J. Phys. Chem. B 1997, 101, 7435-7442
7435
Electron Injection by Photoexcited Ru(bpy)32+ into Colloidal SnO2: Analyses of the Recombination Kinetics Based on Electrochemical and Auger-Capture Models William E. Ford,* Jurina M. Wessels,† and Michael A. J. Rodgers Center for Photochemical Sciences, Bowling Green State UniVersity, Bowling Green, Ohio 43403 ReceiVed: January 7, 1997; In Final Form: June 24, 1997X
The photosensitization of colloidal particles of antimony-doped SnO2 by electrostatically adsorbed Ru(bpy)32+ (bpy ) 2,2′-bipyridine) produced Ru(bpy)33+ and a conduction band electron (ecb-) with a quantum efficiency close to 1. The influence of the injected electron population density on the dynamics of the subsequent recombination between Ru(bpy)33+ and ecb- was investigated by nanosecond laser flash photolysis. The initial average number of Ru(bpy)33+/ecb- pairs per particle (Ne) created by the laser pulse was varied in the range 0.03-7. The rate of recombination (krec) increased nonlinearly by a factor of 250 over this range in Ne. The nonlinear dependence of krec on Ne was analyzed quantitatively using two different models: (1) an electrochemical model based on the Butler-Volmer equation and (2) a model based on the Auger recombination mechanism. Both models gave satisfactory fits to the data with parameters that are physically reasonable. The Butler-Volmer equation was used to relate the rate of recombination to the overpotential driving the reaction by making the assumption that the overpotential is linearly proportional to Ne. That analysis gave values of (2.1 ( 0.7) × 106 s-1 for the intrinsic rate constant and 0.59 ( 0.11 for the transference number, with each excess electron contributing 22 ( 6 mV to the overpotential. In the Auger-capture model, it was assumed that an equilibrium exists between Ru(bpy)33+ and the stannic hydroxyl radical, >SnO•, and that ecb- recombines with >SnO• and not directly with Ru(bpy)33+. The values of n0 and B1 exp(∆E°/kBT) were found to be (1.3 ( 0.1) × 1019 cm-3 and (9 ( 1) × 10-34 cm6 s-1, respectively, where n0 is the equilibrium concentration of free electrons, B1 is the coefficient for the conduction band-to-trap Auger transition, and ∆E° is the difference in reduction potentials of Ru(bpy)33+ and >SnO•. Comparison of these data to those for colloidal particles of undoped SnO2 showed that the doping of SnO2 with antimony does not significantly affect the dynamics of electron recapture by Ru(bpy)33+.
Introduction The recombination of charge carriers in semiconductor materials is important in a wide variety of light-generating and electricity-generating optoelectronic devices, including lightemitting diodes and semiconductor lasers as well as photodetectors and photovoltaic solar cells. The efficiencies of these devices are largely governed by the relative rates of radiative and nonradiative recombination processes.1-4 Heterogeneous photocatalysis is another semiconductor-based process whose efficiency is largely governed by the dynamics of charge-carrier recombination. During the photocatalytic oxidation of pollutants, for example, oxidative pathways initiated by valence band holes (hvb+) or reductive pathways initiated by conduction band electrons (ecb-) generally work in combination with O2 toward the ultimate mineralization of organic compounds.5-7 Recombination between hvb+ and ecb- competes with these reactions and thus is an important factor in determining the overall quantum efficiency. Several recent publications deal with the dynamics of charge-carrier recombination in TiO2.8-10 The recombination process described in this paper is most relevant to the development of electrochemical photovoltaic cells that rely on dye sensitization of wide bandgap semiconductors. Recent developments in that field have led to new solar cells based on nanocrystalline films with energy conversion efficiencies as high as 10%.7,11 Recombination of the photoinjected ecbwith the oxidized sensitizer at the semiconductor surface † Current address: GSF-Flow Cytometry Group, Ingolsta ¨ dter Landstr. 1, 85764 Oberschleissheim, Germany. X Abstract published in AdVance ACS Abstracts, August 1, 1997.
