Physicochemical characterization of passive films on niobium by admittance and electrochemical...

Electrochimica Acta 50 (2005) 5090–5102

Physicochemical characterization of passive filmson niobium by admittance and electrochemical

impedance spectroscopy studies

F. Di Quarto∗, F. La Mantia, M. SantamariaDipartimento di Ingegneria Chimica, Universit`a di Palermo, Viale delle Scienze, 90128 Palermo, Italy

Received 24 November 2004; received in revised form 10 March 2005; accepted 10 March 2005Available online 28 June 2005

Abstract

An analysis of the electronic properties of amorphous semiconductor–electrolyte junction is reported for thin (Dox < 20 nm) passive filmgrown on Nb in acidic electrolyte. It will be shown that the theory of amorphous semiconductor–electrolyte junction (a-SC/El) both in the lowband-bending and high band-bending regime is able to explain the admittance data of a-Nb2O5/El interface in a large range (10 Hz–10 kHz)o

the theoryo ent. Somep mechanismo©

K

1

oagpc

tutepct

rable

anyrmed

omeer-eousandakessferencytrongthe-

ng ifhous

risesfit-

0d

f frequency and electrode potential values.A modelling of experimental EIS data at different potentials and in the frequency range of 0.1 Hz–100 kHz is presented based on

f amorphous semiconductor and compared with the results of the fitting of the admittance data obtained in a different experimreliminary insights on the possible dependence of the density of state (DOS) distribution on the mobile defects concentration andf growth of anodic film on valve metals are suggested.2005 Elsevier Ltd. All rights reserved.

eywords:Physicochemical characterization; Semiconductor–electrolyte junction; Electrochemical impedance spectroscopy

. Introduction

It has been recently suggested that the electronic propertiesf passive films could be one of the key factors in controllinglso the pitting behaviour of metals and alloys. Such a sug-estion is in agreement with the importance of the electronicroperties of electrode surface in determining the kinetics ofharge transfer at the electrode–electrolyte interfaces[1].

Impedance spectroscopy (EIS) and differential capaci-ance study (Mott–Schottky analysis, M–S) are very pop-lar techniques for the characterization of semiconduc-

or/electrolyte interface (SC/El) as well as in studying thelectronic properties of semiconducting passive films. Theseroperties are frequently investigated by measuring theapacitance of the junction as a function of electrode poten-ial and frequency of the superimposed ac signal. Although

∗ Corresponding author. Fax: +39 0916567280.E-mail address:[email protected] (F. Di Quarto).

an investigation in a large range of frequencies is prefefor getting reliable information on the flat band potential,Vfb,and donor or acceptor concentration of passive films, in mcases the study of electronic properties has been perfoby using the M–S analysis at only one frequency.

When such an approach is followed it happens that smeaningful physical information on the electronic propties are lost and more importantly in some cases erronestimate of characteristic levels of the junction (flat bpotential, energy band location, etc.) are obtained. This mmore difficult to understand the kinetics of charge tranat the interface. We stressed years ago that the frequdependence of M–S plots should be considered as a sindication that the simple M–S analysis based on theory of perfect crystalline SC is, in many cases, misleadiused to get information on electronic properties of amorpsemiconducting passive films.

In the case of EIS data analysis, a first challenge ain the choice of the equivalent circuit to be employed in

013-4686/$ – see front matter © 2005 Elsevier Ltd. All rights reserved.
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F. Di Quarto et al. / Electrochimica Acta 50 (2005) 5090–5102 5091

ting experimental data as well as on the physical meaningto attribute to the different passive elements of the electricalequivalent circuit. A further complication in the EIS analysisof amorphous passive films comes out by the fact that usu-ally frequency independent space charge capacitance,CSC,are employed in the equivalent circuit by tacitly assuming thatany frequency dependence derives from more exotic passiveelements, whose physical meaning is sometimes obscure.

In previous papers[2–5], we proposed that a frequencydependence inCSC or, more generally, in the admittance ver-sus electrode potential curves could be a strong indication thata distribution of electronic states within the gap of material ispresent, and that such an occurrence is theoretically foreseenwithin the frame of a-SC theory. According to this we haveshown that some of discrepancies usually encountered in theuse of traditional M–S analysis of passive films/electrolyteinterface could be overcome by a correct analysis of admit-tance data based on the theory of a-SC Schottky barrier[2–5].In these papers a low band bending approach was used aimedto show the ability of the new approach in locating correctlythe energy levels of the film/electrolyte junction and partic-ularly in getting out an almost frequency-independent flatband potential value at different frequencies. In some of theprevious works, it was shown that a density of states (DOS)distribution not constant in energy was present within the filmand that such an important parameter could be also thicknessd

tialc rr eoryo ruc-t laint al-l thatt er thet lud-i

process observed during the permanence of a-WO3 passivefilm into the aqueous solution hampered to get a more quanti-tative information on the existence of spatial as well as energydependence in the DOS distribution. At this aim we decidedto come back to study the electronic properties of thin (20 nm)passive films on niobium owing to the fact that it could behaveas a model system if we consider that:

(a) anodic film on Nb can be grown easily in a large range ofelectrode potential (i.e. thickness) and solution pH[8];

(b) anodic Nb2O5 films are amorphous in all range of thick-ness provided that anodization process is performed inabsence of breakdown[9,10];

(c) the stability of a-Nb2O5 thin film is very good in acidicas well as in strong alkaline solutions[11].

Thus, a detailed analysis of EIS and admittance behaviourof amorphous semiconducting Nb2O5 film was performedbased on the theory of a-SC Schottky barrier aimed to getinformation on DOS distribution both in energy and in space.

