Interpretation of Apparent Activation Energies for Electron Transport in

Department of Chemistry, UniVersity of Bath, Bath BA2 7AY, United Kingdom, Department of ... have reported that the activation energy for electron tra...
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2006, 110, 13694-13699 Published on Web 06/22/2006

Interpretation of Apparent Activation Energies for Electron Transport in Dye-sensitized Nanocrystalline Solar Cells Laurence M. Peter,*,† Alison B. Walker,‡ Gerrit Boschloo,§ and Anders Hagfeldt§ Department of Chemistry, UniVersity of Bath, Bath BA2 7AY, United Kingdom, Department of Physics, UniVersity of Bath, Bath BA2 7AY, United Kingdom, and Center of Molecular DeVices, Department of Chemistry, Royal Institute of Technology, Teknikringen 30, SE-100 44 Stockholm, Sweden ReceiVed: May 6, 2006; In Final Form: June 13, 2006

Electron transport in dye-sensitized nanocrystalline solar cells appears to be a slow diffusion-controlled process. Values of the apparent electron diffusion coefficient are many orders of magnitude smaller than those reported for bulk anatase. The slow transport of electrons has been attributed to multiple trapping (MT) at energy levels distributed exponentially in the band gap of the nanocrystalline oxide. In the MT model, release of immobile electrons from occupied traps to the conduction band is a thermally activated process, and it might therefore be expected that the apparent electron diffusion coefficient should depend strongly on temperature. In fact, rather small activation energies (0.1-0.25 eV) have been derived from time and frequency resolved measurements of the short circuit photocurrent. It is shown that the MT model can give rise to such anomalously low apparent activation energies as a consequence of the boundary conditions imposed by the short circuit condition and the quasi-static relationship between changes in the densities of free and trapped electrons. This conclusion has been confirmed by exact numerical solutions of the time-dependent generation/collection problem for periodic excitation that provide a good fit to experimental data.

Introduction (DSCs)1-3

Dye-sensitized nanocrystalline solar cells continue to excite interest as a potentially low cost alternative to conventional photovoltaics. Several aspects of their behavior, including the mechanisms of photovoltage generation and electron transport, remain the subject of discussion. One striking observation is that transport of photoinjected electrons to the anode via the network of interconnected oxide nanoparticles can require times ranging between several milliseconds and several seconds, depending on illumination intensity.4-8 This contrasts with the situation in conventional p-n junction photovoltaic devices, where carrier collection generally occurs on a sub-microsecond time scale. Early discussions of transport in nanocrystalline TiO2 networks introduced the concept of an intensity-dependent electron diffusion coefficient to model the observed behavior,9 and this was later related to a multiple trapping (MT) model involving an exponential distribution of trapping states in the energy gap.6,10-13 Alternatively, the process can be modeled by a continuous-time random walk (CTRW) process with a distribution of waiting times.10,14 The equivalence between the MT regime and the CTRW process has been reexamined recently by Bisquert.15 Boschloo and Hagfeldt,16 and more recently Kopidakis et al.,17 have reported that the activation energy for electron transport appears to be much lower than that expected on the basis of * To whom correspondence should be addressed. [email protected]. † Department of Chemistry, University of Bath. ‡ Department of Physics, University of Bath. § Department of Chemistry, Royal Institute of Technology.

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the MT model if the majority of electrons are trapped. The experimental approaches taken by the two groups are different. Boschloo and Hagfeldt used intensity modulated photocurrent spectroscopy (IMPS) with bias light, whereas Kopidakis et al. measured transient photocurrents in the absence of additional bias light. Here, we focus discussion on the results of Boschloo et al., since in their work the quasi-Fermi level (QFL), nEF, can be considered as independent of time to a good approximation. This is not the case for large amplitude photocurrent transients. Both sets of measurements gave activation energies in the range 0.1-0.25 eV. These energies are lower than that expected for thermal release of electrons below the QFL, nEF, which is typically calculated to be 0.4-0.5 eV below the conduction band energy Ec at the light intensities used. Greijer-Agrell et al.18 have also measured activation energies for transport in nanocrystalline TiO2 permeated with electrolyte using a conductivity method. In the regime where most of the electrons are trapped, the activation energy was found to be on the order of 0.3 eV. When the electron density was increased so that there were more than 20 electrons per particle, the activation energy was found to be 0-0.15 eV depending on the electrolyte. O’Regan and Durrant19 have questioned the significance of the low values of activation energy reported in refs 16 and 17 and have suggested that measurements made close to open circuit give more reliable information. These authors also reported evidence that the energy difference between the conduction band and the redox Fermi level is temperature dependent, decreasing with a coefficient of -1.7 meV K-1. They attribute the low activation energies for electron diffusion reported in refs 16 and 17 in part to a failure to account for this effect. In the present paper, © 2006 American Chemical Society

