Photoanodic Dissolution of n-InP: An Electrochemical Impedance

Z. Hens, and W. P. Gomes*. Laboratorium voor Fysische Chemie, Universiteit ... Publication Date (Web): July 26, 2000. Copyright © 2000 American Chemi...
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J. Phys. Chem. B 2000, 104, 7725-7734

7725

Photoanodic Dissolution of n-InP: An Electrochemical Impedance Study Z. Hens† and W. P. Gomes* Laboratorium Voor Fysische Chemie, UniVersiteit Gent, Krijgslaan 281 (S12), B-9000 Gent, Belgium ReceiVed: March 22, 2000; In Final Form: May 29, 2000

In this paper, we present a study on the electrochemical impedance of the n-InP photoanode in 1.2 M HCl, investigating the potential range between almost complete recombination and almost complete photoanodic dissolution. Two features characterize the impedance spectra obtained, i.e., the seeming absence of the recombination impedance and the presence of an inductive loop. Combination of the electrochemical impedance spectroscopy results with Mott-Schottky measurements indicates that the seeming absence of the recombination impedance can be accounted for by assuming that recombination occurs on a rapidly oxidizing decomposition intermediate. Moreover it is argued that, in addition, a slow oxidation step must be present in the overall dissolution process. Together with literature results, these hypotheses are implemented in an impedance calculation. It is demonstrated that the resulting impedance provides a good description of the experimental data, including the presence of the inductive loop. Furthermore, the correspondence between theory and experiment enables us to identify the slow step as the oxidation (by hole capture) of either the fourth or the fifth decomposition intermediate. Therefore, this case study demonstrates that, for the investigation of photoanodic dissolution reactions at n-type semiconductor electrodes, EIS is a technique that provides information complementary to that obtained by other experimental methods (such as IMPS).

the equivalent parallel resistance RP of the interface equal to16

1. Introduction Upon illumination, a flux of holes directed towards the semiconductor surface is created in an n-type semiconductor electrode polarized under depletion conditions.1,2 Typical for these so-called photoanodes is the potential-dependent competition between recombination and interfacial transfer of the photogenerated holes.3 At potentials close to flat-band potential, Vfb, the density of conduction-band (CB) electrons in the depletion layer is high enough to ensure that, for many semiconductor electrodes, all these holes are annihilated by recombination, either at the electrode surface or within the spacecharge layer. At more positive potentials, recombination is effectively ruled out due to the depletion of CB electrons. In this case, all photogenerated holes are transferred across the interface, leading to a potential-independent photocurrent density j∞. If the liquid constitutes an indifferent electrolyte, this hole transfer often involves semiconductor dissolution.4,5 Photoanodic dissolution of semiconductors has been investigated intensively for the last 20 years because of the fact either that it hampers the proper functioning of photoelectrochemical solar cells6,7 or that it enables the (photo)etching of n-type semiconductors.7 For these studies, extensive use has been made of time-dependent measurement techniques in general and of impedance techniques in particular.3 Intensity modulated photocurrent spectroscopy (IMPS) has been applied both for the investigation of recombination and of dissolution processes at photoanodes.3,8-10 On the other hand, electrochemical impedance spectroscopy (EIS) has mainly been used for the study of recombination processes (recombination impedance) solely. As demonstrated extensively by various authors in the case of the n-GaAs photoanode,11-15 surface recombination leads both to a capacitive peak in a plot of the interfacial capacitance versus the applied potential and to a high-frequency limiting value of † Research Assistant of the FWO-Vlaanderen [Fund for Scientific Research Flanders (Belgium)].

lim RP(ω) )

ωf∞

k BT e|jREC|

(1)

In contrast to recombination, investigations of semiconductor photodissolution using EIS are rare, probably due to the fact that the technique cannot be applied systematically if a true photocurrent plateau exists. Indeed, in the photocurrent plateau (i.e., j ) j∞), the dissolution process may lead to an infinite impedance, as shown further e.g. by eqs 7, 8, 31, and 32. The photoanodic dissolution of n-InP in 1.2 M HCl is well documented in the literature. Vermeir et al.17 and Preusser et al.18 demonstrated that 6 elementary charges are needed for the dissolution of one formula unit of InP. Evidently, as n-InP does not dissolve anodically in the dark, the first step of this dissolution process corresponds to the capture of a photogenerated hole by an unbroken surface bond (formally written as X0), resulting into a decomposition intermediate (written as X1). Subsequently, an IMPS study of the n-InP photoanode by Erne´ et al.10 showed on the one hand that recombination occurs at the semiconductor surface and on the other hand that, at low light intensity (j∞ < 10-6 A cm-2), three of the six oxidation steps involved in the anodic dissolution process are electroninjection steps. The authors argued that these three injection steps occur successively, having an increasing electron-injection rate constant. Unfortunately, their results did not allow them to locate the injection steps in the overall reaction mechanism. At high light intensities (j∞ > 10-3 A cm-2), the oxidation mechanism of two of these three steps changes into hole capture (due to the increasing hole density at the interface) so that only a single electron-injection step remains. In this paper, we present a case study of the photodissolution of n-InP in 1.2 M HCl using EIS in the potential region in which recombination competes with photodissolution. Two features characterize the electrochemical impedance spectra of this

10.1021/jp0010740 CCC: $19.00 © 2000 American Chemical Society Published on Web 07/26/2000

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Figure 1. Current density vs potential curve obtained at n-InP(hν) (ND ) 6.7 × 1017 cm-3) in 1.2 M HCl with potential sweep (20 mV s-1) from cathodic to anodic potentials. The division in potential ranges (A-C) is discussed in the text.

