9438
J. Phys. Chem. B 2000, 104, 9438-9443
Determination of Rate Constants for Dark Current Reduction at Semiconductor Electrodes Using ZnO Single-Crystal Microelectrodes Sheila Rodman and Mark T. Spitler* Polaroid Corporation, Waltham, Massachusetts, and ChemMotif, Inc., Concord, Massachusetts 01742-2411 ReceiVed: February 3, 2000; In Final Form: July 19, 2000
A method based on microelectrodes of ZnO single crystals was employed to determine the effective reaction velocity for dark reduction of ferrocenium acceptors by electrons from the semiconductor conduction band. Measurement techniques were explored and theoretical models for steady-state voltammetry at semiconductor electrodes developed. The microelectrode method as presented here represents an unambiguous experimental method for ascertaining ideal interfacial behavior even at high current flows found near the flatband potential. These results are compared with reaction velocities derived from extrapolation methods at millimeter-sized ZnO electrodes. It is seen that the extrapolation method is insensitive to nonideal behavior in the reduction reaction whereas the microelectrode method is not.
The experimental determination of the rate constants for electron transfer at semiconductor electrodes has been actively pursued as an area of research with the intent of confirming models for electron transfer at the semiconductor/electrolyte interface. The experimental work includes studies of the dark transfer of charge carriers between the semiconductor and a redox agent, the photoinduced charge transfer from the semiconductor to the redox agent, and the photooxidation of molecules adsorbed to the semiconductor. The measured rates of reactions and the derived rate constants have been compared with the predictions of models for semiconductor electrochemistry presently in use. Of particular interest are mechanisms for hot electron transfer,1 the qualitative models for thermalized semiconductor electrochemistry constructed by Gerischer,2 and the more quantitative formulations for electron transfer obtained when they are grafted to modern theories of electron transfer, such as those of Marcus or Jortner.3,4 These experimental examinations of heterogeneous electron transfer at semiconductor surfaces have approached the problem from many different angles. Each requires a substantiation that the experimental system under study behaves in an ideal manner, one which corresponds to the models and allows the experimental results to be interpreted in terms of the theory. For example, the luminescence lifetime of excited semiconductors has been used to derive an electron transfer rate constant from the conduction band to an acceptor.5 However, a complicated modeling scheme must be used to describe the potential distribution at the interface, and if the system does not rigorously adhere to this scheme, the interpretation of the work can be compromised. Fajardo and Lewis6 have used this in the measurement of slow electron transfer from the conduction band of Si to methyl viologen acceptors. Nozik and coworkers7,8 have employed this approach in reports of faster electron transfer from GaAs to cobaltocenium, but had to quantify the extent of surface adsorption of the acceptor before reliable estimates of the rate constants could be made. Extensive impedance analysis was also required to ensure consistency with ideal polarization behavior of the electrodes. In an elegant and different optical approach, the modeling of transient grating spectroscopy has been employed to estimate the time for hole transfer from the TiO2 valence band to OH-
in solution.9 Once again, the system behavior must conform to the models of transient gratings for the results to be interpreted with confidence. In the femtosecond oxidation of dithiacyanato-bis(dicarboxybipyridyl)Ru(II) at nanocrystalline TiO2 surfaces, the rate constant for electron transfer appears to be shorter than 50 fs.10 Yet, care must be taken to ensure that the femtosecond timeresolved spectroscopy of the oxidized chromophore be analyzed properly.11 Such an activationless electron transfer is expected to be fast, but comparison of this activationless electron transfer with models for electron transfer implies that new models must be developed.12 Great care must be taken that the fluorescence lifetime not be influenced by energy transfer between adsorbed dye molecules. An old but useful tool for determination of rate constants is that of the potential dependence of the dark reduction of acceptors over the conduction band of n-type semiconductors. The use of this technique to examine electron transfer theory at semiconductor electrodes was introduced by Vanden Berghe et al.13 and used in more recent examinations of electron transfer models.6,7 This technique uses a measurement of absolute current that is extrapolated to the flatband potential. The validity of the result depends both on substantiation of the ideal nature of the interface and the assurance that the flatband potential determination is accurate. A solid-state physics perspective on the surface capture of electrons by acceptors can be