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Electrochimica Acta 52 (2007) 6771–6777

Diffusivity of anion vacancies in WO3 passive films

Gerardo Vazquez, Ignacio Gonzalez ∗,1

Universidad Autonoma Metropolitana-Iztapalapa, Depto. de Quımica, Area de Electroquımica,Apdo. 55-534, C.P. 09340, Mexico D.F., Mexico

Received 1 February 2007; received in revised form 25 April 2007; accepted 25 April 2007Available online 6 May 2007

bstract

The WO3 films were grown in 0.1 M HClO4 aqueous solution, at different formation potentials (Ef) in the range of 2.0–7.0 V versus sce, onelectrode. The anion diffusion coefficient (DO•• ) of WO3 films was calculated from EIS spectra, following the surface charge approach (at

igh-field limit approximation), the Point Defect Model and the Mott–Shottky analysis. Among the parameters necessary to evaluate DO•• , thealf-jump distance (a) is very relevant, given that a small variation in a has a great impact in the calculation of DO•• . In this work, it is proposed

he half-jump distance (a) should be evaluated from spectroscopic data (available in the literature). The value of a (∼1.9 A) is taken from latticeonstants of a-WO3 (amorphous-WO3), with different values of N (coordination number), and the lattice constants of m-WO3 (monoclinic-WO3).he calculated value of DO•• was ∼3 × 10−17 cm2/s.2007 Elsevier Ltd. All rights reserved.

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1

eywords: Diffusivity; Passive films; EIS; W oxides; Mott–Schottky

. Introduction

The WO3 has an increasing importance on new technologiesuch as flexible smart windows [1] due to its electrochromicroperties [2–5], WO3 can be used as electrochemical func-ional material, in thermal control applications and active layer,s well as gas sensing applications [6], where WO3 shows anxcellent sensitivity to H2S and other gases, such as: NOx andO [7]; it can be used in the photoelectrochemical degradationf dyes, such as naphthol blue black diazo [8], among otherpplications. The mechanism to describe the growth of passivelms can be explained by the Point Defect Model (PDM) [9–11],

he PDM considers that the oxide film contains a lot of pointefects (e.g. anionic and cationic vacancies and metal intersti-ials). The transport of point defects within the film could be dueo the existence of a high electric field (∼106 − 107 V/cm) andhe movement of point defects within the film is associated with

he growth of the film. This idea was essentially first proposedy Verwey [12], and later by other authors, such as: Mott andabrera [13,14]. The transport of point defects within the film

∗ Corresponding author. Tel.: +52 58044671; fax: +52 58044666.E-mail address: igm@xanum.uam.mx (I. Gonzalez).

1 ISE member.

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013-4686/$ – see front matter © 2007 Elsevier Ltd. All rights reserved.oi:10.1016/j.electacta.2007.04.102

an be quantitatively described by their diffusivity. An effort toalculate the anion vacancies diffusivity (DO•• ) in oxide pas-ive films, has been performed by the surface charge approacheveloped by Bojinov [15]. Among the parameters necessaryo evaluate DO•• in the surface charge approach, the half-jumpistance (a) is very relevant (i.e. 2a equals the lattice constant ofhe unit cell). The values obtained by Bojinov for a, in the WO3lms, in different acid media, are in the range of ∼1.8–2.5 A15]. Since a very small variation in the value of a has a greatmpact in the calculation of DO•• (e.g. differences of 0.5 A inhe value of a changes DO•• in 1 order of magnitude), then its very important to have an accurate value of a, and use a asnown parameter in the calculation of DO•• .

.1. Overview

Diffusivity of the point defects in WO3, has been calculatedy Sikora et al. [16]. The approach of this paper started withhe Nernst–Planck equation and by means of the Point Defect

odel (PDM) computed the anion vacancy diffusivity (DO•• ), a

alue of 10−14–10−15 cm2/s was obtained. However, the surfaceharge approach developed by Bojinov [15], suggests that thequation used by Sikora et al., in calculating DO•• , is associatedo the low-field limit approximation of a more general equation

6 ochim

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1

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dneibftSWvcvbSctat

bWdlfi

1

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C

wmcdE

w

wsT

1a

e(z

wfiv

aa

772 G. Vazquez, I. Gonzalez / Electr

iven by Fromhold [17]. The equation of Fromhold at the high-eld limit approximation seems to be a better option in order toalculate DO•• , due to the assumption of the existence of higheld strength within the oxide passive film.

