Mott-Schottky Analysis of Nanoporous SemiconductorElectrodes in Dielectric State Deposited on SnO„F…

e

is

ld cause aing to

Journal of The Electrochemical Society, 150 ~6! E293-E298~2003!0013-4651/2003/150~6!/E293/6/$7.00 © The Electrochemical Society, Inc.

E293

2Conducting SubstratesFrancisco Fabregat-Santiago,a Germa Garcia-Belmonte,a Juan Bisquert,a,* ,z

Peter Bogdanoff,b and Arie Zabanc,*aDepartment de Cie`ncies Experimentals, Universitat Jaume 1, 12080 Castello´, SpainbDepartment Solare Energetik, Hahn-Meitner Institut, 14109 Berlin, GermanycDepartment of Chemistry, Bar-Ilan University, Ramat-Gan 52900, Israel

This paper analyzes the dark capacitance of nanostructured electrodes in the dielectric state, with particular emphasis on TiO2

electrodes deposited over a transparent conducting substrate of SnO2(F). It is shown that at those potentials where the TiO2

nanostructure is in the dielectric state, the capacitance is controlled by the contact SnO2(F)/~electrolyte, TiO2). The partial or totalcovering of the substrate by a dielectric medium causes a modification of the Mott-Schottky plot of the bare substrate. We provida mapping of the various Mott-Schottky curves that will appear depending on the film characteristics. If the dielectric nanoparticlescompletely block part of the substrate surface, the slope of the Mott-Schottky plot increases~with the same apparent flatbandpotential! as an effect of area reduction. The covering of a significant fraction of the surface by a thin dielectric layer shifts theapparent flatband negatively. Measurements on several TiO2 nanostructured electrodes show that the capacitance contribution ofthe semiconductor network in the dielectric state is very low, indicating that the field lines penetrate little into the TiO2 network,not much further than the first particle. The different surface covering observed for rutile-anatase and pure anatase colloidsexplained by lattice matching rules with the substrate. By comparing different electrodes, the Helmholtz capacitance at theSnO2(F)/solution interface was calculated and the apparent flatband potential was corrected for the effect of band unpinning.© 2003 The Electrochemical Society.@DOI: 10.1149/1.1568741# All rights reserved.

Manuscript submitted May 1, 2002; revised manuscript received December 8, 2003. Available electronically April 10, 2003.

Nanoscaled materials have become very important for recenttechnological developments in broad fields such as photovoltaics,

able way, because charge transfer from the substrate cousignificant loss of photogenerated electrons.12 It is also interest

ctrm

s,paanernof nanin

ses

rotricew

thes,ntiuccnd

ntaor2-

dcitnssue

ecpe

esa i

evaluate the influence of charge stored in the capacitor at theeasure-mation

electricalents,1 soFinally,particles6

. An anal-,t the

te isim of thisf nanopo-ning meth-g of nano-field willthis re-

ependingization of

possiblech effects, etc.

es theup to theed tin ox-way ispect toevelop acitance persameis anmi levelsum-S plot isThe effectbserved

photocatalysis, energy storage~batteries, supercapacitors!, elechromic devices, etc. Nanostructured electrode materialslong, for example, to the groups of porous semiconductornanocomposites, or sintered nanoporous metal oxides. Innanostructured semiconductor electrodes, and especially nTiO2 , have raised widespread attention for their use in sevtoelectrochemical applications such as dye-sensitized naline solar cells~DSSCs!.1 This kind of electrodes consists oparticulate semiconductor networks composed of smalldoped metal-oxide crystallites~in the 10 nm range!, sinteredconnected structure on top of a conducting substrate. Theinvolve three different phases: the transparent conductingof F-doped SnO2 , the porous TiO2 network, and the electBecause of the small size of the nanoparticles and the inometry of the porous network permeated with electrolyte, nods and models are required to describe these systems.