found in earlier work by Morrison14 at ZnO surfaces using the same approach. In this case, many different oxidants were employed which had different reorganization energies for the electron transfer. In more recent work, Koval et al.15 have demonstrated that the use of structurally similar ferrocenes as redox agents with similar reorganization energies allows for a more thorough examination of electron transfer theory at semiconductor electrodes. Subsequent work has followed this model.6-8 A technique that offers an unambiguous evaluation of velocities for electron transfer reactions at electrodes is that of steady-state voltammetry, because it compares in an inherent fashion the competition between the rate of electron transfer with the rate of diffusion of the reactant species to and from the electrode surface. These diffusion problems are well-known
10.1021/jp0004258 CCC: $19.00 © 2000 American Chemical Society Published on Web 09/15/2000
Dark Current Reduction at Semiconductor Electrodes and can be described analytically or numerically and have been explored in recent work studying electrochemical reactions at microelectrodes in the 1-100 µm range.16 For a hemispherical microelectrode, the diffusion velocity kd of reactants to the electrode surface is given as kd ) 4D/πr where D is the diffusion constant of the reactant and r is the radius of the electrode.17 This serves as known reference point when evaluating the rate constant for electron transfer at that electrode. At the half wave potential E1/2 of the steady-state voltammetric current-voltage curve the velocity of the electron transfer reaction ket equals the diffusion velocity, kd ) 4D/πr ) ket. We will use this technique in this work to obtain unambiguous measures of the reaction velocity for the reduction of selected ferrocenium ions at ZnO semiconductor microelectrodes. This includes ferricenium (FC+), dimethyl-ferricenium (diM+), and decamethyl-ferricenium (DM+). This determination of ket can be made whether the electrode is behaving in an ideal fashion, and indeed, the microelectrodes of this work exhibited nonideal fashion. It is found to be a practical way to obtain rate constants for reactions at a real electrode. FC+, diM+, and DM+ ions are used in this work so that a correlation might be made between the measured rates and the thermodynamics of the reaction. With these results in hand, the validity of the dark current extrapolation method of Morrison and Vanden Berghe et al.13 for determination of rate constants is examined. In this technique, reduction currents over the conduction band of n-type semiconductor electrodes to an acceptor in solution are extrapolated to their value at the flat band potential, and a value for the ket is extracted and compared with electron transfer theory. We have performed these measurements with larger electrodes of the same ZnO needle crystals and the same ferricenium acceptors as the microelectrode studies and have derived ket values for these reactions in the standard manner. Given the results from the microelectrode measurements, our conclusion is that the extrapolation technique is unreliable. A linear extrapolation can be obtained even when the electrode is damaged and defective, and even the qualification measurements, such as flatband potential measurements, exhibit ideal behavior. We conclude that these extrapolation measurements can only be valid if their corresponding microelectrode steadystate voltammetry exhibits ideal behavior. Experimental Section The needle crystals of ZnO used in this study were approximately 5 mm in length with a hexagonal cross section of 0.2-0.3 mm, coming to points at each end. The crystals were mounted in glass tubing with commercial 5-ton epoxy with a ZnO point protruding. Electrical contacts were made at the opposite end using GaIn eutectic and conductive silver epoxy. The point of the electrode was polished to a flat surface to a 0.05 µm finish using diamond grit; the microelectrodes thus obtained began with a 50 µm diameter and became increasingly larger with each polish. Both Zn and O faces were used in these studies. No significant differences were found in their behavior. The ferrocene, dimethylferrocene, and decamethyl ferrocene were obtained from Lancaster Chemicals and used as obtained. The oxidized forms were prepared as the BF4- salts from these reagents through a chemical oxidation with hydroquinone and HBF4. HPLC grade acetonitrile was dried with activated 5 Å molecular sieves before use; the TBABF4 salts were purchased from Southwestern Analytical. SSV was performed using a computer-controlled Solartron 1286 potentiostat. Mott-Schottky analyses were made with the
J. Phys. Chem. B, Vol. 104, No. 40, 2000 9439
Figure 1. The SSV of a 22/28 mM diM+/diM solution for a 50 µm Pt diameter microelectrode and a ZnO microelectrode. The plateau current relative to that of Pt gives the diameter of the ZnO electrode, a result which corresponds to the radius obtained from optical microscopy. The E1/2 value is shifted to -540 mV (vs FC+/FC) and the E1/4-E3/4 span broadens to 90 mV. The electrolyte was 0.10 M TBABF4 in acetonitrile.