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J. Phys. Chem. B, Vol. 110, No. 28, 2006 13695

we show that the intuitive identification of the activation energies for electron transport measured under short circuit conditions with the difference between quasi-Fermi energy (nEF) and the conduction band energy (Ec) is incorrect. Furthermore, we also predict that the apparent activation energies for electron transport measured under short circuit conditions will not be influenced by changes in the conduction band position. The starting point for the discussion is the work of Bisquert and Vikhrenko.20 These authors have demonstrated that the “apparent” diffusion coefficient derived from small amplitude perturbation measurements is in fact determined by the relaxation time for conduction band electrons, which depends not only on transport but also on variations in the occupancy of electron traps as the electron QFL changes with time. Following Bisquert and Vikhrenko, this relationship can be described by the quasi-static approximation, from which it follows that the apparent diffusion coefficient of electrons, Dn, is related to the diffusion coefficient of electrons in the conduction band, D0, by

Dn )

( )

∂nc D ∂nt 0

for

∂nc ,1 ∂nt

(1)

Here, the (∂nc/∂nt) term relates the QFL-dependent changes in the conduction band and trapped electron densities (nc and nt, respectively). Recent numerical calculations to examine the validity of the quasi-static approximation have revealed that it applies over a wide range of experimental conditions.21 The analysis presented here, which is based on the approach of Bisquert and Vikhrenko, shows that the apparent activation energies for electron diffusion predicted by the MT model are much smaller than (Ec - nEF). The reason for this is the fact that the activation energy reflects temperature-dependent changes in nEF (imposed by the short circuit boundary condition) as well as the dynamic relationship between trapped and free electrons in the system. In the present work, the validity of conclusions based on the quasi-static approach has been confirmed by exact numerical calculations of the small amplitude photocurrent response in the frequency domain (intensity modulated photocurrent spectroscopy, IMPS5,22). The conclusions should be generally valid for small amplitude perturbations since, as we have shown elsewhere, the IMPS response can be related by inverse Fourier transformation to the small amplitude photocurrent transient response.23 Values of apparent activation energies for electron diffusion have been calculated as a function of trap distribution functions, and the results agree well with values obtained by IMPS measurements.16 Transport and Interfacial Transfer of Electrons in the DSC. Electron transport to the collecting anode in dye-sensitized nanocrystalline solar cells is driven by the concentration gradient of electrons in the conduction band of the oxide, since the electrical field in the bulk of the nanostructured electrode is negligible as a consequence of screening by the high ionic strength electrolyte. Under photostationary conditions, where the rates of electron trapping and detrapping are equal, the electron density profiles and current can be obtained by solving the continuity equation

∂2nc(x) ∂nc(x) -Rx - k0[nc(x) - neq] ) RI0e + D0 ∂t ∂x2

(2)

Here, nc(x) is the position-dependent density of conduction band electrons, neq is the density of conduction band electrons when the system is in equilibrium in the dark (i.e., the Fermi level in

the oxide is equal to the redox Fermi level EF,redox), R is the absorption coefficient of the sensitized layer, I0 is the incident photon flux corrected for reflection losses, D0 is the diffusion coefficient of electrons in the conduction band, and k0, the pseudo-first-order rate constant for the transfer of electrons from the conduction band to redox species (usually I3-), is the reciprocal of the conduction band electron lifetime, τ0 (electron transfer via surface states has been neglected for simplicity). For illumination from the anode side (x ) 0), the boundary conditions are

dnc(x) ) 0 at x ) d dx

kextnc(0) ) Dn

dnc(x) at x ) 0 (3) dx

where kext (cm s-1) is the first-order heterogeneous rate constant for electron extraction at the anode5 and d is the thickness of the nanocrystalline layer. For sufficiently large kext, electron extraction becomes diffusion controlled. The efficiency of electron collection at the anode is determined by competition between electron transport and transfer to I3-. The electron diffusion length is given by