S1089-5647(97)00094-1 CCC: $14.00
competes with regeneration of the sensitizer by reaction with an electron donor in the liquid phase and thus limits the overall photocurrent efficiency. The success of Gra¨tzel and co-workers11 with these new solar cells has spurred on a great deal of research activity in the subject of interfacial charge-transfer processes in colloidal semiconductor systems.7,11,12 Despite these efforts, more precise data about the details of photosensitized electron injection and recapture processes are required. In this paper we describe a laser flash photolysis investigation of the dynamics of electron recapture in colloidal tin dioxide and its dependence on the initial number of ecb- photoinjected into each particle. Previous studies have shown that the oxidant Ru(bpy)33+ (bpy ) 2,2′-bipyridine) is produced with near unit quantum efficiency with visible light when Ru(bpy)32+ is electrostatically adsorbed to colloidal tin dioxide particles in aqueous suspension (eq 1).13-15 Electron injection in eq 1 is characterized by a rate constant kinj, the order of magnitude of which is 109 s-1.16
Ru(bpy)32+ + hν (532nm) f Ru(bpy)32+* f Ru(bpy)33+ + ecb-(SnO2) (1) Ru(bpy)33+ is subsequently reduced back to Ru(bpy)32+ by the conduction band electron, ecb- (eq 2). This electron
Ru(bpy)33+ + ecb-(SnO2) f Ru(bpy)32+
(2)
recapture process is characterized by a rate constant krec, the order of magnitude of which is 106 s-1 and is slow enough for the Ru(bpy)33+ to oxidize other molecules in solution.13,15,17 © 1997 American Chemical Society
7436 J. Phys. Chem. B, Vol. 101, No. 38, 1997 The present study mainly employs antimony-doped tin dioxide, Sb/SnO2, as the colloidal semiconductor. The properties of this semiconductor and its interactions with Ru(bpy)32+* are similar to those of undoped SnO2 used in earlier experiments.13,18,19 This paper focuses on two aspects of the kinetic behavior of electron recapture: (1) the nonexponential timedependence and (2) the dependence of krec on the average number of Ru(bpy)33+/ecb- pairs per particle, Ne, produced during the laser pulse. We13,15,17 and others20,21 have observed that the time-dependence of electron recapture following injection is well-described by the stretched-exponential (or Kohlrausch) relaxation function, but the origin of this kinetic behavior and the significance of the parameters used to describe it have remained obscure.22-24 Although the stretched-exponential formulation is an empirical approximation, its use facilitates extraction of krec values from the experimental data so that the dependence of krec on Ne can be analyzed. The main purpose of this paper is to develop a quantitative understanding of how the rate constant for electron recapture by Ru(bpy)33+, krec, depends on Ne. Here we analyze the functional relationship between krec and Ne by using two different models. One model is based on the Butler-Volmer equation for electrode kinetics.25-28 The Butler-Volmer equation is a semiempirical one derived from a macroscopic point of view. The other model is based on the Auger recombination mechanism in semiconductors.1-4 Auger transitions are quantum mechanical phenomena, so that this model treats the recombination from a microscopic viewpoint. Equations derived from both the Butler-Volmer and Auger-capture models are shown to fit the data satisfactorily with physically reasonable parameters and thus provide a means of quantitative analysis of factors that determine krec. Experimental Section Materials. Colloidal antimony-doped SnO2 containing 19.0 ( 0.1 g of hydrated Sb/SnO2 per 100 g of sol and colloidal SnO2 containing 20.0 ( 0.1 g of hydrated SnO2 per 100 g of sol were purchased from Alfa. Electron microscopic and gravimetric characterizations of the dispersions were described previously.13,14,18 The Sb/SnO2 dispersion has NH4+ as counterion and contains 5 parts Sb/100 parts Sn; the Sb is present as both Sb5+ and Sb3+ ions.29 The undoped SnO2 dispersion has K+ as counterion. Particle concentrations were calculated on the basis of a diameter of 4 nm and density of 6.95 g cm-3.30 The calculated particle concentrations in the undiluted dispersions of Sb/SnO2 and SnO2 were 1.39 × 10-3 and 1.46 × 10-3 M, respectively. Tris(2,2′-bipyridyl)ruthenium(II) chloride hexahydrate (from Aldrich) was used as received. Concentrations of Ru(bpy)32+ in H2O were calculated from absorption spectra on the basis of an extinction coefficient of 1.43 × 104 M-1 cm-1 at 453 nm.31,32 Water was distilled and then passed through a Sybron/Barnstead nanopure II purification system (resistivity 17 MΩ or greater). Sample Preparation. Generally, 2.20 ( 0.01 mL portions of a stock solution of [Ru(bpy)3]Cl2 (23 ( 1 µM) in water were transferred to borosilicate glass test tubes, to which 0.5-200 µL portions of either Sb/SnO2 or SnO2 colloidal dispersions were added. An equivalent volume of water was added to one of the test tubes to provide a reference solution. For samples with Sb/SnO2 or SnO2 particle concentrations below 0.3 µM, the colloidal dispersions were diluted with water before adding them to the stock solution of [Ru(bpy)3]Cl2. After mixing, the samples were sonicated in a bath (Bransonic, Model 1200) for ca. 30 s. The sample pH was 9 ( 1, depending on the degree