system, i.e., the seeming absence of the recombination impedance and the presence of an inductive loop. Starting from the results of Mott-Schottky measurements, it is shown that the seeming absence of the recombination impedance can be explained by assuming that recombination occurs on a decomposition intermediate. This idea leads to a set of model equations from which an impedance expression for the n-InP(hν)|1.2 M HCl system is derived. This impedance accounts for the inductive loop appearing in the impedance spectra. Moreover, it yields additional insight into the overall dissolution mechanism. In this way, this case study demonstrates that EIS is a valuable experimental technique for the investigation of the mechanism of dissolution reactions, providing information complementary to that obtained by IMPS. 2. Experimental Section All electrodes used in the experiments were (100) faces of single-crystal n-InP. The doping density of the crystals was either 5.0 × 1015 cm-3 (dopant unknown, MCP Electronics) or 6.7 × 1017 cm-3 (Sn-doped, MCP Electronics). The samples were mounted as disk electrodes with a diameter of 3 or 4 mm. Prior to measurements, they were polished on 50 nm Al2O3 powder and etched in 12 M HCl for 20 s. This etch removes a surface layer of at least 1 µm in thickness,19 resulting in a homogeneous but roughened surface. The 1.2 M HCl solutions were prepared using reagent grade concentrated HCl (37%). Experiments were carried out in a conventional, sealed threeelectrode electrochemical cell using a saturated calomel electrode (SCE) as the reference electrode. To keep the cell oxygenfree, nitrogen was bubbled through the electrolyte prior to and between all measurements while it was blown over the solution during the measurements. The electrode was illuminated using light (λ ) 488 nm) from an argon ion laser (Ion Laser Technology, Ltd.). The cell potential was controlled by an EG&G potentiostat (model 273) whereas a Solartron FRA (model 1250) was used for the impedance measurements. The amplitude of the small-signal potential perturbation (rms) was set equal to 10 mV. 3. Experimental Results 3.1. Current Density vs Potential Measurements. A photocurrent density vs potential curve obtained at n-InP (ND ) 6.7 × 1017 cm-3) in 1.2 M HCl under constant illumination (denoted as n-InP(hν)|1.2 M HCl) is shown in Figure 1. Three different regions may be discerned. In region A, the current density is effectively zero due to recombination of photogener-

Hens and Gomes

Figure 2. Mott-Schottky plots determined at a dark (dark circles) and an illuminated (open squares, j∞ ) 1.5 mA cm-2) n-InP (ND ) 6.7 × 1017 cm-3) electrode in 1.2 M HCl. The potential sweep is from anodic to cathodic potentials, and the measuring frequency is 8.2 kHz. ∆V0 indicates the shift of the extrapolation point with the V-axis for the data in the anodic potential range due to illumination.

ated holes and CB electrons. In region C, recombination is ruled out resulting in a current density (j∞) proportional to the current density of the photogenerated holes. Region B constitutes the transition between both regimes. Current density vs potential curves obtained at different light intensities are similar to the one shown in Figure 1, although the different potential regions shift slightly as a function of the light intensity. Current density vs potential curves recorded at the lower-doped material are similar to the one shown in Figure 1. 3.2. Mott-Schottky Measurements. Figure 2 shows, as an example, Mott-Schottky plots for the higher-doped n-InP in 1.2 M HCl, obtained by plotting the inverse square of the equivalent parallel capacitance of the interface (CP), registered at a high frequency, vs the applied electrode potential. The black dots and the open squares represent the results of a measurement at a dark and at an illuminated (j∞ ) 1.5 mA cm-2) electrode, respectively. The extrapolated Mott-Schottky plot for the dark electrode intersects the potential axis at -550 mV vs SCE, leading to an estimated flat-band potential of -575 mV vs SCE. This value agrees reasonably well with the more elaborate results presented in ref 20. Under illumination, the extrapolation of the Mott-Schottky plot from potentials more positive than 100 mV vs SCE intersects the potential axis at -350 mV vs SCE. The difference ∆V0 (cf. Figure 2) reflects a shift of the band edges at the semiconductor surface, indicating that positive charges are accumulated at the interface under illumination. Within the current-density range we investigated (0.3 mA cm-2 0.8, r∞/r0 becomes larger than unity, especially at higher light intensities. This is tantamount to the disappearance of the inductive loops in the spectra recorded close to the photocurrent plateau. Figure 6B shows that the dependence of r∞ and r0 on y at the lower-doped material resembles qualitatively to that at the higher-doped material (only one light intensity was used in this case). Quantitatively, r∞ and r0 are smaller by a factor 1.21.3 for the lower-doped material. The characteristic frequency of the inductive loop is plotted versus y in Figure 7A (two different light intensities). For a given light intensity, ω0 is largely independent of y in the range y > 0.5. Figure 7B shows the relation between ω0 and j∞ (loglog plot) determined at a fixed value of y (y = 0.7) (open circles, ND ) 6.7 × 1017 cm-3; open diamonds, ND ) 5.0 × 1015 cm-3). As the slope of the fitting line through the data points obtained at the higher doped material equals 1, a linear relation between ω0 and j∞ exists:

ω0 ) Rj∞

(6)

At y = 0.7, R is about equal to 5 × 105 C-1 cm2. It should

Figure 7. (A) Spectrum parameter ω0 obtained at n-InP(hν) (ND ) 6.7 × 1017 cm-3): open circles, j∞ ) 0.38 mA cm-2; black upward triangle, j∞ ) 1.5 mA cm-2. (B) ω0 as a function of j∞ for n-InP (ND ) 6.7 × 1017 cm-3) (circles) and n-InP (ND ) 5.0 × 1015 cm-3) (diamond).

