Electrical properties of ITO/benzylated cyclodextrins (β-CDs (Bz))/Al diode structures
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Electrical properties of ITO/benzylated cyclodextrins (β-CDs (Bz))/Al diode structures
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2006 Sci. Technol. Adv. Mater. 7 772
(http://iopscience.iop.org/1468-6996/7/8/A02)
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Science and Technology of Advanced Materials 7 (2006) 772–779
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Electrical properties of ITO/benzylated cyclodextrins(b-CDs (Bz))/Al diode structures
M. Bouzitouna, C. Dridia,�, R. Ben Chaabanea, H. Ben Ouadaa, H. Gamb, M. Majdoubb
aLaboratoire de Physique et Chimie des Interfaces, Departement de Physique, Faculte des Sciences de Monastir, 5019 Monastir, TunisiabLaboratoire des polymeres et des Biopolymeres, Departement de Chimie, Faculte des Sciences de Monastir, 5019 Monastir, Tunisia
Received 31 May 2006; received in revised form 27 September 2006; accepted 2 November 2006
Available online 19 January 2007
Abstract
Investigations of the electrical characteristics of benzylated cyclodextrins (b-CDs (Bz)) diodes are reported. We present current–voltage
characteristics and impedance spectroscopy measurements performed on partially benzylated cyclodextrins b-CDs (Bz) thin films in
sandwich structures ITO/b-CDs (Bz)/Al. The static electrical characterizations show a space charge limited conduction (SCLC) and a
conductivity with power low frequency behavior characteristic of a hopping transport in disordered materials. The impedance spectra
can be discussed in terms of an equivalent circuit model designed as a parallel resistor RP and capacitor CP network in series with resistor
RS. We extract numerical values of these parameters by fitting experimental data. Their evolution with bias voltages has shown that the
SCLC mechanism is characterized by an exponential trap distribution. We estimated from the capacitance voltage characteristic an
acceptor concentration of about 1011 cm�3 due to trap states.
r 2006 NIMS and Elsevier Ltd. All rights reserved.
Keywords: ITO/b-CDs (Bz)/Al diode; I–V characteristics; Space charge limited current (SCLC); Impedance spectroscopy (IS)
1. Introduction
Organic materials have recently attracted a great interestas semi conducting materials [1–3] and have been success-fully applied to electroluminescent devices [3,4], field-effecttransistors [5,6], solar cells [7,8] and chemical sensors [9,10],in particular, molecular host–guest recognition systemssuch as calixarenes and cyclodextrins [10–16]. The latterhave good physical and chemical stability, compatibilitywith large field applications and lead to low-cost devices.Among synthetic host molecules, cyclodextrins (CDs) haveemerged as an ideal candidate for this pursuit due to itswell-defined molecular cavity and its ability to accommo-date a variety of guest molecules [17]. Nevertheless, thestrong solubility of CDs in water makes their use difficultin the development of chemical sensors working in aqueousmedia. In our group, different immobilization methods
e front matter r 2006 NIMS and Elsevier Ltd. All rights rese
am.2006.11.006
ing author. Institut Superieur des Sciences Appliquees et
de Sousse, Cite Ettafala, 4003 Ibn Khaldoun Sousse,
216 98 940 256; fax: +216 73 332 258.
ess: [email protected] (C. Dridi).
have been used to overcome this problem: the b-CDsmolecules were fixed to silica insulator surfaces; either bychemically grafting polymethyl-hydrosiloxanes (PMHS)chains as a coupling agent [18] or by physically incorpor-ating CDs in plasticized poly(vinyl chloride) (PVC) [19]. Inprevious work, we have chemically modified b-CDs (b-CDs(Bz)) and used them as sensitive membranes for heavymetal cations detection (such as Pb2þ and Cd2þÞ [20]. Thishas given us a great stimulus to the deeper study of theirelectrical and dielectric properties. Impedance spectroscopy(IS) has proved to be a powerful tool for studying thetransport mechanisms and relaxation processes in organicdevices [21,22]. The objective is to establish a correlationbetween chemical structure, in terms of number of benzylmoieties, and electrical performances toward an ion sensorwith optimized sensitivity. Indeed, we present in anotherwork [23] a comparison between CD grafted with 5 and 10benzyl moieties in terms of heavy metal detection.In this paper, we have investigated the electrical proper-
ties of this partially benzylated b-cyclodextrin whosesynthesis was presented in the previous work [20].We investigated charge-carrier injection and transport
rved.