When a nanoporous electrode reaches equilibrium inthe three phases involved, having different work functionbrate at a unique value of the Fermi level and redox potecause in equilibrium most of the semiconductor nanostrinert electrically, charge separation and electrical fields areto the contact between the substrate and the electrolyte, athe nanoparticles closest to the substrate.2 This coSnO2(F)/~electrolyte, TiO2) has been discussed in several w

Here we analyze the information that can be obtaineSnO2(F)/~electrolyte, TiO2) contact properties from capameasurements applying Mott-Schottky~MS! analysis. We cothe capacitance-voltage relationship of the conducting SnO2(F)strate partially or totally covered by a dielectric medium. Thtive is to compare the capacitance of the nanoporous elthat of the bare substrate in order to characterize the prothe former.

With respect to DSSCs, there are several points of interstudy. It is useful to determine the exposed substrate are

* Electrochemical Society Active Member.z E-mail: bisquert@uji.es

o-ay be-

organicrticular,oporousal pho-crystal-ano-d low-

asystemsubstratelyte.ate ge-meth-

dark,2

equili-al. Be-ture isonfined

also toct

ks.12

on theanceiderb-objec-trode torties of

t to thisn a reli-

SnO2(F)/~electrolyte, TiO2) contact in electrochemical mment techniques.13,14 An alternative way to obtain the inforsupplied by capacitance measurements is to monitor thefields in the substrate by interference reflection measurem1

it is important to develop comparison tools for future work.a possible influence of the dark potential drop in the nanoon the photovoltage obtained in DSSC should be exploredogy with thep-n junction mechanism has been suggested6 buissue is still controversial.7,9,10

Although nanoporous TiO2 on conducting SnO2 substrataken here as the specific example of reference, the aresearch is more general. So far, a wide variety of types orous electrodes has been reported in the literature concerods of preparation, materials and mixtures, size and dopinparticles, surface covering treatments, etc., and thepresumably grow in the future. Accordingly, we provide inport a mapping of the various MS curves that will appear don the film characteristics. This will enable good characterthe electrode, not only as prepared, but also following thechanges during the electrochemical manipulations, by suas intercalation, change of contact area, change of doping

Background

The substrate capacitance.—In the nanoporous electrodelectrolyte permeates the semiconductor network voidssubstrate. Therefore the conducting substrate of fluor-dopide (SnO2(F)) is in contact with the electrolyte~electrical pathin Fig. 1!. At stationary potentials that are positive with rethe flatband potential, highly doped semiconductor films dspace-charge region at the surface, which provides a capaarea unitCb ~band bending!. The capacitanceCb is about theorder of the Helmholtz capacitanceCH ; thus, the situationintermediate to the limits of band-edge pinning and Ferpinning. The effect of the band unpinning can be treated15 asmarized in the Appendix. According to Eq. A-10, a linear Mpredicted, and this is observed as reported in Ref. 16-18.of the variation of Helmholtz potential is to shift the o

s nngntiat

@

nds itaer

eh

inuselyte

ic

tor network. The theoretical analysis2 indicates that the electricalfield at the SnO2(F)/TiO2 junction penetrates the porous network to

ports4,6,10

on state,viously inc, and ine can be

ends overto thoseelectri-k imped-here.0

porousde theand the

vide dif-junc-stribution

substrateal pathwayhere thecontact.

th ii thembinationance/contact,ssibletric, withic. Since itaterial is

t theendent of

he porousition,oporous

d noter on.

@2#

so thatHelm-

over theitor withe sub-

@3#

lectrolytease Eq.