Ford et al. of dilution of the Sb/SnO2 or SnO2 (undiluted pH = 10). Laser flash photolysis experiments were performed within ca. 3 h after preparation of the samples. It should be noted that aging of the samples for periods of the order of 1 day resulted in a decrease in quenching of Ru(bpy)32+* by the particles. A possible explanation of this aging effect is that silica leached slowly from the glass or quartz container and eventually deposited onto the surface of the semiconductor particles, creating an insulating barrier for electron injection. For this reason, the stock sols of SnO2 and Sb/SnO2 were stored in inert plastic containers. Measurements. All spectroscopic measurements were conducted at the ambient temperature, 23 ( 2 °C, with 1 cm2 quartz cuvettes. The ground-state absorption spectra were measured with a GBC Model 918 UV-visible spectrophotometer. Laser flash photolysis experiments were carried out using the frequencydoubled output (532 nm) of a Q-switched Nd:YAG laser (Continuum Surelite I) as the excitation source (ca. 7 ns pulse width), as described earlier.13 The interference filter used for time-resolved emission was a narrow band-pass filter centered at 610 nm (Oriel 53925). Typically, the kinetic data obtained from 15 laser shots were signal-averaged for each measurement. An optically matched solution of [Ru(bpy)3]Cl2 in water was used for actinometry.33 The samples were air-saturated in most cases; dissolved O2 shortens the emission lifetime of Ru(bpy)32+ in aqueous solution but has no other obvious effect on the system being investigated (vide infra). Data Handling and Calculations. Time-resolved absorption data were obtained with a monitoring wavelength of 450 nm, where the production of either Ru(bpy)32+* or Ru(bpy)33+ results in a net negative absorbance change, ∆A, due to bleaching of the MLCT absorption band of Ru(bpy)32+.31,32 Electron injection from Ru(bpy)32+* into the conduction band of the semiconductor particle results in luminescence quenching predominantly within the laser pulse when the complex is bound to the particle (eq 1), whereas the luminescence of the unbound complex decays exponentially with a lifetime of several hundred microseconds.14 Since binding of Ru(bpy)32+ to the semiconductor particles is an equilibrium process,14 the ∆A observed immediately after pulsed laser excitation, ∆A(0), contains contributions from both Ru(bpy)33+ and unquenched Ru(bpy)32+* (eq 1). On the basis of the observation that the decay of Ru(bpy)33+ via recombination with ecb- (eq 2) is adequately described by the stretched-exponential relaxation function,13,15,17,20,21 the time dependence of ∆A, ∆A(t), following laser excitation is given by eq 3, where l is the path length (1.0 cm), c+(0) and c*(0)
∆A(t) ) lc+(0)∆+ exp[-(krect)β] + lc* - (0)∆* exp(-k*t) (3) represent the concentrations of Ru(bpy)33+ and unquenched Ru(bpy)32+* immediately after laser excitation (t ) 0), respectively, and ∆+ and ∆* are the extinction coefficient changes associated with the formation of these two species. The decay of Ru(bpy)33+ is characterized by the rate constant krec and stretching parameter β (where 0 < β e 1), while the decay of Ru(bpy)32+* is characterized by the rate constant k*. The values of ∆+ and ∆* are -1.3 × 104 and -1.0 × 104 M-1 cm-1, respectively.31-33 Equation 3 was used previously to describe the decay kinetics in colloidal composites containing SnO2, surfactant bilayers, and an amphiphilic ruthenium(II) polypyridine complex.18,19 Since both Ru(bpy)33+ and Ru(bpy)32+* contribute to ∆A, three different fitting routines were employed to fit the
Electron Injection by Photoexcited Ru(bpy)32+
J. Phys. Chem. B, Vol. 101, No. 38, 1997 7437
Figure 1. Time-resolved absorbance changes at 450 nm in air-saturated and argon-saturated solutions of Ru(bpy)32+ (23.6 µM) and Sb/SnO2 colloidal particles (2.21 µM. The experimental data points are represented by dots, and fits of the data to eq 3 (algorithm A3) are represented by solid lines. The residue of the fits are shown as insets and the contributions to the absorbance changes made by the unbound population of Ru(bpy)32+* are shown as a dashed lines. The parameters for decay of Ru(bpy)33+ in the air-saturated sample were ∆A(0) ) -0.132 ( 0.002, krec ) (1.80 ( 0.09) × 107 s-1, and β ) 0.299 ( 0.003. In the argon-saturated sample, they were ∆A(0)) -0.124 ( 0.002, krec ) (1.46 ( 0.07) × 107 s-1, and β ) 0.317 ( 0.003.