.2. Anion vacancies in the WO3

The exact nature of the WO3 films with regard to their beingrystalline or amorphous is not easy to ascertain. Apparently,ery thin films on valve-metals tend to be amorphous or micro-rystalline [18]. Thermodynamically, the tungsten cations inhe oxide film are found in a very high oxidation state (+6);hen the formation energy of cation vacancies is expected toe much higher than that for the oxygen vacancies, this is truenly in the bulk of the oxide. However, the reactions of genera-ion of defects at the metal/film and film/solution interfaces are

uch more energetically favorable, and the activation energiesor the electrochemical reactions for the generation of positiveefects at the metal/film interface and negative defects at thelm/solution interface might not be so different. The dominantoint defects in the passive films can be elucidated directly byetermining the oxygen/metal ratio through the films [19] andndirectly by Mott–Schottky plots (M–S analysis). The use ofhese plots has been object of controversy, due to the frequentlybserved non-linearity for the metal/oxide film/electrolyte junc-ions; this behavior has been attributed according to Tomkiewicz20] to: non-homogeneus doping levels, deep doping levels,mong others. However, none of these appears to provide aiable explanation of the entire form of the M–S plots, found forany metal oxides in presence of different electrolytic media.nother issue in M–S plots is the change of the measured capac-

tance with frequency. The change with the frequency for thinnodic WO3 films, was associated with the amorphous naturef the oxide [18]. Nevertheless, Pajkossy [21] attributes the

ispersion of the capacitance, to the roughness of the surface (notecessarily of an amorphous material). The semiconductor prop-rties are more clearly manifested in crystalline materials thann amorphous ones, because in amorphous materials the recom-ination could be a major issue. Since the WO3 films have beenound clearly behave as a n-type semiconductor [18,22–24],hen the crystallinity of WO3 films should not be discarded.emiconductor [18,22–24] and conduction studies [25] on bulkO3 have shown it to be oxygen deficient. If it is assumed that

acancies act as dopants (oxygen vacancies imparting n-typeharacter and cation vacancies p-type character), then oxygenacancies are considered to be electron donors. However, it cane other kinds of defect-charge compensation different from thechottky-type defect occur in solid oxides (e.g., low-valency

CO(Lss) = CO(0) exp

[−

(Lss

a

)th

(zFaE0

RT

)− a

DO••sh

ations, [26], surface states [27–29], a non-uniform distribu-ion of charge carriers and dielectric relaxation phenomena [27],mong others). In this study, the diffusivity is considered to behat for the combination of these point defects, and the VO•• i

ica Acta 52 (2007) 6771–6777

e the dominant defect (majority carriers), which gives to theO3 the n-type semiconductor character, then the density of

onors (ND) could be measured by the M–S analysis and corre-ated with the concentration of donors near the interface metal/lm [16].

.3. The density of donors (ND) according to PDM

The detailed explanation of PDM can be reviewed in severalapers [9–11], this paper is concerned only with the ND as aunction of Ef (Eq. (1)) given by Sikora et al. [16]. Sikora etl. started from the Nernst–Plank equation, concluding that theensity of donors (ND) is given by the following relationshipEq. (1)).

O(Lss) = ND = w1 exp�−bEf� + w2 (1)

here CO(Lss) is the concentration of oxygen vacancies at theetal/film interface, at which they are generated (therefore, their

oncentration at this interface is the highest), ND the density ofonors in the passive film, w1 and b the unknown constants andf is the film formation potential.w2 is given by [16]:

2 = JORT

zFE0DO••(2)

here Jo is the steady-state flux of donors, F the Faraday con-tant, E0 the mean electric field strength, R the gas constant andis temperature (K).

.4. The density of donors (ND) by the surface chargepproach

Bojinov [15] started with Eq. (3), which is a more generalquation for CO(Lss), given first by Fromhold [17].

FaE0

RT

)]+ JO

(1 − exp

[−

(Lss

a

)th

(zFaE0

RT

)])(3)

here CO(0) is the concentration of oxygen vacancies at thelm/solution interface, CO(Lss) the concentration of oxygenacancies, at the metal/film interface, a is the half-jump distance.