It will be shown that such a complex analysis, carried outboth in a large domain of frequency and in a large range ofpotential, could provide more insight on DOS distribution andelectronic properties of passive niobium helpful to a deepercomprehension of the mechanisms of growth and breakdownof passive films.

2s

theb se ofc m-p . InF rgyf theg ciallyi and

F r amo uctor; (b)C hed are ceb

ependent[1–5].More recently in order to account for the differen

apacitance behaviour of anodic a-WO3 films in a largeange of electrode potential, it was suggested that the thf inverse metal–insulator–semiconductor (M–I–S) st

ure, valid for crystalline material, could be able to exphe observed behaviour[6]. Such an interpretation was chenged by the present authors who were able to showhe observed behaviour of a-WO3 passive film in a very largange of electrode potential values was compatible withheory of amorphous semiconductor Schottky barrier incng the high band-bending regime (HBB)[7]. The dissolution

ig. 1. Density of electronic states as a function of energy foohen–Fritzsche–Ovishinsky model; (c) Mott and Davis model. Hatcand mobility edges, respectively.

. Theoretical background on amorphousemiconductor Schottky barrier

In order to understand the main differences inehaviour of a-SC Schottky barrier with respect to the carystalline semiconductor (c-SC) it may be helpful to coare preliminarily the DOS distribution in both materialsig. 1, we report the model of DOS distribution versus ene

or generic amorphous and crystalline material. Althougheneral features of DOS distribution are preserved, spe

f we compareFig. 1a with c, some differences are evident

rphous and crystalline semiconductor: (a) crystalline semicondas represent the localized gap states,EC andEV are the conduction and valen

5092 F. Di Quarto et al. / Electrochimica Acta 50 (2005) 5090–5102

Fig. 2. (a) Schematic DOS distribution in a-SC and (b) energetics at ana-SC/El junction. The dashed area evidences localized electronic states.

they strongly affect the response of a-SC/El junction underac electrical as well under light stimulus[12].

With respect to the impedance measurements the exis-tence of deep tails of electronic states within the gap has anoticeable influence on the shape of space charge barrier aswell as on the frequency response of such barrier[13]. Thisis better evidenced inFig. 2where the energetics of n-type a-SC/El interface is reported, under the simplifying hypothesisof constant DOS and spatially homogeneous material.

The main difference with respect to the case of crystallineSC is that the net space charge depends on both the ionizedimpurities and the localized states within the mobility gap.The electronic states lying into the gap (seeFig. 2) do notfollow instantaneously the imposed ac signal, but they needa finite response time. This response time depends on theirenergy position with respect to the Fermi level and it can bemuch longer than the period of the ac signal having angularfrequencyω. In fact an exponential relaxation time,τ, for thecapture/emission of electrons from electronic statesE belowEF is assumed to hold according to the relationship:

τ = τ0 exp

(EC − E

kT

)(1)

where at constant temperature,τ0 is a constant characteristicof each material usually ranging between 10−14 and 10−10 s.T gingf too.O g theb lyingb acs hel eps .

con-da ingf from

the condition:ωτ = 1, as:

EC − Eω = −kT ln(ωτ0) (2)

According toFig. 2the intersection ofEω with the Fermi levelof material determines a characteristic point in the barrier,xC,at which corresponds a band bendingψC given by:

|e|ψC = |e|ψ(xC) = −kT ln(ωτ0) − EF (3)

whereEF = (EC −EF)bulk.xC is now a distance in the barrierwhich changes with changing frequency,ω, and band bendingψS. In particular,xC increases with increasing frequency, atconstant polarization, or with increasing polarization at con-stant frequency. From the theory it comes out that the totalcapacitance is sum of two series contribution coming fromthex<xC andx>xC regions of the a-SC. The contribution tothe conductance comes mainly from the region aroundx=xCdividing the total response from null response regions. In thehypothesis of a constant DOS the total capacitance is givenby the sum of the two contributions:

1

C(ψS, ω)= 1

C(ψC,0)+ xC

εε0(4)

whereC(ψC,0) =√εε0e2N andxC =

√εε0e2N

ln ψSψC

. After

substitution the following relationship is obtained for the totalc

C

w y:

G

T entso thatψ owb .t

ar-a ithint gesi OSv -t ftt

orψ att Refs.[ bem

his means that, at constant band bending, with chanrequency the levels which can follow the signal changen the other hand, at constant ac frequency, by changinand bending the more deep levels of depleted regionelow a critical level will not change the occupancy withignal. According to Eq.(1), by decreasing the energy of t

ocalized state in the gap,τ increases sharply so that detates (for whichωτ � 1) do not respond to the ac signal

By assuming a full response for states satisfying theition ωτ � 1 and a null response for states havingωτ � 1,sharp cutoff energy level,Eω, separating states respond

rom those not responding to the signal, can be defined

apacitance:

LBB(ψS, ω) =√εε0e2N

(1 + ln

ψS

ψC

)−1

(5)

hilst the parallel conductance of the junction is given b

LBB(ψS, ω) =(ωπkT

2|e|ψC

)√εε0e2N

(1+ ln

ψS

ψC

)−2

(6)

he analytical solutions for the admittance componf the junction have been derived under conditionsS >ψC ≥ 3kT/e and at not too high band bending (land-bending regime,eψS < (Eg/2 –EF)) ([1–5] and Refs

herein).It has been shown thatG(ψS,ω) has a spectroscopic ch

cter with respect to the distribution of electronic states whe gap, whilst variations in DOS cause only minor chann theC(ψS, ω) versus potential plots provided that the Daries little over an energy range ofkT. In previous equaions ε0 is the vacuum permittivity,e the absolute value ohe electronic charge,ε the film dielectric constant andψS ishe potential drop inside the a-SC as above.

