4930
J. Phys. Chem. 1996, 100, 4930-4936
Investigation of the Kinetics of Redox Reactions at GaAs Electrodes by Impedance Spectroscopy I. Uhlendorf,†,‡ R. Reineke-Koch,‡ and R. Memming*,‡,§ Institut fu¨ r Solarenergieforschung, Sokelantstr.5, 30165 HannoVer, FRG, Fachbereich Physik d. UniVersita¨ t Oldenburg, FRG ReceiVed: August 4, 1995; In Final Form: December 1, 1995X
The electrochemical oxidation and reduction of the Cu1+/Cu2+ redox system at p-GaAs electrodes have been investigated by impedance spectroscopy. This reaction is a diffusion-controlled process via the valence band, which competes well with the anodic decomposition reaction. The method of impedance spectroscopy makes it possible to determine kinetic parameters such as the rate constants for the oxidation and the reduction of the redox system. In addition the shift of the flatband potential and of the corresponding energy bands at the GaAs-electrolyte interface could be followed during current flow. In the case of this redox system the potential dependence of the rates and of the interfacial current are in agreement with the theory on charge transfer kinetics at semiconductor electrodes. Above a Cu1+ concentration of 4 mM the oxidation current increased linearly with the concentration, and the second-order rate constant was determined to be k+ v ) 5.10-19 cm4 s-1.
jv ) ekv Nvcox ) constant
1. Introduction The modern theories on charge-transfer processes introduced by Marcus1 and Gerischer2,3 have led to a principal understanding of the energetics of corresponding reactions at metal and semiconductor electrodes. Many processes at a large variety of semiconductors have been investigated, and during the last decade researchers concentrated mainly on photoelectrochemical effects because of their interest in solar energy applications. This has led in some way to a neglect of dark reactions. They are of importance because one can obtain very valuable information from studies of majority carrier processes, the kinetics of which is usually dependent on the potential. From pure photocurrent measurements one obtains only limited information on the kinetics because photocurrents are determined by light excitation. Recently, we have derived a quasi-Fermi level concept that makes it possible to compare quantitatively majority carrier processes at p-type electrodes to minority carrier processes at n-type electrodes of the same material or vice versa.4,5 These investigations have shown quite clearly that certain kinetic factors involved in a specific reaction become more evident in majority carrier processes. According to the basic theories mentioned above, the interfacial currents in the presence of a redox system are given for a conduction band process by6
(2b)
in which Nc and Nv represent the effective density of states (given in cm-3) at the conduction and valence band edges of a semiconductor, respectively, cred and cox are the concentrations of the reduced and oxidized species of the redox couple (also given in cm-3), respectively, and the different ki’s are the second-order rate constants (given in cm4 s-1). According to eqs 1 and 2, only the reduction current via the conduction band (eq 1b) and the oxidation current via the valence band (eq 2a) depend on the surface concentration of electrons and holes, respectively. Using an n-type electrode for the first reaction (reduction via the conduction band) and a p-type electrode for the second reaction (oxidation via the valence band), we have here majority carrier processes. The surface concentrations of electrons and holes, ns and ps, depend on the band bending φsc at the interface and are related to the corresponding bulk densities no and po by
( (
) )
ns ) no exp -
eφsc kT
(3a)
ps ) po exp +
eφsc kT
(3b)
+ j+ c ) ekc Nccred ) constant
(1a)
In these cases the currents are expected to depend exponentially on the potential across the space charge layer, which is given by
jc ) ekc nscox
(1b)
φsc ) UE - Ufb
and for a valence band process by + j+ v ) ekv pscred
(2a)
* Author to whom correspondence should be addressed. † Present address: Institut fu ¨ r Angewandte Photovoltaik, c/o Flachglas LBO, Haydnstr. 19, D-45884 Gelsenkirchen. ‡ Institut fu ¨ r Solarenergieforschung. § Fachbereich Physik d. Universita ¨ t Oldenburg. X Abstract published in AdVance ACS Abstracts, February 15, 1996.