Figure 8. Equivalent circuit related to surface recombination of photogenerated minority charge carriers (cf. ref 15). The impedances of the different circuit elements are given in the text.

be noted that the ω0 value obtained at n-InP (ND ) 5.0 × 1015 cm-3) does not fit the line simulating the data obtained at n-InP (ND ) 6.7 × 1017 cm-3). In the region y < 0.5, ω0 decreases steadily as the current density approaches 0, both at the lower and at the higher doped material. Since the data obtained at different light intensities remain equidistant, the linear dependence between ω0 and j∞ holds in the region y < 0.5 as well. 4. Discussion 4.1. Recombination Impedance. It was demonstrated by Erne´ et al. that recombination in n-InP photoanodes occurs at the semiconductor surface.10 Assuming the oxidation rate to be proportional to the hole density at the semiconductor surface, Vanmaekelbergh et al.15 showed that the electrochemical impedance related to surface recombination corresponds to the equivalent circuit shown in Figure 8. Writing the flux of electrons transferred from the conduction band to the recombination centers as25 βnnSs(1 - θ) and that of holes transferred from the valence band to the recombination centers as βppSsθ and neglecting electron excitation from the surface states, the impedances of the different circuit elements read (jREC, current density associated with electron-hole recombination; jPH,

Photoanodic Dissolution of n-InP

J. Phys. Chem. B, Vol. 104, No. 32, 2000 7729

Figure 9. (A) Simulated impedance spectra based upon the parallel connection of the interfacial capacitor (500 nF cm-2) and the recombination impedance [R1, 50 Ω cm2; R2, 250 Ω cm2; C, 40 nF cm-2 (open squares) and 10 µF cm-2 (open circles)]. (B) Equivalent parallel resistance corresponding to the impedance spectra shown in (A) (same symbols used).

current density associated with photogenerated hole flux)15

kBT e|jREC|

(7)

kBT βnnS 1 e|jREC| βppS 1 - |jREC/jPH|

(8)

e|jREC| 1 e2 s(1 - θ) ) kBT βnnS kBT

(9)

R1 ) R2 ) C)

As demonstrated by Vanmaekelbergh et al.,14,15 the capacitance C accounts for the capacitive peak observed in the MottSchottky plots of n-GaAs under illumination, whereas R1 corresponds to the high-frequency limit of the equivalent parallel of the recombination impedance. resistance RREC P Our measurements indicate that the typical properties of the recombination impedance are absent at n-InP(hν)|1.2 M HCl. Considering e.g. the results for the case y ) 0.5, the parallel resistance (i.e., R∞) constituting the high-frequency capacitive semicircle is at least 3-4 times larger than R1 since R∞/R1 = r∞(1 - y). In addition, the capacitive peak observed in a CP vs V plot is far too low as compared to the theoretical value expected for surface recombination. To understand these measurement results, let us first of all analyze the recombination impedance as a function of the capacitance C. In Figure 9A, we have plotted simulated impedance spectra (50 kHz > f > 2 Hz) corresponding to the parallel connection of the interfacial capacitor (500 nF cm-2) and the recombination impedance [R1, 50 Ω cm2; R2, 250 Ω cm2; C, 40 nF cm-2 (open squares) and 10 µF cm-2 (open circles)]. Clearly, taking C equal to 40 nF cm-2, only a single semicircle is resolved in the Nyquist plot whereas two semicircles appear if C equals 10 µF cm-2. Plotting RREC for both P cases (Figure 9B) demonstrates that only in the latter case the high-frequency limiting value is reached within the frequency range of the simulation. Since realistic values for the different circuit elements were used, this example indicates that the typical features of the recombination impedance will disappear if the capacitance C is sufficiently small. Unfortunately, if recombination occurs at fixed recombination centers, C cannot remain small throughout the whole potential region. Taking e.g. a density of surface states equal to 1012 cm-2, C will equal 6.4 µF cm-2 when all surface states are occupied by holes, a typical situation at potentials where dissolution competes with recom-

bination. Therefore, at more positive potentials, one would expect a considerable capacitive peak in a CP vs V plot and two capacitive semicircles at high frequencies (cf. Figure 9A, open circles) in the impedance spectrum. This contradiction with our measurements may be accounted for by assuming that recombination does not occur at existing recombination centers (i.e., recombination centers that exist regardless of the electrode process) but rather at intermediates of the dissolution reaction. In the case of n-GaP and n-GaAs photoanodes, this idea has been suggested before by Vanmaekelbergh et al., basing upon results of photoanodic stabilization measurements.26-28 As C equals the product of e2/kBT and the density of unoccupied recombination centers (cf. eq 9), it is given by e2xrec/kBT if recombination occurs at decomposition intermediates (xrec being equal to the surface concentration of these intermediates). This capacitance may be low throughout the whole potential region if the (pseudo-) first-order rate constant of the oxidation of Xrec (i.e., the intermediate at which recombination occurs) is large. For simplicity, we will assume recombination to occur on X1, i.e. the first decomposition intermediate. Hence, the oxidation of X1 must be a reaction step, fast enough to maintain the surface concentration of X1 (x1) low throughout the whole potential region, thus accounting for the absence of the features of the recombination impedance. In the case of hole injection at n-InP by Fe(CN)63- in 1.3 M KOH, a similar absence of the recombination impedance was found.29 This similarity between recombination at n-InP in 1.2 M HCl and in 1.3 M KOH provides additional support for the hypothesis formulated, including the assumption that X1 is indeed the intermediate at which recombination occurs, as this intermediate is probably affected least by changing the composition of the electrolyte solution. As considered by Oskam et al.,30 electron excitation from the surface states to the conduction band might be an alternative explanation for the absence of the recombination-impedance features. However, this assumption does not lead to a satisfactory description of the experimental spectrum parameters as a function of the normalized current density and is therefore not considered further. 4.2. Qualitative Picture of the Reaction Mechanism. In section 4.1, the hypothesis was formulated that recombination takes place at intermediates of the anodic decomposition (e.g. X1). If so, the absence of a significant capacitive peak indicates that the surface concentration of this intermediate (x1) must be low. However, Mott-Schottky measurements reveal a shift