Eq. (3) has two limit cases: the high-field limit, zFaE0/RT � 1nd the low-field limit zFaE0/RT � 1. Considering that in almostll cases Lss � a, it is obtained that:

At high-field limit:

CO(Lss) = CO(0) exp

[−

(Lss

a

)]− 2aJO

DO••exp

[zFaE0

RT

]

(4)

And at low-field limit:[ (zFE0Lss

)]RTJO

CO(Lss) = CO(0) exp −RT

−DO••zFE0

(5)

Comparing Eq. (1) with Eq. (4) and (5) and solving for DO•• ,t is found that:

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1

s

L

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Foafic

ct

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as(t

2

pd(bum

Nssr

2

Amts

G. Vazquez, I. Gonzalez / Electr

At high-field limit:

DO•• = 2aJO

w2exp

[zFaE0

RT

](6)

And at low-field limit:

DO•• = RTJO

w2zFE0(7)

here R is the universal gas constant, T the temperature (K),the Faraday’s constant, a the half-jump distance and z is the

harge of the anionic vacancy. Considering the current is mainlyue to the flux of oxygen vacancies, then JO = −iss/2e, where isss the steady-state current density and e is the charge of thelectron.

The value of DO•• can be calculated with Eq. (2), which isdentical to Eq. (7) (at the low-field limit). However, Eq. (6)high-field limit) will be used in this work in the calculationf DO•• , because the high limit approximation agrees with theupposition of the existence of a high electric field within thexide film.

.5. Calculation of E0 and w2

The value of E0 can be calculated from Eq. (8) [16] using thelope of Lss versus Ef, but to get E0, α must be known

ss = 1

E0(1 − α)Ef + B. (8)

here α is the polarizability of the oxide film/solution interfacend B is a constant related with the thickness of the film at a for-ation potential of 0 V, it depends on pH, on the rate constants

or film formation at the metal/film interface and film dissolu-ion at the film/solution interface. The surface charge approachrovides a way to calculate the value of α, by means of Eq. (9)15]. The values of Rb and Rsc are obtained from fitting the EIS

pectra using the equivalent electric circuit (Fig. 1) proposedy Bojinov [15]. The circuit contains: Rel which is the solutionesistance; Cs associated to the Faradaic pseudocapacitance, Rbhe resistance of the oxide film, C is associated to the barrier film

ig. 1. Equivalent electric circuit (eec) used to simulate the impedance behaviorf the WO3 films, in 0.1 M HClO4 solution. Rel is the solution resistance; Cs

ssociated to the Faradaic pseudocapacitance; Rb the resistance of the oxidelm; C the barrier film capacitance; Rsc and Lsc are associated with the surfaceharge at the film/solution interface [15].

wotwuieK

2

afiit0s

ica Acta 52 (2007) 6771–6777 6773

apacitance, Rsc and Lsc are associated to the surface charge athe film/solution interface.

Rb

Rsc= α

1 − α(9)

The value of E0 is calculated by comparing Eq. (8) with thelope of the curve of Lss versus Ef (Lss is obtained from theapacitance measurements of the oxide film). Finally, w2 can beound by fitting ND versus Ef and comparing with Eq. (1).

Then, in this work, a combination of the surface chargepproach (at the high-field limit), the Mott–Schottky analy-is, the PDM and the value of the half-jump distance ∼1.90 Aobtained from the literature [30,31]), has been used to calculatehe DO•• of the WO3 films.

. Experimental procedure

The working electrode is a W bar (Alfa&Aesar, 99.95%urity), embedded in Teflon, where only the base of the cylin-er (φ = 6.3 mm) is exposed. The reference electrode is calomelsce), in a Luggin probe. The counter electrode is a graphitear. The potentiostat/galvanostat, E&GG, PAR, model 283, wassed to apply potentials and a frequencies analyzer, Solartron,odel IF 1260, for impedance measurements.The W electrode was polished with SiC sandpaper, 1200.

ext, it was polished with alumina (0.05 �m), until a mirrorurface is obtained, then it was rinsed with deionized water;ubsequently, it was placed in an ultrasonic bath for 5 min andinsed again with deionized water.