0022-3654/96/20100-4930$12.00/0
(4)
in which UE is the electrode potential and Ufb the flatband potential. Accordingly, the interfacial currents representing a majority carrier process should depend strongly on the electrode potential. Assuming that the externally applied voltage occurs entirely across the space charge layer (Ufb ) constant), the theoretical slope of corresponding current-potential curves is expected to be 60 mV per decade of current. Unfortunately, there are nearly no papers in which corresponding investigations have been reported. There is only one © 1996 American Chemical Society
Redox Reactions at GaAs Electrodes
J. Phys. Chem., Vol. 100, No. 12, 1996 4931
paper published by Morrison in 1969 in which currents were plotted vs the potential across the space charge layer φsc for n-type ZnO electrodes in the presence of various redox systems.7 In this case the slopes of the curves mostly meet the theoretical condition of 60 mV per decade. Other examples of currentpotential curves with a slope of 60 mV/decade are interestingly the anodic dissolution of p-type silicon in HF8 and of p-type germanium9 and GaAs in acid solutions.4 However, by the evaluation of current potential curves of other majority carrier processes, found scattered in the literature, with respect to their slopes, mostly rather large values of >100 mV/decade were determined, i.e., data which are in contradiction to the theory. In the case of redox reactions a slope of 60 mV/decade does not necessarily mean that the kinetics given by eqs 1b, 2a, and 3 are fulfilled because such a slope is also obtained if a reaction is diffusion controlled (reversible reaction). Under these circumstances the 60 mV slope originates from the Nernst law, which determines the concentrations of the oxidized and reduced species at the interface, as we have found for instance for the redox system Cu+/Cu2+ at a p-type electrode of GaAs4 and with Fe(phen)32+/Fe(phen)33+ at a p-type WSe2 electrode.5 The origin of a slope of more than 60 mV/decade as found for many redox reactions is unknown in most cases because most researchers did not analyze the current potential curves in detail. One reason may be the formation of a surface layer and another the change of the Helmholtz layer when scanning across a certain potential range. In the case of a fast diffusion-controlled reaction the kinetically controlled current can be obtained in principle by measuring the interfacial current at different rotation speeds of the semiconductor electrode. Since the diffusioncontrolled current increases with the square root of the rotation speed, the kinetic current jkin can be obtained from the relation4
1 1 1 + ) j jkin jdiff
(5)
by extrapolating 1/j to infinite rotation speeds. In many cases, however, the value of 1/jkin is nearly zero, i.e., the determination of the kinetically controlled current becomes very inaccurate. In the present paper we apply the method of impedance spectroscopy in order to obtain data on the charge transfer kinetics by investigating the oxidation and reduction of redox systems (mainly of Cu1+/Cu2+) at p-type GaAs electrodes. 2. Electrochemical Procedures The single crystalline GaAs samples, purchased from Wacker, Burghausen (Germany), had a doping of 2.2 × 1018 cm-3 if not stated otherwise. The crystals were ground on one face and etched in a solution of NH3 (25%), H2O2 (30%), and H2O in the ratio 3:1:15 and afterward rinsed in 6M HCl. This surface was used for making an ohmic contact on p-GaAs by depositing a sequence of metal layers such as gold (5-20 nm)/zinc (70400 nm)/gold (200-400 nm) on the etched sample by vacuum deposition and subsequent heating in a H2 atmosphere. Here, the first Au layer acts as an adhesion layer through which the Zn diffused during heating. Electrically, the GaAs samples were connected with a copper wire by using silver epoxy. The electrodes (surface area, 0.125 cm2) were finally glued in a circular tubing so that rotation was possible under defined conditions. Before the measurements each electrode was carefully polished (final polishing with 50 nm Al2O3). Before each individual measurement the electrodes were etched in the same solutions as mentioned above. The electrochemical measurements were performed in a typical potentiostatic three-electrode arrangement in which the reference electrode was dipped into a solution in a separate
Figure 1. Current-potential curves (vs SCE from ref 4) at a rotating p-GaAs electrode in 6 M HCl with 2.5 mM CuCl at 1000 rpm. jcorr is the current of the anodic decomposition, jox the partial current of the Cu1+ oxidation, and jtot the total current.