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Hens and Gomes

(-e∆V0) of the band edges at the surface of 150-200 meV upon illumination in potential region C. Assuming the Helmholtzlayer capacitance CH to be equal to 5 µF cm-2, this band-edge shift implies a surface concentration of decomposition intermediates between 1012 and 1013 cm-2 (=CH∆V0/e). If this surface concentration were due to X1, a capacitive peak of the order of a few tens of µF cm-2 would occur31 (C ) e2x1/kBT). As the value of the experimental peak is far below this value, the accumulation of charge cannot be related to X1. Consequently, there must be (at least) one slowly oxidizing decomposition intermediatesdifferent from X1sin the dissolution process, causing the accumulation of charge at the interface. Assumingsfor simplicitysthere is only one, we will denote this intermediate as Xm. If one writes the oxidation rate by hole capture or electron injection of an intermediate Xr (1 e n e 5) as krpSxr [(kr: rate constant (cm2 s-1)] and as kr,injxr [kr,injxr: rate constant (s-1)] respectively, the (pseudo) first-order rate constant κm (i.e., either kmpS or km,inj) may be estimated from the surface concentration of Xm (xm). Since xm ranges between 1012 and 1013 cm-2, κm ranges32 between 102 and 103 s-1 when j∞ equals 1 mA cm-2. Let us investigate whether this slow oxidation step can be related to the oxidation of one of the three decomposition intermediates injecting electrons at low light intensity (abbreviated in what follows as DIsIELLI or, in singular, as DIELLI) as observed by Erne´ et al.10 The first-order rate constants for electron injection of these intermediates were estimated in ref 10 to be 6 × 102 s-1, 6 × 105 s-1, and larger than 6 × 105 s-1, respectively. Hence, it follows that only the first DIIELLI may possibly correspond to Xm. However, from the results of Erne´ et al.,10 it also follows that, when j∞ is of the order of 10-3 mA cm-2, electron injection and hole capture by the first DIIELLI have comparable rates. Therefore, the pseudo-first-order rate constant for hole capture by this intermediate will be approximately 6 × 105 s-1 when j∞ equals 1 mA cm-2. Consequently, none of the DIsIELLI can correspond to Xm, which has a pseudo-first-order rate constant between 102 and 103 s-1 when j∞ equals 1 mA cm-2. Since the DIsIELLI occur successively, only three intermediates remain as possible candidates for the slow oxidation step: X2 (DIsIELLI: X3 to X5), X4 (DIsIELLI: X1 to X3) or X5 (DIsIELLI: X1 to X3 or X2 to X4). 4.3. Impedance of the n-InP(hν)|1.2 M HCl System. On the basis of the dissolution model as presented by Erne´ et al.,10 six dynamic equations describing the rate of change of the surface concentration of the decomposition intermediates must be considered:

dx0 ) βnnSx1 + (k5pS + k5,inj)x5 - k0x0pS dt

(10)

dx1 ) k0x0pS - βnnSx1 - (k1pS + k1,inj)x1 dt

(11)

dxr ) (kr-1pS + kr-1,inj)xr-1 - (krpS + kr,inj)xr 1 < r e 5 dt (12) The term (k5pS + k5,inj)x5 (eq 10) expresses the fact that the oxidation of X5 results into reaction products and into a new InP unit (X0) exposed to the solution. The term βnnSx1 in eqs 10 and 11 takes into account that recombination occurs at the first decomposition intermediate. In all equations, the oxidation rates are written as the sum krpSxr + kr,injxr, thus acknowledging that it is not a priori known which oxidation steps occur by electron injection and which by hole capture.

The current density (ac) at the semiconductor side of the interface consists of a part related to the (dis)charging of the space-charge layer [CSC(dφSC/dt)] - φSC denoting the potential drop across the semiconductor side of the interface and of a part associated to the oxidation reaction. The latter part is the sum of the current density jPH (positive) corresponding to the photogenerated hole flux, the recombination current density jREC (negative), and the electron-injection current density jINJ (positive). Thus

j ) CSC

dφSC dt

+ jPH + jREC + jINJ ) CSC

dφSC dt

+ jPH 5

eβnx1nS + e

kr,injxr ∑ r)1

(13)

A final equation follows from the rate of change of the total number of photogenerated holes in the valence band (P). Valence-band holes are generated at a rate given by jPH/e and consumed by the hole capture steps of the dissolution reaction. Therefore, dP/dt reads

dP dt

)

jPH e

5

- k0x0pS -

krxrpS ∑ r)1

(14)

Assuming that the pseudo-first-order rate constant for the oxidation of either33 X4 or X5 is significantly lower than that of the other oxidation steps, all concentrations of intermediates but that related to the slow oxidation step may considered to be quasi-stationary (i.e., dxr/dt = 0, where r denotes a rapidly oxidizing intermediate). To simplify the calculations, we will assume the first DIIELLI to be oxidized solely by hole capture, a reasonable assumption given the light intensities used for our experiments. By consequenceswriting the small-signal variation of a quantity a as a˜ exp(iωt)sthe small-signal approximations of eqs 10-14 read34

iωx˜ 0 ) δ(k5x5pS) + δ(βnx1nS) - δ(k0x0pS)

(15)

0 ) δ(k0x0pS) - δ(k1x1pS) - δ(βnx1nS)

(16)

0 ) δ(k1x1pS) - δ(krxrpS) - δ(kr,injxr) 1 < r < m

(17)

iωx˜ m ) δ(km-1xm-1pS) + δ(km-1,injxm-1) - δ(kmxmpS) (18) 0 ) δ(kmxmpS) - δ(k5x5pS)

(19)

If m ) 5, eq 19 is trivial whereas if m ) 4, it expresses the quasi-stationarity of x5. In the literature, it has been argued that the number of holes in the valence band (P) of an illuminated n-type semiconductor may be considered as quasi-stationary within the experimental frequency range if a depletion layer exists near the surface.35 Therefore, eq 14 yields the relation 5

0 ) δ(k0x0pS) +

δ(krxrpS) ∑ r)1

(20)

Furthermore, the small-signal variation of the current density passing through the interface may be written as (cf. eq 13) m-1

˜j ) iωCSCφ˜ SC - eδ(βnx1nS) +

δ(kr,injxr) ∑ r)2

(21)

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Finally, to simplify the calculation somewhat, we will neglect the influence of a changing concentration of intermediates on the potential drop across the double layer. This means that the relation between φ˜ SC and φ˜ (i.e. the variation of the interfacial potential drop) will be written as36

u) 6kBT25(1 - y) + y[5(m - 1) + ((6 - m) - 4(m - 1))y] eγj∞ 5(1 - y)y[5m + (2(6 - m) - 4m)y] (28) V)