Ln ) xD0τ0

(4)

For efficient electron collection, Ln should exceed d. This condition is usually fulfilled for dye-sensitized cells utilizing the iodide/triiodide redox couple. Measured Electron Diffusion Coefficients and Lifetime: the Quasi-static Approximation. Apparent diffusion coefficients (Dn) and apparent lifetimes (τn) of electrons have been derived from measurements using small amplitude perturbations of light intensity superimposed on larger steady background illumination to linearize the system response. The experimental responses can be modeled satisfactorily (without taking trapping/ detrapping explicitly into account) by using solutions of the continuity equation with an appropriate transient or periodic illumination function.5,24 The values of Dn and τn obtained in this way are found to vary in opposite senses with light intensity, whereas their product remains constant over five decades of light intensity.6,25 This remarkable observation has been explained elegantly by Bisquert and Vikhrenko,20 who have shown that the values Dn and τn measured by small amplitude relaxation methods are determined by the coupling of the time-dependent densities of conduction band electrons and trapped electrons via trapping and detrapping involving a distribution of trapping states. These authors have introduced the quasi-equilibrium assumption, which links the rates at which the trapped electron density and free electron density change under dynamic conditions by the expression

∂nt ∂nt ∂nc ) ∂t ∂nc ∂t

(5)

Here, ∂nc/∂t and ∂nt/∂t are the rates of change of conduction band and trapped electron densities, respectively, and ∂nt/∂nc can be obtained from the density of states function for the trap states, g(ET) at the trap energy ET by noting that

nt )

∫ g(ET)fFD(ET,nEF) dET

(6)

where fFD is the Fermi-Dirac function, which, for the sake of illustration, we can approximate by its degenerate form θ(ET - nEF), where the step function θ(x) is 0 for x < 0 and 1 for x > 0. (Note that this convenient approximation, which introduces

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an error of ground 10%, is not used in the exact numerical calculations of the apparent activation energy shown in Figure 5.) Using this approximation

∂nt ∂nt ∂nEF kBT ) ≈ g(nEF) ∂nc ∂nEF ∂nc nc

(7)

The variation of Dn with light intensity can be explained in terms of the MT model. The work of Bisquert and Vikhrenko20 has shown that the quasi-static approximation predicts that the apparent diffusion coefficients and apparent electron lifetimes derived from the relaxation time constant of the photocurrent and photovoltage, respectively, are related to the real diffusion coefficient and lifetime of conduction band electrons by

Dn )

( )

τn )

( )

and

Figure 1. Calculated profile of electron density and corresponding quasi-Fermi level (QFL) under short circuit conditions. Incident photon flux is 1016 cm-2 s-1. Illumination is from the anode side: R ) 520 cm-1, Ec - EF,redox ) 1.0 eV, NC ) 1020 cm-3, Nt,0 ) 1019 cm-3, D0(298) ) 0.4 cm2 s-1, R ) 520 cm-1, and d ) 10 µm.

∂nc D ∂nt 0

∂nt τ ∂nc 0

(8)

under conditions where (∂nc/∂nt) , 1. It follows that the quasistatic approximation predicts that the values of Ln derived from Dn and τn should be approximately constant provided that the values of (∂nc/∂nt) are reasonably similar under short circuit and open circuit conditions at a given intensity. Here, we consider an exponential distribution of trapping states

g(ET) )

Nt,0 (-(Ec-ET)/kbTc) e kBTc

(9)

where kbTc ) mc corresponds to a characteristic energy describing the width of the distribution and Nt,0 is the total density of traps. It follows from eq 7 that the dependence of the apparent electron diffusion coefficient of conduction band electrons on the electron quasi-Fermi level (QFL), nEF, is given in this case by

( ) (

)

