Physica B 425 (2013) 48–57

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Physica B

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On the variation in the electrical properties and ac conductivityof through-thickness nano-porous anodic alumina with temperature

Mahmood Tahir a, Mazhar Mehmood a,n, Muhammad Nadeemb,Abdul Waheed a,1, Muhammad Tauseef Tanvir a

a National Centre for Nanotechnology & Department of Metallurgy and Materials Engineering, Pakistan Institute of Engineering and Applied Sciences (PIEAS),Islamabad, Pakistanb EMMG, Physics Division, PINSTECH, Islamabad, Pakistan

a r t i c l e i n f o

Article history:Received 24 September 2012Received in revised form9 May 2013Accepted 12 May 2013Available online 28 May 2013

Keywords:Porous anodic aluminaCharge carriersAdsorbed waterHoppingTunneling

26/$ - see front matter & 2013 Published by Ex.doi.org/10.1016/j.physb.2013.05.018

esponding author. Tel.: +92 512207813; fax: +ail addresses: mazhar@pieas.edu.pk, fac129@pesent address: Islamia College, Peshawar.

a b s t r a c t

The electrical response of self-organized through-thickness anodic alumina with hexagonal arrangementof cylindrical pores has been studied as a function of temperature. Mechanically stable thick porousanodic alumina was prepared, by through-thickness anodic oxidation of aluminum sheet in sulfuric acid,with extremely high aspect ratio pores exhibiting fairly uniform diameter and interpore distance. It wasobserved that the electrical properties of through-thickness anodic alumina are very sensitive to minutechanges in temperature and the role of surface conductivity in governing its electrical response cannot beoverlooked. At high frequencies, intrinsic dielectric response of anodic alumina was dominant. Thefrequency-dependent conductivity behavior at low and intermediate frequencies was explained on thebasis of correlated barrier hopping (CBH) and quantum mechanical tunneling (QMT) models, respec-tively. Experimental data was modeled using an equivalent circuit consisting of Debye circuit, for bulkalumina, parallel to surface conduction path. The surface conduction was primarily based on two circuitsin series, each with a parallel arrangement of a resistor and a constant phase element. This suggestedheterogeneity in alumina pore surface, possibly related with islands of physisorbed water separated bythe regions of chemisorbed water. Temperature dependence of some circuit elements has been analyzedto express different charge migration phenomena occurring in nano-porous anodic alumina.

& 2013 Published by Elsevier B.V.

1. Introduction

Owing to high dielectric strength, resistance against hostileenvironments, mechanical and thermal stability, porous anodicalumina (PAA) has attracted great attention of researchers [1,2].These characteristics make it a suitable candidate for a variety ofapplications including fabrication of electronic devices, magneticstorage discs, capacitors with barrier-type dielectric layers, gassensors, heat sinks in IC's and biological membranes etc. [3–5]. Asit is almost transparent to ordinary light [6], its photo- and electro-luminescence properties have been studied for various applica-tions such as photonic crystals (polarizers) [7,8] and host for otherfluorescent materials etc. [9,10]. It bears extremely large surfacearea due to high aspect ratio of ordered nano-channels [11]rendering it novel catalytic, magnetic and electronic properties.

Pore diameter, interpore spacing and mutual arrangement ofthe pores in PAA can be easily controlled by appropriate choice of

lsevier B.V.

92 512208070ieas.edu.pk (M. Mehmood).

anodizing conditions, and pre- and post-treatments [12]. Nano-structured surface obtained by appropriate choice of electropol-ishing pretreatment as well as suitable anodizing conditions mayhave excellent hexagonal ordering of the pores with uniform porediameter and extraordinarily high aspect ratio of straight pores.One of the reasons for widespread use of PAA is ease in itspreparation with a variety of architectures [13]. These include,for example, templates with ordered arrangement of taperednanopores [14], and bi-layer structures in which pores grownfrom the opposite sides meet at a partially pierced barrier typeoxide formed by through-thickness oxidation of alumina sheet[15].