CH φ˜ SC ) φ˜ ∆ γφ˜ CH + CSC )

(22)

Using eqs 15-22, ˜j (eq 23) may be expressed as a function of φ˜ , thus enabling the calculation of the electrochemical impedance of the n-InP(hν)|1.2 M HCl system. This constitutes a straightforward but rather dull exercise in algebra. In the case that only one electron-injection step occurs (an acceptable assumption in our case), the resulting impedance (m ) 4 or m ) 5) reads

kBT

1 {βnnS(k0x0 + 4k1x1)(iω + kmpS) + k1pS e2γ βnx1nSk1pS [iω((m - 1)k0x0 + (6 - m)k1x1) + 5kmpSk0x0 + 5k0pSk1x1]}

Z)

{iω[mk0x0 + 2(6 - m)k1x1] + 6kmpSk0x0 + 6k0pSk1x1}-1 (23)

6kBT25(1 - y) + 5y(5 - 4y) eγj∞ 5(1 - y)6y(5 - 4y)

5 - 4y ωs ) 6kmpS m(5 - 4y) + 2(6 - m)y

∆ γr ) 65(1 - y) + y[(m - 1) + (2 - m)y] r∞,id ) ∞ (1 - y)y[5m + (12 - 6m)y]

This impedance has the general form

)(

)

ω ω V+i u 1+i ωs ωs

-1

(24)

Since k0pSk1x1 equals k0pSkmxm, the term in k0pSk1x1 (cf. eq 24) is much smaller than the term in k0pSkmx0. Hence, the impedance parameters u, V, and ωs introduced in eq 24 read

u) kBT βnnS[k0x0 + 4k1x1] + k1pS[(m - 1)k0x0 + (6 - m)k1x1] βnx1nSk1pS[mk0x0 + 2(6 - m)k1x1]

e2γ

(25) V)

kBT βnnS[k0x0 + 4k1x1] + 5k1pSk0x0 6βnx1nSk1pSk0x0 e2γ ωs )

6kmpSk0x0 mk0x0 + 2(6 - m)k1x1

(26) (27)

It may be easily shown that the impedance as expressed by eq 24 corresponds to a semicircle in the Nyquist plane between the extreme values u at high frequencies and V at low frequencies and with characteristic frequency ωs. If V < u, the semicircle is inductive. Considering the model equations (eqs 10-12) and the current density (eq 13) under dc conditions, the three elements constituting u, V, and ωs (i.e., k0x0pS, k1x1pS, and βnx1nS) may be written as a function of the current density j and the limiting current density j∞. This allows us to write these quantities as a function of y ) j/j∞:

(30)

Taking the limit y f 0, the ratio V/u equals m/6, whereas it equals 4/3 if y f 1. Since m < 6, both limiting values demonstrate that the impedance associated to the dissolution model will show an inductive loop close to y ) 0 and a capacitive loop close to y ) 1. The transition occurs at y = 0.68, both for m ) 4 and m ) 5. Therefore, when y < 0.68, one could try to identify the parameters u, V, and ωs with the spectrum parameters R∞, R0, and ω0sused to analyze the impedance spectra of the n-InP(hν)|1.2 M HCl systems respectively. Hence, doping-density independent values of r∞ and r0 (written as r∞,id and r0,id) may be obtained from eqs 28 and 29 as

∆ γr ) r0,id ) 0

(

(29)

5 - 4y2 y(5 - 9y + 4y2)

(31) (32)

In the case that X2 would be the slowly oxidizing surface intermediate, the impedance calculation may be carried out similarly. Doing so, one will find that the reduced impedance parameters are now given by (1 electron injection step assumed)

5 - 3y - y2 m)2 r∞,id )3 (1 - y)y(5 - 3y) rm)2 0,id )

5 - 4y2 y(5 - 9y + 4y2)

(33) (34)

It should be noted that the impedance expression obtained contains a single time constant only, which belongs to the inductive loop. Evidently, the high-frequency capacitive semicircle in the impedance spectra (cf. Figure 4) follows from the parallel connection of R∞ and CSC. On the other hand, the model does not account for the additional capacitive semicircle at low frequencies that is present at the highest illumination levels. This indicates that the model cannot be applied at the lowest measuring frequencies, which is by no means surprising since, e.g., the accumulation of reaction products at the interface and their transport through the Helmholtz layer have not been taken into account. 4.4. Calculated vs Experimental Impedance. In the preceding section, we have calculated the electrochemical impedance of the n-InP photoanode. This calculation resulted into expressions for the reduced impedance parameters r∞,id and r0,id and for the characteristic frequency ω0 as a function of y and m. In Figure 10A,B, we have plotted the spectrum parameters r∞,id and r0,id as obtained from the measurements at n-InP (ND ) 6.7 × 1017 cm-3) and n-InP (ND ) 5.0 × 1015 cm-3), after correction for the finite value of CH. This correction is calculated using the normalized value of the CPE coefficient Q, as obtained by simulating the capacitive semicircle using an R(RQ) circuit (cf. section 3.3). The coincidence of the data points obtained at

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Figure 10. (A) Reduced circuit parameter r∞,id as obtained from the experiments at different values of j∞ (same symbols as in Figure 6A,B) compared with the predicted behavior, calculated for m ) 2, m ) 4, and m ) 5, assuming a one electron injection step. (B) Reduced circuit parameter r0,id as obtained from the experiments at different values of j∞ and as calculated by assuming a one electron injection step. For both figures, the parameters are calculated assuming CH ) 2 µF cm-2.