.1. The WO3 formation and EIS characterization

The W electrode was placed in 0.1 M HClO4 (J.T. Baker,.R.) aqueous solution, not bubbled with nitrogen. Then a for-ation potential (Ef) was imposed for 1800 s, in order to form

he oxide film. The Ef was within the range of 2.0–7.0 V versusce. The current response was monitored during the experiments.

The characterization of the formed films was done by EIS,ith the following parameters: the impedance spectra werebtained at the same potential that they were formed; the ampli-ude of perturbation was ±30 mV, the frequency range scannedas from 100 kHz to 10 mHz. The validity of EIS data was eval-ated through the analysis of error distribution of the real andmaginary impedance components. The random distribution ofrrors obtained in the data fitting indicates their consistency withramers–Kronig transforms (see below).

.2. Differential capacitance measurements

The differential capacitance measurements were performed,t constant frequency (1 kHz), after the film was formed. Thisrequency was selected from a previous study, where the capac-tance of the oxide film was measured at different frequencies,

n the range 10 Hz to 10 kHz. The spectra were obtained withhe following parameters: the initial potential was Ef, the final.1 V. In order to avoid any variation of the film thickness, thecan rate was 0.5 V/s and the amplitude of the ac perturbation

6 ochimica Acta 52 (2007) 6771–6777

±ii

3

ditssTewotWisstfttp

3

iosTtF

aTi

Fo

Fig. 3. Nyquist diagrams of the WO3 films, previously formed in the 0.1 MHClO4 solution, at different formation voltages Ef, within the range from 2.0teu

Ks

T

wsHv

3

twfilm thickness, since the film thickness is expected to increasewith the Ef, then Cb diminishes with the increment of Ef. Rsc

774 G. Vazquez, I. Gonzalez / Electr

10 mV. The space charge capacitance was calculated assum-ng Csc = −1/(ωZim), where Zim is the imaginary component ofmpedance and ω is the angular frequency.

. Results and discussion

The oxide films on W were formed applying Ef for 1800 s,uring all the experiment the current density was monitored,t was observed that the current density diminished with theime until a constant value was reached (i.e. steady state). Fig. 2hows the values of the current density at steady state (iss) ver-us the Ef. Since iss is nearly constant, then d(ln iss)/dEf ∼ 0.he PDM’s diagnostic criteria [11], proposes that the interfacialquilibrium is governed by anion transmission on passive films,hen: d(ln iss)/dEf = 0 and χ = δ, where χ is the oxidation statef the metal atom in the oxide net, and δ is the oxidation state ofhe dissolved metal atom. Presumably, the dissolution product of

is the WO22+ cation, in very acid solutions, and the tungstate

on (WO42−) in solutions with higher pH [32]. The oxidation

tate in the oxide film is VI [32], and is also VI when it is dis-olved, thus satisfying the condition χ = δ. Therefore, accordingo the PDM, it can be concluded that the interfacial equilibriumor the system, is governed by anion transmission, attributed tohe transport of the VO•• (oxygen vacancies) through the film,his fact is in agreement with the assumption that the dominantoint defects in the oxide film are the oxygen vacancies.

.1. EIS spectra of the WO3 films

The impedance spectra of the WO3 in 0.1 M HClO4 are shownn Fig. 3. From the spectra, it could be seen that the impedancef the WO3 films depends on Ef. The spectra comprise flattenedemicircles, with an inductive loop at middle to low frequencies.he flattened semicircles indicate the existence of at least two

ime constants. This requirement is satisfied by the circuit inig. 1.

The impedance spectra for the Ef’s of: 2.0, 3.0, 4.0, 5.0, 6.0nd 7.0 V/sce, were fitted using the program Boukamp [33].he typical random distribution of errors (Fig. 4) obtained

n the data fitting spectra indicates their consistency with the

ig. 2. Steady-state current density (iss) obtained during the potentiostatic grownf the WO3 films, at different formation potential Ef in 0.1 M HClO4 solution.

ai

Fiddt

o 7.0 V, indicated in the figure. The spectra in symbols represent the spectraxperimentally obtained. The spectra in lines belong to the calculated spectrasing the fitting values of the boukamp program and the eec in Fig. 1.

ramers–Kronig transforms. This distribution is similar for thepectra of Fig. 3.