compartment, the latter being in contact with the working solution by a salt bridge. We used a Hg/HgCl2/6M HCl reference electrode (HCl-calomel), its potential being -0.088 V, vs a Hg/HgCl2/saturated KCl (SCE). In addition a Pt wire as a second reference electrode was used, which was dipped directly into the working solution. It was short-circuited to the SCE via a 10 nF capacitor. This was necessary for minimizing errors in the potential measurements at high frequencies.9 A frequency analyzer solartron FRA 1255 (Schlumberger Technologies, GB) was used for the impedance measurements. An ac signal of 2-5 mV was superimposed on the dc voltage. Impedance measurements performed at different frequencies were always started at low frequencies. After a certain potential was set, the first impedance measurement at low frequency was performed after about 30-180 s in order to reach a stationary state. 3. Results A typical current-potential dependence as measured with a p-type GaAs electrode in a 6 M HCl solution with and without 2.5 mM Cu1+ is given in Figure 1.4 The impedance spectrum without Cu1+ is shown in Figure 2. This rather complex impedance spectrum showing several time constants changes considerably upon addition of Cu1+, and above about 1 mM Cu1+ the positive imaginary part Z′′ disappears (Figure 2). Simultaneously, the total current decreases at first and then increases with rising Cu1+ concentrations (Figure 3). All further impedance measurements were performed at rather high Cu1+ concentrations. The potential dependence of the impedance spectra was then measured at a Cu1+ concentration of 8 mM. The results are given in parts a-d of Figure 4. The current potential curves measured simultaneously under stationary conditions are shown at the top of Figure 4. The extrapolation of the spectra to a frequency of ω f 0 yields more or less the slope of the current-potential curve. The absolute values of the impedance at low frequencies decrease with increasing anodic potentials. The typical shape of the spectra remained. Typically, two half circles occur, in which the high-frequency half circle is governed by the space charge capacity and the charge transfer resistance, whereas the other shows a 45° slope at its high-frequency end, which is typical for a diffusioncontrolled reaction. Besides the investigation of the impedance spectra, typical dynamic Mott-Schottky measurements were also performed
4932 J. Phys. Chem., Vol. 100, No. 12, 1996
Uhlendorf et al. from eq 6 by using eqs 7a, 3b, and 4 and is given by
∆I s e s - k-∆csox ) k+cred ∆U + k+∆cred eA kT E
(8)
The ∆ terms are the complex amplitudes of the corresponding time-dependent modulations. In the following the other terms, s such as cred in eq 8, are mean values of the corresponding timedependent values. The relation between the time-dependent s and ∆csox and the current variation concentration changes ∆cred ∆I has been derived by Gabrielli.13 In the case of a rotating electrode he has obtained
s -∆cred
Figure 2. Impedance spectra for a rotating p-GaAs electrode (420 rpm) at an electrode potential of UE ) +0.4 V (vs HCl-calomel) for different Cu1+ concentrations in 6 M HCl.
)
∆csox
∆I ) eA
(x) iω D
tanh δN
(9)
xiωD
in which δN is the thickness of the Nernst diffusion layer, ω the modulation frequency, and D the diffusion coefficient (Dox ) Dred). Inserting eq 9 into eq 8, one obtains
(
(x)
iω tanh δN ∆UE kT 1 D 1 ) 2 + s + + s (k+ + k-) ZF ) ∆I e A k cred k cred xiωD
)
(10) Accordingly, the impedance ZF of the Faradaic process consists of two parts. One is given by the charge-transfer resistance Rct and the other by the so-called Warburg impedance Zw so that we have Figure 3. Interfacial current vs Cu1+ concentration in HCl at UE ) +0.4 V. Data were taken simultaneously with the impedance measurements given in Figure 2.
under cathodic polarization (reverse bias) at a high frequency (30 kHz) at various Cu1+ concentrations. The Mott-Schottky plots and the flatband potential are slightly shifted upon increasing the Cu1+ concentration (∆Ufb ) 60 mV for a Cu1+ concentration of 16 mM).