TABLE 1: Sum of Squared Relative Errors for the Parameters r∞,id and r0,id, Calculated for One Injection Step in the Reaction Mechanism ∑(r∞,id) ∑(r0,id)

m)2

m)4

m)5

26.9 0.27

0.36 0.27

0.31 0.27

n-InP of both doping densities as shown in Figure 10A,B was obtained by equating CH to about 2 µF cm-2. Although this seems a relatively low value, numerical values reported in the literature for CH at illuminated electrodes have approximately the same magnitude: 4.0 µF cm-2 at n-Si(111) in37 aqueous 1 M NH4F + 0.5 M KCl + 0.1 M K3(FeCN)6 and 3.0 µF cm-2 at38 n-Si(111) in 0.5 M KCl + 0.1 M K3(FeCN)6. Together with the corrected experimental data, Figure 10A,B m)2 m)4 m)5 shows the functions r∞,id (y), r∞,id (y), r∞,id (y) (Figure 10A), and r0,id(y) (Figure 10B). Clearly, for r0,id, a good correspondence exists between the experimental data and the calculated curve. m)4 m)5 On the other hand, only the curves r∞,id (y) and r∞,id (y) agree well with the experimental data (deviations < 25%) whereas large systematic deviations are observed between the data and m)2 the r∞,id (y) curve. These observations are confirmed by the sum of squared relative errors39 Σ, as shown in Table 1. Consequently, X2 is not likely to be the slowly oxidizing intermediate. In the region y < 0.5, correspondence with the m ) 4 curve seems to be better whereas if y > 0.5, the data apparently correspond to the m ) 5 curve. However, considering the approximations made in the impedance calculation, i.e., the assumption of one electron-injection step and the limitations of the method used to determine R∞, these differences should not be overinterpreted. Consequently, we think Figure 10A,B does not allow one to discriminate reliably between the cases m ) 4 and m ) 5. An argument in favor of the hypothesis m ) 4 may follow from the ratio between r0,id and r∞,id as y f 0, which equals m/6. Thus, in the case m ) 5, this ratio is given by 5/6, i.e., 0.833, whereas in the case m ) 4, this lower limit equals 4/6, i.e. 0.667. Experimentally, we found values as low as 0.71 for the ratio ro,id/r∞,id, suggesting that X4 is a more probable candidate for the slowly oxidizing intermediate than X5. In the cases m ) 4 and m ) 5, the calculated impedance predicts an inductive loop in the range 0 e y < 0.68. This prediction agrees reasonably well with our measurements since, at the higher values of the limiting photocurrent density used (j∞ > 1 mA cm-2), the inductive loop has indeed disappeared at y = 0.8 (cf. section 3.3). At values of j∞ smaller than 1 mA

cm-2, the inductive loop still persists at y = 0.8, the difference R∞ - R0 however being small [(R∞ - R0)/R∞ = 0.04]. At given value of y, eq 30 indicates that ω0 should be proportional to the density of photogenerated holes at the semiconductor surface (pS). Since at the light intensities used, ∆V0 is roughly independent of light intensity (hence of j∞), the concentration of intermediates at the surface is independent of j∞. By consequence, pS is proportional to j∞, within the experimental range of light intensities. Hence, eq 30 agrees qualitatively with the experimentally observed linear dependence of ω0 on j∞, at given value of y. Since we obtained a proportionality constant R between ω0 and j∞ of about 5 × 105 C-1 cm2 at y ) 0.7 (cf. section 3.3), eq 3140 yields

kmpS = 5 × 105j∞

(35)

Consequently, the pseudo-first-order rate constant of the slowest oxidation step is about 5 × 102 s-1 at y ) 0.7 if j∞ equals 1 mA cm-2, a value that corresponds well with the originally estimated rate (cf. section 4.2). In the limiting cases y f 0 and y f 1, the relations between ω0 and kmpS (using eq 30) read respectively

6 6 km j limω0 ) kmpS ) xf0 m m k0x0 PH limω0 ) xf1

km jPH 6 6 k p ) 12 - m m S 12 - m k0x0 5

(36) (37)

If one assumessas we did while calculating the impedancess that the rate constant km is independent of the potential drop φEL across the Helmholtz layer, these two limits indicate that ω0 decreases as y is increased from 0 to 1. This prediction is in contradiction with the experimental results (cf. Figure 7A). Presumably, the observed dependence of ω0 on y can only be explained within the model proposed if it is assumed that km depends on φEL. Since the oxidation of Xm is an anodic process, one would expect it to increase as φEL increases, in agreement with the observed increase of ω0 with increasing y. 4.5. Origin of the Inductive Loop. The impedance expression given in eq 23 predicts the presence of an inductive loop in the impedance spectrum of the n-InP photoanode, especially if y f 0. We think it is worthwhile to give a simplified calculation explaining how this loop emerges from the dynamic equations describing the dissolution process. This is done most easily by considering the case y f 0 (i.e., close to the potential region of complete recombination). Assuming a single electroninjection step, eq 20 reads

0 ) k0x0p˜ S + k0pSx˜ 0 + (m - 2)k1x1p˜ S + (m - 2)k1pSx˜ 1 + (6 - m)kmxmp˜ S + (6 - m)kmp˜ Sx˜ m (38) If y f 0, the third and the fifth terms in eq 38 are negligible as compared to the first one. Consequentlyswith recognition that k0pSkmx0 is much larger than k0pSkrxr (1 e r e 5)sthe variation δ(krxrpS) reads

δ(krxrpS) = krpSx˜ r

(39)

Equation 39 indicates that, in the limit y f 0, a change of the oxidation rate of the decomposition intermediates is mainly due to a change of their surface concentration and not to a change of pS. This means that, at high frequencies, only the oxidation rate of the intermediates up to Xm changes whereas at low frequencies, the oxidation rate of all 6 intermediates will vary.