The capacitance Cb has been calculated according to Eq. (10).he complete set of fitted results is shown on Table 1.

1

C= 1

Cb+ 1

CH(10)

here C is the capacitance value evaluated from the fit of EISpectra, Cb the capacitance of the oxide film and CH is theelmholtz layer capacitance which is assumed to have a constantalue of 20 F cm−2 [34].

.2. Analysis of Table 1

The oxide film resistance (Rb) increases as the Ef is greater,his could be due to the fact that the film thickness is increasedith the Ef. The oxide film capacitance is inversely related to the

nd Lsc are associated with the surface charge at the film/solutionnterface.

ig. 4. Typical error distribution of the real and imaginary components of thempedance spectra, of the WO3 films, formed in the 0.1 M HClO4 solution, atifferent formation voltages Ef, within the range from 2.0 to 7.0 V. The randomistribution of errors obtained in the data fitting (for film grown at 4.0 V) indicatesheir consistency with Kramers–Kronig transforms.

G. Vazquez, I. Gonzalez / Electrochimica Acta 52 (2007) 6771–6777 6775

Table 1Values of the elements of the electrical circuit (Fig. 1), obtained by fitting the corresponding transfer function and experimental data, of the impedance spectra ofWO3 films, in a 0.1 M HClO4 aqueous solution

Ef (V) Cs (F cm−2) Rb (ohm cm2) Cb (F cm−2) Rsc (ohm cm2) Lsc (H cm2)

2.0 1.70 × 10−3 1.83 × 104 6.07 × 10−6 6.79 × 104 1.43 × 105

3.0 1.40 × 10−3 2.48 × 104 4.92 × 10−6 7.96 × 104 2.36 × 105

4.0 1.20 × 10−3 3.24 × 104 4.23 × 10−6 8.90 × 104 2.77 × 105

5.0 1.15 × 10−3 3.90 × 104 3.75 × 10−6 10.00 × 104 4.11 × 105

6 −3 4 −6 4 5

7

R

throbin(ri

3

t1EoCigibIhit

Fa(d

icsaEt

wdε

tp(

(o[apF

N

.0 1.11 × 10 4.80 × 10

.0 1.05 × 10−3 5.15 × 104

el = 7 ohm cm−2.

The value of α can be calculated using Eq. (9), the values ofhe resistances Rb and Rsc are taken from Table 1. The value of α

as been averaged, in order to obtain a single value which mayepresent the polarization at the M/S interface, the average valuef α is ∼0.3, this value is smaller than the 0.5 value assumedy Sikora et al. [15], and that obtained by Bojinov (0.42–0.58)n sulphuric and phosphoric solutions with 1, 2 and 4 M [16];evertheless, the concentration in this work is very different0.1 M). Macdonald et al. [35] obtained values for α within theange of 0.2–0.8 for several metals (Fe, Ni, Cr, Fe–Ni, Fe–Cr)n borates and phosphates buffers.

.3. Results of the Mott–Schottky analysis

In this work, the differential capacitance measurements ofhe oxide films were measured at high frequency (typicallykHz), immediately after the oxide film was grown, at specificf. The M–S plots were traced, assuming constant thicknessf the oxide film during the measurement and considering thatsc = −1/(ωZim) is valid. The typical behavior of the M–S curves

s shown in Fig. 5. Similar curves were obtained for the filmsrown at different Ef. Fig. 5 shows two linear regions, one regions for Ec > 0.9 and the second for Ec < 0.9 V. This behavior haseen observed for different media and metal oxides [16,36,37].

n Fig. 5, the measured capacitance at the range of 0.9 < Ec < 4 V,ave other contributions different to the space charge capac-tance, even Faradaic reactions in Ec near to Ef. Meanwhile,he Faradaic reaction is not present for Ec < 0.9 V, then the

ig. 5. Typical behavior of the space charge capacitance Csc of the WO3 films,fter 1800 s of growth at Ef = 4.0 V, in 0.1 M HClO4, with the applied potentialEc) varied at 0.5 V/s. The Csc was evaluated assuming Csc = −1/(ωZim). Theashed line represents the M–S interval of the fitting with Eq. (11).

wie

Ffl

3.43 × 10 12.11 × 10 4.83 × 103.33 × 10−6 12.50 × 104 8.60 × 105

mpedance response is only capacitive, attributed to the spaceharge capacitance. The curve in Fig. 5, clearly shows an n-typeemiconductor behavior of the oxide film, which is expected formaterial doped with electron donors (VO), then the ND and thefb can be estimated using the M–S relationship, in Eq. (11), in

he region of Ec < 0.9 V.

