Frumkin corrections for heterogeneous rate constants at semiconducting electrodes

c~ B

E L S E V I E R Journal of Electroanalytical Chemistry 438 { 1997) 191-197

Frumkin corrections for heterogeneous rate constants at semiconducting electrodes t

William J. Royea, Olaf Kriiger, Nathan S. Lewis Dicision of Chemist~' and Chemical Engineering, California Institute of Technology, Pasadena. CA 91125. USA

Received 20 September 1996; received in revised form 9 Dccem~r 1996

A~t ruc t

Fmmkin corrections for semic~,~,2uctor electrodes have been evaluated in both depletion and accumulation comlRh~s. In conjuuction with ~he Gouy--Chapmap. Stem m~xiel, a finite difference approach was used to calculate the potential drop in a d e ~ e d seraicomluct~" and in the compact and diffuse layers of the contacting solution as a function of the potential applied to the solid~kluid interface. At potentials greater than 30 mV positive of the flat-band potential Era the potentizd drop across the solution accounts for less than 3% of the total potential drop across an n-type semiconductor of dopant density I x 10 ~s cm-3 in a medmnolh: solution of i.0 M LiC1. Urg~er these con~',.itions, the concentration of a non-adsorbing, dipositively-charged redox species at the outer HelmhoRz plane does not vary from its cgucentration in the bulk of the solution by more than 2%. This relatively small concentration gradient and potemial drop acia~s the t4,.~-nholtz layer combine to produce negligible Framkin correction terms for kinetic data at depleted semiconductor eiectrodcs c~4pa~d to those for metallic electrodes at the same appfied potential relative m the potential of zero charge. Lh~ler accumulation condifons, th~ potential drop across the solution is more significant, and the concentration of redox species at the surface can be as much as t~%-¢ as great as that in tim bulk of the solution. However, these conditions require an applied potential of - 1 V relative to Eeo. Add/¢kmally, under all conditions that were simulated, the correction to the driving force used to evaluate the heterogeneous rate constam dc, es not exceed 2% of the uncorrected heterogeneous rate constant. © 1997 Elsevie: Science S.A.

Keywords: Heterogeneous rate constants; Frumkin corrections: Semiconducting electrodes; Depletion condi~i.,n; Accumulalion cor~dition

1. Introduction

In contrast to the well-known equations that have been developed by Frumkin and others for correction of heterogeneous rate constants at metallic electrodes [1,2], few efforts have been made to determine the magnitude or form of such corrections for semiconducting electrodes. To date, experimental rate constant data for semiconductor electrodes have not been corrected for double-layer effects [3-9]. Additionally, theoretical models that have been developed to estimate the maximum inteffacial charge- transfer rate constant value under optimal exnergicity at a semiconductorlliquid contact have not considered such correction terms in their analysis of experimental data [ 10-12].

• Corresponding author. E-mail: [email protected]. This paper was presented at the International Symposium on Electron

Transfer in Protein and Supramolecular Assemblies a! Interfaces held in Shonan Village, Kanagawa, Japan on 17 to 20 March 1996.

0022-0728/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved. PII S0022-0728(96)05074-7

Such corrections might, however, be important when com- paring the theoretically predicted charge-transfer rate constants to experimental kinetic dala for charge-transfer pro- cesses across scmiconductorlliquid interfaces. In fact, a recent analysis has asserted that experimental charge-transfer rate constants for reaction of non-adsorix~, outer-sphere redox species at semiconductor elecmxtes may require Fmmkin-like double layer corrections of an order of magnitude to facilitate comparison to theoretical l:~ed~:tions [13]. A quantitative evaluation, of the Frumkin a~ncctioas for semiconducting electrodes is therefore the focus of this

paper. Two cases arc considered herein: a semiconductor dec -

trode in de'~letion and one in accumulation [14,15]. These two cases ~ "e treated separately, because there is a large difference in Lhe magnitude of the differential capacitance of the semiconductor relative to tim differential c~pacitance of the double layer under accumulation or depletion conditions [2]. For depletion conditions, we have u~_d an analytical treatment to evaluate tbe Frun~ih correctiem. In

192 W.J. Roye,,z et al. / Journal of Electroanalytical Chemist D' 438 (1997) 191-1t,:7

accumulation, the Frumkin corrections were evaluated nu- merically, using digital-simulation methods to solve for the potential dropped across the ,.lectrode as a function of the potential applied to the solidlliquid contact.