The evaluation will be performed on the basis of the kinetic model given by eqs 1 and 2. Since both the oxidation and the reduction of the Cu1+/Cu2+ redox system occur via the valence band,4 the Faradaic current is given according to eqs 2a and 2b by
(6)
s and csox are in which A is the surface area of the electrode, cred the concentrations of the reduced and oxidized species of the redox system at the surface, respectively, and ps is given by eq 3b. As an abbreviation, we introduce
k+ ) k+ v ps
(7a)
k- ) kv Nv
(7b)
in which only k+ is expected to be potential-dependent provided that the energy bands remain pinned. By the modulation of the electrode potential by ∆UE, the corresponding time-dependent current modulation can be derived
(11)
in which
Rct )
( ) { ∂U ∂I
)
ci
e2A + s k cred kT
}
-1
(12)
and
(x)
tanh δN
Zw ) Rct(k+ + k-)
4. Evaluation of the Impedance Data
s s I ) eA (j+ - j-) ) eA (k+ v pscred - kv Nvcox)
Z ) Rct + Zw
xiωD
iω D
) σ0
(x)
tanh δN
xiωD
iω D
(13)
In eqs 12 and 13 only the redox reaction is considered. In the same potential range the anodic decomposition of GaAs also occurs (see Figure 1). The kinetics of the latter reaction is a very complex reaction in which six holes are required for the dissolution of one GaAs surface unit.14-16 This also becomes evident from the impedance spectra measured with solutions free from any redox system (Figure 2). At present there is no unique reaction mechanism that would explain the impedance data of this reaction. To include this process, a resistance RD is introduced in the equivalent circuit as a first approximation (Figure 5). At larger concentrations of Cu1+(cred > 4 mM), however, the current of the anodic dissolution is small compared to that of the oxidation of Cu1+ and the influence of the anodic dissolution on the impedance spectra is negligible. In this case the evaluation of the data can be performed by using the equivalent circuit given in Figure 5 and by neglecting RD. Besides these two Faradaic processes, the space charge capacity Csc also has to be considered, the latter being in parallel to RD and Rct as shown in Figure 5. The resistances of the semiconductor and of the electrolyte are represented by Rs in Figure 5.
Redox Reactions at GaAs Electrodes
J. Phys. Chem., Vol. 100, No. 12, 1996 4933
Figure 4. Impedance spectra of a rotating p-GaAs electrode (400 rpm) in 6 M HCl containing 8 mM Cu1+ ions at different electrode potentials (parts a-d) as also indicated in the stationary current-potential curve.
Figure 5. Equivalent circuit diagram.
Figure 7. Mott-Schottky plot of the space charge capacity for a rotating p-GaAs electrode (400 rpm) in 6 M HCl with 8 mM Cu1+. The slope of the solid lines corresponds to the doping, as proven by dynamic measurements (see text).
Figure 6. Absolute value of the impedance |Z| (O) and phase angle φ (0) vs frequency. Solid lines are theoretical curves, calculated by using the following fitting parameters: Rs ) 1.1 Ω, Rct ) 99 Ω, RD ) 1132 Ω, Csc ) 2.19 × 10-7 F, and σo ) 3.64 Ω cm s-1.
In general one obtains a good fit of the experimental impedance spectra by using the equivalent circuit in Figure 5 and eqs 12 and 13. The best way of illustrating the quality of the fit is given by a plot of the absolute value of the impedance |Z| and of the phase angle R vs frequency at a fixed potential (Figure 6). Larger deviations occur only at very high frequencies. They are due to measuring errors caused by the apparatus itself. It is interesting to note that RD is by a factor of 10 larger than Rct (see figure caption of Figure 6), i.e., the anodic dissolution process does not influence the evaluation. Similar results have been obtained for other potentials provided that the Cu1+ concentration is above about 4 mM. Since a good fit was obtained, Csc, Rct, and Zw could be determined from the impedance data. The potential dependence
of Csc for c(Cu1+) ) 8 mM is given in Figure 7. There is a slight concentration dependence for c(Cu+) < 2 mM (Figure 8). From Rct and Zw the sum of the rate constants, k+ + k-, was calculated by using eqs 12 and 13. Figure 9 shows that the rate constants decrease with increasing Cu1+ concentrations and become constant for c(Cu1+) > 4 mM. Since any influence of the anodic dissolution on the evaluation of the rate constants can only be avoided for sufficiently high Cu1+ concentrations, only for c(Cu1+) > 4 mM can reliable results be expected. The result is given in Figure 10. For comparison, a theoretical curve is given in this figure. Unfortunately, it was also impossible to measure impedance spectra at electrode potentials of UE > 0.47 V because of interference with the anodic decomposition process. It should be further emphasized here that k+ itself cannot be derived directly from Rct according to eq 12 because s of the reduced species of the the surface concentration cred redox system is considerably lower than the bulk concentration for a diffusion-controlled reaction. This quantity can be calculated from currents, but the error is very large, especially for low values. 5. Discussion Earlier investigations have already shown that the oxidation and reduction of the Cu1+/Cu2+ redox system at GaAs electrodes are diffusion-controlled reactions over the whole potential range.4 In general, the total interfacial current depends on the kinetically and diffusion controlled currents jkin and jdiff,
4934 J. Phys. Chem., Vol. 100, No. 12, 1996
Uhlendorf et al.