ARTICLE IN PRESS
Glass substrate
ITO
β -CD (Be)
Al
+V
-V
Fig. 1. ITO/b-CD(Bz)/Al diode heterostructure.
M. Bouzitoun et al. / Science and Technology of Advanced Materials 7 (2006) 772–779 773
processes of b-CDs (Bz) diodes via I–V characteristicsas well as impedance spectroscopy. The diode structurestudied is a metal/polymer/metal consisting of indiumtin oxide as positive contact and an aluminum electrode asthe negative contact as shown in Fig. 1. The IS measure-ments at different bias voltages have given a deeper insightinto transport mechanism of the device. Moreover, theobtained results were fitted to an electrical equivalentcircuit in order to extract the relaxation time in thesematerials.
Fig. 2. AFM topography of a spin-coated partially benzylated cyclodex-
trins (b-CDs (Bz), image at 3D.
2. Experimental details
Thin films were obtained by spin-coating b-CDs(Bz)/chloroform solutions on ITO glass substrates which wereprecleaned by successive ultrasonic treatment for 20min inacetone and methanol followed by drying with nitrogen gas(ITO-thickness 100 nm, sheet resistance 20O=cm2Þ. Fol-lowing the spin-coating process, the films were annealed for30min in an oven at 80 �C under ambient atmosphere. Theindium tin oxide (ITO) thin films have been used, due totheir good efficiency as a hole injecting material intoorganic thin film. The aluminum top electrode wasdeposited on b-CDs (Bz) thin film by thermal evaporationthrough appropriately shadow masks in high vacuumconditions ð10�6 TorrÞ. The current–voltage characteristicsof the ITO/b-CDs (Bz)/Al devices were measured from anapplied bias of �5–5V by using a Keithley 236 sourcemeasure unit. In general, the excitation potential for ACmeasurements is given by:
V ¼ V0 þ Vmod cosðotÞ, (1)
with ðV0 is the DC bias and Vmod is the oscillation leveland o=2p is the frequency. In our case, the measurementswere performed in the following conditions V0 : 023:5Vand Vmod of 50mV over a frequency range of500Hz–13MHz using a computer controlled HP 4192ALF. All electrical measurements were performed in darkand at room temperature.
The UV–Visible absorption spectra were recorded witha Perkin-Elmer UV–Vis spectrophotometer (Lambda 35)on b-CDs (Bz) thin films spin-coated on spectrosilglass substrates to avoid the strong absorption of ITO inthe UV.
3. Results and discussion
3.1. Surface morphology
The produced films are smooth and homogeneous in surfacetopography as shown in the AFM image in Fig. 2. The surfaceshows a very low roughness (RMS) of�0:5 nm measured overa 500� 500 nm2 area. The thickness was estimated by theAFM-scratching technique to be about 20nm.
3.2. UV–Vis study
UV–Vis spectra of b-CDs (Bz) performed in solution andon thin film samples, exhibit one significant absorptionband ðlmax ¼ 260 nmÞ as shown in Fig. 3a. The estimatedoptical band gap is about 4.4 eV giving a device electronicstructure shown in Fig. 3b.
3.3. Current–voltage characteristics
The measured I–V characteristics show typical diodebehavior with a threshold bias voltage of 2.5V. As it can beseen in Fig. 4a, the current was observed in both forwardand reverse bias voltage. The electrically symmetric I–V
characteristic, for low voltages, can be explained by thelocalized-state theory with defects providing the localizedgap states [24]. Indeed, if the defects are amphoteric, it ispossible to observe electrically symmetric behavior at lowvoltages. The asymmetric I–V characteristic at highervoltages is attributed to the difference of injection barriersto electron and hole due to different work functions for theITO anode (4.7 eV) and the Al cathode (4.3 eV). The I–V
characteristic in log–log plots of ITO/b-CDs (Bz)/Al isshown in Fig. 4b. The current dependence of appliedvoltage appears to follow power law behavior J / V m withtwo regimes corresponding to:
(1)
An ohmic region with J / V at low voltage. Indeed, forthe case of existence of a single set of shallow traps inARTICLE IN PRESS
4003002000.0
0.3
0.6
0.9
Ab
so
rban
ce
CD-Bz (Solution)
CD-Bz (thin film)
Solvent:Chloroform
EGAP
HOMO
LUMO
Holes
β-CDs (Bz),EGAP= 4.4eV
Anode (ITO, 4.7 eV) Cathode (Al, 4.3eV)
Electrons
(nm)
Fig. 3. (a) UV–Vis spectra of b-CDs (Bz) and (b) energy levels diagram of
ITO/b-CDs (Bz)/Al.