In both cases, we have used the Gouy-Chapman-Stem (GCS) model to describe the double layer in the electrolyte [2]. Although the GCS treatment is only approximately in accord with experimental data [2], the GCS model was adopted because it performs satisfactorily at the high electrolyte concentrations and low inteffacial potential drops that are characteristic of most experimet~tai situations en- countered in interfacial kinetic measurements at semiconductorlliquid interfaces [3-5,8.9]. In addition, the GCS model allows formulation of a numerical ct,:~parison between the results presented in this paper and the Frumkin corrections for metallic electrodes.

2. Theory

For a semiconductor electrode in depletion, the equations governing the Frumkin corrections can be expressed as follows [2]:

-:q'bz

[Alo .p = [A]b,ake k.r (1)

rA 1 .qE. . 1 Jbulk k T

k , , . ~ = ~0,.~ [A]o.-----~ e " (2 )

In these equations, [A]onp/moleculescm -~ is the concentration of oxidized species A at the outer Helmholtz plane (Oi-lP). [A],.~] k is the concentration of A in the bulk of the solution, z is the charge on A. t~ 2 is the potential drop

vacuum level

E =Efb Esc m EHm Eom 0

I Semiconductor I

! i

k~

across the diffuse layer, q is the electronic charge; k , is the Boitzmann constant, T is the temperfi'i,fe, ket.Ej/cm 4 s- I is the heterogeneous rate constant [10] ~'or interfacial charge transfer at the flat-band potential Em of the semiconductor, k~t.~, is the experimentally observed heterogeneous rate constant at a specific poteutial E of the semiconductor electrode, a is the transfer coefficient, and E . is the potential dropped across the Hehnholtz layer at the applied potential of interest. Although an n-type sere!- conductor is used a,; an example throughout this paper, analogous equations are readily derived for p-type semiconductor electrodes.

Eq. ( l ) acce,mts for the effect of the ch~ge density in the electrode on the equt:ibrium concentration orofile of a charged ion that underg(,es faradaic cha:,?e trarlsfer at the electrode surface. This equation is identic;~t ;~ Ihe conventional Frumkin concentration-correction term fc: a metallic electrode [2]. Eq. (2) is similar in form to the Frumkin correction of the rate constant for a metallic electrode, but significant modifications are required in order to apply this correction to semiconducting electrodes. Since each semiconductor has a unique position of its energy bands relative to the Nernstian potential of the solution, various semiconductor electrodes will produce various driving forces for interfacial charge transfer even when the electrodes are maintained at identical potentials relative to the Nenlst potential of the electrolyte. In addition, the Butler- Voimer expression with a = 0.5 cannot be used to describe the del:,endence of the observed current density on the potential of a semiconductor electrode [10-12]. The conventional Fi,amkin correction, which uses the value of the overpotential and the Butler-Volmer relationship to relate the value of the experimentally observed rate con-

vacuum level

!

k G ~ ! m

J [A]0Hp ~ [A]bulk Ef

t E = [A]oHP ~ [~!bulk

J

x~ x 2

(a) (b)

Scheme 1. Potential drops at a se~'ficonductodliquid interlace (a) at the Itat-band potential and (b) in depletion conditions: E is the applied potential, E~: is the potential dropped across the. space-charge region of the semiconductor. E n is the potential dropped across the Helmholtz layer, and E D is the potential across the diffuse layer of the sotution (i.e. the potential at the OHP, xz). The position of the local vacuum level reflects the change in electrostatic potential at various positions in sp~ce perpendicular to the plane of the semiconductedliquid contact. The relative potential distribution is exaggerated in the scheme due to the large diffc,,'ences in the magnitudes of the variou~s potential drops at the semiconductorlliquid contact.