@4#

themedium,

@5#

tee lthwtati

Journal of The Electrochemical Society, 150 ~6! E293-E298~2003!E294

MS plot negatively, so that:~i! the slope ofC22 vs. V doechange with respect to that predicted in band-edge pinnitions, 2/(NDe« r«0), and ~ii ! the apparent flatband pote2(U0 1 U1), which is shifted negatively from the real flpotential (2U0) by an amount15,18

U1 5NDe« r«0

2CH2

Electronic properties of the porous semiconetwork.—The TiO2 used in nanostructured photoelectrodeally very lightly doped, and in equilibrium in the dark it connegligible number of free electrons. Thus, it can be considTiO2 matrix contains no space charge of any kind, and it ba dielectric.2,4

When the TiO2 is in the dielectric state the electrical fieldthe particles is governed by the Laplace equation. Becasmall size of the nanoparticles surrounded with electroimpossible to sustain large-scale electrical fields in the sem

Figure 1. Scheme of the three-phase contact between the substraing SnO2 film, the porous TiO2 network, and the electrolyte. Thdiagrams show the electrical potential distribution in pa(SnO2/solution! and ii (SnO2 /TiO2/solution!. Note that this represeninverted with respect to band bending~electron energy!.

otcondi-

al isband

1#

uctors usu-ins aed thataves as

sideof the

, it isonduc-

a small distance, in the order of a few particles. Several reare in agreement with this result.

The TiO2 phase can be taken to electron accumulatieither by negative potential bias or by photogeneration. Obthat state the TiO2 cannot be considered at all as a dielectrifact the impedance behavior of the nanoporous electroddescribed as a transmission line equivalent circuit that extthe whole porous structure.19 The present study is restrictedsteady states where the TiO2 porous network is basically ancal insulator. An experimental study of the porous networance in accumulation conditions has been reported elsew2

Models

Capacitance and potential distribution of a nanoelectrode in the dielectric state.—In the nanoporous electroconducting substrate is in contact both with the electrolyteTiO2 particles. These two pathways, shown in Fig. 1, proferent capacitive contributions. For path i, SnO2(F)/electrolytetion, the behavior of the capacitance and the potential diare discussed in Appendix A.

In the sintering process the nanoparticles stick to thesurface and cover a part of the surface; hence, the electricii in Fig. 1 relates to the part of the substrate surface wSnO2(F)/TiO2 junction replaces the substrate/electrolyteFrom the point of view of the electrical response, in paHelmholtz layer over the bare substrate is replaced by a coof two capacitances: a dielectric medium (TiO2) with capacitareaCd , and a Helmholtz layer at the dielectric-electrolytehaving a capacitance/areagCH . The factorg accounts for a poincrease of the area of the Helmholtz layer over the dielecrespect to the substrate area that is covered by the dielectris assumed that the amount of dopant in the covering mnegligible, it is expected according to our previous study2 tharegion the field penetrates into the nanostructure is indepthe applied potential. This understanding justifies treating tsemiconductor network as a constant capacitance,Cd . In addwe expect2 that the electrical field lines penetrate the nanstructure not much further than the first particle; hence,g shoulbe much larger than 1. This point is further considered lat

The dielectric capacitance for TiO2 can be estimated as

Cd 5« r2«0

t

where« r2 ' 50 andt is the thickness of the dielectric layer,Cd ranges from 4 to 1mF/cm2 for t 5 10-50 nm. Usually theholtz capacitance is aboutCH 5 10mF/cm2.

Hence, in path ii, the dielectric and the Helmholtz layerdielectric, considered jointly, constitute a constant capaccapacitance~per TiO2 covered unit area of contact with thstrate!

Cns 5 @1/Cd 1 1/~gCH!#21

Therefore, the analysis of Appendix A for the substrate/ejunction~path i! can be repeated for path ii of Fig. 1. In this cA-10 gives

1

C2 52

NDe« r«0~Va 1 U0! 1 S 1

Cd1

1

gCHD 2

Combining the two branches i and ii, when a fractionf ofsurface of the conducting film is covered by the dielectricwe find the expression of the total capacitance