experimentally determined ∆A(t) data to eq 3. Nonlinear regression using the Marquardt-Levenberg algorithm was performed with SigmaPlot. If the fraction of Ru(bpy)32+* quenched by Sb/SnO2, fq ) c+(0)/(c+(0) + c*(0)), was g0.9, the exponential decay of Ru(bpy)32+* (second term of eq 3) could be neglected and the data were fitted to the stretched exponential component of eq 3. This algorithm was designated A1. The fraction quenched, fq, was obtained from time-resolved emission measurements according to eq 4, where LS(0) is the initial intensity of the emission due to unquenched Ru(bpy)32+ in the sample with Sb/SnO2 and LR(0) is the initial intensity of the emission in a reference solution of Ru(bpy)32+ without Sb/ SnO2.14,15
due to the oxidation of Ru(bpy)32+ into Ru(bpy)33+ (eq 1). The recovery of the absorption loss was essentially 100%, although the stretched-exponential nature of the recovery meant that it often extended to times beyond the window of the kinetic measurement. Complete recovery was evident from the fact that there was no net loss of Ru(bpy)32+, which would have been manifested by a decrease in the initial absorbance change at 450 nm during the experiment or by a permanent change in the absorption spectrum of the sample. Neither of these manifestations of irreversibility were observed. The reduction of O2 by ecb- could provide alternate channels for the disappearance of Ru(bpy)33+, such as the one described by eqs 5 and 6, in which O2 serves as an electron-transfer
fq ) 1 - LS(0)/LR(0)
O2 + ecb-(SnO2) f O2•-
(5)
Ru(bpy)33+ + O2•- f Ru(bpy)32+
(6)
(4)
The other two algorithms made use of transient absorption data from the reference solution of Ru(bpy)32+ without Sb/SnO2 to take into account the fraction of unquenched Ru(bpy)32+* (second term of eq 3). These two algorithms differed in the manner in which the values of c+(0) and c*(0) were obtained. In one of them, designated A2, the values of c+(0) and c*(0) were allowed to float, while in the other, designated A3, the values of c+(0) and c*(0) were constrained by the value of fq determined by eq 4. Algorithm A1, which employed the stretched exponential decay function only, was found to give values of c+(0), krec, and β that were indistinguishable, within the standard error, from the ones obtained with algorithms A2 and A3 when fq g 0.9 Although the exclusive use of algorithm A3 instead of A2 would be preferable since it constrains c+(0) and c*(0) to values consistent with the emission data, the fitting routine using A3 sometimes converged on untenable values of β ≈ 1 (i.e., singleexponential decay) for samples having fq e 0.5. To avoid biasing the values reported in this paper, results obtained from more than one algorithm are presented for most of the samples, but these will be indicated by different symbols (circles for A1, squares for A2, and triangles for A3). Results and Discussion General Considerations. Electron injection by photoexcited Ru(bpy)32+ resulted in a net loss of light absorption at 450 nm
mediator via the superoxide radical anion. An earlier study with undoped SnO2 particles and Ru(bpy)32+ showed that dissolved O2 has no observable effect on krec.13 The implication was that the reduction of O2 by ecb- occurred at a rate that was much slower than the rate of recombination between Ru(bpy)33+ and ecb-. Since the driving force for the reduction of O2 by ecb- increases as the initial average number of Ru(bpy)33+/ecb- pairs per particle, Ne, increases (vide infra), we reexamined the possible involvement of O2. The decay curves in Figure 1 show the disappearance of Ru(bpy)33+ following electron injection into Sb/SnO2 under airsaturated and argon-saturated conditions. The decay kinetics were analyzed using the fitting routine which includes the component to the observed absorbance change which arises from unbound Ru(bpy)32+* (fq ) 0.90) (algorithm A3 in the Experimental Section). The contributions from the unbound population of Ru(bpy)32+* are shown in each panel (dashed lines). The fits (solid lines) of the data are shown in Figure 1 superimposed on the data points, and the residuals of the fits are shown in the insets. The value of Ne in this experiment was 4.5 ( 0.2. The values of krec obtained from the fits were (1.8 ( 0.1) × 107 s-1 under air and (1.5 ( 0.1) × 107 s-1 under argon. The influence of dissolved O2 (ca. 3 × 10-4 M),34 on krec is negligible compared to the influence of Ne (vide infra).