W.J. Royea et al. / Journal of Electroanalytical Chemistm. 438 { 1¢97) 191-197 193

stant to the value of the standard rate congtant [2]. is therefore not a particularly informative quantity for a semiconductorlliquid contact.

Instead, a more convenient epprouch is to set the reference potential of the semiconductorlliquid contact to be equal to the tiat-band potential of the semiconductor electrode Efb. At this potential, zero net charge exists on the semiconductor electrode, so the Frumkin correction term of Eq. ( l ) vanishes (Scheme 1). Additionally, at the flat- , "d potential $2 = 0 and E H = 0, so the redox acceptor experiences the full driving force for the interfacial charge transfer process due to the potential difference between the conduction band edge of.the semiconductor electrode and the formal putent~ai of the redox species. At other applied potentials, some of the potential will drop across the Helmholtz layer (Scheme l). This potential drop will change the value of the inteffacial driving force (by an, amount qE H ) experienced by a redox species located at the outer Helmholtz plane. We assume for simplicity that a Butler-Voimer relationship with a = 0.5 can be used to describe the variation in this rate constant over small changes in interfacial driving force, although a more rigor- ous treatment would utilize the Marcus-Gerischer formal- ism to perform the relevant analysis [12]. Within these constraints, Eq. (2) incorporates both the concentration gradient correction term of Eq. (I) and the kinetic correction term due to apparent band-edge movement, in order to relate kct.E to the value of the rate constant that wmfid bc measured at the tint-band r~t.ential of the semiconductor, /%t,E,~ [2].

In accumulation conditions, Eq. (1) also describes the Frumkin correction for the concentration of oxidized redox species at the OHP. However, in contrast to depletion conditions, the rate constant is not a simple function of the incremental change in interfacial driving force across the solidlliquid interface, becau,qe the distribution in energy of the occupied electronic states of the semiconductor changes significantly as the electrode is',',.~iased into accumulation. The interfacial rate constant is thus a complicated function of potential in this region. For accumulation conditions, we have therefore only computed the Frumkin-like correction that is required to describe the potential actually experienced by the redox species at the OHP at a given potential of the semiconductor electrode.

Within the GCS theory, the differential capacitance of the electrolyte is given by the reciprocal relationship [2]

1 1 ! (3 )

C~o,, C a C n

where C a is the differentiM capacitance of the Helmholtz layer, C D is the differential capacitance of the diffuse layer, and C~ot, is the total differential capacitance of the solution.

At a semiconductorlliquid interface, the differential capacitance of the semiconductor space-charge region C~ and the differential capacitance of the solution C~oD, are

represented by capacitors connected eteclr~cally in series [12,14-16]. Thus, the total differential capacitance of the semiconductorlliquid junction can be expressed as

I I I I . . . . + o + _ _ ( 4 ) C,o,~, C~ C. Co

For an incrementa| ~pp!ied potenti~ A E, the potentials across the various capacitive elements are readily corn- puted to be

a E = aE,~ + a E . + a E o ( 5 )

:vith

( C,o,o, %, t aE,~faex i co c. ] (6a)

aE. f a E x { l c,o,~, C,o~ I , c~ ~ ! (~) ( C,o,~, c,~,,)

AE o = AE X ! C,~ C . (6c)

The actual applied potential £ is most conveniently de- fined for computational r:;r:', ses relative to tim flat-band ~otential~ at which E n = E o = 0. From this reference po- iential, Eqs. (6a), (6b) and (6c) can be integrated numeri- cally, using a finite difference approach for steps AE, to produce the de.sir-~i potential E. This process yields values oi E~, E o, and E n, hhe integral potential drops across the semiconductor, the diffuse layer, and the Helmholtz layer respectively, as a function of the electrode po~ntial. In this notation, 02 = Eo = E - E~ - E . . The dis~bufion o f electrons in the space charge region of the semicomh~ctor was assumed to be insensitive to the presence of a f a r a d ~ current across the solidlliquid contact because the transpcnt of charge carriers in the solid is much more rapid than mass transport or diffusive motion of redox ions in the solution.

