Influence of Electrical Potential Distribution, Charge Transport, and

Feb 10, 2000 - The role of electrical potential, charge transport, and recombination in determining the photopotential and photocurrent conversion eff...
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J. Phys. Chem. B 2000, 104, 2044-2052

Influence of Electrical Potential Distribution, Charge Transport, and Recombination on the Photopotential and Photocurrent Conversion Efficiency of Dye-Sensitized Nanocrystalline TiO2 Solar Cells: A Study by Electrical Impedance and Optical Modulation Techniques J. van de Lagemaat,* N.-G. Park, and A. J. Frank* National Renewable Energy Laboratory, Golden, Colorado 80401 ReceiVed: September 7, 1999; In Final Form: NoVember 5, 1999

The role of electrical potential, charge transport, and recombination in determining the photopotential and photocurrent conversion efficiency (IPCE) of dye-sensitized nanocrystalline solar cells was studied. Electrostatic arguments and electrical impedance spectroscopy (EIS) are used to obtain information on the electrical and electrochemical potential distribution in the cell. It is shown that on the macroscopic level, no significant electrical potential drop exists within the porous TiO2 when it contacts the electrolyte and that the electrical potential drop at the transparent conducting oxide substrate (TCO)/TiO2 interface occurs over a narrow region, one or two layers of TiO2. Analyses of EIS and other data indicate that both the photopotential of the cell and the decrease of the electrical potential drop across the TCO/TiO2 interface are caused by the buildup of photoinjected electrons in the TiO2 film. The time constants for the recombination and collection of the photoinjected electrons are measured by EIS and intensity-modulated photocurrent spectroscopy (IMPS). As the applied bias is varied from short-circuit to open-circuit conditions at 1 sun light intensity, recombination becomes faster, the collection of electrons becomes slower, and the IPCE decreases. The decrease of IPCE correlates directly with the decline of the charge-collection efficiency ηcc, which is obtained from the time constants for the recombination and collection of the photoinjected electrons. Significantly, at open circuit, ηcc is only 45% of its short-circuit value, indicating that the dye-sensitized nanocrystalline TiO2 solar cell behaves as a nonideal photodiode.

Introduction Photochemical solar cells based on dye-sensitized nanocrystalline porous films of TiO2 are presently under active investigation as a low-cost alternative to conventional silicon solar cells.1-3 Under bright sunlight (AM 1.5), high-efficiency dye-sensitized TiO2 solar cells with a Ru-bipyridyl-based chargetransfer dye can supply a current density above 18 mA cm-2 at a voltage of about 0.5 V.3 However, most cells deliver a current density much less than 18 mA cm-2. A low photocurrent density generally indicates a low incident photon-to-current conversion efficiency (IPCE) in the spectral region, where the dye absorbs strongly. The IPCE can be expressed as the product of the light harvesting efficiency of the dye (ηlh), the quantum yield of electron injection (ηinj), and the efficiency of collecting the injected electrons (ηcc) at the transparent back contact. A low IPCE is therefore the result of either inefficient light harvesting by the dye, inefficient charge injection into TiO2, or inefficient collection of injected electrons. A low ηlh can be due to, for example, a low dye concentration,4,5 a TiO2 film too thin to absorb a significant fraction of the incident light, insufficient light scattering within the film,5-7 absorption of light by TiO2 or the redox electrolyte, and dye degradation.8 A low ηinj can be due to, for instance, dye desorption, dye aggregation, or band edge movement.9 A low ηcc can be attributed to competition between fast recombination of photoinjected electrons with the redox electrolyte or oxidized dye and electron collection. It is difficult experimentally to determine which of the three efficiencies (light absorption, charge injection, and charge collection) limit the IPCE. From a practical perspective, if one could, for example, eliminate one of the three factors as a cause

of the low IPCE, effort could be directed toward modifying the cell conditions to improve the other two. Recently, we developed a theoretical model,10 based on the continuity equation and other considerations, that relates the ratio of the time constants for the collection and recombination of photoinjected electrons to the charge-collection efficiency of the cell. This model utilized results from intensity-modulated photovoltage spectroscopy (IMVS) and intensity-modulated photocurrent spectroscopy (IMPS) measurements,9,10 which showed that the rate of recombination (rate ∝ n2.2) and the electron concentration in the conduction band (ncb ∝ n2.7) depend on the concentration (n) of photoinjected electrons. A method was introduced for calculating the time constant at short circuit (τsc) and at open circuit (τoc) from the theoretical electron concentration profile. The time constant for recombination at short circuit was then calculated using the electron concentration at short circuit. The charge-collection efficiency was estimated from the ratio τoc/τsc at short circuit obtained from combined IMVS and IMPS measurements. IMPS and IMVS have proven useful for studying charge transport10-16 and charge recombination9 in nanocrystalline films. In principle, IMVS measurements can be applied at any constant current.9 In practice, however, it is limited to the special case in which no current flows through the external circuit (i.e., open circuit). It is important, however, to understand how recombination and charge transport in dyesensitized nanocrystalline solar cells are affected by the applied potential. The electrical potential distribution, the location of charge separation, and the mechanism for the photopotential in dyesensitized nanocrystalline TiO2 solar cells are current issues of debate.9,17-19 It has been suggested that electrons move by