In high band-bending region, i.e. fS >Vg = (Eg/2−EF)/e, when a deep depletion region

he surface of a-SC/El junctions appears, according to14–16] (seeFig. 3) the total capacitance of barrier canodelled as a two series capacitance:

1

C(ψS, ω)= 1

C(ψg, ω)+ xg

εε0(7)


Fig. 3. Energy band diagram for an a-SC Schottky barrier under depletionconditions in the high band-bending region.

where the first term of Eq.(7) represents the low band bend-ing contribution calculated at the electrode potential wherethe Fermi level crosses the mid gap energy and coincides withEq.(4)after notation substitution. The second term representsthe capacitance of the deep depletion region going from thesurface to the pointxg of the junction (seeFig. 3), where aparabolic potential distribution exists, and is frequency inde-pendent.

In the hypothesis of constant DOS an analytical expressionhas been derived forxg by Da Fonseca et al.[17], whichallows to get the following analytical expression for the totalcapacitance:

1

CHBB(ω,ψS)= 1√

εε0e2N

(ln

ψg

ψC+√

1+ 2

ψg(ψS − ψg)

)

(8)

It comes out from Eq.(8) that at high band bending the capac-itance changes as the square root of the band bending as soonas the term 2ψS/ψg becomes much higher than 1. If thisoccurs a square root dependence from electrode potential canappear in the capacitance behaviour which could account forthe M–S-like behaviour observed, sometimes, for a-SC/Eljunctions at high band bending.

Under the hypothesis of constant DOS,N(E) =N, it isa n ofGe

G

E ndi-t itht

Fig. 4. Equivalent circuit by assuming an ideally polarizable interface andin absence of surface states contribution.

3. Choice of the equivalent circuit of a-SC/Elinterface

By assuming an ideally polarizable interface and inabsence of surface states contribution to the measuredimpedance, it has been shown[2–5] for a series of amorphoussemiconducting passive films (a-WO3, a-Nb2O5, a-TiO2) thatthe equivalent circuit reported inFig. 4was able to provide afitting of the admittance behaviour in a large range of thick-ness and ac frequency. Moreover, such studies performed inthe low band-bending region suggested that the DOS distri-bution was slightly changing in energy and decreasing withincreasing film thickness[2–5].

More recently, in an attempt to fit the admittance curvesof a-WO3 anodic films, we became aware[1,7] that thesimple circuit ofFig. 4 was unable to explain the overallimpedance behaviour of a-SC/El interface in a very largerange of electrode potential (including the deep depletionregion) and investigated frequencies (0.1 Hz–100 kHz). It isworth to mention that in some cases with a-WO3, at not toohigh electrode potential and after long term stabilisation, agood fitting of EIS spectra in the 0.1 Hz–100 kHz range wasobtained by substituting a CPE element (withn= 0.98) to theadmittance of a-SC barrier.

In order to reduce the arbitrariness of the choice of theequivalent circuit and to account for possible charge trans-fp tf e inp ationo essesc pec-t ntiala wider ge,U ngt

i-

lso straightforward to derive from the general expressio(ψS, ω) (seeAppendix Aand Refs.[15,16]) the followingquation for the conductance of the barrier[1]:

HBB(ω,ψS) = π2fkT

|e|ψC

√εε0e2N

×(

lnψg

ψC+√

1 + 2

ψg(ψS − ψg)

)−2

(9)

qs. (8) and (9) have been derived under the same coions valid for the LBB expression and they coincide whe previous ones forψS≤ψg.

er processes, we have chosen the equivalent circuit ofFig. 5,roposed by Gomes and Vanmaekelbergh[18,19], to accoun

or the impedance behaviour of crystalline SC/El interfacresence of electron charge transfer and/or recombinf electrons and holes trough surface states. Both procould occur in the different potential regions where EIS sra were obtained and by considering that the differedmittance measurements were performed in a veryegion of electrode potential, from the formation voltaF, close to the estimatedVfb potential, so encompassi

he HBB and LBB regime.A modification to the original circuit was made by:

(a) adding the conductanceGSC(ω, ψS) of a-SC barrier inparallel withCSC(ω, ψS) according to the theory prevously reported;


Fig. 5. Equivalent circuit for a-SC/El interface in presence of electron charge transfer and/or recombination of electrons and holes trough surfacestates.

Fig. 6. Equivalent circuit employed in fitting differential admittance curves.

(b) substituting to the originalR1 resistance[19], in the armof the circuit accounting for the charge transfer via local-ized states at the semiconductor surface, a CPE elementwhich could account for the presence of a continuous dis-tribution in energy of electronic states also at the surfaceof a-SC.

The equivalent circuit used for the analysis and fitting (seebelow) of the admittance plots is reported inFig. 6, and it wasderived from the previous one by means of series to paralleltransformation of the arm in parallel to a-SC Schottky-barrier.The fitting procedure used to get the different elements ofequivalent circuit will be described in a following section.

4. Experimental

Niobium (purity 99.9) rod from Goodfellow Metals (Cam-bridge) was used for the experiments. It was sealed into ateflon cylinder with an epoxy resin leaving a flat circular sur-

face (0.5 cm diameter) in contact with the electrolyte. Beforeeach experiment the metal surface was treated mechanicallywith abrasive paper followed by polishing with alumina pow-der (0.5�m), then electropolished in HF/H2SO4 solution(V= 15 V for 45 min at 1500 rpm) and final cleaning in soni-cated distilled water. The metal was then anodized at constantgrowth rate (10 mV s−1) in 0.5 M H2SO4 (UF = 5 V(SCE))using a potentiostat (EG&G PAR 173) equipped with a signalgenerator (EG&G PAR 175), and kept for 2 h to the formationvoltage (stabilisation).

The capacitance of the junction during the growth of pas-sive films was acquired by using a lock in amplifier (EG&G7260) technique by superimposing an ac signal of 10 mV rmsat 1 kHz.