7438 J. Phys. Chem. B, Vol. 101, No. 38, 1997
Figure 2. Time-resolved absorbance change at 450 nm in a solution of Ru(bpy)32+ (20.8 µM) and Sb/SnO2 colloidal particles (116 µM) with three laser intensities, increasing from (a) to (c). The experimental data points are represented by dots, and fits of the data to the stretched exponential equation (algorithm A1) are represented by solid lines. The parameters obtained from the fits were (a), ∆A(0) ) -0.0445 ( 0.0002, krec ) (1.49 ( 0.01) × 105 s-1, and β ) 0.703 ( 0.003; (b), ∆A(0) ) -0.0934 ( 0.0002, krec ) (1.74 ( 0.01) × 105 s-1, and β ) 0.680 ( 0.002; (c), ∆A(0) ) -0.159 ( 0.001, krec ) (1.90 ( 0.01) × 105 s-1, and β ) 0.666 ( 0.001. Inset: Dependence of krec on Ne.
The crux of this paper is to model the dependence of krec on Ne. Since the values of krec were determined by fitting the recombination kinetics to the stretched exponential equation, the validity of applying this equation has to be scrutinized. The stretched exponential function models a system that relaxes with a continuous distribution of exponential decay times whose peak value is near the characteristic lifetime (e.g., krec-1) and whose width is inversely related to the stretching parameter β.22,24,35-37 The function was introduced on purely empirical grounds, but there are now several derivations for systems in three dimensions, involving diverse concepts such as percolation, hierarchical relaxation of constraints, multipolar interaction transitions, and dispersive transport of defects.36 Common to these processes is their first-order nature. Therefore, the stretched exponential equation should describe the recombination kinetics under conditions when second-order effects are negligible. Figure 2 presents kinetic data obtained with three samples having Ne 99.5% of the particles had either one pair or none.38 The fits of the data to the stretched exponential function are superimposed as solid lines on the data points. These results demonstrate that the recombination is distinctly nonexponential when the possibility of more than one pair being created on a single particle is negligible. It is known that an elementary first-order reaction may exhibit kinetic behavior characteristic of a stretched exponential when the rate constant can be described by a Gaussian distribution.22-24 In the case of surface state kinetics on colloidal semiconductors, Albery et al. concluded that the dispersion in the kinetic transients is caused by the dispersion in radii of the particles.23 This factor should be at play in our experiments as well, not only kinetically, but also because Ne is an ensemble-averaged value. Similar effects can arise when the rate constants are determined by a Gaussian distribution of activation energies, as discussed mathematically by Richert.22 This situation can
Ford et al.
Figure 3. Dependence of the stretching parameter, β, on Ne. The filled data points are for Sb/SnO2 particles and the unfilled symbols are for SnO2 particles. The different symbols represent the fitting algorithms employed: circles for A1, squares for A2, and triangles for A3. Inset: Dependence of β on the width of the Gaussian distribution, σ, of decay functions analyzed by Richert.22
arise from charge-carrier trapping-detrapping.39-44 Trapping and its repercussions for the Auger-capture model will be discussed later in this paper. A good indication that dispersion in krec is the underlying source of the stretched exponential character in the recombination is the dependence of the stretching parameter β on Ne (Figure 3). β shows an asymptotic dependence on Ne, approaching a limit of ca. 0.3 at the higher values of Ne from an initial value of ca. 0.7 at the lower values of Ne. Richert fitted a series of curves derived by numerical integration of the explicit expression for dispersive kinetics to a stretched exponential decay function.22 He found that β did not vary linearly with the width of the Gaussian distribution, σ, but had an asymptotic dependence, decreasing from a value of ca. 1 at the lower σ toward a limit of ca. 0.2 at the higher σ (Figure 3, inset).45 Although the two plots in Figure 3 have different limiting values of β, the trend is clearly the same. Thus the dependence of β on Ne that we observe most likely results from the fact that we are treating a dispersive kinetic system with the stretched exponential approximation. Since the stretched exponential equation is an empirical relation, the physical significance of the rate constant and stretching parameter (e.g., krec and β in eq 3) is obscure.22-24 However, rate constants obtained using this equation differed by less than a factor of 2 from those obtained by numerical integration of the expression for dispersive kinetics.46 The stretched exponential decay function has also proven to be useful in obtaining physically reasonable bimolecular rate constants for reactions occurring at the surfaces of colloidal particles.13,15,17,47 These results indicate that values of krec obtained using eq 3 are physically significant. Electrochemical Model Based on the Butler-Volmer Equation. The Butler-Volmer equation is a parametrization of the potential dependence of electrochemical rates on electrode surfaces25-28 which has also been applied to colloidal semiconductor particle “microelectrodes.”12,25,48-54 Our analysis of the dependence of krec on Ne, is based on the premise that the overpotential for electron transfer, E - E°′, varies linearly with Ne, i.e., E - E°′ ) gNe, where g is a constant. The latter
Electron Injection by Photoexcited Ru(bpy)32+
Figure 4. Semilogarithmic plot of the dependence of krec on Ne. The symbols have the same meanings as in Figure 3. The curve was obtained from a least-squares fit of the data for Sb/SnO2 to eq 7. The inset shows the same data for values of Ne in the range 0-1.4 Ru(bpy)33+/ecb- pairs per particle. The curve corresponds to values of k°, R, and g of (2.1 ( 0.7) × 106 s-1, 0.59 ( 0.11, and 0.022 ( 0.006 V, respectively.