1

C2sc

= 2NA

NDFεrε0

(E − Efb − RT

F

)(11)

here NA is the Avogrado’s number (6.02 × 1023 1/mol), ND theensity of donors, F the Faraday constant (∼9.65 × 104 C/mol),r the relative permittivity (∼43 in perchloric acid [38]), ε0he permittivity of vacuum (8.8542 × 10−12 F/m), E the appliedotential, Efb the flat band potential, R the constant of the gases8.314 J/K mol) and T is temperature (K) (∼298 K).

The values of ND estimated using Eq. (11) are shown in Fig. 6the values of ND are in the same order of magnitude as thosebtained by Sikora et al. [16]) and that obtained by Biaggio et al.38]. In Fig. 6 it is observed that the ND diminishes exponentiallys the Ef increases, the fit of data in Fig. 6 with Eq. (1) waserformed, and Eq. (12) represents the best fit of the curve inig. 6.

D = 5.7 × 1018 + 2.1 × 1020 e−0.4204Ef (12)

hen Ef = 0 in Eq. (12) and ND is 2.16 × 1020 cm−3. This values smaller than that obtained (ND = 3.2 × 1020 cm−3) by Sikorat al. [16] in 1 M H3PO4 and that obtained by Biaggio et al. [38]

ig. 6. Variation of density of donors (ND) of the WO3 films, previously grownor 1800 s, in 0.1 M HClO4 at different film formation potentials (Ef). The dashedine is the fitted curve given by the equation showed in the figure.

6776 G. Vazquez, I. Gonzalez / Electrochim

FW

(tcoiba

3

tscLasai(tle

Fit(

3

ihir

adaUtooria

otnteietpoTacecbv

ig. 7. Variation of the inverse of the barrier film capacitance (1/Cb), of theO3 films, previously grown for 1800 s, in 0.1 M HClO4 at different Ef.

∼1.1 × 1022 cm−3), nevertheless, the media and/or concentra-ion are different than the used in this work. The value of w2an be obtained by comparing Eq. (12) with Eq. (1), the valuebtained in this way is 5.7 × 1018 cm−3. The value of w2 usedn the previous study of the semiconductor properties of WO3y the surface charge approach [15] was taken from Sikora etl.’s work [16].

.4. Film thickness

The behavior of 1/Cb versus Ef (Fig. 7) has a linear rela-ionship (except the point at 7 V, indicating a possible chargeaturation at this potential). The linear relationship of 1/Cban be correlated with the thickness of the oxide film, takingss = εrε0/Cb, which corresponds to the relationship of a par-llel plate capacitor. The values of Lss as function of Ef arehown in Fig. 8. Fig. 8 shows that the film thickness increasess the Ef is increased, the comparison of the slope of the curven the Lss versus Ef graph, with Eq. (8), gives the value of E0∼5.9 × 106 V/cm). This value agrees with the assumption of

he high field strength within the oxide film, the value is simi-ar to those reported by other authors, for different oxides andlectrolytic media [16,34,39].

ig. 8. Variation of the thickness of the WO3 films (Lss), grown for 1800 s,n 0.1 M HClO4 with different Ef. The Lss was calculated from the values ofhe oxide film capacitance (Cb), the equation of the linear regression is showndashed line) in the figure.

itwa

o∼t∼vtt

TC

iα

Ew

ica Acta 52 (2007) 6771–6777

.5. The half-jump distance (a)

Since a very small variation in the value of a have a greatmpact in the calculation of DO•• , then it is very important toave an accurate value of a, and to use a as known parametern the calculation of DO•• . In this work, a is obtained from dataeported in the literature.