The impedance spectra and differential admittance curveswere recorded by using a Parstat 2263 (PAR), connected toa computer for the data acquisition. The data were then pro-cessed according to the theory of a-SC and as reported inAppendix A.

For all the experiments a Pt net having a very high surfacearea was used as counter electrode.

5. Results and discussion

5

pi lot,i fa :

.1. Film formation

In Fig. 7we report theC−1 versus formation voltage,UF,lot for anodic film on Nb grown at constant rate (10 mV s−1)

n 0.5 M sulphuric acid solution. From the slope of such a pt is possible to derive the anodizing ratioA, i.e. the inverse onodizing electric fieldA= 1/Fel, according to the relation

d(

1C

)dUE

= A

εε0(10)


Fig. 7. C−1 vs. formation voltage,UF, for an anodic film grown on Nb atconstant rate (10 mV s−1) in 0.5 M sulphuric acid solution.

whereε is the dielectric constant of a-Nb2O5, assumed equalto 42 [3,20] andε0 is the vacuum permittivity. An anodiz-ing ratio of 26.8A V−1 was obtained in agreement with datareported in Refs.[3,9–11,21]. On the other hand a slightlylower anodizing ratio (A= 26.5A V−1) was estimated byassuming an unitary efficiency of film formation and the rela-tionship valid for growth under constant electric field:

Fel = 1

A= zFρv

ηiM(11)

wherev is voltage scan rate,ρ = 4.7 g cm−3 the film density[11],M the molecular weight and the other symbols have theirusual meaning. The difference in the value of anodizing ratioobtained with the two methods are within the 1% and it couldbe attributed to the presence of a very small electronic currentwhich reduces the film formation efficiency,η, to 99%.

It is worth to note that an anodizing ratio (A= 26.7A V−1)almost coincident with the previous one was measured underotherwise identical conditions by using theC−1 versusUFslope and identical dielectric constant for film grown in 0.1 MNaOH solution. A lower efficiency of film formation (96%)was derived by using Eq.(10) in this last solution so that itseems reasonable to attribute such differences in film forma-tion efficiency to differences in kinetics of oxygen evolutionreaction in the two investigated electrolytes.

t in them ationp atea da essw dur-i w inti ss( ee-m tantp

5.2. Electrochemical impedance spectroscopy (EIS)data analysis

After stabilisation the anodic film were investigated in0.5 M H2SO4 electrolyte by means of EIS from the opencircuit potentialUOC up to 5 V(SCE) in step of 1 V above 1 Vand at 0.5 V(SCE). For a better reliability in the calculation ofDOS distribution a further EIS measurement was performedat the fixed potential of 0.18 V(SCE). This was the lowestvalue of electrode potential at which the constraintψS >ψC(see Section2) still holds at the lowest frequency (0.1 Hz)employed in EIS analysis (seeAppendix A).

In separate experiments the curves of differential capaci-tance and conductance of a-SC/El junction as a function ofUE at different frequencies (10 Hz–10 kHz) were obtained ina range of electrode potential starting from 5 V(SCE) downto −0.25 V(SCE).

In Fig. 8we report the impedance spectra, in the Bode rep-resentation, at different potential of a-Nb2O5/H2SO4 junctionfor a film grown in the same electrolyte according to the pro-cedure described above. Regardless of the electrode potentialthe Bode diagram displays a similar behaviour with a char-acteristics inflection point in the module of the impedance

Fig. 8. Impedance spectra in the Bode representation at different potentialsof a-Nb2O5/H2SO4 junction for a film grown in the same electrolyte to5 V(SCE) at 10 mV s−1 after stabilisation.

After reaching the chosen final voltage,UF = 5 V(SCE),he potential remained constant and a rapid decreaseeasured current was observed. During such stabiliseriod (∼=2 h) the film continued to grow at much lower rnd a very low current value (0.3�A cm−2) was measuret the end of stabilisation. An estimate of the final thicknas obtained by measuring the total charge circulated

ng the stabilisation period and by using the Faraday lahe hypothesis of negligible electronic current (η≥ 99%) andn absence of film dissolution[11]. The estimated thickneDox = 193A) at the end of this procedure is quite in agrent with data reported in literature for film grown at consotential[9–11,21].


Fig. 9. DOS distribution vs. distance in energy (or frequency) fromEF,obtained by fitting the impedance spectra at differentUE for the film ofFig. 8.

and a corresponding minimum in the phase versus frequencyplots. We like to stress that the minimum in theφ◦ versuslog f plots shifts toward higher frequency and becomes lesspronounced with increasing electrode potential. This find-ing seems to exclude as source of phase decreasing any filmdissolution hypothesis, which could introduce in the equiv-alent circuit an inductance element. These features in Bodediagram remained unchanged also in presence of differentredox couples (ferro/ferricyanide and Ce4+/Ce3+). In partic-ular, although the phase value at the minimum was a littledepending on the redox couple, its position as a function offrequency was almost independent from the nature of redoxcouple in solution.

In Fig. 9we report the DOS distribution, as a function ofdistance in energy (or frequency) fromEF, obtained by fit-ting the impedance spectra of the sample at variousUE. Moredetails on this procedure can be found inAppendix A. Thedifferent DOS distribution, obtained at each potential, showsthat a minimum in DOS is present at around 1.5 kHz corre-sponding to a distance in energy fromEF of about 0.12 eV.The maximum in DOS appears located close to the lowestlimit of frequency range investigated and precisely at around0.4 Hz, which corresponds to a distance in energy fromEF ofabout 0.33 eV. From such a figure it comes out also that DOSdistribution changes with the potential in an appreciable wayin the range of frequency comprised between 10 and 10 kHz

and in agreement with a variation of DOS along the film. Infact, although the shape of the DOS does not change withchanging potential (seeFig. 9), it is possible to see that thefitting of impedance spectra at more anodic potentials waspossible by assuming increasing values in DOS distributionas clearly evidenced in the same figure. This last aspect willbe better evidenced in the following section where the fittingof differential admittance curves (CSC(ω, ψS) andGSC(ω,ψS) versusUE) will be presented and discussed.