Figure 10. Sum of rate constants (k+ + k-) vs electrode potential for various Cu1+ concentrations. The theoretical curve (model) was calculated assuming k+ ) k- ) 0.02 cm s-1 at the standard redox potential and assuming that the energy bands remain pinned during polarization.
Figure 8. Dependence of space charge capacity on Cu1+ concentration for p-GaAs at different electrode potentials: (2) 0.37 V; (0) 0.37 V; (O) 0.4 V; (~) 0.39 V. Curves 1-3 were obtained with a doping of 1.1 × 1017 cm-3 and curve 4 with 2.2 × 1018 cm-3.
Figure 9. Sum of rate constants (k+ + k-) vs Cu1+ concentration (evaluated from impedance spectra): (~) 0.37 V; (]) 0.37 V; (O) 0.4 V; (2) 0.39 V.
respectively, as described by eq 5. These two currents are given by4
jkin )
b e(k+cred
-
k-cbox)
(14)
1 D b (k+cred - k-cbox) jdiff ) e δN (k+ + k-)
(15)
δN ) 1.6D1/3ν1/6ωr-1/2
(16)
in which
ν is the viscosity and ωr the angular velocity of the electrode. Since 1/jkin , 1/jdiff for a diffusion-controlled current, it is expected, according to eqs 14 and 15, that k+ + k- . D/δN. With a rotation speed of 400 rpm one obtains δN ) 2.2 × 10-3 cm, and together with D ) 6 × 10-6 cm2 s1 we have D/δN ) 2.73 × 10-3 cm s-1. Since, according to Figure 9, the lowest value of k+ + k- ) 1.8 × 10-2 cm s-1, the above condition is
fulfilled. In the case of a Pt electrode it is also a reversible reaction, i.e., the same condition is fulfilled. Concerning the potential dependence of k+ + k-, it seems to follow the theoretical expectation (Figure 10). According to eqs 3, 4, and 7, only k+ is expected to rise exponentially with increasing anodic potential, whereas k-, which dominates over k+ at potentials negative with respect to the standard redox potential of 0.42 V, is expected to be constant. The theoretical curve in Figure 10 has been calculated assuming that the energy bands remain pinned at the surface. In this case the rate constant k+ should increase by 1 order of magnitude when varying the electrode potential by 60 mV. In order to check this more accurately, it would have been necessary to measure the impedance at more anodic potentials, i.e., beyond 0.5 V. As already mentioned in the previous section, however, this is not possible because one passes the saturation current and faces strong competition by the anodic dissolution reaction. In addition it can be recognized in Figure 10 that k+ + k- does not remain constant but increases when varying the electrode potential toward cathodic potentials. Since just in this range the Mott-Schottky plot is not a straight line (see Figure 7), this increase can be due to a shift of energy bands at the GaAs surface. This problem will be analyzed below. The rate constants k+ v and kv defined in eqs 2a and 2b depend on the density of energy states in the redox system at the valence band edge. According to the Marcus-Gerischer model, we have then with eqs 7a and 7b + k+ ) kv,max Dredps
(17a)
DoxNv k- ) kv,max
(17a)
in which k+ v,max and kv,max are the maximum rate constants and Dred and Dox are the distributions of densities of the occupied and empty states in the redox system, respectively. Introducing equations of Dred and Dox into eq 17, one obtains17 (the maxima of Dox and Dred are normalized to unity)
( (
) )
+ ps exp k+ ) kv,max
(EVs - E°F,redox - λ)2 4kTλ
(18a)
k- ) kv,max Nv exp -
(Esv - E°F,redox + λ)2 4kTλ
(18b)
According to eqs 6 and 7, it follows that the rate constants k+
Redox Reactions at GaAs Electrodes
J. Phys. Chem., Vol. 100, No. 12, 1996 4935
Figure 12. Potential dependence of the rate constants (a) and of their sum (b). Experimental data (O) are taken from Figure 9 (16 mM Cu1+). Solid curves were calculated by using eqs 20 and 21 with ko ) 1 × 102 cm s-1 and Ufb values from Figure 7.