use of a Stanford Research Systems 551 lock-in amplifier. Both metal and ZnO electrodes were polished in 0.05 µm diamond paste, rinsed in water, etched briefly in concentrated H3PO4 or 0.10 M HCl, and rinsed in deionized water and acetonitrile. The H3PO4 etch gave the most reproducible results. An HCl etch resulted in a less-controlled result with significant shifts of the CV curve along the potential axis. Commercial Pt microelectrodes were used to calibrate the ZnO electrodes and were themselves calibrated daily through SSV with ferrocene solutions of known concentration. In this calibration, a half dozen such curves were recorded and an average radius was derived from the plateau current. For the purposes of these experiments a radius within 10% of the specified value was considered acceptable. The ZnO microelectrodes have the cross section of a regular hexagon, but should have currents within 20% of that of an electrode of radius equal to the side of the hexagon. The acetonitrile electrolyte solution was bubbled with Ar prior to use to decrease the O2 concentration. A Ag+/AgNO3 reference electrode was used in these experiments which was calibrated against ferrocene before each use. All experiments were performed with poised redox couples at 5-10:1 ratios of oxidized to reduced forms. In mixed experiments, the reduced form of only the more positive couple was added. Results Steady-state voltammetry for the reduction of diM+ at ZnO electrodes produced current voltage curves as shown in Figure 1 where a diffusion-limited plateau was attained. A CV curve obtained with a 100 µm diameter Pt microelectrode is also given in Figure 1 and served as a calibration of the solution concentration and the size of the ZnO electrode. The size calibration from the Pt microelectrode measurements matched that found from examination of the ZnO with a stereoscopic optical microscope at 255× magnification. No anodic currents distinguishable from noise were observed at ZnO with any of the reduced forms FC, diM, or DM of the redox couples.
9440 J. Phys. Chem. B, Vol. 104, No. 40, 2000
Rodman and Spitler
Figure 2. The SSV curves for 2/15 mM FC+/FC and 1/5 mM DM+/ DM couples are shown with currents normalized. Conditions are as in Figure 1, but with a 100 µm diameter ZnO electrode.
Figure 3. An example of a concentration study is given here for diM+ in which [diM+] is increased from 0.16 to 3.2 mM. The plateau current at a 100 µm diameter ZnO electrode was found to be linear with [diM+] and the range of E1/2 values was within 30 mV of the average of 510 mV, indicating that the rate constant did not change significantly with concentration.
Similar CV curves were found for FC+ and DM+ and are shown in Figure 2a,b for a 100 µm diameter ZnO electrode. Concentration studies were performed with each acceptor to confirm that the plateau current of the SSV was in direct proportion to the concentration of the acceptor in solution. An example of such a study for diM is given in Figure 3 spanning a range from 160 µM to 3.2 mM. With solution concentrations of diM above 400 µM, the E1/2 values were found not to vary by more than (30 mV during these concentration studies, indicating that the rate constant for electron transfer also did not vary. As has been well documented for ZnO18 there is an equilibration period of some tens of minutes for the double layer of a ZnO electrode following an acid etch, with HCl requiring a much greater recovery period than concentrated H3PO4. We observe an initial shift in the CV curves of Figure 1 amounting to a negative shift of about 100 mV for the first CV curve immediately following an acid etch. With time, this shift lessened to approached the curves of Figures 1 and 2; it could also be followed with impedance measurements.
Figure 4. (a) A double wave is observed for the reduction of a mixed solution of FC+/DM+. The difference between the E1/2 values of the two curves provides an accurate measure of the reduction rate constant ket for the two compounds. (b) A double wave is shown for a mixed diM+/DM+ solution. (c) A mixed reduction of FC+/diM+ must be done in two steps. First, a FC+ curve is recorded, followed by addition of diM+ and a repeat measurement. The difference between the two curves represents the diM+ SSV wave.