Alumina can host a variety of nanostructures grown in-situ forthe formation of electrical, magnetic, optical, catalytic and sensingdevices, etc. [16]. Magnetic materials, such as Ni, Co, Fe and CoFealloy etc., embedded in porous alumina matrix have their applica-tions for the fabrication of high-density magnetic recording media[17,18]. PAA templated ferromagnetic nanowires, Au/Ni wires forexample, have been used in the field of biomagnetics, in whichthese nanowires sense the biomolecules, sort the cells, and per-form required biological manipulations [19]. Similarly, the fabrica-tion of electronic devices at nanoscale using PAA templates has

M. Tahir et al. / Physica B 425 (2013) 48–57 49

also been reported [20,21]. Syntheses of one-dimensional semi-conductor wires (CdS, CdxZn1−xS, CdSxSe1−x, GaAs etc.) and one-dimensional superlattices have been proposed [22,23]. Nanotubearrays of carbon and metals have also been suggested [24,25].Routkevitch et al. [26] have reported a comprehensive overview ofPAA-templated electronic device applications. Alumina itself isextensively used for sensor applications [27,28]. The contributionof electrical and electrochemical response of anodic alumina to theoverall response of the devices, in these cases, cannot be ignored.An appropriate understanding of electrical properties of PAA isthus extremely useful for its enhanced use.

Impedance spectroscopy is a very powerful analytical techni-que used in material research for electrical characterization [29]. Itcan be used to study the temperature dependence of resistanceand capacitance of materials [30]. In this technique, an ac signal isapplied to measure the response of the material in terms of acomplex electrical quantity as a function of applied frequency.Different complex electrical quantities include electrical impe-dance (Zn), dielectric permittivity (ϵn), electrical modulus (Mn)and admittance (Yn). These quantities are interrelated [31]:

Mn ¼ 1=ϵn ¼ jωCoZn ¼ jωCoð1=YnÞ ð1Þ

where Co is empty cell capacitance, ω is angular frequency andj2¼−1. Dielectric properties have often been represented in termsof complex admittance, which is given by [32]:

Yn ¼ Gðω; TÞ þ jωCðω; TÞ ð2Þwhere G(ω,T) is the conductance and C(ω,T) is the capacitance,which are functions of frequency and temperature (T). Aboverelationship can also be expressed as:

sn ¼ s=ðω; TÞ þ js==ðω; TÞ ð3Þwhere sn(¼Yn/(S/d), where S is the contact area (of the electrode),d is thickness of the sample and their ratio i.e. S/d is the geometricfactor) is the complex conductivity and s/ (¼G(ω,T)/(S/d)) ands//(¼ωC(ω,T)/(S/d)) are its real and imaginary parts.

In this manuscript, we have thoroughly investigated the elec-trical response of porous anodic alumina by impedance spectro-scopy. Extremely thick two-layer anodic alumina has been grownby two-step anodic oxidation of aluminum sheet at 25 V. Effect oftemperature on conduction mechanism and charge carrier trans-port has been studied at different frequencies. The electricalconduction mechanisms have been further explored by analyzingconductivity data and employing different theoretical models, viz.correlated barrier hopping (CBH) and quantum mechanical tun-neling (QMT) models, to explain the dependence of electricalresponse on measurement temperature. An equivalent electricalcircuit has also been proposed to interpret and analyze theexperimental data.

Fig. 1. Typical cross-sectional: (a) and top view (b) SEM images of through-thickness anodic alumina.

2. Experimental

Through-thickness anodic alumina structure comprising ofporous layers grown from opposite sides and separated from eachother at the center by a partially pierced barrier layer containing avery small fraction of residual aluminum pieces was synthesized.The method opted for its preparation was the same as reported inRef. [15] except 0.3 M sulfuric acid was used and anodizing wasperformed at 25 V. This resulted in fabrication of bi-layer anodicalumina. The thickness for each layer is approximately ∼250 μm.

Scanning electron microscopic images were acquired usingSEM (JSM-5910, JEOL) and FESEM (Camscan Apollo 300). Compleximpedance spectroscopy was done in the frequency range of10−1–106 Hz using Alpha-N analyzer (Novocontrol, Germany) byapplying 0.2 V ac signal. Water permeable thin gold electrodes

were deposited on both sides of the sample through sputteringusing 99.99% pure gold target. Silver paste was used in a very smallquantity to make electrical contacts. The sample was placed insidea homemade sample holder which was connected to a dc powersupply in order to control the temperature. The measurementswere carried out in the temperature range of 29–70 1C withan accuracy of 70.5 1C. Temperature was stabilized for about10 minutes prior to each reading. The absence of any extraneousinductive and/or conductive coupling was ensured in the givenfrequency range by monitoring the dispersive behavior of theconnecting cables using a standard sample. WINDETA software wasused for data acquisition while ZView software was used for curvefitting.