10.1021/jp993172v CCC: $19.00 © 2000 American Chemical Society Published on Web 02/10/2000

Conversion Efficiency of Nanocrystalline TiO2 Solar Cells diffusion in nanoporous TiO2 films11,13,15,16,20 because the major part of the films appears to be free of electric field. Currentvoltage (J-V) studies in our laboratory21 place the applied potential at the transparent charge-collecting back contact (TCO)/TiO2 interface and suggest the presence of an electrical potential barrier at the TCO/TiO2 interface that hinders the dark electron injection from TCO to TiO2 and promotes the transfer of photoinjected electrons from TiO2 to TCO. The presence of an electrical potential drop inside the TiO2 layer close to the TCO/TiO2 interface has been established by time-resolved photocharge22 and electroreflection measurements.23 The existence of an electrical field near the TCO surface was later confirmed by dark electrical impedance measurements.24 In the electrical impedance study, it is argued that when the film is “doped” by injecting electrons from the TCO layer into TiO2, the electric field extends progressively, as a function of applied bias, into the TiO2 layer. This reasoning contradicts, however, the electroreflectance study, which shows that at negative applied potentials (corresponding to strong electron injection from TCO into the TiO2 film) a strong electroreflectance signal is observed when TiO2 is illuminated through the TCO layer, whereas none is seen when the cell is illuminated from the electrolyte side. These observations suggest that varying the electrical potential of the TCO layer does not change the electrical potential inside the bulk of the TiO2 layer but only alters it close to the TCO/TiO2 interface. Recently, it was argued18,19 on electrostatic grounds that the electrical potential drop occurs within the first 1 or 2 layers of TiO2 at the TCO/ TiO2 interface and that no electrical field can exist in the bulk of TiO2. It has been suggested18 that the dye-sensitized nanocrystalline TiO2 solar cell behaves analogously to a solid-state p-n junction. According to the authors of this paper,18 the photopotential of the cell arises from the compensation of the dark potential drop at the TCO/TiO2 interface by the buildup of photoinjected charge at the interface. They suggest that the dark potential drop therefore limits the maximum attainable photopotential of the cell. In their model, charge separation occurs at the TCO/TiO2 interface owing to the presence of the electrical field. Their model contradicts, however, IMVS studies9,21 that show that the photopotential of the cell is caused by the buildup of photoinjected charges in the film, which raises the Fermi level of the TiO2 and therefore that of the TCO. Furthermore, their contention that the dark barrier at the TCO/TiO2 interface limits the photopotential is contrary to studies that show that exposing the TiO2 surface to various amines (in the dark) improves considerably the photopotential of the cell.3,9,21 In this article, the location of the electrical potential drops in the cell, the mechanism for the origin of the photovoltage, and the consequences of applying a bias voltage are examined. A simple statistical model is developed relating the time constants for recombination and charge collection to the charge-collection efficiency ηcc and the incident photon-to-current conversion efficiency IPCE of the cell over a wide range of applied bias. Electrical impedance spectroscopy (EIS) and IMPS are used to measure the respective time constants for charge recombination (τEIS) and for the combined processes of charge collection and charge recombination (τIMPS). EIS is also shown to provide information on the electrical and electrochemical potential distribution in the dye-sensitized nanocrystalline TiO2 solar cell and the energy distribution of surface states. Theory Electron Transport and Recombination. Figure 1 shows a schematic of the major kinetic parameters of a dye-sensitized

J. Phys. Chem. B, Vol. 104, No. 9, 2000 2045

Figure 1. Scheme showing the important kinetic parameters. τinj is the time constant for electron injection from the excited dye to the conduction band of a TiO2 particle. τt and τdt are the time constants for trapping and detrapping photoinjected electrons. τcc is the time constant for the collection of the photoinjected electrons at the back contact (TCO), and τr is the time constant for the recombination of photoinjected electrons with electron acceptors (redox species or oxidized dye) at the TiO2/redox electrolyte interface.