This time dependence for equilibration induced a scatter of (50 mV in the E1/2 values obtained for single CV curves from measurement to measurement. However, the relative E1/2 values for the various ferriceniums could be established with greater accuracy through experiments with mixed solutions of ferriceniums. CV curves are given in Figure 4 for three separate mixtures. For the cases of Figure 4, parts a and b, a clear double wave is observed for mixtures of FC+/FC and DM and diM/diM+ and DM. The corresponding experiments with an HCl etch were also performed. A difference was found in the FC/DM curves, an example of which is given in Figure 4a where the reduction of DM+ is shifted by about 150 mV positive of the H3PO4 sample. For diM+ and FC+/FC mixtures, the E1/2 potentials are too close for a distinct double wave as is shown in Figure 4c. In this case, a FC+/FC CV curve was recorded first, then diM+ was added and the CV curve recorded again. The difference between the two curves allowed an accurate assessment of the difference between the two couples. Using an E1/2 for FC+/FC
Dark Current Reduction at Semiconductor Electrodes
J. Phys. Chem. B, Vol. 104, No. 40, 2000 9441
TABLE 1: Tabulation of Rate Constants
FC+ diM+ DM+ DM+
E1/2 (vs FC/FC+) ket ) 4 × 10-4cm/s
ket (extrapolation) (cm/s)
k° (extrapolation) (E ) Ec) cm4/s
A (extrapolation) cm4/s
-440 (( 30 mV) -510 -870 (H3PO4) -720 (HCl)
(at -440) 2 × 10-4
4 × 10-18 (0.8-8)
3 × 10-17 (0.8-6)
(at -720) 5 × 10-4
7 × 10-23
3 × 10-20
Range of standard deviation given in parentheses. λ is taken to be 0.55 eV.
obtained from single-wave experiments as a reference, the resultant E1/2 values for diM and DM extracted from such double wave experiments with a 100 µm diameter electrode are listed in Table 1. At E1/2, ket is equal to the rate of diffusion away from the electrode. The ratio of these two rates is given by the parameter κ ) πketr/4D. The E1/2 values represent the potential where κ ) 1; for these electrodes, where r ) 50 µ, this corresponds to a first-order reaction velocity ket of 4 × 10-4 cm/s. The reduction of the ferriceniums is a second-order reaction dependent on the acceptor concentration and the electron concentration at the ZnO surface. In an examination of electron transfer theory, one should employ electrochemical models for current flow of a rigor equal to that of the Marcus model. In this case, where the maximum second-order reaction rate constant is less than 10-16 cm4/s, the exact analytical expression19 describing the current-voltage curve at semiconductor electrodes reduces to that of the ad hoc model:
j) Fket[FC+] ) Fko(esf)[FC+]
(1)
where F is Faraday’s number. The value of (esf) in #/cc is related to the flatband potential Vfb by the relation
(esf) ) Nd exp(V-Vfb)e/kT
(2)
where Nd is the doping density of the ZnO and V is the applied bias on the ZnO electrode. The expression of eq 1 can be used to construct the form of SSV waves at semiconductor microelectrodes following the manner of previous work.17 In summary, the extensive modeling of SSV at metal microelectrodes can be carried over to semiconductor electrochemistry easily since the bulk of the modeling pertains to the relation between reactant diffusion and current flux. At the end of such modeling, one plugs in the electrochemical relation between current flux and potential for the electrode of interest. In this case, the relation of eq 1 works well. Horrocks et al.20 come to a similar conclusion through a different approach. In the regime of validity for eq 1, the potential dependence of j, through κ, is given for semiconductors by
(E-E1/2) ) (kT/q){ln κ - 0.06985}
(3)
