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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

共Received 8 April 2003; accepted 10 November 2003兲

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. 关DOI: 10.1063/1.1638619兴

I. INTRODUCTION

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.

II. THEORY

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 Ec⫽e2/2(CL⫹CR) 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 关I(V)兴curves and is known as the Coulomb block-

ade effect.

Let us now calculate the variation of differential capaci-

tance 共DCA兲associated with this effect. In the following, we

will consider the case presented in Fig. 1, with CR⬎CL.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

VL⫽(CRV⫺Q)/(CL⫹CR) and VR⫽(CLV⫹Q)/(CL⫹CR).

In the Coulomb blockade regime, Qis zero below the thresh-

old voltage VT⫽e/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

Qtot⫽CLCR

CL⫹CRV⫹CR

CL⫹CRNe.

The DCA is thus

C共V兲⫽dQtot

dV ⫽CLCR

CL⫹CR

⫹CRe

CL⫹CR

dN

dV .共1兲

The ﬁrst term is the ‘‘normal’’ capacitance Ceq

⫽CLCR/(CL⫹CR), 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 V⫾⌬V/2. When V⬍VT⫺⌬V/2,

the number of charges on the island is always zero so that

C⫽Ceq . When Vis swept between VT⫺⌬V/2 and VT

⫹⌬V/2, the island is charged and discharged during every

oscillation of the voltage. According to Eq. 共1兲, this increases

the capacitance above Ceq by ⌬C, with

⌬C

Ceq

⫽e

CL

dN

dV .共2兲

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 ⌬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 关C(V)兴expected for an island assem-

bly having a Gaussian distribution of areas. The following

assumptions were made: 共i兲the mutual capacitance between

an island and one of the electrodes is C⫽S/d, where dis

the insulating barrier thickness, Sthe island area, and the

dielectric constant of the barrier; 共ii兲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 (VT) of an island contributes signiﬁcantly to the ef-

fect 共which is strictly true if the junctions are weakly

asymmetric2兲.

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 Ni80Fe20– 150 Å/Al2O3–25Å/

Co-tCo /Al2O3–25 Å/Co–150 Å//Si, where tCo is the nomi-

nal thickness of deposited Co. In this paper we present re-

sults on two samples with tCo⫽4 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

and10ÅofCoonAl

2O3produces 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 1000⫻1000

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 tCo⫽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

tCo⫽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 tCo⫽4 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 共see Fig. 2兲. Coulomb

blockade effects disappear around 100 K for the tCo⫽4Å

sample and around 70 K for the tCo⫽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 fin two-probe

mode. Data were analyzed assuming that the samples can be

modeled with the equivalent circuit shown in Fig. 3共a兲. The

capacitance values are then extracted from Z,⌰, and the

resistance of the leads RSusing the equation

C⫽

⫺Zsin

2

f共RS

2⫹Z2⫺2RSZcos

兲共3兲

where

苸关⫺

/2,0兴.Zand

are functions of voltage and

frequency.

In samples without a cluster between the two alumina

barriers 共classic tunnel junctions兲, we do not ﬁnd a signiﬁ-

cant 共⬍1%兲variation of the DCA with voltage. On the other

hand, we present in Fig. 4共a兲aC(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

2.26⫻1016 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 CLand CR.

FIG. 2. Resistance of the samples with tCo⫽10 Å 共a兲and4Å共b兲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. 共a兲From lower to upper curves, the voltages are 100, 50, 25, 20, 15, 10,

5, and 2 mV. 共b兲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

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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. 4共b兲, 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 共from 30

to 50 K兲, 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. 3共b兲. In such

a case, the total charge stored in the system is Qtot

⫽关CLRL/(RL⫹RR)兴V共assuming that CRRR⬍CLRL). 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:

C共V兲⫽dQtot

dV ⫽RLCL

RL⫹RR

⫹CLVd

dV

RL

RL⫹RR.共4兲

This equation can be put in a simpler form if we assume

that RLⰇRR:

C共V兲⬇CL

冉

1⫺RR

RL

⫺Vd

dV

RR

RL

冊

.共5兲

As RLⰇRR, 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

becomes

C共V兲⬇CL

冉

1⫺RR

RL

⫹VRR

RL

2

dRL

dV

冊

.共6兲

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. 4共b兲.

In Fig. 5, we present the C(V) curves obtained at4Kfor

the tCo⫽10 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 共dis兲charging characteristic time constant of the

islands. The effect decreases when the time constant is

greater than the oscillation half period. The 共dis兲charging

FIG. 3. 共a兲Equivalent 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 ⍀.共b兲Equivalent circuit representation of an

island outside the Coulomb blockade regime. CL(CR)andRL(RR) are the

capacitance and the resistance of the left 共right兲tunnel junction.

FIG. 4. C(V) curve measurement for the sample with tCo⫽4 Å. The fre-

quency of measurement is 2 kHz, and the amplitude of the voltage oscilla-

tion is ⌬V⫽1 mV. 共a兲Curve at 10 K. 共b兲Evolution of the curves with

temperature. From lower to upper curves, the temperature are 4, 10, 30, 50,

100, and 300 K.

FIG. 5. 共a兲C(V) measurement at 4 K of the sample with tCo⫽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. 共b兲A 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

⫽RCisland⫽RS/d, where RS⫽15 ⍀cm2is 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 共dis兲charging 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 共see the inset in

Fig. 5兲. 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 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.

ACKNOWLEDGMENTS

Many thanks to M. Bowen and E. Price for their careful

reading of the manuscript.

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