C 5 ~1 2 f !Ci 1 f Cii

conduct-oweray i

on is

itan ojunle

heplo

fac.—smC

ise

@

pactruitm

The dielectric layer covers the substratecompletely.—Sometimes a thin dielectric layer is fabricated on the

order toAlterna-

se poresanipula-

ered com-d gt for or-S plot of

sed as

@7#

@8#

arallel tont flatband

mentionhicknt. In thisasinghe space-layera MS plot.

mixedc ca-

contri-5, whereultingthe par-

s. We can,ive!,ecause asial dropfor bare

ently posi-, at large

@9#

@10#

Eq. 7,ever, foruse the

rtant mayof mea-

heless, Eq.covered

ace cov-neg-

ine for thebare elec-y a con-ata in the

O2(F, wTh

straf tf t

akend

1.5ge coroct. ~sio

Journal of The Electrochemical Society, 150 ~6! E293-E298~2003! E295

A main purpose of this formula is to compare the capacthe nanoporous electrode to that of the bare substrate iobtain information about the real state of the triplesubstrate/~film, electrolyte!. Since there are different possibfigurations, particles sizes, materials, etc., we examine in ting a number of cases and their manifestation in the MS

The dielectric covering completely blocks part of the surthe specific capacitance of the dielectric covering is very! CH , theCii can be neglected in Eq. 5. The capacitance

mined by the substrate area exposed to the electrolyte. Thof the covered electrode is then described by

1

C2 52

~1 2 f !2NDe« r«0~Va 1 U0 1 U1!

In this case the covered electrode shows the same apband as the bare electrode,2(U0 1 U1), but the covered eleexhibits a larger slope, as shown in Fig. 2a. This case is qesting because it allows determination of the real fraction afree substrate area, (12 f ), in a very simple way.

Figure 2. Simulations of the capacitance of bare and TiO2 covered Snelectrodes in contact with solution.~s! The bare SnO2(F) substrateND 5 1021 cm23, « r 5 9, CH 5 10mF cm22, and U0 5 0 V. ~,!electrode with a dielectric that covers totally or partially the subface.~a! Covering by a TiO2 nanoporous film, with a fractionf 5 0.2 osubstrate covered from the electrolyte. The specific capacitance oparticles is very small,Cd 5 0.05CH , so that the part covered mcontribution to the overall capacitance~see text!. The apparent flatbatential 2(U0 1 U1) is indicated.~b! Complete covering (f 5 1, g 5with a thin dielectric layer withCd 5 5CH . ~c! Half covering (f 5 05 1) with dielectric capacitanceCd 5 0.5CH . ~d! Half coverin5 0.5,g 5 1) with dielectric capacitanceCd 5 0.1CH . ~ ! Th

pacitance of the 50% free substrate area.~e! Covering by a TiO2 nanopfilm, with a fraction f 5 0.2 of the substrate covered from the elearea enhancement factorg 5 5, and dielectric capacitanceCd 5 0.5CH

Same as~e! with g 5 50,Cd 5 10CH . ~ ! The asymptotic expresthe capacitance, Eq. 9.

nce ofrder toction,con-follow-ts.

eIfall,d

deter-MS plot

6#

rent flat-odee inter-ount of

substrate prior to the deposition of the nanostructure inprevent the contact of the substrate with the electrolyte.tively, an insulating polymer is grown in the bottom of thein order to block the pinholes. In connection with these mtions, we now consider that the substrate has been covpletely with a dielectric layer of thicknesst, i.e., f 5 1 an5 1 in the notation introduced. This case is also relevan

ganic layers deposited on semiconductor surfaces. The Mthe covered electrode follows Eq. 4, which can be expres

1

C2 52

NDe« r«0~Va 1 U0 1 U2!

where

U2 5NDe« r«0

2Cns2 5

NDe« r«0

2 S 1

CH1

1

CdD 2

In this case the MS plot of the covered electrode is pthat of the bare substrate, as shown in Fig. 2b. The appareshifts negatively, with respect to the substrate, by (U2 2 U1).