3. Resul t s

3.1. Depletion conditions

Under conditions that produce a depletion of majodvy carriers in the space-charge region of a nonMegenerate semiconductor electrode, the differential capacitance of the semiconducting phase is given by the Mott-Schottky equation [16]:

I/2 - 2

C.~ = E - e ~ - ( 7 )

where e~ and N a are the dielectric constant and dopant density of the semiconductor respectively, and 60 is the permittivity of free space. Eq. (7), which was obtained uslag th~ deplcdo, avv~oximadon [16], has ~el , shown to describe the value of C~ to within a few percent under

194 w.J. Royea et aL / Journal of Electroanalytical Chcmistry 438 (1997) 191-197

depletion conditions of a semiconductor electrode [17]. A more exact treatment is available in the literature that includes the differential capacitance arising from mobile minority carriers under inversion conditions, but the sim- plified expression of Eq. (7) will be used in the discussion herein because this expression is a very accurate descrip- tion of C~,_. under moderate depletion conditions.

An expression for the differential capacitance of the solution containing a z': z' electrolyte is given by [2]

1 Xone ! - - +

e , e o 2 e h e z ,2q2[E]/kl t T I/2 cosh z'qdP2

where Xou e is the width of the Helmhohz layer, ¢~ is the dielectric constant of the solvent in the Helmhoitz layer, ~b i~ the dielecwic con~t:lnt of the bulk solvent, z'q is the charge on the ions of the electrolyte, and [El is the concentration of electrolyte in the solution. The first term in Eq. (8) describes the differential capacitance of the Helmhoitz layer, and the second term accounts for the differential capacitance of the diffuse layer.

It is useful (although not required) to introduce an approximation to aid in the evaluation of ~2 as a function of E. Since the dopant density of non-degenerate semiconducting electrodes is typically only l014 to 1017cm -3, a first-order ca!cma,'ion using F.q~ (7) an~' (~) at an electrolyte concentration > 0.1 M indicates thai C~ <~: C~,,t . under depletion conditions. As a result, most of the potential drop will occur across the space charge region of the semiconductor. Thus ~b 2 will be sufficiently small that z 'q~b2/2kaT"¢: 1, so c o s h ( z ' q d ~ 2 / 2 k a T ) = 1.0. This approximation can be checked for consistency after E~, ED, and E u are determined. Once the values of E~ and E H are known, ~b z can be computed, and the Frumkin terms of Eqs. (1) and (2) can then be calculated for the system of interest.

Fig. l(a)-Fig, l(c) plot the values of E~, E , , and ED as a function of E - Efb for various dopant densities of an n-type semiconductor electrode in contact with a 1 M solution of a I:1 electrolyte in CH3OH. Since the values of E u and ED were very small under all conditions of interest, the quantities 1 - ( E H / E ) and 1 - ( E D / E ) have been plotted in Fig. l(b) and Fig. l(c) respectively. For these calculations, a Helmhoitz layer thickness of x 2 = 5 × 10 -s cm and a Helmholtz layer differential capacitance of C a = 5 p,F cm-2, representative of methanolic solutions of 1.0M LiCl [18], were used in Eq. (8). These values produce a dielecLOc constant of 3 for the solvent layer near the surface of the electrode, which is in agreement with experimental data on the differential capacitance of the Heimholtz layer for a 1 M electrolyte composition [18]. Using these ve!u~'~: the computed value of E H is generally on the order of 1 to 10mV l'or sem.;c~ndu~io~, i~dvh,~ a dopant density -4 1 × 1016cm-~, while E n is ca. 20 to

1 .GO0

0.800

0.000

IlJ 0.400

0.200

0.000 0.00 0.20 0.40 0.00 0.00 1.00

S - F<,. (V)

"t.000 ~

, , ' _ . 0.~C " ' " " • . . . . . . . . . . . .

O.BO0 0.00 0.20 0.40 0.60 0.80 1.00

S - F.,~ (V)

'.= ;¢=:.. ....... :::::::::::::::::::::::::: 0.090

0.092 0.00 0.20 0.40 0.60 0.00 1 .~

E - F~b (Vl

Fig. I. Plots of (a) E,~, (b) I - ( E n / E ) , and (c) I - ( E D / E ) as a function of E - Ero for an n-Si electrode in depletion. Potentials were computed for the various dopant densities indicated m the legends.