3. Results and discussion

Fig. 1(a) and (b) shows typical cross-sectional and top view SEMimages, respectively, of the PAA sample prepared by two-stepanodizing in 0.3 M H2SO4 at 25 V. Cylindrical parallel pores withuniform interpore distance are successfully formed at this voltage.Domains with well-defined hexagonal pore arrangement are seenalong with some pentagons (with missing pores at the center) andother irregular arrangements at the domain boundaries (Fig. 1(b)).The average interpore distance is ∼62 nm. Apparently, the poresseem to cover about 22% of the surface (Fig. 1b).

In order to view the sandwiched structure between oppositelygrown porous oxide layers, cross-sectional views of the middle oftypical PAA sample are shown in Fig. 2. The pores are relatively

Fig. 2. Cross-sectional views of the middle portion of a typical PAA sample.

M. Tahir et al. / Physica B 425 (2013) 48–5750

less organized at the middle part and tend to turn (or adopt wavypattern) deviating from their straight path. Non-uniform electro-striction stresses around the pore tips seem responsible for thistypical behavior when the metal-oxide interface advancing fromopposite sides approach each other. It appears from these SEMimages that aluminum is mostly consumed at the middle, exceptat some islands. It is interesting to note that a number of pores ofthe opposite sides tend to meet each other, similar to theobservations made in Ref. [15] for the through-thickness porousanodic alumina grown in oxalic acid at ordering voltages of 40 and50 V. It may, therefore, be suggested that the barrier layer presentin the middle of the sample is not compact and is mostly pierced,resulting in continuity of the pores grown from opposite sides.

Fig. 3 shows complex impedance plane plots of the sample atsome selected temperatures. For each plot, two arcs are observed,the larger one observed in the low frequency range is not fullytransformed into a semicircle due to the limit in the availablefrequency range. The two arcs observed in the given frequencyrange suggests the presence of at least two relaxation mechan-isms. The radii of both the arcs increase with measurementtemperature.

Fig. 4 shows typical Bode plots (impedance vs. frequency)obtained at different temperatures. The frequency fo (defined asthe frequency where the linear relationship of the curve is justobtained while moving towards higher frequencies) decreases withan increase in measurement temperature. At higher frequencies(f4fo), all curves of |Z|−f obtained at different measurementtemperatures follow straight lines of slope ‘–1’with a correspondingphase angle of –901 (not shown). It may be assumed that thedielectric capacitance of oxide dominates the overall impedance inthis frequency range. The overall impedance of PAA sample atrelatively low frequencies shows a rise of up to about one order ofmagnitude with the measurement temperature. It is generallyobserved that the conductivity of semiconductors and insulatorsincreases with rise in measurement temperature due to increase inthe number density of charge carriers, i.e., free electrons and/orholes. On the other hand, metals show a decrease in electricalconductivity with temperature. Conductivity of metals is commonlyfrequency-independent, although, at high frequencies, skin effectcauses frequency dependent changes. One possibility may be theretention of metallic clusters in the anodic alumina that is respon-sible for its metallic like conductivity behavior. Other possibility

Fig. 3. Typical complex impedance plane plots obtained at: (a) 29 1C, (b) 40 1C, (c) 50 1C and (d) 70 1C; insets show corresponding magnified views of the plots in the highfrequency region. Frequencies, in Hz on decimal logarithm scale, for some selected data points are indicated by arrows.

Fig. 4. Changes in total impedance of the sample with frequency, at differenttemperatures.

M. Tahir et al. / Physica B 425 (2013) 48–57 51

may be the surface effects along the pore walls. As the pore densityis very high, surface conduction along pore walls may have asignificant contribution. Surface conduction may be significantlyenhanced by the presence of adsorbed water on the pore surface,

for instance, allowing a Grotthuss chain reaction for conduction atroom temperature [33].