because for Figure 11. Schematic presentation of the valence band with respect to the energy states of the redox system at two different electrode potentials (EF,1 and EF,2) assuming a shift of the flatband potential by 0.1 V.
and k- are equal at the standard redox potential, i.e., +
-
(19)
Assuming that the energy bands are not completely pinned but are shifted upon polarization, one obtains from eqs 18a, 18b, and 19
(
)
ps (E°v,s - E°F,redox - λ)2 - (Esv - E°F,redox - λ)2 exp p°s 4kT λ
(
k- ) ko exp
eqs 20a and 20b can be approximated by
(20a)
)
(E°v,s - E°F,redox + λ)2 - (Esv - E°F,redox + λ)2 4kT λ
(20b) in which p°s is the hole density and E°v,s the position of the valence band, both at UE ) U°redox ) - E°F,redox/e. As already mentioned before, the potential distribution changes in the potential range considered here. Such a change has been observed before with GaAs electrodes (see, e.g., ref 18). The shift of bands with respect to the energy levels of the redox system is illustrated in Figure 11. From Figure 7 the flatband potential and consequently the position of the valence band Esv and the potential across the space charge layer φsc can be determined for any electrode potential. In this procedure it is assumed that each experimental value of 1/Csc2 in Figure 7 (also in the transition range) lies on an individual Mott-Schottky curve with a slope given by the doping. The slopes of the extreme cases are illustrated by the solid lines in Figure 7. The slopes are identical to those obtained by dynamic MottSchottky measurements under reverse bias. From φsc the corresponding ps value can be calculated by using eq 3b. By use of these data, the individual rate constants k+ and k- and their sum have been calculated from eqs 20a and 20b for a Cu2+ concentration of 16 mM as shown in parts a and b of Figure 12. In the calculation a reorientation energy of λ ) 0.5 eV has been used because such a value has been determined experimentally for hexacyanoferrate, i.e., a complex with strongly bonded ligands similar to the case of the copper chloride complex. The best fit has been obtained for ko ) 1 × 10-2 cm s-1. Much higher values of λ, such as λ ) 1 eV, do not change much the theoretical curves in Figure 12. This is not surprising
( (
)
E°v,s - Esv ps k ) ko exp p°s 2kT +
k ) k ) ko at UE ) U°redox
k+ ) ko
Esv - E°F,redox , λ
k- ) ko exp
)
E°v,s - Esv 2kT
(21a)
(21b)
i.e., the potential dependence of k+ and k- becomes independent of λ. Only the absolute values of these rate constants depend on λ because ko varies with λ as given by the Marcus theory. Since, however, ko is an adjusted parameter, the evaluation is not affected by λ. From the ko value one can also determine the second-order rate constant k+ v after having evaluated the surface hole density at the standard redox potential (U°redox ) 0.42 V). For copper ion concentrations greater than 8 mM we -19 cm4 have obtained p°s ) 2 × 1016 cm-3 and k+ v ) 5 × 10 -1 s . Assuming again λ ) 0.5 eV, we also have determined + -17 cm4 k+ v,max by using eq 18a, and we obtained kv,max ) 1 × 10 -1 s . In the case of a much larger λ value such as 1 eV, one -17 cm4 s-1. Both values are in obtains k+ v,max ) 6 × 10 agreement with the theoretical values derived by Lewis19 who predicted maximum rate constants on the order of 10-17-10-16 cm4 s-1. At present we cannot give more accurate values because we did not find any data on the reorientation energy for the Cu1+/Cu2+ system in the literature. The evaluation of the charge transfer between the valence band of GaAs and the Cu1+/Cu2+ redox system is based on the assumption that the oxidation of Cu1+ occurs independently from the anodic decomposition process. This is fulfilled for sufficiently large Cu1+ concentrations because the current then increases linearly with the concentration. The situation is different at very low concentrations for which the total current even decreases with increasing concentrations (Figure 3). These results indicate that the oxidation of Cu1+ and the decomposition do not occur independently in this concentration range. Similar observations have been made earlier by various authors.12,14,15,20,21 This phenomenon can be interpreted by assuming the formation of an intermediate surface state in the decomposition process (see, e.g., ref 21 and literature cited there). With the neglect of direct hole transfer from the valence band to Cu1+, the corresponding mechanism is given in Figure 13. According to this reaction scheme, a GaAs+ radical is formed in the first reaction step. The anodic shift of the flatband potential in the anodic potential range was interpreted by the formation of these radicals.22 The latter reacts in a pure chemical step with hydroxyl or chloride ions (X-) in the solution, leading to the
4936 J. Phys. Chem., Vol. 100, No. 12, 1996
Uhlendorf et al.