30mV for a semiconductor with a dopant density of I × 1017cm -3 (Fig. l(b)). The value of ~2 is typically computed to be < I mV for all dopant densities investi- gated in this work (Fig. l(c)).

Fig. 2 depicts the value of the Frumkin correction term that describes the com, entration profile of redox species in the electrolyte, for a semiconductor with dopant density I × l0 '~ cm -3, resulting from the potential drop computations that are presented in Fig. 1. In these computations, the redox species was assumed to have a charge of z = + 2, so that the Fr , mkin corrections of Eqs. (1) and (2) could be evaluated for some of the most highly charged outer- sphere redox species that have been used to date in kinetic measurements at semiconductor electrodes [3-9]. As displayed in Fig. 2, [A]om, differed by no more than 2% from its value in the bulk of the solution, even at potentials greaiel" than + 1.0 V vs. Efb. Even extreme cases which were net likely to be established experimentally, such as a

W.J. Royea et al. / Journal of Electroanalytical Chemistry 438 (1997) 191-197

0.012 [

0.00:3

0 % " " ' " o . , . • ~ ~ " .00 o.ae 0.~o 0.~o 0 1 ~ 1.00

e - E,,, 00

Fig. 2. The ratio of the concentration of redox species at the OHP relative to the concentration of redox species in the bulk of the solution ptotted vs. E - Eeo for an n-type semiconductor electrode in depletion. A dopant density of I × 10'scra -'~ was used for these computations.

semiconductor of dopant densi ty 1 × 1 0 ~ c m -~ at an applied bias of + 1 0 V vs. E~,, only produced a 27% devia- t ion o f [A]oa P relative to [A]b~n~.

For a semiconductor of dopant densi ty I × 10~Scm -3, Fig. 3 depicts the value of k : t . r J k ~ o e that results f rom Eq. (2). In no case was this correct ion s ignif icant in magni tude, with computed devia t ions o f ke~.r relative to ke~.r,~ be ing < 10% tbr all condi t ions likely to be encoun- tc :ed exper imental ly . The cor rec t iom of Eq. (2) were < 30% of ke~ r for semiconductors hav ing dopant densi-

ties ~ 1 × 10ng'cm -3 (for 0 . 0 3 < E < 1.0V), and were < 100% of k , r for a semiconductor of dopant densi ty 1 × i0 cm ( fo r0 .0 . E - . 0 . 6 V ) .

3.2. Accumulation conditions

In accumulat ion, c losed-form expressions have not been der ived for C~ as a funct ion of E. A digital simu~afiou was therefore util ized in order to evaluate the F rumkin

1.10.

t.o?

? =I /

0.00 0.20 0.40 0.60 0.80 1.00 E-Ero (VI

Fig. 3. A plot of the Frumkin correction term for the heterogeneous rate constant for electron transfer from an n4ype semiconductor electrode in depletion (N o = I × 10nScm 3) to a ~ o x species in tl~ solution phase. The expe,imentally observed rate constant ket.E is corrected to its value at the point of zero net charge in the semiconductor, k,~.r . The correction accounts for a potential drop across the Helmholtz layer affecting the rate constant as well as for effects of electrostatically-in- daced concentration gradient~ between the OHP and the bulk of the electrolyte solution.

195

e ' °e [ (a) /

I~. - F.~ l v l

0.oo

-O.lO

-~.2G

-0.39

~ 4).40

~ , ~

,IMm

4 .O0 -0.80 - 0 . N -0.40 .0.20 0.N s - s ~ 00

0 . ~

.0.005

U~'0.010

-0.015

-0.020 - I .N -0.N -0.m -0.40 -0.20 0.00

Z" Z., 00 Fig. 4. Potential dropped across (a) the semiconductor, (b) the Heimho~ layer, and (c) the diffuse layer as a function of E - F~ fox ~ n-Si electrode in accunmlation. All Wxan~els ~e given in the text.

correct ions for the semiconductor~iquid interface in this appfied potential regime. T h e G C S theory was again used to provide an approximate ( ~ r i p t i o n o f the potential dis t r ibut ion in the electrolyte. T h e s imulat ion was per- fo rmed wi th the T o S C A program, which self-consistently solves Po i s son ' s equat ions in a semiconductor electrode as a funct ion o f potential , subject to var ious user-specif ied b o u n d ~ condi t ions and initial condi t ions [19-21] . The potential drops across the double layer o f the solut ion were calculated f rom the conduct ion band-edge movements that were computed using the digital s imulat ion program.