-5 -4 -3 -2 -1 0 1 4 5
-4.0x10-6
-3.0x10-6
-2.0x10-6
-1.0x10-6
0.0
1.0x10-6
2.0x10-6
3.0x10-6
I (A
)
V (Volts)
10.1
10-8
10-7
10-6
1x10-5
Lo
g I (
A)
LogV (Volts)
2.0
C
B
1.3
A
32
Fig. 4. (a) Current–voltage characteristics of the ITO/b-CD (Bz)/Al and
(b) log–log plot of the I–V curve of Fig. 4a. The slopes of AB and BC lines
indicate two regions ohmic and SCLC with exponents 1.22 and 2.04,
respectively.
M. Bouzitoun et al. / Science and Technology of Advanced Materials 7 (2006) 772–779774
solids, at low voltages, electrical conduction is Ohmicand the current density is described by [25]
J ¼ qmn0V
d, (2)
where q: is the electronic charge, m: the charge carriermobility, n0: the free carrier density, V: the appliedvoltage and d: the film thickness.
(2)
A trapped charge limited region with J / V2. As thevoltage increases, the current is controlled by spacecharge and follow the relation [25,26]:J ¼8
9eym
V 2
d3, (3)
where e is the organic material permittivity, y relatingthe proportion of trapped charges ðnt; ptÞ to freecharges ðn; pÞ then y ¼ n=ðnþ ntÞ ¼ p=ðpþ ptÞ where n
and nt are the density of free and trapped electronrespectively, p and pt the density of free and trappedhole respectively.
These two regimes show clearly a space charge limitedcurrent (SCLC) process [27,28]. This is a similar behavior asobserved in previous works for several ITO/organic film/Aldiode structures [29–36]. The organic layers used were a
conjugated polymer (PPV), a small molecule ðAlq3Þ, analternating block copolymer and other different macro-molecules (diazoamino glyoxime (DAG) and anthylvinylbenzene (AVB)) [29–36]. These reports have considered thatthis SCLC mechanism is attributed to a bulk-limitedconduction process. However, de Boer and Morpurgo [37]have shown recently; that surface traps cause a large changein the electrostatic profile of the bulk material therebyprofoundly affecting the behavior of SCLC. Their calcula-tions illustrate how the combined effect of surface and bulktraps results in features in the I–V curves that have been sofar attributed to different physical mechanisms. The crucialpoint is that in high-purity samples, the total amount ofsurface traps can dominate over the total amount of bulktraps, even for sizable contact separations [37].
3.4. Impedance spectroscopy
3.4.1. GðoÞ and CðoÞ measurements
Impedance spectroscopy was applied to understandthe dielectric behavior of the ITO/b-CDs (Bz)/Al device.
ARTICLE IN PRESS
1 10 100 1000 10 0001E-12
1E-11
1E-10
1E-9
1E-8
10 100 1000 10 000
E1-11
f (KHz)
Ca
pa
cita
nce
(F
)
f (KHz)
Ca
pa
cita
nce
(F
)
0V0.5V1V1.5V2V2.5V3V3.5V
Fig. 6. Frequency dependent capacitance of the ITO/b-CD(Bz)5/Al diode
at different bias voltage.Th respective capacitance traces are off-set by
1 pF for clarity.
M. Bouzitoun et al. / Science and Technology of Advanced Materials 7 (2006) 772–779 775
The variation of conductance and capacitance, of b-CDs(Bz) derivative based diodes versus frequency at differentbias voltages are plotted as shown in Figs. 5 and 6,respectively.
The conductance characteristic remains constant at lowfrequencies and power-law behavior was observed athigher ones. Indeed, a hopping transport mechanism canbe postulated. In general, the trend of the conductance withfrequency, in disordered materials, obeys the followingrelation [38]:
GðoÞ ¼ Gdc þ GacðoÞ where GacðoÞ ¼ Aos, (4)
where Gdc is the dc conductance, o the angular frequencyof the applied excitation, A constant, s the critical exponent0oso1.