nanocrystalline solar cell. Upon excitation of adsorbed dye molecules, electrons are injected into the conduction band of TiO2 with a time constant (τinj) that is in the femtosecond-topicosecond time regime.25-27 The injected electron can be captured by trap states or can be thermally emitted back to the conduction band9,11,13,28 with respective time constants of τt and τdt. The photoinjected electrons spend most of their time in traps. Thus, as the electrons traverse the interconnecting network of TiO2 particles to the transparent charge-collecting back contact TCO with a time constant τcc, they undergo many trapping and detrapping events.11,13,28 Alternatively, the electrons can react (recombine) with electron acceptors (redox species or oxidized dye molecules) with a time constant (τr) at the semiconductor/ redox electrolyte interface. Usually, the recapture of an electron by the oxidized dye is prevented by an even faster neutralization of the oxidized dye by the redox electrolyte.29 The recombination of electrons occurs principally via trap states rather than via the conduction band.9 In this model, it is assumed that the probability that an injected electron reaches the collecting TCO layer depends on how far away from the TCO layer the electron is injected and the relative value of τr. If an electron is created far from the conducting TCO layer, the time constant for its collection will be relatively large. If an electron is injected near the TCO layer, the time constant for its collection will be relatively small. Thus, because of the competing recombination reaction, the closer the electron is generated to the collecting TCO layer, the higher its probability of being collected. The average of the time constants for charge collection τcc is given by

∫0dτcc(x)G(x)ηcc(x) dx τcc ) ∫0dG(x)ηcc(x) dx

(1)

where τcc(x) and ηcc(x) are the respective time constant and probability that an individual electron injected into a TiO2 particle by a dye at x will reach the collecting TCO layer. The charge-generation term G(x) is the rate of electron injection from the excited dye to the TiO2. In turn, ηcc(x) is given by the ratio of τr and τcc(x).

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ηcc(x) )

τr τr + τcc(x)

van de Lagemaat et al.

(2)

To a first approximation, it is assumed that τr does not depend strongly on x. This is a reasonable assumption because the only region in the TiO2 film where the electron concentration (τr ∝ n)9 is much different than its average (or bulk) value is in the vicinity of the TCO/TiO2 interface. Moreover, electrons injected close to the TCO/TiO2 interface have a probability of collection close to unity (τcc(x) , τr; eq 2). The overall collection efficiency (ηcc) is given as the rate at which the electrons are collected divided by the rate at which they are generated in the film.

∫0dG(x)ηcc(x) dx ηcc ) ∫0dG(x) dx

(3)

where integration occurs over the film thickness (d). Rearrangement of eqs 1-3 yields the ratio of the time constants for recombination and charge collection.

τr ) τcc

ηcc

∫0dG(x) dx

τcc(x)

∫0 τ + τ d

r

(4)

G(x) dx

cc(x)

Recognizing from eq 2 that τcc(x)/(τr + τcc(x)) ) 1 - ηcc(x), eq 4 can be simplified to

ηcc τr ) τcc 1 - ηcc

(5)

Substituting into eq 5 the relation10 τIMPS-1 ) τr-1 + τcc-1, one obtains

τr τIMPS

) (1 - ηcc)-1

Figure 2. Gaussian box enclosing the back contact (TCO) and a representative part of the porous TiO2 structure in contact with a high ionic strength electrolyte. The charge of photoinjected electrons (-) inside or in surface traps of TiO2 is compensated by cations (+) within a few angstroms of the surface. Also, electrons in the TiO2 layer at the TCO/TiO2 interface are compensated by equal and opposite charges in the TCO layer at the interface. Thus, there is no net charge in the volume enclosed by the Gaussian surface, and the electric field is zero at all points on the surface of the box.

(6)

Equation 6 describes a simple relation for determining ηcc from the time constant extracted from IMPS (τIMPS) and the lifetime of electrons for recombination τr. At open circuit, τr can be obtained by IMVS measurements.9 A direct method for obtaining τr at any applied bias, using electrical impedance spectroscopy (EIS), will be introduced below. Electrical Potential Distribution. The existence of an electrical potential drop inside the TiO2 layer close to the TCO/ TiO2 interface has been established by time-resolved photocharge22 and electroreflectance measurements.23 Moreover, in the former study, it is shown that downward band bending from the TiO2 to the TCO occurs, implying the presence of an accumulation layer in the TiO2 at the TCO/TiO2 interface. The existence of such a space-charge layer seems to be accepted by others.17,18,21 Figure 2 shows a schematic of a porous TiO2 film in contact with both TCO and the redox electrolyte. Because of the high ionic strength of the electrolyte (1 M LiI), any charge residing inside the porous TiO2 structure or on its surface is compensated by the electrolyte within a few angstroms of the surface.30 From the Gauss law of electrostatics, the density of field lines leaving a closed surface is proportional to the charge inside it E‚n b dA ) Q/0, where B E is the electric field vector, b n is (∫∫SB the vector normal from the surface, and Q is the charge residing in the volume enclosed by the surface). Figure 2 shows a