proportionality is based on the capacitor model for the charging of small particles with electrons.55-57 These considerations lead to eq 7, where k° is the rate constant at the formal potential E°′, R the transfer coefficient (0 q/(kBT)), when the back
J. Phys. Chem. B, Vol. 101, No. 38, 1997 7439 reaction can be ignored. As such, it neglects the negative deviation from linearity that occurs as the overpotential approaches zero. The value ket° varies over almost 7 orders of magnitude, from 10-7 cm s-1 for the reduction of O2 to 0.4 cm s-1 for the reduction of Rh(bpy)33+. The value of ket° estimated for the present system (10-2-10-1 cm s-1) lies well within that range. The transfer coefficient R obtained from the fit (0.59 ( 0.11) lies also within the range of values obtained from the TiO2 systems (0.5-0.85). The overpotential for heterogeneous electron transfer on semiconductor particles has generally been varied by changing the pH of the aqueous solution. The solution pH affects the electrochemical potential by affecting the ζ potential at the particle-water interface.52 In other cases, the overpotential was changed by generating excess conduction band electrons in the particles either photochemically,50 radiolytically,51,59,60 or electrochemically.11,12,61 These latter cases provide a comparison for the contribution made by each Ru(bpy)33+/ecb- pair to the overpotential in eq 7, g ) 22 ( 6 mV. Nenadovic´ et al.59 used a pulse radiolysis technique to inject electrons into 45-nm diameter WO3 particles and found that the flat-band potential became about 15 mV more negative for a decade increase in electron concentration, corresponding to ≈3 × 10-3 mV/electron. Rajh et al.60 also used pulse radiolysis to inject electrons into 5-nm diameter TiO2 particles and found that the flat-band potential became ≈65 mV more negative upon injection of five electrons, corresponding to about 13 mV/ electron. Hoyer and Weller61 performed cyclovoltammetry with a nanoporous electrode of 5.9-nm diameter ZnO particles and found that increasing the applied potential above a critical value of -540 to -1000 mV resulted in the accumulation of ca. six electrons, corresponding to ≈80 mV/electron. Finally, it should be noted that the charging with electrons of colloidal-sized particles of Sn or Ag embedded in metal oxide matrixes results in incremental potential increases on the order of 10 mV.55,57 Clearly, the potential change per electron depends on a number of factors including the size and composition of the particle as well as the dielectric properties of the medium, but the value of g obtained here appears to be a physically reasonable one. The Butler-Volmer equation is a semiempirical one derived from a macroscopic point of view. As such, it is based on several assumptions and approximations that weaken when electrochemical reactions are scrutinized from a microscopic, quantum mechanical point of view.7,25-28,62 Nevertheless, it has proven to be a good starting point for modeling a wide variety of heterogeneous electron-transfer reactions, including, as shown in this study, electron recapture on colloidal tin dioxide particles. In the following section, it is shown that recapture can also be interpreted using a model based on the Auger mechanism. Model Based on Auger Capture. The Auger-capture process is one of the nonradiative mechanisms by which charge carriers recombine in semiconductors.1-4 The energy released when an electron-hole pair recombines is transferred to a third carrier (electron or hole). This mechanism can be especially important in colloidal semiconductors, where the small volume of the particle results in a strong interaction between electronhole pairs.42 The appearance of Auger recombination on going from monophotonic to multiphotonic excitation regimes in semiconductor sols drastically increases the rate of recombination and can be accompanied by photoionization.42 The recombination kinetics in the colloidal tin dioxide system may be pictured in terms of the model in Figure 5, which is analogous to a general model for recombination between holes and electrons in semiconductors.1-4 In this model, it is assumed that Ru(bpy)33+ and the stannic hydroxyl radical, >SnO•, are
7440 J. Phys. Chem. B, Vol. 101, No. 38, 1997
Ford et al. injection event (eq 1). On the basis of the assumption that the semiconductor is nondegenerate, the rate of recombination, krec, is given by eq 11, where a′ ) BS + B1 n0 + B2 r0, b′ ) B1 + B2 exp(∆E°/kBT), and c′ ) n0 exp(∆E°/kBT) + r0.