Kuzmin et al. [30] reported a study of the local environmentround tungsten ions for the a-WO3 (amorphous-WO3), withifferent values of N (coordination number), which is an aver-ged value of the number of neighbors around the tungsten ions.sing EXAFS and XDR and performing a fitting procedure,

he coordination number and the Debye–Waller factors werebtained, with them the lattice constant a was calculated, thebtained value is 1.9 A. In the other hand, Hjelm et al. [31]eported that the lattice parameter for m-WO3 (monoclinic-WO3s the most stable form of WO3 at room temperature) is 3.78 A,nd then a ∼ 1.89 A.

The M–S analysis is valid for crystalline or (micro)crystallinexide films and would vary if they are amorphous. Investiga-ions using M–S plots, have found WO3 films to behave as a-type semiconductor [16,37–39], which found the ND to be inhe order of ∼1019–20, proving to have reproducibility. Consid-ring that the dispersion in the capacitance of the WO3 filmss due to: (i) the roughness of the film and (ii) the probablyxistence of a synergic effect due to the non-uniform distribu-ion of charge carriers, low-valency cations, dielectric relaxationhenomena, among others mentioned before. The crystalliner microcrystalline nature of the WO3 is taken into account.his consideration is important because the transport numbernd conduction mechanism in the growth of the oxide filmsould vary between amorphous and crystalline. Then, consid-ring the crystallinity of WO3, the density of donors can bealculated by M–S; the evaluation of DO•• could be performedy means of Eq. (6) (taking a equal to 1.9 A), the calculatedalue for DO•• is ∼3 × 10−17 cm2/s (as an example to see themportance of the parameter a, if the value of a was takeno be equal to 2.7 A, then DO•• would be ∼7 × 10−19 cm2/s,hich is 100 times smaller than the calculated with= 1.9 A).

Table 2 shows the parameters calculated in this work tobtain DO•• with Eq. (6), the DO•• was calculated to be3 × 10−17 cm2/s, which is ∼2 orders of magnitude smaller

han the value obtained by Eq. (7) (using the low-field limit10−15); and it is ∼1 order of magnitude greater than the

alue obtained with the surface charge approach [15], where

he half-jump distance values are in the range of 1.8–2.5 A, inhe calculation of DO•• .

able 2alculated parameters to obtain the DO•• of the WO3 films

ss 8.2 × 10−6 A/cm2

0.3

0 5.9 × 106 V/cm

2 5.7 × 1018 1/cm3

ochim

4

wvsttvttSEtt1t(wwdD

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[[[[[[[

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[[[

[[

[[[[[[[[[

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G. Vazquez, I. Gonzalez / Electr

. Conclusions

By means of the table from the PDM, for diagnostic criteria, itas concluded that the interfacial equilibrium due the transportacancies within the WO3 film is governed by anion transmis-ion, attributed to the transport of the VO•• (oxygen vacancies)hrough the film, this fact is in agreement with the assumptionhat the dominant point defect in the oxide film are the oxygenacancies (majority carriers). The obtained values of ND are inhe order of ∼1019–20, it was found that ND depends exponen-ially with the Ef, the ND values are similar to those obtained byikora et al. Additionally, the value of α was calculated usingq. (9), the average value for α was ∼0.3, this value is smaller

han that obtained by other authors; nevertheless, the concentra-ion of the acidic solution, in this work is very different (0.1 M)./Cb allows calculating the thickness of the film, which is func-ion of the Ef. With the slope of Lss versus Ef the value of E0∼5.9 × 106 V/cm) was obtained, the values are in agreementith the assumption of the existence of the high field strengthithin the oxide film. On the other hand, taking a ∼ 1.9 A, theiffusivity of the anion vacancies was calculated, the value ofO•• was ∼3 × 10−17 cm2/s, which is at least 2 orders of mag-

itude smaller than the value obtained by Eq. (7) (using theow-field limit ∼10−15); and it is ∼1 order of magnitude greaterhan that obtained by the surface charge approach, the differencen the values of the diffusivities is attributed either to the varia-ion of the half-jump distance values evaluated in the range of.8–2.5 A.

cknowledgements

This work has been carried out with the financial aid, fromONACYT (SEP-2004-C01-47162). G. Vazquez is grateful toONACyT for the postgraduate grant.

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