With respect to full coverage of the substrate we brieflyanother possible behavior which was described elsewhere21 for tsurface covering layer and non-negligible amount of dopacase as the applied potential increases from2U0 , first an incredepletion region is formed in the surface layer, and when tcharge region reaches the whole thickness,t, the depletioncontinues to grow into the substrate. The consequence iswith two different consecutive slopes as shown in Ref. 21

Partial covering of the substrate surface, withcontribution from free and covered surface.—If the dielectripacitanceCd is not too small, then one must consider bothbutions from paths i and ii using the general formula Eq.Eq. A-10 and 4 describeCi and Cii , respectively. The resexpression cannot be written in a simple way, apart fromticular cases already examined in the two previous sectionhowever, see that when the applied potentialVa is large~positthe mixed capacitor in the surface is no longer important, bthe depletion capacitorCb in Eq. A-1 decreases, the potentwill be mainly into the substrate. Therefore the MS plotssubstrate and covered electrode must be parallel at sufficitive potential. In fact it can be shown that Eq. 5 reducespositiveVa, to

1

C2 52

NDe«1~Va 1 U0 1 U3!

where

U3 5NDe«1

2 F ~1 2 f !1

CH2 1 f S 1

Cd1

1

gCHD 2G

In the particular case off 5 1, Eq. 9 reduces trivially towhich gives the case shown in Fig. 2b of parallel lines. Howpartial covering Eq. 9 is not a very strict criterion, becapotential required to make the surface capacitors unimpobe impractically large. Consequently, in the practical rangesurement one may indeed observe different slopes. Nonet9 is useful in describing the trends of the data for a partiallyelectrode, as illustrated in the following.

In Fig. 2c and d we show the simulations for 50% surferage. In Fig. 2c the dielectric capacitanceCd 5 0.5CH is notligible; hence, it may be appreciated that the capacitance lcovered electrode tends to become parallel to that of thetrode. However, the behavior predicted by Eq. 9, shown btinuous line in Fig. 2c, is not reached by the capacitance d

)ithete sur-

hehe nano-s nopo-)

,g(fa-us

rolyte,f!n of

practical range. In Fig. 2d, the dielectric capacitance is small,Cd

5 0.1CH ; thus, at the potentials close to the apparent flatband theeto

fluow

nontre p

thityblesena0.te

thencecnd

calec

S

an. Theanes

enoenina cectrisnqu

@1

etb

strboEpe

spwish

e caronfr

cs

-endse

OH andth an Au-

analyzer

ained waselec-thesetive and,ce of thescribes

mples waseasuredFig. 3.

r the bareFig. 3b

f sampleurementsplots fol-

l but have

Table I. Main characteristics of the samples used in this study.

x.ition

anatasenatasenatase,tile

substrates1 andBareat pH

Journal of The Electrochemical Society, 150 ~6! E293-E298~2003!E296

free substrate surface determines the capacitance of thelectrode, as shown by the dotted line, that correspondsNonetheless, the dielectric covering exerts an increasing inthe potential becomes more positive, giving rise to the dcurvature of the 1/C2 line.

The dielectric capacitance spreads through the nanetwork.—So far it has been assumed that the dielectric coto the capacitance consists of a thin layer, or a set of singlcontacting the substrate; hence, in the previous analysisenhancing factor wasg 5 1. Finally, we consider the possibilthe electrical field enters the nanostructure to a consideraalthough emphasizing that this situation is unlikely becaueffective shielding by the liquid phase surrounding theticles. In Fig. 2e and f we take a low covering fractionf 5which is expected in a nanoporous electrode in which no incovering of the substrate was performed. In Fig. 2eenhancing factor isg 5 5, and the behavior of the capacitato this event is similar to that already commented in connFig. 2c. In Fig. 2f we assume a large enhancement ofg 5 50 adielectric capacitanceCd which is larger than the Helmholtztance. As a result the capacitance of the nanoporous elarger than that of the bare electrode, so in this case the Mthe covered electrode is below the substrate line.