As shown in Fig. 6 the capacitance decreases as thefrequency increases (capacitance curves are off-set by 1 pFfor clarity). This effect can be ascribed to traps inside thedevice [39] as observed also in polymer light emitting diodesby Blom and coworkers [40]. In our case, it appears that thecapacitance C exhibits distinct frequency dependence:
�
Fig
vol
At low frequencies C decreases and the frequency atwhich this decrease stops is about 10–100 kHz depend-ing on the applied bias. Indeed, since the device is spacecharge limited, the ac frequency modulates the numberof charge carriers in the device. Nevertheless, the timescale for the built-up of charge carriers is given by thetransit time tt of the injected carriers. At low frequenciesthe built-up of the space charge is fast enough to followthe ac modulation. Given that tt is finite the correspond-ing current lags behind the ac modulation excitation andthis result in an additional inductive (negative) con-tribution to the capacitance.
� For higher frequencies the space charge cannot beredistributed in a period of the applied voltage and themeasured capacitance equals to the geometrical value,C0 ¼ e0erA=d.0.1 1 10 100 1000 100001E-8
1E-7
1E-6
1E-5
bias 0.0V s=0.97bias 0.5V s=0.93bias 1.0V s=0.89bias 1.5V s=0.78bias 2.0V s=0.67bias 2.5V s=0.60bias 3.0V s=0.54bias 3.5V s=0.50
f (kHz)
G (
Ω−1
)
. 5. Conductance plot as a function of the frequency in different bias
tages.
Moreover, the slow decrease of C and strong increase ofG with frequency reflect the relaxation of permanentdipoles present in the material. The disordered nature ofthe organic material gives rise to a distribution of dipolarrelaxation times, (the evolution of the latter with biasvoltage will be discussed in Section 3.4.3) and the dielectricresponse can be accurately described by the Cole–Colerepresentation [41].
3.4.2. Cole–cole representation
The IS technique consists in the measurement of theelectrical impedance Z as a function of the frequency of theinput signal over a wide frequency range. The collecteddata may be visualized as a Nyquist diagram or cole–coleplot, represented by the imaginary component Z00 of theimpedance as a function of Z0 the real component. TheCole–Cole plots of ITO/b-CDs (Bz)/Al device at several dcbias voltages as seen in Fig. 7, display a single semicircleand a net decrease in diameter with increasing applied dcbias. The strong dependence of the impedance with the biasindicates the presence of a depletion layer in the samplewith a resistance decreasing with increasing bias. Capaci-tance voltage measurements (see Section 3.5) confirm thepresence of a depletion layer in forward bias before theinjection become significant. In this case two semicirclesshould be obtained on the plot of ImðZÞ vs Re(Z), onerepresenting junction and the other representing the bulkbehaviour. In our case, only one semicircle is observedbecause the bulk and the junction capacitances are notsufficiently different. Thus, the semicircle diameter corre-sponds to the sum of bulk and junction resistances. Analogresults has been obtained for PPV [42] and AVB [36] baseddevices.A single semicircle in a Nyquist plot suggests an
equivalent electrical circuit, shown in Fig. 8, composedof a single parallel resistor RP and a capacitor CP networkplaced in series with resistance RS [21]. We plotted the
ARTICLE IN PRESS
0 1x107 2x107 3x107
0.0
6.0x106
1.2x107
Bias 0.0VBias 0.5VBias 1.0VBias 1.5VBias 2.0VBias 2.5VBias 3.0V
Z' (Ω)
Z "
(Ω)
Fig. 7. Cole–cole plots of the ITO/b-CD(bZ)/Al device at several bias
voltage.
Rp
Cp
Rs
ITO / β-CD(bZ) β-CD(bZ) bulk and β -CD(bZ) /Al junction
Fig. 8. The electrical equivalent circuit of the ITO=b�CDðBzÞ5=Al diode
which consist of a single parallel resistor RP and capacitor CP network
with a series resistance RS.