Fig. 5(a) shows capacitance (C(ω,T)) versus frequency forvarious temperatures. About one order of magnitude decrease incapacitance with temperature is observed at low frequencies. Thecapacitive contribution of the water like network largely dependson the bonding that exists between adsorbed water molecules andadjacent hydroxyl groups present on the alumina pores [34]. Atlow temperatures, water molecules are free to reorient withapplied field and, hence, high capacitance may be observed. Whentemperature is increased, enhanced bonding may occur possiblydue to gradual conversion of single bonding into double bonding.As a result, the tendency of free reorientation may become moreand more difficult with temperature leading to decrease incapacitance. The overall contribution of water molecules to capa-citance may also partly depend on the amount of adsorbed waterthat may vary with temperature. As can be seen in subsequentsections, the overall impedance has been analyzed using equiva-lent circuit taking into account surface conduction mechanismsinvolving adsorbed water.

As mentioned earlier, impedance follows slope of ‘−1’ in highfrequency region of Bode plots (Fig. 4). Corresponding to thisregion, horizontal lines are seen when capacitance is plottedagainst frequency (Fig. 5(a) and (b)). This frequency-independentpart extends to relatively lower frequency range with increase inmeasurement temperature. Thus, high frequency capacitance(horizontal part) seems predominantly related with intrinsicdielectric response of anodic alumina. As shown in Fig. 5(c), this

Fig. 5. (a) Variation in capacitance, C(ω,T), of the sample with frequency at differentmeasurement temperatures; (b) shows the same, magnifying high frequencyregion, (dotted lines are shown as an eye guide to indicate some horizontal partof the curves at high frequencies); (c) plot of high frequency capacitance(corresponding to horizontal dotted lines in (b)).

Fig. 6. Real part of the conductivity vs. frequency at different measurementtemperatures.

Fig. 7. The proposed equivalent electrical circuit.

M. Tahir et al. / Physica B 425 (2013) 48–5752

high frequency capacitance decreases up to about 10% withincrease in measurement temperature.

Fig. 6 shows real part of conductivity (s/(ω)) against frequencyon log–log scale at various measurement temperatures. For thematter of accuracy, it seems worth-mentioning here that the units

of conductivity have been used as S cm−1, which are normally validfor bulk conductivity. However, some of the contributions inconductivity are related with surface conduction phenomena. Inthis connection, the surface area S in the geometrical factor couldbe replaced by ‘the number of pores in the sample’ times ‘thecircumference of one pore’. Based on this, the conductivityreported in the units of S cm−1 (Fig. 6) can also be read in theunits of 3.47�10−6 S (for our sample) if it is termed to be surfaceconductivity of the pores. We have, however, not preferred so inour plots because the conductivity has been measured as the bulkconductivity of the sample, although explained on the basis ofsurface contribution in some part.

Conductivity strongly depends on measurement temperature.Two straight line regions with a slope of less than unity are clearlyseen, in different frequency regimes depending on temperature. Inmaterials, conductivity often follows a vastly used power law [32]:

s=ðωÞ ¼ so þ Bωs ð4Þwhere so is frequency-independent part of the conductivity, B isfrequency coefficient and s is the frequency exponent. A straight lineregion in the log s/(ω) vs. logf should correspond to a typical

Fig. 8. (a–i) Real and imaginary parts of admittance versus frequency at different measurement temperatures. Solid lines represent the best-fit simulation using in-housemodel to the experimental data (points). Fitting parameters are given in Fig. 10 and Table 1.

M. Tahir et al. / Physica B 425 (2013) 48–57 53

Fig. 9. (a–i) Real and imaginary parts of impedance against frequency at differentmeasurement temperatures. Solid lines represent the best-fit simulation usingZView software to the experimental data (points). Fitting parameters are given inFig. 10 and Table 1. Typical complex impedance plane plot (j) and Bode (phaseangle) plot (k) obtained at 50 1C are also shown.

M. Tahir et al. / Physica B 425 (2013) 48–5754

conduction mechanism following the above power law. In Fig. 6, twostraight line regions may be related to two different conductionmechanisms, following above power law, as predominant in differentfrequency ranges. Had these two mechanisms operative in parallel,their sum (s/ov(ω)) would have exhibited much higher conductivitiesas compare to the observed values. Same is true for the vice versa.Hence, the observed changes in conductivity with temperature arenot in agreement with a presumption of parallel conductionmechanisms alone. In addition to above changes in conductivity, adeflection with increased slope is also observed at higher frequencyside, which may be justified by considering another mechanisminvolving frequency-dependent changes in conductivity. Thisrequires to be considered as parallel to (at least) the intermediatefrequency mechanism (explained above).