Potential distribution.—Let us consider semiconductor nticles attached to a conducting substrate as shown in Fig. 1potential difference between the bulk of the substrate and tthe solution,Vtot , is distributed differently in the two paths, ias indicated in Fig. 1. The potential distribution in path i is din the Appendix. Let us consider path ii. We denoteVns the potdrop from the junction SnO2 /TiO2 to the bulk solution, sVtot 5 Vb 1 Vns and Vns 5 Vd 1 VH , where Vd is the potdrop in the TiO2 nanoparticles~see Fig. 1!. We adopt agaassumption that the contribution of the TiO2 nanoparticles isstant capacitanceCd ~assuming that the TiO2 is in the dielstate!, in which case the total differential capacitance compnanoparticle and the Helmholtz layer is a constant,Cns, giveEq. 3. The space charge in the substrate,Qb in Eq. A-4, eQns 5 CnsVns. Hence, we get

Vns 5 2U2F S 1 1Va 1 U0

U2D 1/2

2 1GwhereU2 is defined in Eq. 8. The result in Eq. 11 allows dnation of the three potential drops~Helmholtz, dielectric, andbending! in which Vtot is divided in path ii. This potential dition is consistent with the analysis developed in Ref. 2 ajunction potential, with the different notationb 5 2eU2 /kBT in54 of Ref. 2. Equation 11 is even more general than theanalysis because it accounts explicitly for the effect of thholtz layer over the nanoparticles.

Experimental

The effect of the coverage was studied by impedancecopy over two different SnO2(F) substrates, labeled FTO1,approximate sheet resistance of 20V/h, and FTO2, with aresistance between 10 and 15V/h. Two pieces of FTO1 werered with TiO2 colloids of different average diameter, prepreported in Ref. 22, conforming samples 100 and 102, andof FTO2 was covered in a similar way with TiO2 nanoparticlesDegussa P25 resulting on sample P25III. The characteristisamples are summarized in Table I.

Electrochemical measurements were done in a threecell using a platinum wire as counter electrode and a staAgCl in 3 M KCl as reference electrode. The electrolytes u

coveredEq. 6.

ence asnward

porousibutionarticlese area-thatextent,of the

nopar-2,ntional

area-e duetion toa

paci-trode isline of

opar-he totalbulk ofd ii,cribedtialthattialtheon-ricing thebyals

1#

ermi-andibu-ut the

q.reviousHelm-

ectros-th aneetov-ed ase piece

omof these

lectrodeard Ag/d were

solutions of distilled water adjusted to pH 11 and 2 with KH2SO4 , respectively. Impedance spectra were obtained witolab PGSTAT-30 potentiostat equipped with an impedanceand controlled by a PC.

The range of potentials where these spectra were obtchosen to avoid both charge transfer from the SnO2(F) to thetrolyte and electron accumulation in the TiO2 nanoparticles. Atconditions the behavior of the samples was fully capacitherefore, it was only necessary to consider the resistanelectrolyte~in series! to obtain an equivalent circuit that dewell all the impedance spectra. The capacitance of the saobtained by fitting this simple RC series circuit to the mimpedance spectra. The calculated MS plots are shown in

Results and Discussion

Figure 3a compares the capacitance results obtained foFTO1 with those of samples 100 and 102 at pH 11, andcompares the capacitance of the bare FTO2 with that oP25III at two pH values. It can be seen that in the meascorresponding to a given substrate and pH value, the twolow straight lines with the same apparent flatband potentia

Bare FTOFTO covered with

nanocrystalline TiO2

Samplename

Sheetresistance~V/h!

Samplename

Colloid diameter~nm!