krec ) a′c′ + [a′ exp(∆E°/kBT) + b′c′]∆n + [b′ exp(∆E°/kBT)](∆n)2 (11)
Figure 5. Radiative and Auger transitions for recombination between conduction band electrons and stannic hydroxyl radicals. The stannic hydroxyl radical (>SnO•) is assumed to be in equilibrium with the Ru(bpy)33+/Ru(bpy)32+ redox couple at the semiconductor surface. ∆E° is the difference in reduction potentials of the Ru(bpy)33+/Ru(bpy)32+ and >SnO•/>SnO- redox couples. The rates of the transitions per unit volume are BSnr, B1n2r, and B2nr2, where n and r are the concentrations of free electrons and stannic hydroxyl radicals, respectively.
in equilibrium according to eq 8 and that the conduction band electron recombines with >SnO• (eq 9) and not directly with Ru(bpy)33+, since the >SnO• radical is directly coupled to the semiconductor lattice while Ru(bpy)33+ is electrostatically bound to the surface.
Ru(bpy)33+ + >SnO- h Ru(bpy)32+ + >SnO•
(8)
>SnO• + ecb- f >SnO-
(9)
Figure 5 illustrates three transitions: radiative single-electron recombination, which is characterized by the transition coefficient BS, and two types of Auger recombination. B1 is the coefficient for Auger-capture when the third carrier is an electron, and B2 is the coefficient when the third carrier is a stannic hydroxyl radical. The rates of the transitions per unit Volume are BSnr, B1n2r, and B2nr2, where n and r represent the concentrations of free electrons and stannic hydroxyl radicals, respectively. Mulvaney et al.63 postulated that photogenerated valence band holes in colloidal SnO2 are rapidly trapped by surface states. In analogy to TiO2 systems,5-12,25,64-66 the trapped hole may be the stannic hydroxyl radical (>SnO•), in which an oxygen surface anion radical is covalently bound to tin atoms. The reduction potential, E°, of the trapped hole on TiO2 can be used to estimate the electrochemical potential of >SnO•. E° of the trapped hole on TiO2 is estimated to be +1.5 V (vs the normal hydrogen electrode, NHE) at pH 3.66 Holes in the valence band of SnO2 are ca. 0.5 V more oxidizing than holes in TiO2.67 Thus E° of the trapped hole on colloidal SnO2 can be expected to be ca. +2.0 V vs NHE at pH 3. Subtracting 6 × 0.06 V from this value to adjust for pH25 gives E° = +1.65 V vs NHE at pH 9. This level is indicated in Figure 5 as a dashed line. Since E° of the Ru(bpy)33+/Ru(bpy)32+ couple is +1.26 V vs NHE,31,32 the difference in reduction potentials of Ru(bpy)33+ and >SnO•, ∆E°, for eq 8 is estimated to be -0.4 ((0.1) V. The corresponding energy gap between the conduction band and >SnO• is 2.3 ((0.1) eV. According to the Boltzmann equation, the equilibrium concentration of >SnO• (∆req) is determined by the concentration of injected electrons, ∆n, and ∆E° (eq 10).
∆req ) ∆n exp(∆E°/kBT)
(10)
Consequently, the total concentration of >SnO•, r ) r0 + ∆req ) r0 + ∆n exp(∆E°/kBT), where r0 is the equilibrium concentration of >SnO• before the photoinitiated electron
The two Auger coefficients, B1 and B2, are expected to be of the same order of magnitude.1-4 Since there is no apparent fluorescence from the Sb/SnO2 and SnO2 sols used in this study and recombination between Ru(bpy)33+ and ecb- does not result in any apparent fluorescence, the radiative term BS in a′ can be neglected. The value of r0 can be estimated by considering the reverse of eq 9 as a thermal ionization of an impurity and using eq 12, where Nd ) 1022 cm-3 represents the concentration of the impurity and Ed ) 2.3 eV represents the ionization energy.70
r0 = (n0Nd)1/2 exp(-Ed/2kBT)
(12)
The equilibrium concentration of free electrons, n0, in SnO2 and Sb/SnO2 ranges between 1018 and 1020 cm-3,67,71,72 and hence the upper limit for r0 is ≈102 cm-3. Thus n0 >> r0 and B2r0 SnO- couple is +1.5 ((0.2 V) vs NHE at pH 9. Equation 11 was derived on the assumption that the semiconductor is nondegenerate. When the semiconductor becomes degenerate, the Maxwell-Boltzmann statistics no longer apply to the energy distribution of the free carriers, and the electrostatic interaction of the carriers can result in a narrowing of the energy bandgap and alterations of the band shapes.2,77 The assumption of nondegeneracy may not be valid at the higher electron injection levels, where free electron concentrations of ca. 1020 cm-3 can be reached (Figure 6).78 It has been suggested that degeneracy can cause a weakness of the quadratic dependence of recombination rate on free carrier concentration.73 This effect is not apparent in our results, however (Figure 6).