102 103 104 105 106
104
105
106
Re Z(Ω)
Im Z
(Ω)
0.54
Fig. 9. The cole–cole plot of the ITO/b-CD(bZ)/Al device at 2.5V bias
voltage (in log–log representation). The slope of 0.5 means the curvature is
semicircle.
M. Bouzitoun et al. / Science and Technology of Advanced Materials 7 (2006) 772–779776
data in log–log scale (Fig. 9). The slope of 0.5 meansthe curvature is a semicircle [21,41]. The obtainedslopes are in the range of 0.46–0.68 for high and lowbias voltages. It suggests a single characteristic relaxationtime.
We presented in Fig. 10(a) the real part of the impedanceof ITO/b-CDs (Bz)/Al device as a function of frequency at0, 1.5 and 3.5 bias voltages. The maximum Re(Z) valuecorresponds to the sum of the device resistance and theresistance of the ITO substrate. At higher frequencies,Re(Z) (Fig. 10a) and Im(Z) (Fig. 10b) are more frequency-independent.
3.4.3. Simulation of impedance spectroscopy results
The impedance of the equivalent circuit is given by
ZðoÞ ¼ RS þRP
1þ ðo=o0Þ2
" #� j
RPo=o0
1þ ðo=o0Þ
� �
¼ ReðZÞj ImðZÞ, ð5Þ
where o is the angular frequency of the ac excitation ando0 ¼ 1=RPCP the proper angular frequency of the circuit.Eliminating the angular frequency we obtain the followingequation relation:
ReðZÞ � RS þRP
2
� �� �2þ ImðZÞ2 ¼
R2P
4. (6)
The minimum Re(Z) value observed at high frequenciescorresponds to the RS value which is bias and frequencyindependent and should be attributed to the hole injectinginterface ITO/b-CDs (Bz) which can be regarded as ohmic.In this device RS is about 289.5 O (for 0V bias). Themaximum Re(Z) value at low frequencies corresponds tothe sum of RS and the b-CDs (Bz) resistance RP which isstrongly bias dependent. Fig. 11 shows the measured realand imaginary parts of the impedance as a function of thefrequency for the diode ITO/b-CDs (Bz)/Al at dc biasvoltage of 0V. Symbols are measured data and the solidlines are the fitting results using the correspondentequivalent circuit of Fig. 8. The fit of experimental resultsby Eq. (5) give the parameters RP and CP of the circuitwhich are summarized in Table 1. The interfacial capaci-tance is negligible, which suggests that there are negligibleinsulating barriers such as oxide or air gap formed betweenthe b-CDs (Bz) thin film and the top and/or bottomelectrodes.The variation of the fitting parameters of the ITO/b-CDs
(Bz)/Al device are shown in Fig. 12. The RP decreases asthe dc bias voltage increases while CP is almost indepen-dent of bias voltage. The independence of CP with appliedbias is exactly the behavior expected for the SCLCmechanism, where the total charge in the device is equalto CPV .
ARTICLE IN PRESS
1 10 100 1000 10 000
0.0
5.0x106
1.0x107
1.5x107
2.0x107
2.5x107
3.0x107
3.5x107
Z '(Ω
)
f (KHz)
0.0
5.0x106
1.0x107
1.5x107
0 V1.5V3.5V
0 V1.5V3.5V
Z "
(Ω)
f (KHz)
1 10 100 1000 10 000
Fig. 10. Frequency dependence of (a) real part and (b) imaginary part of
the impedance for different bias: 0, 1.5 and 3.5V. Symbols are
experimental data and the solid lines are the fitting data using the
equivalent circuit represented in Fig. 8.
1 10 100 1000 10 000
0.0
5.0x106
1.0x107
1.5x107
2.0x107
2.5x107
3.0x107
(Z', Z
") (
Ω)
Bias = 0V
f (KHz)
ITO / β-CDs(Bz)/Al
Z' Z"
Fig. 11. An example of fitting the impedance data of the
ITO=b�CDðBzÞ5=Al diode using a single parallel resistor RP and capacitor
CP with a series resistance RS. The data are well fitted for over than four
decades. Symbols are experimental data and the solid lines the fitting ones.
-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
0
5
10
15
20
25
30
11
101.9
Ln V
Ln
RP
Bias (V)
RP (MΩ)
CP (pF)
Fig. 12. The variation of the fitting parameters of the ITO=b�CDðBzÞ5=Al
diode using the equivalent circuit of Fig. 8. The inset shows the plot of
logRP vs logV , and m is obtained as about 1.9 by the linear fitting of the
data.