Photoanodic Dissolution of n-InP Consequently, a change of the surface concentration of X1 will lead, at high frequencies, to a change of the current density equal to mek1pSx˜ 1 (only the first m oxidation steps can follow the variation). At low frequencies, this change will be 6ek1pSx˜ 1 (all 6 oxidation steps follow the variation). Therefore, the amplitude of the current density variation is higher at low frequencies than that at high frequencies, their ratio being equal to 6/m. This situationswhich originates from the slow oxidation step hampering the dissolutionsis typical for a circuit involving an inductor, as this element only conducts electric current at low frequencies. In two recent papers,41,42 it has been stressed that inductive behavior is often observed in impedance spectra of anodically oxidizing metal and p-type semiconductor electrodes. The characteristic frequency of these inductive loops is for all systems about equal and proportional to the current density. Therefore, a common origin for this inductive behavior, in casu ion transfer through the Helmholtz layer, has been suggested. However, in the present case, the equations describing the dynamics of the reaction intermediate surface concentrations clearly suffice to account for the inductive loop present. Moreover, the characteristic frequency of the inductive loops observed at n-InP(hν)|1.2 M HCl is about 100 times larger than that generally observed at metal and p-type semiconductor electrodes. This demonstrates that the inductive behavior observed here does not fit the scheme proposed in refs 41 and 42swhich was based upon a typical characteristic frequency of the inductive loop. Evidently, the alternative explanation proposed here for the appearance of an inductive loop does not invalidate the model of Erne´ et al.41,42 since major differences exist between the decomposition of metal and p-type semiconductor electrodes analyzed by these authors and the n-InP photoanode studied here. For example, a variation of the applied potential leads in the latter case almost solely to a variation of the potential drop across the semiconductor side of the interface. On the other hand, in the former case, it results almost exclusively into a variation of the potential drop across the Helmholtz layersa hypothesis which is essential to the modeling proposed in ref 42. 4.6. Dissolution Mechanism of n-InP in 1.2 M HCl. The impedance calculation performed in section 4.3 was based upon two main assumptions, i.e., quasi-stationarity for all but one decomposition step and recombination on the first decomposition intermediate. The position of the slowly oxidizing intermediate in the reaction sequence was a priori unknown. However, the absence of the typical features of the recombination impedance eliminates X1 as a possible candidate. Moreover, since none of the DIsIELLI can correspond to the slow step, also X3 must be rejected. Finally, comparison between the calculated and the experimental impedance parameters eliminates X2 as a possible candidate. Unfortunately, no decisive distinction between X4 and X5 is possible (cf. Figure 10), although X4 seems a more probable candidate. Close to the photocurrent plateau, the pseudo-first-order rate constant of this slow oxidation step is about 5 × 102 s-1 (j∞ ) 1 mA cm-2). With combination of these results with the conclusions deduced by Erne´ et al.10 from optoelectric impedance measurements, a more detailed picture of the dissolution reaction emerges, as shown in Figure 11. The photoanodic dissolution is initiated by hole capture in an unbroken surface bond X0, creating a decomposition intermediate X1. At this first intermediate, a competition exists between recombinationsrestoring the unbroken surface bondsand further oxidation. The applied

J. Phys. Chem. B, Vol. 104, No. 32, 2000 7733

Figure 11. Diagram showing the two possible sequences of electrochemical steps leading to the dissolution of one formula unit InP. The arrows on the left and the right of an intermediate indicate reaction steps involving VB holes and CB electrons, respectively. (A) The three successive DIsIELLI correspond to the oxidation of the intermediates X1 to X3, and the slow step in the reaction mechanism corresponds to the oxidation of either X4 or X5. (B) The three successive DIsIELLI correspond to the oxidation of the intermediates X2 to X4, and the slow step in the reaction mechanism corresponds to the oxidation of X5.

electrode potential determines whether recombination or oxidation dominates (cf. current density vs potential behavior, Figure 1). The decomposition reaction continues with three successive DIsIELLI,10 which may correspond either to the sequence (X1, X2, X3) (Figure 11A) or to the sequence (X2, X3, X4) (Figure 11B). Finally, one slow hole-capture step follows which may be either the oxidation of X4 [in that case, the DIsIELLI correspond to the sequence (X1, X2, X3)] or the oxidation of X5. 5. Conclusions In this paper, we have analyzed the electrochemical impedance spectrum of the n-InP photoanode. Two aspects characterize the impedance measured, i.e., the absence of the typical features of the recombination impedance and the presence of an inductive loop at low frequencies. We have argued that the former aspect (i.e., the absence of recombination impedance features) can be explained by assuming that recombination occurs at a decomposition intermediate, competing with the (fast) subsequent oxidation of this intermediate. Still, a shift of the band edges of approximately 150-200 meV is observed in the potential region of the photocurrent plateau. As this points to the accumulation of intermediates at the surface, we have assumed in addition the occurrence of a slow oxidation step in the dissolution reaction. Starting from this picture of the dissolution reaction, an impedance calculation has been performed, the result of which depends on the position m of the slow oxidation step in the reaction mechanism. A good correspondence between the experimental data and the calculated impedance follows both for m ) 4 or m ) 5. In addition, we could estimate the rate of the slow oxidation step as 5 × 102 s-1 at j∞ ) 1 mA cm-2. The results obtained fit very well in the decomposition scheme which Erne´ et al.10 deduced from the optoelectrical admittance of the n-InP photoanode. Therefore, this case study shows that electrochemical impedance spectroscopy provides information on the mechanism and the kinetics of photodissolution reactions complementary to that provided by IMPS measurements.