Approcompos

FTO1 20 100 91.1 100%102 31.4 100% a

FTO2 10-15 P25111 25.0 70% a30% ru

Figure 3. MS plots of the measured capacitance of the conductingand the nanostructured TiO2 electrodes listed in Table I.~a! Bare FTO~s! nanoporous TiO2 samples~h! 100 and~,! 102 at pH 11.~b!FTO2, at pH~s! 2 and~h! 11, and nanoporous TiO2 sample P25III~,! 2 and~L! 11.

different slopes, depending on the type of colloids deposited on thesurface of the SnO2(F). This behavior is similar to the case de-

o tligcaanth

urtna2

esthacto10th

e

dicre

k oomsis, thtratruthd

etrthch

aponwMS

c1

etdiof11q.anA

ese32/I

rand bwelecco

Ife se,ctierpon d

cedicmurin

served for rutile-anatase and pure anatase colloids is explained bylattice matching rules with the substrate.

e Helm-f the ap-

omisio´n-1045.undations.

ts of this

ctor

spondsrfaces the

-1#

tivityIf weof the

ge e, as

A-2#

e SnO2in the

o theolution

-3#

-4#

fferentialce equalside of the

-5#

-6#

A-7#

-8#

series,

-9#

Journal of The Electrochemical Society, 150 ~6! E293-E298~2003! E297

scribed in the previous section in which the contribution tpacitance due to the addition of the TiO2 was considered neg

Therefore, for nanoporous TiO2 electrodes the specifictanceCns in Eq. 3 is much lower thanCH , which means thatCdthe factorg are small. This result supports the assumptionfield lines penetrate little into the TiO2 network, not much fthan the first particle, as suggested by several theoretical a

Because the deposition of the nanoparticles just reducfective substrate area that contributes to the capacity,5 (1 2 f )Ci , it is possible to calculate the coverage fataining the values of 16% for sample 100, 10% for sample70% for sample P25III. The difference of covering betweenfirst samples is related to the different average sizes of thfrom which the films are made~see Table I!.

The higher covering factor found for sample P25III is inof a greater affinity of the Degussa powder, which is a mixtucrystalline structures anatase, 70%, and rutile, 30%, to sticSnO2(F) surface than that of samples 100 and 102 made franatase colloids. This fact may be explained on the balattice matching rules. As observed by X ray diffraction~XRD!SnO2(F) surface is formed by cassiterite crystals with teunit cell with lattice parametersa 5 4.7 andc 5 3.2 Å. The sture of rutile TiO2 phase contained in Degussa powder hasunit cell and very similar lattice parameters,a 5 4.6 an5 3.0 Å, while the anatase unit cell is body centered twith parametersa 5 3.9 andc 5 9.5 Å, very different tofrom the substrate. Therefore, it is easier for rutile to attacover more fully the SnO2(F) surface than for anatase.

In Fig. 3b it is shown that changing the pH modifies theflatband potential by 570 mV@for the measurements donethe bare and covered SnO2(F)], which is in good agreement59 mV displacement per unit of pH. From the slopes of theof the two uncovered SnO2(F) samples, donor density waslated, obtainingND 5 1.1 3 1021 cm23 for FTO1 andND 53 1021 cm23 for FTO2, which agrees with the lower shetance of the second SnO2(F) sample. According to Eq. 1 thisence on the donor density~slope in the MS plot! is the causedifferent values of the apparent flatband potential at pHobserved in Fig. 3a and b. By using this difference and EHelmholtz capacity was calculated to beCH 5 9.3mF cm22

thus, the flatband potential obtained is2U0 5 20.18 V vs.AgCl ~0.03 V vs.NHE!.

Finally, we note that an earlier version of the model dhas been used recently23 to analyze the characteristics of thIredox couple in TiO2-covered tin dioxide.

Conclusions

We have studied the capacitance characteristics of tconducting substrates such as SnO2 , totally or partially coveredielectric medium, in contact with an electrolyte. We shothe triple contact, conducting substrate/~nanoporous film, elyte!, can be studied by dark capacitance measurements,the nanoporous film electrode with the bare substrate.