NcTc -((Ec-nEF)/k)((1/T)-(1/Tc)) ∂nc ∂nc ∂nEF Dn(nEF) e ) ) ) D0 ∂nt ∂nEF ∂nt Nt,0T (10) There is considerable uncertainty about the value of the electron diffusion coefficient D0 for nanocrystalline anatase permeated by an electrolyte. For single-crystal anatase, the Hall mobility measurements reported by Forro et al.26 correspond to a room-temperature value of D0 ≈ 0.4 cm2 s-1. For dry 9 nm diameter nanocrystalline anatase, Kroeze et al.27 have derived a much lower value of the electron mobility from time-resolved microwave measurements that corresponds to D0 ) 10-3 cm2 s-1, compared with earlier measurements by the same group, which gave D0 ) 10-2 cm2 s-1 for smooth anatase films and D0 ) 8 × 10-3 cm2 s-1 for nanocrystalline TiO2. Recent measurements on dry films using teraherz spectroscopy also give lower values of D0,28 although it appears from this work that the large mismatch in dielectric constant between anatase and air is responsible. It seems probable that diffusion of electrons in the nanocrystalline TiO2 network will be influenced by dynamical Coulomb screening involving the image charge in the electrolyte,29 although the analogy with ambipolar diffusion in semiconductors suggests the effects may be small since the

electrolyte ions are present in large excess.11 Here, we have chosen to use the value of D0 for bulk anatase, but the choice of D0 value does not affect the general conclusions. Temperature Dependence of Dn under Short Circuit Conditions. Interpretation of the effects of temperature on electron transport in the DSC is potentially complicated by the fact that many of the parameters in eq 9 are temperature dependent. The redox Fermi level and the conduction band density of states both depend on temperature, and the conduction band energy may also change as the result of changes in the surface dipole potential.28 In addition, the mobility of conduction band electrons may also change not only as a consequence of optical phonon scattering by the TiO2 lattice26 but also because of changes in electrolyte screening effects.29 However, to gain an initial qualitative understanding of the meaning of the thermal activation energy for electron transport in the multiple trapping regime, we begin by neglecting these effects. Provided that Ln > d, solution of the continuity equations shows that the conduction band electron profile under short circuit conditions is monotonic as illustrated in Figure 1. The figure also shows that the electron QFL, which is related to the conduction band electron density by

nc ) Nce-(Ec-nEF/kBT)

(11)

is relatively flat in the bulk of the film but decreases rapidly near the anode. At any position in the nanocrystalline layer, the flux of electrons Jn(x) is related to the local gradient of free electron density dnc(x)/dx by

Jn(x) ) D0

dnc(x) dx

(12)

If all photoinjected electrons are collected under short circuit conditions (Ln > d), then the electron flux at x ) 0 must remain constant when the temperature is changed, provided that the generation rate is independent of temperature. For the purposes of illustration, we begin by neglecting the temperature dependences of D0 and NC and assume diffusion-controlled extraction of electron at the anode (kext f ∞). In this case, calculations confirm that imposition of the boundary condition J(x ) 0) ) 0 implies that the electron density profile n(x) should remain unchanged when the temperature is raised. This condition can

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Figure 2. Left-hand part of the figure illustrates how the Fermi-Dirac function changes with temperature when the electron QFL is held constant. Note that raising the temperature leads to a substantial increase in conduction band electron density (nc ) fFDNC). The right-hand part of the figure shows how under short circuit conditions the electron QFL moves downward to maintain constant conduction band electron density (fFDNC ) constant) when the temperature is raised.

only be met if the electron QFL moves down with temperature to ensure that the right-hand side of eq 11 remains constant. This means that the local electron QFL should decrease as the temperature increases. For this simplified case, it is easy to show that the decrease in local QFL is given by the linear expression

Ec - nEF,T2(x) )

()

T2 (Ec - nEF,T1(x)) T1

(13)

The downward shift in the QFL required to maintain the value of the Fermi-Dirac function fFD constant at E - Ec as the temperature increases is illustrated in Figure 2. The left-hand plot indicates how the Fermi-Dirac function (and hence the conduction band electron density nc ) fFDNC) is affected by increasing the temperature from 273 to 373 K if the electron QFL is constant. By contrast, the right-hand plot shows the downward shift of the QFL required to hold fFD (and hence nc) constant when the temperature is changed (cf. eq 13). At first sight, it might appear that lowering the electron QFL would reduce the value of the apparent electron diffusion coefficient Dn, giving rise to a negative activation energy for transport. However, this is not the case as can be seen by examination of the term ∂nc/∂nt ) ∂nc/∂nEF ∂nEF/∂nt in eq 9. ∂nc/∂nEF is given by the Boltzmann limit of the Fermi-Dirac function, so that it changes with temperature. By contrast, ∂nt/ ∂nEF is independent of temperature to a good approximation (cf. eq 7). The way in which temperature-induced changes in the QFL affect Dn/D0 ) ∂nc/∂nt is illustrated by the construction in Figure 3 for an initial value of nE - EF,redox ) 0.5 eV. It can be seen that the increase in temperature lowers the QFL but at the same time increases ∂nc/∂nt, so that the net effect is to increase Dn. This figure makes it clear why the apparent activation energy measured under short circuit conditions cannot be equated with the energy difference (EC - nEF). Figure 4 illustrates the temperature dependence of Dn predicted using the approach outlined above for different initial values of the electron QFL at 300 K. The activation energies obtained by fitting the temperature dependence of Dn are very