This predicts an ideal E1/4-E3/4 potential span for the SSV wave of 59 mV. E1/4-E3/4 is the difference between the potentials where the current equals 1/4 and 3/4 of its plateau value. However, the SSV of Figures 1 through 4 that is positive of Vfb show an E1/4-E3/4 separation that approaches 90 mV. Horrocks et al.20 observe such deviations in the current voltage curves of Si and CdS using SECM as the scanning electrochemical tip is moved across the surface of the electrode. There are areas of the electrode where ideal behavior is observed and other, damaged areas where it is not. Evidently, it was not possible to obtain a completely ideal surface with these ZnO microelectrodes, despite all of the
various etching techniques that were explored. Surface states or adsorption are most likely involved in this broadening of E1/4-E3/4. A distribution of surface states, for example, with energies in the band gap below the conduction band edge would have a distribution of ∆G values for the reduction which could cause such an effect. These results for ket from the microelectrode method will now be compared with evaluations of ket derived from the extrapolation method. From eq 1 it is seen that ket at E1/2 is equal to ket ) k°(esf)E1/2. According to eq 2, ket should be dependent upon (V - Vfb) in an exponential fashion. To obtain k°, one extrapolates to Vfb the absolute current for reduction of an acceptor at the surface of a ZnO electrode in order to obtain the current at flatband potential. The flatband potential Vfb for these electrodes was determined through impedance measurements during the course of CV measurements, which for FC+ reached the plateau current. Vfb was found to be the same for all redox couples and independent of concentration up to 5 mM. The electrode in electrolytes containing different redox couples might differ in flatband potential if surface states interact with the redox couple as Koval et al.15 have found for ferrocenes at InP. The redox agents charged the surface and shifted band-edge energies. However, Koval and Olsen21 have also found that for WSe2, the interfacial energetics are unaffected by reduction of various ferroceniums. Extensive impedance analysis was used in drawing this conclusion. Therefore, it is not unprecedented that the various ferriceniums should have no distinctive charging effects upon ZnO. A Vfb of -0.585 V ( 0.035 (vs FC+/FC) was found as was a doping density of (2-3) × 1016/cc. These values held constant over the 500-3000 hz modulation range. The same donor density was found for these electrodes in an aqueous electrolyte of 1.0 M KCl. Representative curves for these extrapolations of CV data to Vfb are shown in Figure 5 for reduction of FC+ and DM+ following a 0.1 M HCl etch. Use of concentrated H3PO4 as an etchant resulted in smaller linear regions for these plots. Using ZnO electrodes with a 0.2 mm cross section and an HCl etch, the CV curves of Figure 5 for reduction of FC+ and DM+ were obtained in solutions where the concentration varied from 1 to 30 mM. With an Nd of 3 × 1016/cc and a Vfb of -0.585, the ko values for reduction can be calculated from eq 1; they are listed in Table 1 in units of cm4/s. With this information, a comparison of the extrapolation and microelectrode methods is possible. The calculated reaction velocity at E1/2 for these reduction reactions is then derived from ko by multiplying by (esf)E1/2. These values for ket from the extrapolation method are listed in Table 1. The exponential current voltage curves of Figure 5 for FC+ and DM+ for an HCl etch can be analyzed through extrapolation to obtain the preexponential factor A in a Marcus interpretation3 of the electron transfer reaction:
ko ) A exp(-(∆G + λ)2/4λkT)
(4)
9442 J. Phys. Chem. B, Vol. 104, No. 40, 2000
Figure 5. The approach of Vanden Berghe et al.13 is taken to determine k° for FC+ and DM+ reduction following a 0.1 M HCl etch of the 500 µm diameter ZnO electrode. The reduction current is extrapolated to Vfb and k° is extracted from the result through the use of eqs 1 and 2. Solution concentrations of FC+ and DM+ are 15.2 mM and 9.8 mM with 0.10 M TBABF4 in acetonitrile.