Table 1
Electrical parameters of the device calculated from the fit of the
experimental data
Bias (V) 0 0.5 1 1.5 2 2.5 3 3.5
Rs (O) 289.5 292.7 292.7 289.7 300.7 271.1 276.5 311.4
Rp (M O) 29.01 24.40 13.75 3.92 2.69 1.96 1.61 1.37
Cp (pF) 2.24 1.79 1.70 2.72 2.02 2.07 2.07 1.86
t (ms) 0.400 0.290 0.150 0.070 0.040 0.025 0.021 0.016
M. Bouzitoun et al. / Science and Technology of Advanced Materials 7 (2006) 772–779 777
From the SCLC with an exponential trap distri-bution theory, voltage dependent current for one carrierdominated transport, hole in this case, is given by [21]
J ¼ KVmþ1=d2mþ1, (7)
where d and K are the thickness of the film and a constant,respectively.The voltage dependence of RP is thus described by
RPaV=JaV�m. (8)
The plot of logRP vs logV is shown in the inset of Fig. 12.The linear behavior of logRP vs logV implies thatconductance of the hole in b-CDs (Bz) thin film isconsistent with a SCLC mechanism with exponential trapdistribution where the corresponding m is about 1.9. Thecarriers may be trapped by chemical impurities and/orstructural defects.The range of relaxation times, for different bias voltage,
is indicative of dipolar relaxation.
3.5. Capacitance–voltage characteristics
More quantitative information about trap concentrationin our device can be obtained from capacitance–voltage
ARTICLE IN PRESS
0.0 0.5 1.0 1.5 2.0 2.5 3.08.0x1022
1.0x1023
1.2x1023
1.4x1023
1.6x1023
1.8x1023
2.0x1023
2.2x1023
2.4x1023 75 KHz
Bias (Volts)
1/C
2 (
F -2
)
Fig. 13. 1=C2 vs bias voltage of ITO/b-CDs (Bz)/Al diode (the inset shows
the C–V characteristic).
M. Bouzitoun et al. / Science and Technology of Advanced Materials 7 (2006) 772–779778
spectroscopy. We show in the inset of Fig. 13 the biasvoltage dependent capacitance which indicate the fullydepleted b-CDs (Bz) thin film.
We observe from the evolution of 1=C2 versus biasvoltage (Fig. 13), a locally linear behavior between 1 and3V which confirm the presence of surface traps acting asacceptor in b-CDs (Bz) and allows as to estimate theirconcentration NA. Therefore, from the curve of 1=C2 vs Vfor an abrupt junction, assuming that the traps areuniformly distributed in space, the trap concentration NA
can be calculated using the following relation [42]:
1
C2¼
2ðVD � V Þ
A2qe0eNA
, (9)
where VD: the diffusion voltage and NA: the ionizedacceptor concentration. The ionized acceptor concentra-tion NA have been estimated to be 4:9� 1011 cm�3 and thediffusion voltage VD equal to 4.3V.
Indeed, involving an exponential trap distribution isremarkably in good agreement with the experimentalobservation listed above. However, this model cannot beattributed only to bulk limited conduction in b-CDs (Bz)active layer but trap states should be considered to explainC�2 vs V local linear behavior.
4. Conclusion
In this paper, current–voltage and electric impedancemeasurements were used to investigate the transportcharacteristics of holes in ITO/b-CDs (Bz)/Al diodestructures. The fabrication and testing of a polymer diodehas been reported and discussed. The device exhibitedsymmetric IðV Þ characteristics in both forward and reversebias. The conduction of holes in a film of b-CDs (Bz) isgoverned by space charge limited current characterized bya presence of exponential trap distribution. From fits offrequency dependent experimental data, the exponential
trap distribution and relaxation time can be extracted fromthe space charge limited current theory with traps and theimpedance spectroscopy formalism. Moreover, the slowdecrease of C and strong increase of G with frequencyreflect the relaxation of permanent dipoles present in thematerial. The disordered nature of the organic materialgives rise to a distribution of dipolar relaxation times.From the plot of 1=C2 vs bias voltage of ITO/b-CDs (Bz)/Al diode we deduce the acceptor concentration NA.
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