Figure 3. Construction showing how the apparent electron diffusion coefficient Dn ) (∂nc/∂nt)D0 changes with temperature. The plot shows the different dependences of the ratio Dn/D0 ) ∂nc/∂nt on the QFL at 273 and 373 K for an exponential trap distribution. The initial value of the QFL (at 273 K) ) 0.5 eV above EF,redox. This defines the value of Dn/D0 ) ∂nc/∂nt as shown by the construction. When the temperature is increased to 373 K, the QFL falls to maintain constant electron flux. Nevertheless, the new value of Dn/D0 ) ∂nc/∂nt is higher than the initial value as the construction shows. This change corresponds to a positive apparent energy of activation.

similar to those reported in refs 16 and 17. Calculations show that the activation energy is expected to decrease as the QFL increases due to higher illumination intensity (see Figure S1 in the Supporting Information). Since the electron QFL varies with distance under short circuit conditions (cf. Figure 1), an exact solution requires calculation of the small amplitude response as described in the next section. The activation energy of Dn depends on the width of the trap distribution, which is determined by the value of Tc in eq 8: for a constant value of nEF - EF,redox, the activation energy decreases with increasing Tc (see Figure S2 in the Supporting Information). The width of the distribution is frequently reported

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Figure 4. Temperature dependence of the apparent diffusion coefficient Dn calculated for three values of the electron QFL. Note that the apparent activation energies obtained from the Arrhenius plots are much lower than the corresponding values of Ec - nEF. Since Ec - EF,redox ) 1.0 eV, the values of Ec - nEF from top to bottom are 0.45, 0.50, and 0.55 eV. Note that Ea decreases as the QFL moves toward the conduction band. See Figure 1 for parameter values.

in terms of the parameter β ) T/Tc where T is room temperature or in terms of a characteristic energy mc ) kbTc. The values of β found in our own work correspond to Tc ) 900-1500 K30,31 or mc ) 0.075-0.125 eV. The preceding simplified treatment assumed that the NC and D0 values are independent of temperature. In fact, Nc varies with T3/2 and D0 is also temperature dependent. Forro et al.26 have reported that, for T > 160 K, the mobility, µ0, of electrons in anatase single crystals decreases with temperature in proportion to exp(T0/T), with T0 ) 850 K. It follows from the Einstein relationship (D0 ) (kbT/q)µ0) that D0 should vary with T exp(T0/T). For the purposes of illustration, we have used the temperature dependence of µ0 reported by Forro et al. Incorporation of the temperature dependences of Nc and D0 into the model is straightforward. The increase in Nc with temperature will cause a small downward displacement of the QFL, whereas the decrease in D0 with temperature should lead to a small upward displacement of the QFL. Both displacements can be calculated using the condition that the product ncD0 remains constant under diffusion-controlled short circuit conditions. (It can be shown that this condition holds for the bulk of the film but breaks down near the anode for the finite extraction rate constant, kext.) The calculations show that taking the temperature dependences of Nc and D0 into account has only a small effect on the activation energy. O’Regan and Durrant19 have suggested that measurements of activation energy for electron transport can be affected by a shift in the conduction band edge of TiO2 with temperature. The following argument shows that this is not true for short circuit conditions. Consider a downward shift of the conduction band energy at constant temperature brought about by change in the surface dipole potential. To maintain the constant electron flux condition imposed by the short circuit boundary condition, the electron QFL must also move down by an identical amount. Since the trap DOS is defined with respect to the conduction band edge, the net result is that the effective diffusion coefficient is insensitive to changes in Ec. Incorporation of a temperaturedependent value of Ec into the model confirms that the apparent activation energies are indeed unaffected. Numerical Calculation of the IMPS as a Function of Temperature. Since the electron QFL varies with distance (cf. Figure 1), Dn is not constant across the film. It is necessary to