The theoretical analysis indicates two extreme cases.electric nanoparticles completely block part of the substratwhile the rest of the surface is exposed to the electrolytcrease of the MS slope is interpreted in terms of area reducriterion for determining simply the fraction of surface covto obtain linear MS plots with the same apparent flatbandThe covering of a significant fraction of the surface by a thitric layer shifts the apparent flatband negatively.

In the case of nanoporous TiO2 electrodes, the capacitantribution of the network in the dielectric state is very low, inthat the field lines penetrate little into the TiO2 network, notfurther than the first particle. The different surface cove

he ca-ible.paci-dat the

herlyses.,4

the ef-t isCr, ob-2, ande two

colloids

ativeof thever the100%of theegonalc-

e samecagonaloseto and

parentboth

ith aplots

alcu-.7resis-ffer-thethat is1, thed

g/

cribed2

sparenty ad thattro-mparing

the di-urface,the in-on. Theage istential.ielec-

con-atingchg ob-

The use of differently doped FTO substrates allowed thholtz capacity to be calculated, and therefore, correction oparent flatband potential for the effect of band unpinning.

Acknowledgments

The authors acknowledge support of this work by la CInterministerial de Ciencia y Tecnologı´a under project PB98A.Z. is thankful for the support of the Israel Science Fofounded by The Israel Academy of Science and Humanitie

Universitat Jaume I assisted in meeting the publication cosarticle.

Appendix: Mott-Schottky Plots of Degenerate SemiconduElectrodes

The capacitance of the SnO2(F)/electrolyte contact correto path i in Fig. 1. The space-charge region at the SnO2 suprovides a capacitance/areaCb ~band bending!16-18 which haexpression

Cb 5 « r«0 /w @A

where« r is the dielectric constant of the SnO2 , «0 the permitof free space, andw is the width of the depletion region.denoteND the dopant density,w can be expressed in termsband bending potentialVb and the positive elementary char

w 5 S 2« r«0Vb

NDe D 1/2

@

Let us denote -U0 the electrical potential in the bulk of thfilm, with respect to a reference in solution, in equilibriumdark. As shown in Fig. 1, when a potentialVa is applied tsubstrate, the total potential dropVtot across the substrate/sinterface is

Vtot 5 U0 1 Va @A

The charge/area in the depletion layer is

Qb 5 NDwe @A

In the Helmholtz layer it can be assumed that the dicapacitanceCH is a constant; hence, the integral capacitanthe differential capacitance and the charge in the solution ssurface is

QH 5 CHVH @A

The charge-neutrality constraintQb 5 QH can be written

V1Vb 5 VH2 @A

whereU1 is defined in Eq. 1. Therefore

Vtot 5 Vb 1 VH 5 Vb 1 ~U1Vb!1/2 @

and solving these last two equations forVH we get

VH 5 2U1F S 1 1Va 1 U0

U1D 1/2

2 1G @A

The potential dropVtot is shared by the two capacitors inCb , and the Helmholtz capacitance/areaCH , hence

1

C5

1

Cb1

1

CH@A

Using these expressions forCb andVb , Eq. A-9 can be expressed inthe form

-1

MS

lec

60

em

20

11. M. Turrion, B. Macht, H. Tributsch, and P. Salvador,J. Phys. Chem. B,105, 9732~2001!.

12. H. Tributsch,Appl. Phys. A: Mater. Sci. Process.,73, 305 ~2001!.d S. So¨-

,ectro-

5, 711

enges.

, 741

n,Appl.

. Bogdanoff,

P. Salvador,

1723

em. B,

oanal.

Journal of The Electrochemical Society, 150 ~6! E293-E298~2003!E298

1

C2 52

NDe« r«0~Va 1 U0! 1

1

CH2 @A

Thus, the apparent flatband potential determined from the2(U0 1 U1).15,18

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