On the basis of above discussion, an equivalent circuit isproposed which is shown in Fig. 7. It consists of a Debye equivalentcircuit (Cb||(Rbo−Cbo)) (marked as sub-circuit 1) in parallel with twoVoight elements (RH−CPEH and RL−CPEL,) placed in series withresistance R1 (marked as sub-circuit 2). Here CPE stands forconstant phase element.

It appears that (bulk) anodic alumina exhibits a dielectric disper-sion. It is known that the dielectric dispersion for a single timeconstant can occur due to time-dependent polarization, for whichsub-circuit 1 can be used, commonly known as Debye equivalentcircuit [32]. In such cases, the instantaneous polarization is relatedwith capacitance Cb. As the polarization saturates (at longer times), thecapacitance increases to (Cb+Cbo). Between these two, time-dependentchanges in polarization take place with a time constant of τ(¼RboCbo).It may be worth-mentioning here that the Rbo−Cbo extension of thissub-circuit (parallel to Cb) can take into account the high frequencychanges in (real part of) conductivity as noticed in Fig. 6.

The sub-circuit 2 may account for surface conduction phenom-ena. CPE is often used in the circuit instead of capacitor to accountfor the non-ideal Debye-like behavior [30]. The resistance RH andcorresponding constant phase element, CPEH, belong to relativelyhigh (intermediate) frequency surface conduction mechanismwhile RL and CPEL are presumed to be dominant at low frequen-cies. The two Voight elements, placed in series may be relatedwith heterogeneous structure of the pore surface and/or variationin the level of water adsorption, for instance, due to the formationof islands of physisorbed adsorbed water separated by the regionswhere only chemisorbed water is present [35].

CPE is an empirical function whose admittance (YnCPE) can beexpressed as [32]:

YnCPE ¼ AðjωÞn ¼ Aωn½ cos ðnπ=2Þ þ j sin ðnπ=2Þ� ð5Þ

where A and n are frequency-independent parameters. The valueof n depends on deviation from ideal behavior [36] arising as aresult of heterogeneity of the system (chemical heterogeneity,surface roughness, non-uniform distribution of current etc.) caus-ing frequency dispersion and 0≤n≤1. CPE behaves like an idealresistor for n¼0 and an ideal capacitor for n¼1. Anyway, alimitation is that the CPE is well-approximated only over a limitedfrequency range [32]. It may be worth-mentioning that real part ofthe admittance, Aωn½ cos ðnπ=2Þ� in Eq. (5), is almost same as Bωs inEq. (4), for s¼n and B¼ A½ cos ðnπ=2Þ�=ðS=dÞ. R1 has been incorpo-rated in the model to establish relative contribution of bulk andsurface conduction. Nitsch et al. [37] have also incorporated it intheir proposed ac equivalent circuit used for analyzing the thickfilm humidity sensors, where this accounts for an overall resis-tance to all the (surface) conduction mechanisms.

Using Microsoft Excel, we have calculated/simulated the admit-tance of equivalent circuit and compared with the experimentaldata in order to assess various parameters providing best fit.

M. Tahir et al. / Physica B 425 (2013) 48–57 55

The admittance of the sub-circuit 1 can be calculated by:

Y1 ¼ ω2RboC2bo=ð1þ ω2R2

boC2boÞ þ j½ωCbo=ð1þ ω2R2

boC2boÞ þ ωCb� ð6Þ

The admittance of a single R||CPE circuit is given by:

YRjjCPE ¼ 1=Rþ Aωn cos ðnπ=2Þ þ jAωn sin ðnπ=2Þ ð7ÞFrom above equation, the impedance of single R||CPE circuit can becalculated as follows:

ZRjjCPE ¼ ðYRjjCPEÞ−1 ð8Þwhich yields:

ZRjjCPE ¼ Z=RjjCPE þ jZ==

RjjCPE ð9Þ

where

Z=RjjCPE ¼ Rð1þ ARωn cos ðnπ=2ÞÞ=ð1þ 2ARωn cos ðnπ=2Þ þ A2R2ω2nÞ

ð9aÞand

Z==RjjCPE ¼ −A Rωn sin ðnπ=2Þ=ð1þ 2A Rωn cos ðnπ=2Þ þ A2R2ω2nÞ ð9bÞ

The overall impedance of the sub-circuit 2 is thus given by:

Z2 ¼ Z=2 þ jZ==

2 ¼ ðR1 þ Z=L þ Z=

HÞ þ jðZ==L þ Z==

H Þ ð10Þwhere Z=

L and Z=H are defined by Eq. (9a) and Z==

L and Z==H are defined

by Eq. (9b) for respective Voight elements. The admittance of sub-circuit 2 (Y2) can, accordingly, be determined as:

Y2 ¼ Y=2 þ jY==

2 ¼ Z=2=���Z2

���

2þ jð−Z==

2 =���Z2

���

2Þ ð11Þ

The overall admittance (Y) of the proposed equivalent circuit isthus:

Y ¼ Y= þ jY== ¼ ðY=1 þ Y=

2Þ þ jðY==2 þ Y==

2 Þ ð12ÞBased on above model (form here onwards referred as in-house

model), real, imaginary and absolute admittance were calculated(simulated) and compared with the experimental data. The bestfits to the experimental data obtained at different temperaturesare shown in Fig. 8(a–i). Here the experimental data is shown bysymbols while the simulated data is represented by solid lines. It isevident that the simulated curves agree well with the experi-mental data at all temperatures. From the simulated curves, valuesof different elements used in equivalent circuits were estimated, asshown later.

The same equivalent electrical circuit was also used to fit theexperimental data using ZView software. The n parameters werekept the same as determined on the basis of in-house model(Fig. 8), while the other parameters were allowed to vary for thebest fit. Fig. 9(a–k) compare the measured impedance and thesimulated one for the best fit parameters of equivalent circuit. Typicalcomplex impedance plane and Bode (phase angle) plots at 50 1C(as representative temperature) are also shown in Fig. 9(j) and (k). It

Table 1The best-fit values of parameters of sub-circuit 1 as obtained from two fitting approach

Temperature (oC) Fitted values obtained using in-house model

Cb (F) Cbo (F) Rbo (Ω) τ (s)

29 1.12�10−12 6.80�10−14 1.50�106 1.02�35 1.10�10−12 6.00�10−14 2.50�106 1.50�40 1.07�10−12 6.00�10−14 1.00�106 6.0045 1.05�10−12 5.50�10−14 1.00�106 5.5050 1.03�10−12 6.00�10−14 1.50�106 9.0055 1.03�10−12 6.80�10−14 1.00�106 6.8060 1.02�10−12 6.80�10−14 1.00�106 6.8065 1.02�10−12 6.80�10−14 1.00�106 6.8070 1.01�10−12 6.20�10−14 1.00�106 6.20

may be noticed that the fitted curves are in good agreement with theexperimental data.

The values of parameters of sub-circuit 1 as obtained fromsimulation are shown in Table 1. They do not change significantlywith measurement temperature. The value of Cb (as well as Cb+Cbo)obtained from equivalent circuit fitting is of the order of 10−12 F,comparable to the capacitance observed in the high frequencydomain (Fig. 10(a) and (b)). The time constant, τ, is of the order of∼10−8 s in our case (Table 1). As has been suggested earlier, thiscapacitance seems related with intrinsic response of alumina(walls/bulk). The overall area of the contact (S) is about 0.08 cm2,including the pores. As the thickness (d) of the sample is ∼500 μm,the value of relative permittivity (ϵr) comes out to be 7.65. If thecross-sectional surface area of the pores (about 22%) is excluded,effective S (for the walls) would become about 0.0625 cm2. Basedon this presumption, relative permittivity of anodic alumina (ϵr) hasbeen estimated to be 9.8, which is very close to that of bulk alumina(8≤ϵr≤12) [38].

The best-fit values of various other parameters of the equiva-lent circuit are shown in Fig. 10(a)–(c). As can be noticed that theparameters obtained from two different approaches are close toeach other, exhibiting similar trends as a function of temperature.The resistances R1, RL and RH increase while both AL and AH

decrease with increase in measurement temperature. Similarly, nLdecreases with measurement temperature, while nH remainsalmost temperature independent.