where ∆G ) F(Ec - EoA/D) and λ is the reorganization energy in electron transfer theory. With the structurally similar ferrocenes used in this work, a common λ can be used for all three compounds. A value of 0.55 eV best fits the data and is not an unreasonable λ for the ferroceniums. The results are listed in Table 1 with uncertainties that reflect the influence from such a strong etchant. They are both below 10-16 cm4/s, which is a requirement for the validity of eq 1. Discussion The shapes of the SSV curves for reduction of the various ferroceniums in Figures 1-4 are not ideal. According to eq 3 there should be a 59 mV potential difference between 10% and 90% of the plateau value of the current, but the best that could be obtained was 90 mV. Experimentally, we have observed that the sharpness of the SSV curve can vary greatly with electrode preparation and have concluded that this breadth is indicative of defect behavior in the surface or adsorption. There appear to be no theoretical grounds for such a broadening and ideal SSV curves with a 59 mV breadth have been reported using SECM techniques. It is possible to model the 90 mV width of the curve with a distribution of surface states about the E1/2 value. However, recent work8 has demonstrated that adsorption of the redox agents by the electrode can complicate the interpretation of results. We note that the resultant values of ket for FC+ and DM+ in Table 1 derived from the extrapolation method are comparable to those from the microelectrode experiments even though the latter are consistent with those expected from an electrode where a surface state or adsorption may be involved in the electron transfer. It is evident that the extrapolation procedure is blind to the role of complications in the electron transfer reaction, whereas the microelectrode method is not, and that the use of the extrapolation procedure must be accompanied by assurances that surface states or adsorption do not exist or play no role. The microelectrode technique as presented here represents an unambiguous experimental method for ascertaining ideal interfacial behavior even at high current flows found near Vfb. The microelectrode SSV curves provide a pragmatic means for assessment of this ideality. They also provide an effective rate constant for the reduction reaction at high current such as would
Rodman and Spitler be found in any solar conversion applications of semiconductor electrodes. This method also indicated complications such as the dependence of the DM+ current-voltage curve upon etchant. Surface states might be responsible for this 150 mV shift in the half-wave potential for reduction of DM+ when it is etched with HCl as opposed to H3PO4. However, with this large of a change in the rate constant for reduction, adsorption is the more likely explanation. A H3PO4 etch may lead to adsorption of a layer of DM+/DM couple that would slow the electron transfer reaction through a separation of semiconductor from electrolyte. Meier et al.7,8 have reported in detail on a related cobaltocenium adsorption on GaAs surfaces. It can also be seen in Figure 4 that the E1/4-E3/4 for reduction of DM+ following an H3PO4 etch is much smaller than that for FC+, by as much as 30 mV and can approach the ideal 59 mV limit. With a measured flatband potential of -585 mV (vs Fe(Cp)2+/0), the half-wave potential for the DM+ reduction falls negative of the flatband potential, in an accumulation regime for the semiconductor. Recent work of Meier et al.7 has shown that adsorption of cobaltocenium on semiconductor surfaces can influence its reduction rate. In that case, the presence of adsorbed cobaltocenium accelerated the rate to produce an apparent A value that approaches 10-14cm4/s. Such an acceleration is not observed here. The A values are all below 10-16 cm4/s. Thus, it is possible that the electron transfer could be direct from conduction band to the ferrocenium acceptor ion in solution or it could be mediated by an adsorbed acceptor which has a fast exchange reaction with the corresponding solution ferrocene. The latter situation would prove the most simple explanation of the electron transfer processes of this work because an adsorption of ferrocenium on different surface sites would certainly lead to different λ values for the different adsorbate molecules.22 This would result in a spread of ko values which would broaden the E1/4-E3/4 width of the SSV curves. Analysis of the reduction curve for DM+ following an H3PO4 etch is an involved exercise in the fundamentals of semiconductor electrochemistry, because at bias potentials negative of Vfb, the electrode acquires an accumulation layer at the surface and begins the transition to metallic