In Table 1we report the parameters used for fitting theimpedance spectra at different potentials corresponding tothe DOS distribution displayed in previousFig. 9. We liketo mention that a very good fitting of impedance spectrawas possible by keeping constant the double layer capaci-tance (Cdl = 18.8�F cm−2) and equal to values expected forthis parameter. An almost constant differential charge trans-fer resistance was obtained in a large range of electrodepotentials (2–5 V(SCE)) where a-Nb2O5/El junction is biasedin deep depletion region. As for the other elements of theequivalent circuit representing the arm in parallel with theadmittance of a-SC, we like to stress that CPE (Ycpe=Q(iω)n)element is almost coincident with a resistor (n= 0.15) inall range of investigated electrode potentials, whilst theCSSelement changes noticeably with changing band-bending byincreasing two order of magnitude on going from deep deple-tion condition down toU .

armo s oft ral-l eam ofG i-td arma alentc

C ther ren-t0 mi-n mit-t

TP rent po

U cm2)

00012345

able 1arameters employed for the fitting of the impedance spectra at diffe

E (V(SCE)) Rs (� cm2) Cdl (×10−5 F cm−2) Rct (×103 �

.185 0.55 1.88 1.03

.38 0.55 1.88 1.86

.5 0.55 1.88 2.100.55 1.88 2.550.55 1.88 2.160.55 1.88 2.170.55 1.88 2.120.55 1.88 2.18

OCIn order to have a better view of the influence of this

f equivalent circuit on the differential admittance curvehe junction, it is convenient to convert this arm to the pael scheme reported inFig. 6. According to this scheme thdmittance of the junction, after correction for the (Zel) ele-ents ofFig. 5, consists of a total conductance sum upSC,P(ω, ψS) andGSS,P(ω, ψS), and a total parallel capac

ance given by the sum ofCSC,P(ω,ψS) plus theCSS,P(ω,ψS),erived from the series to parallel transformation of thend accounting for the role of surface states in the equivircuit.

By looking to the data ofTable 2 containing theSS,P and GSS,P values as a function of frequency in

ange 10 Hz–10 kHz, the same employed in the diffeial admittance studies, but at constant potential,UE = 5 and.185 V(SCE), it is possible to reach the following preliary conclusions useful for the analysis of differential ad

ance data:

tentials corresponding to the DOS distribution displayed inFig. 9

CSS (F cm−2) QSS (×10−5 S cm−2) nSS GSS (S cm−2)

4.75× 10−4 6.25 0.15 3.20× 10−5

1.05× 10−4 5.40 0.15 1.80× 10−5

7.50× 10−5 4.50 0.15 8.00× 10−6

2.95× 10−5 3.85 0.15 3.00× 10−6

1.25× 10−5 3.20 0.15 1.30× 10−6

7.50× 10−6 3.20 0.15 8.00× 10−7

5.50× 10−6 3.20 0.15 5.70× 10−7

4.70× 10−6 3.20 0.15 5.70× 10−7


Table 2GSS,PandCSS,Pvalues(seeFig. 6) derived from EIS as a function of frequency and potential

f (Hz) UE = 0.185 V(SCE) UE = 5 V(SCE)

GSS,P(×10−4 S cm−2) CSS,P(�F cm−2) GSS,P(×10−5 S cm−2) CSS,P(�F cm−2)

10 1.13 0.439 5.10 0.363100 1.60 0.061 8.07 0.035

1000 2.26 0.009 11.5 0.04510000 3.19 0.001 16.3 0.006

(a) The total parallel capacitance of equivalent circuitdepicted inFig. 6 can be assumed as coincident withCSC,P(ω, ψS) in the all range of investigated potentialsand within the limits of validity of the theory of a-SC(ψS >ψC) for f≥ 100 Hz. At this frequency the neglectofCSS,Pintroduces an error in theCSC,P(ω,ψS) value lessthan 2%. Such an error becomes less than 1% at 200 Hzand is completely negligible at still higher frequencies.

(b) The contribution ofGSS,Pto the total conductance is rele-vant at lower frequencies but not negligible neither at thehighest frequency used in our measurements (10 kHz)and specially at the high electrode potentials where thiscontribution reaches its maximum. However, a detailedinspection ofGSS,Pdata, in the all frequency and elec-trode potential range, suggests that a simple and rea-sonably good correction procedure of the differentialadmittance curves for deriving theGSC,Pcurves is to sub-tract to the total parallel conductance theGSS,P valueobtained atUE = 5 V(SCE). We have to mention that,owing to a constancy ofGSS,P in a rather large rangeof potential (2≤UE ≤ 5 V(SCE)), the procedure abovementioned allowed to get reliable value of DOS for eachfrequency as a function of band bending in a large regionof film thickness. More details on these aspects will bediscussed in the following section.

5

h int rt inF db on-d und0 ingr e flatb ofs larges nw ntlyt tos pleM allye oni encya anyo rete

deep level of donors. We have to mention that still more puz-zling behaviour, in the frame of classical M–S analysis, couldbe observed in the capacitance behaviour of thicker films aspreviously reported in Refs.[2,3].

According to this an analysis of the differential admittancedata was performed on the basis of the a-SC Schottky barriertheory and by taking into account the results of the EIS study.