Several theoretical models have been proposed to explain ωn

(or ωs)-dependence of conductivity. Quantum mechanical tunnel-ing (QMT) [39] and correlated barrier hopping (CBH) [40] modelsare widely employed. In these models, the frequency exponent, n,behaves differently with temperature. According to CBH model, nis given by the expression [40]:

n¼ 1−6kBT=½WM þ kBTlnðωτ0Þ� ð13Þ

where kB is the Boltzmann's constant, T is the absolute tempera-ture, WM is the binding energy and τo is the characteristicrelaxation time. It is evident from Eq. (13) that n decreases withincrease in measurement temperature in CBH model.

According to QMT model, n is given by [39]:

n¼ 1þ 4=lnðωτoÞ ð14Þ

Thus, if conductivity assumes to follow QMT model, n isexpected to be temperature independent.

It is clear from Fig. 10(c) that the values nL tend to decrease withrise in measurement temperature while nH remain almost unchanged.The former is predominant at low frequencies while the later atrelatively high (or intermediate) frequencies. Hence, it can be sug-gested that, at low frequencies, correlated barrier hopping over theenergy barrier existing between the localized sites is mainly respon-sible for carriers' conduction. At relatively high (intermediate)

es.

Fitted values obtained using ZView Software

Cb (F) Cbo (F) Rbo (Ω) τ (s)

10−7 1.12�10−12 6.00�10−14 1.00�106 6.00�10−8

10−7 1.10�10−12 6.00�10−14 1.00�106 6.00�10−8

�10−8 1.07�10−12 6.02�10−14 1.39�106 8.36�10−8

�10−8 1.05�10−12 5.15�10−14 1.50�106 7.73�10−8

�10−8 1.03�10−12 6.15�10−14 1.99�106 1.22�10−7

�10−8 1.02�10−12 5.15�10−14 1.10�106 5.67�10−8

�10−8 1.00�10−12 4.15�10−14 1.82�106 7.55�10−8

�10−8 1.00�10−12 3.15�10−14 1.99�106 6.27�10−8

�10−8 1.00�10−12 2.05�10−14 2.42�106 4.95�10−8

Fig. 10. The best-fit values of different parameters of the equivalent circuit usingin-house model and ZView software.

Fig. 11. Plots of R−1H and R−1

L against reciprocal of absolute temperature.

M. Tahir et al. / Physica B 425 (2013) 48–5756

frequencies, quantum mechanical tunneling through the energybarrier is the dominant conduction mechanism.

The anodization of aluminum in sulfuric acid results in theformation of amorphous anodic alumina [41]. Surface and thebulk, to some extent, of PAA are doped with different ionic species(SO−2

4 , HSO−4, hydroxyl ions etc.) along with bound water [42]. Due

to amorphous nature of alumina, cationic, anionic and impuritylevels are originated due to dissimilarity in the energy levels of

different lattice sites. There also exist trapping levels and othergross defects in the structure. The overall response of PAA atdifferent temperatures and frequencies relies on the contributionof all these entities. Adsorbed water present on the surface of PAAalso significantly influences its dielectric properties, which, inturn, strongly affect its impedance and capacitance [43].

At most frequencies, surface conduction mechanism due topresence of adsorbed water molecules seems primarily responsi-ble for determining the electrical response of PAA. For physisorbedwater, conduction occurs as a result of proton hopping betweenadjacent molecules at low frequencies. These protons are mainlysupplied by water molecules, Al(OH)3 formed on PAA surface andother impurity anions [35].

Change in conduction mode from hopping to electron tunnel-ing in solids exhibiting ionic conduction is explained by Funke [44]based on jump relaxation model (JRM) by introducing the conceptof successful and unsuccessful hops. According to this model, thereis probability that a jumping charge carrier may jump back(termed as unsuccessful hop). Hopping process is successful onlyif neighborhood becomes relaxed as a result of hopping. Theprobability of successful hops is very high in low frequency region.However, unsuccessful hops tend to increase with increasingfrequency until a stage is reached where hopping mechanism isno more supported by the applied frequency, thus replaced bytunneling mechanism. Thus, conduction mechanism that predo-minantly determines the change in conductivity with frequencymay shift from proton hopping (through physisorbed water) toelectron tunneling (mediated by chemisorbed water) with increas-ing frequency. We suggest that both the mechanisms operate inseries (based on successful fitting of the experimental data). Inother words, heterogeneities in the pore surface possibly exist, forinstance, related with the formation of islands of physisorbedwater separated by the regions with only chemisorbed water. Theformer seems to predominantly determine the frequency depen-dent changes in conductivity at low frequencies while the laterpredominately relates the changes in conductivity at relativelyhigher frequencies before the onset of a predominant intrinsicresponse of the bulk alumina.