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Figure 5. Numerical calculation of the IMPS transit times at three intensities showing fit to experimental data in ref 16 obtained. The wavelength of light was taken as 635 nm giving intensities of 5.89, 1.79, and 0.156 mW cm-2 for the lowest, middle, and highest IMPS transit times, respectively, to match the experimental data in ref 16. Note that these calculations do not rely on the quasi-static approximation (see Supporting Information for details): Nt,0 ) 2 × 1018 cm-3, Tref ) 298 K, NC(T) ) NC(Tref)(T/Tref)3/2, NC(Tref) ) 1020 cm-3, D0(Tref) ) 0.4 cm2 s-1, R ) 500 cm-1, d ) 12 µm, kt ) 1 × 109 s-1, kcb ) 1 × 104 s-1, kext ) 1 × 104 s-1, and TC ) 1300 K.

use a numerical approach to obtain the time- or frequencydependent current response of the system (see Supporting Information for details). In principle, this can be done by assuming that the quasistatic approximation holds locally, but to make a more exact comparison with the experimental data presented by two of the present authors,16 we have solved the exact expressions for the IMPS response of the system without making the steady-state assumption. (See Supporting Information for details.) The calculations also incorporate the temperature dependence of D0 and NC. The IMPS spectra calculated from the exact solution were used to define the mean transit time for the collection of electrons, τIMPS, as the inverse of the radial frequency at which the minimum in the IMPS response occurs. Figure 5 illustrates the calculated temperature dependence of τIMPS for three different illumination intensities and compares the results with the data reported by Boschloo and Hagfeldt.16 It can be seen that the exact periodic IIMPS solution of the continuity equation including trapping/detrapping gives rise to low values of the apparent activation energies that agree well with the experimental observations reported in ref 16. Conclusions We conclude from the analysis presented here that the low apparent activation energies for electron transport measured under short circuit conditions at constant light flux are consistent with the multiple trapping model. However, the work of Kopidakis et al.17 raises some intriguing questions since the transport times were measured for constant trapped electron charge, rather than constant illumination intensity. If bias light had been used, this should in principle correspond to holding the QFL position constant with respect to the conduction band energy. Since bias light is not used, it is difficult to apply the MT model. A definitive conclusion could be obtained by making experimental measurements of the QFL position in the bulk of the nanocrystalline film as a function of temperature. Work on this is in progress. Acknowledgment. This work was supported by U.K. Engineering and Physical Sciences Research Council (under the

Letters Supergen Excitonic Solar Cell Consortium Program) and by the Swedish Energy Agency. Supporting Information Available: Details of the dependence of the apparent activation energy on the electron QFL and on Tc as well as complete expressions and numerical calculations of the temperature dependence of the apparent electron diffusion coefficient are given. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) O’Regan, B.; Gra¨tzel, M. Nature 1991, 353, 737. (2) Gra¨tzel, M. Inorg. Chem. 2005, 44, 6841. (3) Gra¨tzel, M. MRS Bull.2005, 30, 23. (4) Cao, F.; Oskam, G.; Meyer, G. J.; Searson, P. C. J. Phys. Chem. 1996, 100, 17021. (5) Dloczik, L.; Ileperuma, O.; Lauermann, I.; Peter, L. M.; Ponomarev, E. A.; Redmond, G.; Shaw, N. J.; Uhlendorf, I. J. Phys. Chem. B 1997, 101, 10281. (6) Fisher, A. C.; Peter, L. M.; Ponomarev, E. A.; Walker, A. B.; Wijayantha, K. G. U. J. Phys. Chem. B 2000, 104, 949. (7) van de Lagemaat, J.; Frank, A. J. J. Phys. Chem. B 2000, 104, 4292. (8) Peter, L. M.; Duffy, N. W.; Wang, R. L.; Wijayantha, K. G. U. J. Electroanal. Chem. 2002, 524, 127. (9) Cao, F.; Oskam, G.; Searson, P. C. J. Phys. Chem. 1996, 100, 17021. (10) Nelson, J. Phys. ReV. B 1999, 59, 15374. (11) Kopidakis, N.; Schiff, E. A.; Park, N. G.; van de Lagemaat, J.; Frank, A. J. J. Phys. Chem. B 2000, 104, 3930.

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