As previously discussed, the simplified equivalent circuitof Fig. 6 could be employed in the analysis of differentialadmittance data if we limit initially the analysis between 100and 10 kHz, where theCSS,P term is negligible (less than2% in the all potential and frequency ranges). Accordingto this the differential admittance data were initially trans-formed in a series equivalent circuit in order to subtractZelto (Zel +ZSC). Thereafter, the residual impedance was trans-formed back to the parallel configuration and analysed byneglectingCSS,P(ω, UE) contribution to the admittance ofthe a-SC Schottky barrier. Moreover, as anticipated in thelast section, although a good fitting of the conductance plotscould be obtained by a simple subtraction of theGSS,P(ω,UF) value measured atUF, for these frequencies we cor-rected the overall curve by subtracting the values ofGSS,Pobtained from EIS analysis and by interpolating them in theregion where a not constant behaviour was observed. Thislast procedure was preferred because a good interpolatingfunction was found for these frequencies and the resultingc factt mallf e for

.3. Differential admittance (DA) study

In order to highlight the need of a different approache analysis of the differential capacitance data we repoig. 10the M–S plots of a-Nb2O5 film previously investigatey EIS analysis. As previously reported for other semicuctor passive films a very short range of linearity (aro.2 V) is displayed by the curves in the low band-bendegion, which could be used to get a rough estimate of thand potential (around−0.24 V(SCE)). A close inspectionuch plots near the flat band potential showed a rathercattering (0.1 V) ofVfb values if the true straight line regioas used in the extrapolation. However, more importa

han the disturbingly limited straight-line region, we liketress that such a behaviour is not compatible with sim–S analysis of crystalline SC/electrolyte junction usumployed to derive both theVfb and the donor concentrati

n passive films. Moreover, the large frequency dependnd the shape of M–S plots are not easily amenable tof possible behaviour of crystalline SC containing disc

onductance curve was still well behaving owing to thehat the correction terms were at any potential only a sraction of the experimental values. This was not possibl

Fig. 10. C−2 vs.UE plots at different frequencies for the film ofFig. 8.


Fig. 11. Fitting of the experimental admittance curves of a-Nb2O5

(f= 100 Hz; scan rate = 10 mV s−1, sol: 0.5 M H2SO4). (a) GP,SC vs.UE: eψC = 0.20 eV andVfb =−0.245 V(SCE); (b) (CP,SC)−1 vs. UE:eψC = 0.20 eV andVfb =−0.245 V(SCE).

the curve at 10 Hz, because at this frequency theGSS,Pvaluesto be subtracted (determined by EIS in separate experiments)were a larger fraction of the measuredGSC(ω,ψS) and smallerror inGSS,Paffected in a rather unpredictable way the lowband-bending region of measured conductance values in thedifferential admittance experiments. This is not surprisingbecause in the low band-bending region large contributionfrom CSS and CPE elements toGSS,P are expected theo-retically and at low frequencies such contributions becomequantitatively larger than the value ofGSC to be measured.For these reasons in a first quantitative analysis of differen-tial admittance data in a large range of electrode potential wepreferred to analyse the higher frequency region (f≥ 100 Hz).The values ofGSS,P(ω,UE) to be subtracted are those derivedfrom EIS data reported inTable 1 after series to paralleltransformation.

In Figs. 11 and 12we report the differential admittancecurves of the a-Nb2O5/El junction, at different frequenciesas a function of the electrode potential obtained, for the filmpreviously studied by EIS. In the same figures we report thefitting curves as well as the parameters used in the fittingprocedure according to the equations of amorphous Schottkybarrier previously reported (see alsoAppendix A).

Fig. 12. Fitting of the experimental admittance curves of a-Nb2O5

(f= 10 kHz; scan rate = 10 mV s−1, sol: 0.5 M H2SO4). (a) GP,SC vs.UE: eψC = 0.08 eV andVfb =−0.228 V(SCE); (b) (CP,SC)−1 vs. UE:eψC = 0.08 eV andVfb =−0.225 V(SCE).

The fitting of admittance data of a-Nb2O5 thin films, byusing a constant DOS in Eqs.(8) and(9), could be performedin a rather limited potential range (about 0.5 V). This wasat variance with the case of a-WO3 [7], when the use of analmost constant DOS in Eqs.(8)and(9)was able to fit reason-ably well the differential capacitance plot in a 10 V range ofelectrode potential values in high frequency range. We have tomention that differential capacitance curves for thicker filmsof a-Nb2O5 were fitted in the low band bending (up to about2 V) regime by using a constant DOS[2,3]. A possible ratio-nale for such a behaviour will be presented in the followingafter discussing the behaviour of DOS as a function of filmthickness.

In Fig. 13, we report the DOS distribution, derived fromfitting the capacitance data for the three employed frequencies(100 Hz–10 kHz), as a function of the equivalent space chargeregion of a-SC calculated according to the equation:

xSC = εε0

CSC(ψS, ω)(12)

As previously mentioned, the DOS distribution derived fromthe capacitance measurements is a spatially averaged value,whilst a more spectroscopic character is associated to the


Fig. 13. Spatial dependence of DOS derived from the fitting of the differen-tial capacitance curves.

DOS distribution derived from the differential conductance[14–16]. According to this we derived the DOS distribution(seeFig. 14) at xC by using the conductance plots and Eq.(A.5) in Appendix A. xC was calculated as:

xC = xSC − εε0

CSC(ψC,0)(13)

From the data fitting it comes out that for thesefrequencies an almost constant flat band potential(Vfb =−0.24± 0.015 V(SCE)) was measured from bothconductance and capacitance plots. We anticipate that sucha flat band potential was also valid for fitting the differentialadmittance data derived at 10 Hz. This means that for a largerange of frequency encompassing three decades we havebeen able to get an almost constant flat band potential fromadmittance curves of a-Nb2O5. We like to mention that sucha value ofVfb was also in very good agreement with thatobtained from the fitting of photocurrent versus potentialcurves measured for the same junction (Fig. 15a and b).

