurement for the sample with tCo4 Å. The fre-
quency of measurement is 2 kHz, and the amplitude of the voltage oscilla-
tion is V1 mV. aCurve at 10 K. bEvolution of the curves with
temperature. From lower to upper curves, the temperature are 4, 10, 30, 50,
100, and 300 K.
FIG. 5. aC(V) measurement at 4 K of the sample with tCo10 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
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time of an island
can be roughly estimated using the for-
RCislandRS/d, where RS15 cm2is the resis-
tance area product of our alumina layer, d25 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.
In summary, we have shown that DCAmeasurements 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 DCAmeasurements, 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.
Many thanks to M. Bowen and E. Price for their careful
reading of the manuscript.
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... A new concept of variable capacitor has been experimentally tested by Carrey et al. [4]: the device does not potentially suffer from the previously described defaults of MEMS or varactors and is easy to fabricate which is critical for technological integration in industrial circuit fabrication processes. It has also proven to have good performances with relative variation of capacitance higher than 40% for applied bias lower than 3 V allowing low power applications. ...
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Président : Jean-Louis BARRAT (Pr. au LPMCN de l'UCB - Lyon 1) Rapporteurs : Bernard PANNETIER (D.R. au CRTBT de Grenoble - CNRS) Jean-Marc TONNERRE (C.R. au Lab. de Cristal. de Grenoble - CNRS) Examinateurs : Peter LIEVENS (Pr. à la Katholieke Universiteit Leuven - Belgique) Edgar BONET OROZCO (C. R. au LLN de Grenoble - CNRS) Directrice de thèse : Véronique DUPUIS (C. R. au LPMCN de l'UCB - Lyon 1) Invité : Thierry EPICIER (D. R. au Lab. GMPPM de l'INSA de Lyon)
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