7734 J. Phys. Chem. B, Vol. 104, No. 32, 2000 Acknowledgment. The authors wish to thank D. Vanmaekelbergh for supplying the lower doped n-InP samples. References and Notes (1) Morrison, S. R. Electrochemistry of Semiconductors and Oxidized Metal Electrodes; Plenum Press: New York, 1980. (2) Pleskov, Y. V.; Gyurevich, Y. Y. Semiconductor Photoelectrochemistry; Consultants Bureau: New York, 1986. (3) Peter, L. M. Chem. ReV. 1990, 90, 753. (4) Gerischer, H. J. Electroanal. Chem. 1977, 82, 133. (5) Bard, A. J.; Wrighton, M. S. J. Electrochem. Soc. 1977, 124, 1706. (6) Memming, R. In Photoelectrochemistry, Photocatalysis and Photoreactors, Fundamentals and DeVelopments; Schiavello, M., Ed.; NATO ASI Series C, Vol. 146; Reidel Publishing Co.: Dordrecht, The Netherlands, 1985. (7) Gomes, W. P.; Goossens, H. H. In AdVances in Electrochemical Science and Engineering; Gerischer, H., Tobias, C. W., Eds.; VCH: Weinheim, Germany, 1994; Vol. 3. (8) Peter, L. M.; Borazio, A. M.; Lewerenz, H. J.; Stumper, J. J. Electroanal. Chem. 1990, 290, 229. (9) de Wit, A. R.; Vanmaekelbergh, D.; Kelly, J. J. J. Electrochem. Soc. 1992, 139, 2508. (10) Erne´, B. H.; Vanmaekelbergh, D.; Vermeir, I. E. Electrochim. Acta 1993, 38, 2559. (11) Schro¨der, K.; Memming, R. Ber. Bunsen-Ges. Phys. Chem. 1985, 89, 385. (12) van den Meerakker, J. E. A. M.; Kelly, J. J.; Notten, P. H. L. J. Electrochem. Soc. 1985, 132, 638. (13) Vanmaekelbergh, D.; Gomes, W. P.; Cardon, F. Ber. Bunsen-Ges. Phys. Chem. 1985, 89, 994. (14) Vanmaekelbergh, D.; Gomes, W. P.; Cardon, F. Ber. Bunsen-Ges. Phys. Chem. 1986, 90, 431. (15) Vanmaekelbergh, D.; Gomes, W. P.; Cardon, F. J. Electrochem. Soc. 1987, 134, 891. (16) In this equation, kB stands for the Boltzmann constant, T for the absolute temperature, and e for the absolute value of the charge on the electron; jREC is the current density associated with electron-hole recombination (i.e., -e times the electron flux involved in recombination). (17) Vermeir, I. E.; Vanden Kerchove, F.; Gomes, W. P. J. Electroanal. Chem. 1991, 313, 141. (18) Preusser, S.; Herlem, M.; Etcheberry, A.; Jaume, J. Electrochim. Acta 1992, 37, 289. (19) Notten, P. H. L.; van den Meerakker, J. E. A. M.; Kelly, J. J. Etching of III-V Semiconductors, an Electrochemical Approach; Elsevier: Oxford, U.K., 1991. (20) Hens, Z.; Gomes, W. P. Phys. Chem. Chem. Phys. 1999, 1, 3607. (21) The symbols R and Q denote a resistor and a CPE, respectively. Parentheses indicate a parallel connection; hence (RQ) stands for the parallel connection of a resistor and a CPE. (22) Normalization at a frequency ω0 transforms the CPE coefficient Q into a capacitance C according to the equality C ) Qω0n-1. If ω0 is the

Hens and Gomes characteristic frequency of an (RQ) circuit, it is the characteristic frequency of the (RC) circuit as well if C is the normalized value of Q.23 (23) Hens, Z.; Gomes, W. P. J. Phys. Chem. B 1999, 103, 130. (24) This procedure tacitly assumes that the impedance element associated with the inductive loop stands in parallel to the capacitive element causing the high-frequency semicircle. This idea is acceptable since the numerical value of the CPE coefficient indicates that this impedance element correspondssin an empirical waysto the semiconductor space-charge layer. Hence, if CSC , CH (CH: Helmholtz-layer capacitance), the impedance element related to the photodissolution will stand in parallel to the CPE. (25) In these rate expressions, βn and βp are the rate constants for electron and hole transfer to the recombination centre respectively, nS and pS denote the density of CB electrons or VB holes at the surface, respectively, s is the density of recombination centers, and θ is their occupation number. (26) Vanmaekelbergh, D.; Gomes, W. P.; Cardon, F. J. Electrochem. Soc. 1982, 129, 546. (27) Vanmaekelbergh, D.; Rigole, W.; Gomes, W. P. J. Chem. Soc., Faraday Trans. 1983, 79, 2813. (28) Vanmaekelbergh, D.; Gomes, W. P. Ber. Bunsen-Ges. Phys. Chem. 1985, 89, 987. (29) Hens, Z.; Gomes, W. P. J. Phys. Chem. B 1997, 101, 5814. (30) Oskam, G.; Hoffman, P. M.; Schmidt, J. C.; Searson, P. C. J. Phys. Chem. 1996, 100, 1801. (31) Of course, the value of the capacitive peak is also limited by CH. However, this does not invalidate the argument since the observed additional capacitance is much less than CH. (32) In the photocurrent plateau, the current density j∞ equals 6eκmxm (xm: surface concentration of Xm). (33) Taking X2 as the slowly oxidizing intermediate leads to somewhat different equations; the results for this case are summarized at the end of this section (eqs 33 and 34). (34) The expression δ(k0x0pS) denotes the small-signal variation of k0x0pS, divided by exp(iωt), i.e., δ(k0x0pS) ) k0x0p˜ S + k0pSx˜ 0, etc. (35) Vanmaekelbergh, D.; Cardon, F. J. Phys. D: Appl. Phys. 1986, 19, 643. (36) The symbol } means “equal by definition”. (37) Oskam, G.; Hoffmann, P. M.; Searson, P. C. Phys. ReV. Lett. 1996, 76, 1521. (38) Oskam, G.; Schmidt, J. C.; Searson, P. C. J. Electrochem. Soc. 1996, 143, 2539. (39) The quantity ∑ is defined as the sum of the squares of the relative errors between the measured and the predicted value of the reduced spectrum parameters: 2 rpred id ) 1 - meas rid

(

∑ ∑

)

(40) Taking y ) 0.7, the weight of kmpS in eq 31 equals 13.2/11.6 for m ) 4 and 13.2/12.4 for m ) 5. (41) Vanmaekelbergh, D.; Erne´, B. H. J. Electrochem. Soc. 1997, 144, 3385. (42) Vanmaekelbergh, D.; Erne´, B. H. J. Electrochem. Soc. 1999, 146, 2488.