Figure 13. Reaction scheme.
formation of GaAsX at the surface. Another five holes are required for a complete dissolution reaction. Besides this process, a charge transfer between the surface radical formed in the first step and the reduced species of the redox system can occur, leading to the oxidation of Cu1+. By use of the notations given in ref 21, the essential rates are then given by
V1 ) k1ps
(22)
V2 ) k2c(GaAs+)cX-
(23)
V3 ) k3cGaAsXps
(24)
V4 ) k4c(GaAs+)cred
(25)
Under stationary conditions we have
V1 ) V2 + V4
(26)
V2 ) V3
(27)
The total current j is then composed of the decomposition jcorr and the redox current jredox, i.e.,
j ) jcorr + jredox ) e(V4 + 6V3) Inserting eq 26 into eq 28, we obtain
(
)/ (
k4cred j ) eV1 +6 k2cX-
)
k4cred +1 k2cx-
(28)
range, probably some surface layer, similar to the case of Fe2+ oxidation, was formed during anodic polarization.4 This behavior made any analysis impossible. Another example that could be successfully studied by impedance spectroscopy is the reduction of protons (H2 formation) at n-GaAs electrodes. In this case the corresponding transfer rate and current followed exactly the kinetics expected for a reaction at a semiconductor electrode. Interestingly, we found in this case that the electron transfer rate is very large and the current could be described by a thermionic emission model.23 6. Conclusions In the present paper impedance spectroscopy data have been analyzed. In the case of the Cu1+/Cu2+ redox system, in which oxidation and reduction are diffusion-controlled reactions at p-GaAs, the second-order rate constant has been determined. According to these measurements, the flatband potential is shifted during current flow, which has been interpreted to be caused by the formation of surface radicals. Attempts to analyze the oxidation of Eu2+, which is a kinetically controlled process, failed because of layer formation. Since it is difficult to find redox couples suitable for kinetically controlled reactions at GaAs in aqueous solutions, corresponding investigations in nonaqueous electrolytes are under way. Acknowledgment. The authors are indebted to Dr. D. Meissner, ISFH, Hannover for many valuable discussions. We also thank Kerstin Siemoneit, ISFH, Hannover for performing some measurements with the europium redox system at GaAs electrodes. We thank the Deutsche Forschungsgemeinschaft for financial support. References and Notes
(29)
According to eq 29, the total current varies from j ) 6eV1 at cred ) 0 to j ) eV1 at cred ) ∞, i.e., it can decrease by a factor of 6. Experimentally, we found a much smaller decrease (Figure 3) because the direct hole transfer from the valence band to Cu1+, which was neglected in the above derivation, is the dominating process. In the presence of Cu1+ the radical density at the surface is reduced to a lower value, which leads to a corresponding cathodic shift of the flatband potential as proved by dynamic Mott-Schottky measurements. According to the results presented here, the charge-transfer reactions between GaAs and the Cu1+/Cu2+ redox system is well understood. However, the quantitative evaluation of the actual transfer kinetics is rather tedious for diffusion-controlled reactions, such as that of Cu1+/Cu2+ at GaAs. Accordingly, it would be much more feasible to study the essential parameters of a kinetically controlled electron transfer process between a redox system and a semiconductor electrode. In the case of the former Cu system the oxidation current is still not kinetically controlled at 10 000 rpm under anodic bias as can be easily calculated from the data given here. It is very difficult to find another redox couple, the oxidation or reduction of which competes sufficiently well with the anodic corrosion of GaAs and hydrogen formation, respectively. We tested one example, namely the Eu2+/Eu3+ redox system because its oxidation occurs at a p-type GaAs electrode at considerably lower anodic potentials than the anodic decomposition. Surprisingly, the oxidation of Eu2+ is a valence band process although the standard potential (-0.7 V vs SCE) is rather negative with respect to the energy bands at the surface of GaAs.11 Since, however, a pronounced hysteresis occurred in the currentpotential curve when scanning across the corresponding potential
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