J. Phys. Chem. B, Vol. 101, No. 38, 1997 7441 We are aware of the fact that trapping and detrapping processes are involved in the overall process of recombination and may well be a source of the dispersive kinetics underlying the stretched-exponential approximation (vide supra). Colloidal semiconductor particles have a high density of defect sites, mainly at the surface, which can trap charge carriers. Reversible trapping and detrapping of ecb- in colloidal semiconductors generally occurs on a time scale ranging from the femtosecond time regime up to the nanosecond time regime.5,39-44,79 Mulvaney et al.63 examined colloidal SnO2 and found no evidence for deep electron trapping mechanisms that lead to chemical transformations, such as the reduction of Ti4+ sites to Ti3+ in TiO2.5-12,25 Landsberg80 included band-trap Auger recombination processes into a general rate equation and found that they contributed mainly to the linear term. Figure 6 shows that our data fit eq 13, which exhibits the form of the general rate equation used by Landsberg.80 We considered only one kind of Auger-capture process as a first approximation so that the equation could be simplified. This treatment does not allow us to discriminate between the many Auger processes that are possible. Conclusions The nonlinear dependence of recombination rate constant on average number of Ru(bpy)33+ and ecb- pairs created by photoinjection into tin dioxide particles can be treated quantitatively using two different models when the rate constant is determined using the stretched exponential function as a first approximation for an underlying ensemble of statistically distributed rate constants. The models should be of predictive value for Gra¨tzel-type solar cells as well as photocatalytic systems, where excess electrons may accumulate on semiconductor particles. The parameters of the model obtained here are expected to depend on the semiconductor material, but the forms of the underlying rate equations (7 and 13) should be generally applicable. Acknowledgment. Support for this project was provided by NSF Grant CHE-9208551 and by the Center for Photochemical Sciences at Bowling Green State University. J.M.W. thanks the Alexander von Humboldt Foundation for support within the Feodor-Lynen Program. References and Notes (1) Schroder, D. K. Semiconductor Material and DeVice Characterization; Wiley: New York, 1990; pp 359-447. (2) Landsberg, P. T. Recombination in Semiconductors; Cambridge University Press: Cambridge, 1991. (3) Li, S. S. Semiconductor Physical Electronics; Plenum Press: New York, 1993; pp 113-145. (4) Mo¨ller, H. J. Semiconductors for Solar Cells; Artech House: Boston, 1993; pp 9-50, 77-106. (5) Hoffmann, M. R.; Martin, S. T.; Choi, W.; Bahnemann, D. W. Chem. ReV. (Washington, D.C.) 1995, 95, 69-96. (6) Linsebigler, A. L.; Lu, G.; Yates, J. T., Jr. Chem. ReV. (Washington, D.C.) 1995, 95, 735-758. (7) Nozik, A. J.; Memming, R. J. Phys. Chem. 1996, 100, 1306113078. (8) Serpone, N.; Lawless, D.; Khairutdinov, R.; Pelizzetti, E. J. Phys. Chem. 1995, 99, 16655-16661. (9) Grela, M. A.; Colussi, A. J. J. Phys. Chem. 1996, 100, 1821418221. (10) Colombo, D. P., Jr.; Bowman, R. M. J. Phys. Chem. 1996, 100, 18445-18449. (11) Hagfeldt, A.; Gra¨tzel, M. Chem. ReV. (Washington, D.C.) 1995, 95, 49-68. (12) Kamat, P. V. Prog. Reaction Kinet. 1994, 19, 277-316. (13) Ford, W. E.; Rodgers, M. A. J. J. Phys. Chem. 1994, 98, 38223831. (14) Ford, W. E.; Wessels, J. M.; Rodgers, M. A. J. Langmuir 1996, 12, 3449-3453.
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