The fitting of admittance curves at 10 Hz was finally per-formed by using the more general equivalent parallel circuitcontaining also theCSS,Pelement (Fig. 6). In this case thesimplification of neglecting the change inCSS,P andGSS,P

F cec

Fig. 15. Fitting of the experimental admittance curves of a-Nb2O5 (f= 10 Hz;scan rate = 10 mV s−1, sol: 0.5 M H2SO4). (a)GP,SCvs.UE: eψC = 0.26 eVand Vfb =−0.240 V(SCE); (b) (CP,SC)−1 vs. UE: eψC = 0.26 eV andVfb =−0.230 V(SCE).

as a function of potential was introduced. According to this,constant values ofCSS,P andGSS,P, derived from EIS dataat UE = 5 V(SCE), were subtracted to the parallel conduc-tance and capacitance. Such a procedure affects negligiblytheCSC,P values in the all range of potential but still leavesome appreciable errors in theGSC,Pvalues at low potentials(see above). This fact could affect in some way theVfb valuesderived from fittingGSC,Pcurves as well as the value of DOS,deeper in energy, derived at lowxC values (i.e. close to thefilm surface). As forVfb, however, the changes should be inthe order ofkT/e, thus remaining within the usual uncertaintywith which such a parameter is measured. As for the DOS dis-tribution, although its dependence fromxC does not changeappreciably, the values estimated at variousxC could changesensibly depending on the entity of correction performed onGSC at lower potential.

However, our estimate poses such correction in the DOSvalue around 50% atxC = 0. Nevertheless, we have to mentionthat the value of DOS derived fromGSC,Pby assumingGSS,Pconstant and reported inFig. 16agreed very nicely with thoseestimated from EIS and reported in the same figure.

In spite of such a simplification, if we take into accountthat a value ofVfb =−0.245± 0.005 V(SCE) was derived invery good agreement with those previously derived at higher

ig. 14. DOS atxC derived from the fitting of the differential conductanurves.


Fig. 16. Comparison of DOS distribution vs. energy obtained from EIS anddifferential conductance curves atUE = 5 V(SCE) and 10 Hz≤ f≤ 10 kHz.

frequencies, we can reasonably conclude that such a simpli-fication should not heavily affect also the DOS estimate.

By looking to the results of DOS distribution reported inFig. 14, it is possible to explain why Eqs.(8)and(9)obtainedunder the hypothesis of constant DOS (N(E, x) =N) failed infitting the admittance curves in a reasonably larger potentialrange. In fact the behaviour ofN(E, xC) versusxC at any fre-quency (i.e. energy) displays large changes in values on goingfromxC = 0 toxC =Dox (Ncorresponding to this last value wasderived by a polynomial best fitting of the experimental datausually covering between 50% and 70% of total film thick-ness at different thickness) with an almost spatially constantDOS usually limited in a range ofxC/Dox ≤ 0.2. In terms ofband bending this region of constant DOS corresponds topotential drops ranging between 0.2 and 0.4 V fromVfb.

A further confirm to this interpretation comes from a sim-ilar analysis performed on passive a-WO3 film when a fittingof CSC plots in a range of potential of about 9 V was possibleaccording to Eqs.(8) and(9) by using a constant DOS[7].A similar interpretation could account also for the behaviourof thicker films of a-Nb2O5 investigated in the low band-bending regime[2,3] owing to the fact that in that case a smallportion of the film beneath the surface was exploited by theadmittance study. Further measurements are now in progressas a function of different anodizing parameters in order tounderstand such a behaviour as well as how the anodizing pro-c .

iouso perih -o velsd undf ncei stedt tionf lms

in presence of mobile space charge, could be related to theexistence of different types of mobile defects into a growingfilm [7,22,23].

6. Conclusions

It has been shown by a detailed analysis of EIS spectraand differential admittance curves that the theory of amor-phous Schottky barrier is able to explain the overall electricalbehaviour of a-Nb2O5/El interface in the range of electrodepotential fromUF toVfb and ac frequency. By a careful studyin a large span of frequency and voltage, a constantVfb valuefor relatively thin (Dox ≤ 20 nm) anodic amorphous passivefilm has been determined. Such an accomplishment has beenreached by introducing a spatially variable DOS both in mod-elling the EIS spectra as well as in fitting the differentialadmittance plots as a function of frequency and electrodepotential. This extension of theory of a-SC Schottky barrierallowed for getting out a distribution of DOS which mimicsthe distribution of mobile defects during the growth of oxidefilm in presence of a space charge[23].

The results of this study are a further confirm that thesimple Mott–Schottky analysis in presence of amorphous orstrongly disordered films can provide, under special condi-tions, only qualitative information on the location of charac-t vefi veryp orec d ona

dono newi iouro andb

s ino uslys

A

A

stedi -t d toe

N

ess influences the DOS distribution in a-Nb2O5 anodic filmsWe like to comment on a last point related to the prev

ne. Although the variation of DOS for states lying deen energy (10–100 Hz curves ofFig. 14) is similar to thoseigher in energy (1–10 kHz curves ofFig. 14) when extraplated toxC =Dox the number of states at these energy leiffers of about one order of magnitude (as already fo

rom EIS data analysis). The origin of such a differen value is yet not completely clear. We recently suggehat such a DOS, mimicking the mobile defects distribuoreseen theoretically by model of growth of anodic fi

eristic energy levels (Vfb,EC,EV) of semiconducting passilm/El junction. In these last cases we suggest that suchopular approach should be dismissed in favour of a momplex, but physically better grounded, approach base-SC Schottky barrier theory.