Arrhenius plots of R−1L and R−1

H calculated from ZView fittingusing proposed equivalent circuit are shown in Fig. 11. Linear plotwith positive slope is obtained for each data set. Similar behaviorwas also observed by Khanna and Nahar [35] for porous aluminaat temperatures above 45 1C.

M. Tahir et al. / Physica B 425 (2013) 48–57 57

When nano-porous anodic alumina is placed in ambient,equilibrium should be established between the water adsorbedon the surface and vapor pressure of water in ambient. For thereaction:

H2O ðads:Þ⇆H2O ðvap:Þ ð15ÞFree energy change is given by:

ΔG¼ΔGo þ kBTlnðpH2Oðvap:Þ=aH2Oðads:ÞÞ ð16Þwhere ΔG is free energy change and ΔG1 is standard free energychange for the reaction, pH2Oðvap:Þ is partial pressure of water vaporin ambient and aH2Oðads:Þ is the activity of adsorbed water on oxidesurface. For ΔG¼0, Eq. (16) can be written as:

lnðpH2O vap:ð Þ=aðH2O ads:ð Þeq:ÞÞ ¼−ΔGo=kBT ¼−ðΔHo−TΔSoÞ=kBT¼−ΔHo=kBT þ ΔSo=kB ð17Þ

or

lnaH2O ads:ð Þeq: ¼ lnpH2O vap:ð Þ þ ΔHo=kBT−ΔSo=kB ¼ c1þΔHo=kBT ð18Þ

where c1(¼ lnpH2O vap:ð Þ−ΔSo=kB) is a constant, ΔHo is standard

enthalpy change for desorption and ΔSo is standard entropy change.Assuming that the mobility of the charge carriers is not significantlyaffected in the measured temperature range, conductivity could beexpressed in terms of activity of adsorbed water [35]:

s¼ c2aH2Oðads:Þ ð19Þwhere c2 is a constant correlating the two quantities. Above relationis justified for the regime where the charge carrier concentration atthe surface regions can be linearly related with the activity ofadsorbed water. Under equilibrium condition, Eqs. (18) and (19) canbe combined as:

ln s¼ lnso þ ΔHo=kBT ð20Þwhere lnso¼ lnc2+c1. Eq. (20) is in clear agreement with the observedchanges in conductivity with temperature, as shown in Fig. 11. Thepositive slope of the curves may thus partly be related to ΔHo=kB forreaction (15). It may be noted that the slopes for R−1

L and R−1H vs. 1/T

are not exactly same, which may suggest that the states of adsorbedwater primarily contributing to hopping and tunneling mechanismsare different, resulting in two different values of slopes (ΔHo=kBvalues) for these mechanisms. It has been mentioned earlier thathopping is primarily governed by physisorbed water, while tunnelingis mainly associated with chemisorbed water. The estimated value forthe enthalpy of chemisorption of water on the pore surface comesout to be −0.53 eV and that for physisorption is about −0.37 eV,almost 2/3rd of the former. This aspect, although being very inter-esting, needs further clarification, however.

4. Conclusions

(1)

Complex impedance spectroscopic properties of through-thickness anodic alumina change significantly with measure-ment temperature as depicted from complex impedance planeand Bode plots.

(2)

Impedance increases while capacitance decreases at lowfrequencies by about one order of magnitude, with increasein temperature from 29–701C. These changes are attributedto varying degree of water adsorption at pore surface withtemperature.

(3)

The electrical behavior of nano-porous anodic alumina issuccessfully modeled through an equivalent electrical circuit.A comprehensive overview of electrical makeup of PAA is

obtained by analyzing temperature dependence of differentcircuit elements, thus discriminating various charge transportmechanisms.

(4)

The surface conduction with adsorbed water is explained byusing correlated barrier hopping (CBH) and quantum mechan-ical tunneling (QMT) models at low and intermediate frequen-cies, respectively. Inherent capacitive behavior of bulk aluminais dominant at high frequencies, with Debye dielectric relaxa-tion (at single time constant).

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