behavior.18,23 Under an accumulation condition, a negative charge Q builds up in a thin layer at the surface. This charge Q can induce a φ2 potential change in the diffuse double layer and a φHH change in the Helmholtz layer, for which both the applied potential and the cationic [DM+] must be corrected. As has been explored in recent work,23 these changes are modest in the regime of mild accumulation, but appear to accelerate under strong accumulation conditions where the Fermi level of the semiconductor moves into the conduction band at the surface.18 At this potential, which corresponds to about -650 mV for the ZnO electrodes of this study, the accumulation of charge is controlled by a capacitance which is independent of the doping density of the semiconductor and dependent only on its density of states near the conduction band edge:18
Csc ) 6{(5Ncq2/3π1/2kT)(m(N)/m)}(∆φ/kT - θ)1/4 (5) where is the dielectric constant, Nc and Nd are the density of conduction band states and dopants, m(N) is the effective mass of electrons, and θ ) kT ln(Nc/Nd). To extract rate constants from SSV measurements at semiconductor electrodes under this accumulation condition, the potential distribution across the semiconductor, Helmholtz layer, and the diffuse double layer must be deduced. Given models
Dark Current Reduction at Semiconductor Electrodes for accumulation layers at semiconductors18,23 and for diffuse double layers,24 the potential drop across these two layers can be calculated, leaving the difference between their sum and the applied potential as ∆φHH. With this information, one can calculate the electron density at the surface as a function of potential and therefrom the ko for reduction of DM+ following an HCl etch. The result is, however, of limited utility, because it cannot be compared with the corresponding ko results for reduction reactions occurring positive of the flatband potential where the ∆φHH is different. Yet the deduced ko value also cannot be compared with rate constants for reduction at metal surfaces because the potential at which the electron transfer occurs is far from that of the equilibrium potential of the redox couple. One is left in the end with only the very pragmatic result of a known reaction velocity at E1/2. Acknowledgment. The authors acknowledge the advice and assistance of Professor Carl Koval and Dr. David Watts during the course of this work. References and Notes (1) Williams, F.; Nozik, A. J. Nature 1984, 311, 21. (2) Gerischer, H. Z. Phys. Chem. 1960, 26, 223.; Gerischer, H. Z. Phys. Chem. 1961, 27, 48. (3) Marcus, R. A. Annu. ReV. Phys. Chem. 1964, 15, 155. (4) Jortner, J. J. Chem. Phys. 1976, 64, 4860. (5) Forbes, M. D. E.; Lewis, N. J. Am. Chem. Soc. 1990, 112, 3682. (6) Fajardo, A. M.; Lewis, N. S. Science 1996, 274, 969. (7) Meier, A.; Kocha, S. S.; Hanna, M. C.; Nozik, A. J.; Siemoneit, K.; Reinecke-Koch, R.; Memming, R. J. Phys. Chem. B 1997, 101, 7038.
J. Phys. Chem. B, Vol. 104, No. 40, 2000 9443 (8) Meier, A.; Selmarten, D. C.; Siemoneit, K.; Smith, B.; Nozik, A. J. J. Phys. Chem. B 1999, 103, 2122. (9) Wang, D.; Buontempo, J.; Li, Z. W.; Miller, R. J. D. Chem. Phys. Lett. 1995, 232, 7. (10) (a) Hannapel, T.; Burfeindt, B.; Storck, W.; Willig, F. J. Phys. Chem. B 1997, 101, 6799. (b) Ellingson, R. J.; Asbury, J. B.; Ferrere, S.; Ghosh, H. N.; Sprague, J. R.; Lian, T.; Nozik, A. J. J. Phys. Chem. B 1998, 102, 6455. (c) Ellingson, R. J.; Asbury, J. B.; Ferrere, S.; Ghosh, H. N.; Sprague, J. R.; Lian, T.; Nozik, A. J. J. Phys. Chem. B 1999, 103, 3110. (11) Moser, J.; Noukakis, D.; Bach, U.; Tachibana, Y.; Klug, D. R.; Durrant, J. R.; Humphrey-Baker, R., Gratzel, M. J. Phys. Chem. B 1998, 102, 3649. (12) Hannapel, T.; Zimmermann, C.; Meissner, B.; Burfeindt, B.; Storck, W.; Willig, F. J. Phys. Chem. B 1998, 102, 3651. (13) Vanden Berghe, R. A. L.; Cardon F.; Gomes, W. P. Surf. Sci. 1973, 39, 368. (14) Morrison, S. Surf. Sci. 1965, 15, 363. (15) Koval, C. A.; Austermann, R. L.; Turner, J. A.; Parkinson, B. A. J. Electrochem. Soc. 1985, 132, 613. (16) Ultramicroelectrodes; Fleischmann, M., Pons, S., Rolison, D., Schmidt, P. P., Eds.; Datatech Systems, Inc.: Morganton, NC, 1987. (17) Bond, A. M.; Oldham, K. B.; Zoski, C. G. J. Electroanal. Chem. 1988, 245, 71. (18) DeWald, J. F. Bell Sys. Technol. J. 1960, 615. (19) Spitler, M. T. Zeit. Phys. Chem. 1999, 212, 173. (20) Horrocks, B. J.; Mirkin, M. V.; Bard, A. J. J. Phys. Chem. 1994, 98, 9106. (21) Koval, C.; Olsen, J. B. J. Phys. Chem. 1988, 92, 6726. (22) Smith, B. B.; Koval, C. A. J. Electroanal. Chem. 1990, 277, 43. (23) Natarajan, A.; Oskam, G.; Searson, P. C. J. Phys. Chem. B 1998, 102, 7793. (24) Delahay, P. Double Layer and Electrode Kinetics; Interscience: New York, 1965.