Fig. 4 (a ) -F ig . 4(c) depict the dependence o f E= , E H, and E D on E - E r b when an n-Si electrode o f dopant densi ty I × 10XScm -3 is dr iven into accumulat ion. To p e r f o | m these calculat ions, a 1:1 ratio o f oxidized to reduced form of a cobal tecene redox couple was assumed. This redox solut ion produced a barr ier height o f 1 2 0 m V

196 W.J. Royea et al. / Journal of Electroanalytical Chemist O' 438 (1997) 191-197

between the equilibrium conduction-baad edge energy of the semiconductor at the solidlliquid junction and the equilibrium Fermi level of the semiconductor[liquid contact. A charge-transfer rate constant of l × 10-JScm4 s -~ was used in order to produce current density--potential curves that displayed experimentally reasonable current densities. As displayed in Fig. 4(a), the si :ulations with these input parameters reveal that an increasing fraction of the applied po~e~ttial drops across the double layer of the solution as the potential of this n-type electrode becomes more negative. For example, a - 1.0V total applied bias vs. Ero produces a total potential drop across the solution of ca. - 0 .63 V. The value of ~b 2 is also significant in accumulation, and determination of $~ requires the incor- poration of the hyperbolic cosine term in Eq. (5) to compute C n. Values of C o were obtained by computing E o and successively re-evaluating C o, E n, and E , until they converged to within 99.9% of their preceding values.

Fig. 5 illustrates the Frumkin correction term of Eq. ( I ) that results from the potential distribution of Fig. 4. For typical biases into accumulation, with En, - E < 50f' ;aaV, [Ainu r differs from [A]b,lk by up to a factor of !.3. At an applied bias of - 1.0 V vs. Efb, the ratio of [A]oHJ[A]h,lk is 2.2.

Fig. 6 displays the value of ( E - E n ) / E , which represents the fractional deviation of the potential experienced by an acceptor species at the OHP from the value of the potential applied to the semiconductor[liquid interface, as a function of E - E e o . In general, it is not possible to provide an analytical method to extrapolate the corrected rate constants to either the flat-band potential or to the standard Nernstian potential of the redox species, so the correction procedure for this situation was limited to computing the true potential experienced by the redox species at the OHP. In this fashion, the data of Fig. 6 allow correction of experimentally observed ket,E - E data to the actual /%t.E,~- E data to account for the actual potential experienced by the redox species undergoing the interfa- cia! charge transfer event. These rate constants can then be

,s[ 2.0

1.5

1.0 -1.00 -0.80 -0.60 -0.40 -0.20 0.00

S - Ef b (V)

Fig. 5. The ratio of the concentration of redox species at the semiconductor surface to the concentration of re,~ox species in the bulk of the solution vs. E - Etb for an n-St electroa¢ in accumulation.

1.000

,-f /+ / 0'98-0.00 -O.aO -0.60 .OAO ' I , I , i 0 ' O.O0

s . Sfb (V)

Fig. 6. A plot of the driving-force correction influencing the heteroge- neons rate ,':.~stant for e:ectron transfer of a,, I)-')~,J semiconductor driven into accumulation. The quantity ( E - E~)r" L represents the potential experienced by an atceptor species Ioca,:,,~ at the OHP.

subjected to further analysis that might be appropriate to the system under study.