It is also in the scope of this work to show that an abanf simple M–S model of analysis of data could also offers

nsights for understanding the complex electrical behavf passive films as well as their mechanisms of growthreakdown.

Further studies are now in progress on different systemrder to test the validity of model and hypothesis previouggested.

ppendix A

.1. EIS data fitting procedure

In order to fit the EIS data we used the method suggen Ref.[16] to get the DOS distribution,N(E), at each potenial. According to this the following expression was usestimate DOS atxC:

(EF − eψC(ω))

= 2

πkT

G(ω)/ω

C(ω)[1 − xC

εε0C(ω)

] ∫ EF

EF−eψC

N(E) dE

= G(ω)

ωK(ω) (A.1)


where theC(ω) andG(ω) values are those measured at eachpotential, and a trial estimate for DOSN(E) =Nwas initiallyassumed.Eqs.(3) and(A.2):

xC = x(ψC) =∫ ψS

ψC

dψ[2εε0

∫ ψ0

(∫ EFEF−eψ′ N(E) dE

)dψ′] 1

2

(A.2)

were used to obtainψC from ω andxC as a function ofω,ψS andN(E). An iterative technique was used to find thevalue ofN(E) and the iteration was stopped once a differencebetween the (n− 1) andn estimate ofN(E) less than 1% wasreached. The new value of DOS obtained after each iterationwas used to calculate theCSC(ω) andGSC(ω) values to beused in the general equivalent circuit ofFig. 5where constantCH values (around 20�F cm−2) were accepted, whilst allothers elements were derived from the best fitting approach.The procedure was repeated until the error between simulatedEIS spectra and measured EIS spectra was less than 5% in the|Z| versus log(f) plot and±2◦ in theφ(Z) versus log(f) plot.The optimisation ofZel elements (seeFig. 6) was carried outby using the Z-SimpWin software package after subtractingto the equivalent circuit theZSCelements obtained after fittingof EIS data.

The range of frequency investigated (0.1 Hz–100 kHz)a ngeo

E

m-pw rgyr wasUi erf mea-s tedb rvedm ere

rgies( butn s nota ounto of thee

DOSa db so

A

f g

Fig. 17. DOS distribution used for fitting EIS data. Dashed area representsthe experimentally probed region; (- - -): qualitative expected behaviour; (-· - · - · -): hypothetical DOS at which an error of 5% was measured in|Z| vs.log f plot (seeAppendix A).

Eqs.(8) and(9) modified as:

1

C(x, ω,ψS)= 1

C(ω,ψS)fω(x(ψS)) (A.3)

G(x, ω,ψS) = G(ω,ψS)gω(x(ψS)) (A.4)

where fω(x(ψS)) andgω(x(ψS)) are, respectively, two dif-ferent trial functions depending only on electrode potentialbut changing with employed frequency. The term multiply-ing the two trial functions can be considered as coincidentwith the expression of (CSC)−1 andGSC in absence of spatialvariation in DOS (homogeneous film) but averaged in energy(N(E) = N).

The DOS atxC,N(EF −eψC), were obtained by using Eq.(34) of Ref.[15] rewritten as:

N(EF − eψC) = 2

πkT

G(ψS, ω)/ω

C2(ψS, ω)fω(x(ψS))C(ψC,0)

×∫ EF

EF−eψC

N(E) dE (A.5)

and by substituting to the integral the termeψC N(x). As firstapproximation forN(x) that one derived from the fitting ofCSC at the sameω andxC have been chosen.

The other fitting parameters (Vfb, Vg, ψC) were chosent countt

( e ofthethence

( qualle in

llowed to calculate, at each potential, the DOS in a raf energy:

F − eψC (10−1 Hz) < E < EF − eψC (105 Hz)

The minimum value of electrode polarization still coatible with the constraints of the theory (ψS≥ψC ≥ 3kT/e)hich allowed to exploit experimentally the largest ene

ange of DOS to be used in fitting the EIS dataE = 0.18 V(SCE). In fact at this potentialψS = 0.43 V, which

s still 0.05 V higher than theψC value estimated at the lowrequency. Outside the energy range accessible by directurements (seeFig. 17) the real DOS has been approximay assuming a gaussian distribution around the obseaximum in DOS followed by a constant DOS at still lownergies.

The assumption of a constant DOS at the lowest eneexperimentally not accessible region) is a little arbitraryot relevant to our aims, because its exact shape doeffect the response of barrier provided that the total amf deep-lying states is more or less the same than thatxploited DOS distribution.

By assuming in the deep energy region a constantbout three times higher (see inFig. 17) than that estimatey EIS (continuos line) an error in the|Z| around 5% wabtained.

.2. Differential admittance curves fitting

The fitting of CSC(ω, ψS) andGSC(ω, ψS) at differentrequencies (10 Hz≤ f≤ 10 kHz) was carried out by usin

rough an educated guess procedure by taking into achat:

a) as for theψC parameter the constraint of a decreas0.06 eV for decade of frequency was introduced infitting procedure and a check of the acceptability ofψC choice was the prediction of flattening of capacitacurves at the frequency corresponding toψC = 0;

b) theVg parameter was calculated by assuming an eemission-capture time constant for electrons and ho


the conduction and in valence band[14] and by usinga measured mobility gap for a-Nb2O5 film equal to3.40 eV;

(c) theVfb value for fitting theCSC(x, ω, ψS) andGSC(x, ω,ψS) curves was allowed to differ no more than 0.025 V.

References

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W. Archibald, R.A. Abram, Phil. Mag. B 56 (1987) 429.[17] C. Da Fonseca, M. Guerreiro Ferreira, M. Da Cunha Belo, Elec-

trochim. Acta 39 (1994) 2197.[18] W.P. Gomes, D. Vanmaekelbergh, Electrochim. Acta 41 (1996)

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