4. Discussion

l 'he computations presented herein reveal that the Frumkin correction terms for a non-degenerately-doped semiconductor electrode in depletion are negligibly small compared to the typical error associated with experimental determinations of charge-transfer rate constants of semiconductor electrodes. In depletion, the concentration of a non-adsorbing electroactive species at the OHP of a semiconductor[liquid contact is essentially identical to the concentration of the redox species in the bulk of the solution. Furthermore, the potential drop across the Helmholtz layer is a very small fraction of the potential applied to the electrode. Thus, within the framework of the GCS model, the observed rate constant is essentially the same as the Frumkin-corrected rate constant. Most experimental rate constant values that have been quoted previously in the literature for semiconductor[liquid contacts under depletion conditions therefore can be reliably viewed as excellent approximations to the Frumkin-corrected values for these charge transfer rate constants [3-5,8.9],

This behavior can be readily understood from the basic properties of a semiconductor[liquid contact. In depletion, the small value of C~: compared to C a implies that even large positive excursions in the potential applied to an n-type semiconductor[liquid interface produce only small charge densities in the electrode. For instance, at E = + 1.0 V vs. E~. the total ch~ge den,gity in a semiconducting electrode of dopant density 1 × 10~Scm -3 is only 9.3 × 10 -9 Ccm -2, whereas a + 1.OV potential applied to a metallic electrode (relative to its potential of zero charge) in contact with the same electrolyte produces a surface charge density of about 10 -6 Ccm -2. Since the charge density in the semiconductor electrode is so small, the potential dropped across the liquid side of the double layer

W.J. Royea et aL / Jourmd of Electr~mnalytical Chemistry 438 ¢ 1997) 191-107 lcj7

of a semiconductor[liquid junction is very small. There- fore, th~ Ftumkin correction terms of Eqs. ( ! ) and (2) are e s s e n t i a l l y n e g l i g i b l e f o r t h e s e t y p e s o f semiconductorlliquid contact.

The computations underscore an odvao~t~ge of sem,.'cen- ductor electrodes r¢lat]~,: ,o conveational solid metallic electrodes: since C,~ is so much smaller than C, ot., the flat-band poten0al, i.e. the point of zero charge, can be determined directly from differential capacitance vs. potential measurements. This arises because an analytical form of C~ vs. E,¢ is known for the space-charge region of the non-degenerately-doped semicondacting electrode (Eq. (7)), and because C= = Ct,~l under depletion conditions for typical semiconductor[electrolyte contacts.

In accumulation, the situation is somewhat more complicated, because no closed-form expression has been obtained for the differential capacitance of the semiconductor electrode. In contrast to depletion conditions, a significant fraction of the applied potential will drop across the Helmholtz layer, and, under some circumstances, this potential drop may require a significant Frumkin correction to the surface concentration of redox species. The exact partitioning of the applied potential between the semiconductor and the Helmholtz layer will depend on the doping level of the semiconductor, the electrolyte concentration, and the l~adaic interracial charge-transfer kinetics, he- cause rapid charge transfer will prevent accumulation of carriers at the electrode surface, minimizing band-edge movement and minimizing the rest, hing potential drop across the Helmholtz layer. For the conditions simulated in this work, the correction to the driving force used to determine the true heterogeneous rate constant is less than 2% of the applied potential relative to the flat-band potential of the semiconductor electrode. Thus, the combined influence of the Frumkin correction terms will not significantly alter agreement between experiment and theory except at the most extreme applied biases in accumulation.

5. Conclusions

Under moderate depletion conditions, a potential applied to a non-degenerately-doped semiconductor electrode drogs almost entirely across the semiconductor space- charge layer, and produces very little potential drop across the double layer of the electrolyte. For these conditions, the Frumkin corrections for charge-transfer rate constants within the GCS framework are small enough that they

need not be considered except when an accuracy of better than 30% is required in determination of the heterogeneous charge-transt'er rate constant. For ~miconductor electrodes in accumulation, although a considere, hle po t ion of the applied bias ,:an appear across the ~ tmion , significan~ Frumkin CO..'Tec~ions are requi~"d only at tx~tcmiais far removed from the flat-band potential t:f the ~miconductor.

Aclumwledgements

We acknowledge ihe Natienal Scieuce Foundation, grant CHE-9634152, and Kodak Corp. to,- suppo~ of this work, and O.K. acknowledges the DFG for a po~t-doctoral fel- lowship.

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Frumkin corrections for heterogeneous rate constants at semiconducting electrodes