convenient closed surface in the shape of a box to illustrate the electrical potential distribution in the film. The box has its left surface inside the TCO layer and extends outwardly to enclose both TiO2 and the adjoining electrolyte. Because of charge neutrality, the electrons in the TiO2 layer at the TCO/TiO2 interface are compensated by equal and opposite charges in the TCO layer at the interface. Similarly, the charge of electrons in the TiO2 film away from the TCO/TiO2 interface is compensated by cations on the electrolyte side of the interface. Thus, the electric field is zero at all points on the surface of the sides of the box, extending perpendicularly from the TCO layer. Also, since there are no electric field lines inside the conductor TCO (except at very high currents) and there is no net charge in the volume enclosed by the Gaussian surface, the electric field at the surface of the right end of the box is also zero. Because the surface of the porous TiO2 film is at equipotential, the buildup of electrons in the TiO2 due to the equilibration with TCO must be localized near the TCO/TiO2 interface inasmuch as the width of a representative segment of the porous TiO2 structure (e.g., 20 nm diameter string of TiO2 particle) is generally much shorter than its length (e.g., film of thickness d ) 10 µm). Thus, the electrical potential drop must occur over a narrow region of TiO2 (specifically, within either the first layer of TiO2 particles or the thin TiO2 layer used to isolate the TCO surface from the redox electrolyte). This means that a narrow accumulation layer (corresponding to an ohmic contact for photoinjected electrons)31 is present in TiO2 at the TCO/TiO2 interface. The presence of an electrical potential drop across the first layer of TiO2 at the TCO/TiO2 interface is in accord with the conclusion of others.18,19 Thus, on the microscopic (or local) level, an electrical potential drop occurs across an accumulation layer at the TCO/TiO2 interface and the Helmholtz double layer at the TiO2/electrolyte interface. On the macroscopic level, no significant electrical potential drop can exist within the porous TiO2 when it is in contact with an electrolyte. Varying the potential applied to the TCO layer changes the electrical potential drop across the accumulation layer in TiO2 at the TCO/TiO2 interface. Because the Fermi level of the TiO2 layer closest to the interface is in equilibrium with the Fermi level of TCO (i.e., the TCO/TiO2 interface is transparent to

Conversion Efficiency of Nanocrystalline TiO2 Solar Cells

Figure 3. Influence of applying an electrical potential to the back contact (TCO) on the Fermi level (EF) of TiO2. At open circuit, the Fermi level is constant throughout the film for homogeneously absorbed light and equal to that of TCO. At other applied potentials, a Fermi level gradient exists in the film. However, the Fermi level of TiO2 always equals that of TCO at the TCO/TiO2 interface.

electrons), an increase or a decrease of applied bias will cause electrons to flow in either direction across the interface. At constant light intensity, a change in applied bias will produce a gradient in the Fermi level across the TiO2 film, which is illustrated schematically in Figure 3. This gradient provides a driving force for the diffusion of photoinjected electrons toward or away from the TCO/TiO2 interface.14 At open circuit, the Fermi level at the TCO/TiO2 interface and across the TiO2 film is constant for homogeneously absorbed light. Experimental Section Materials. Anatase TiO2 slurries, consisting of 15-20 nm sized particles, were prepared by hydrolyzing titanium tetraisopropoxide (Aldrich, 99.999%) in the presence of distilled acetic acid, followed by autoclaving at 230 °C for 12 h.17 Conducting glass plates (1.2 × 1.25 cm; Asahi Glass Co.; F-doped SnO2 overlayer, 80% transmittance in the visible, 5% haze, 10 Ω/sq) were used as the substrate for depositing TiO2 films. To control the thickness of the deposited films and to protect the electrical contacts, transparent adhesive tape was used (nominal thickness 40 µm). A drop of 10 mM titanium tetrabutoxide (Aldrich 99%) in 2-propanol was spread on the conducting glass surface to produce a thin layer of TiO232 to isolate the conducting glass surface from the redox electrolyte. The TiO2 slurry was then spread on top of it. The TiO2 covered glass was heated in air at 450 °C for 30 min and then allowed to cool. The thickness of the films was about 8 µm, as measured with a Tencor AlphaStep profiler. For photosensitization studies, the TiO2 electrodes were immersed in acetonitrile/tert-butyl alcohol (50:50 v/v%) containing 3 × 10-4 M Ru[LL′(NCS)2] (L ) 2,2′-bypyridine4,4′-dicarboxylic acid, L′ ) bis(tetrabutylammonium) 2,2′bipyridine-4,4′-dicarboxylate) for 24 h at room temperature. The dye-covered electrodes were then rinsed with the acetonitrile/ tert-butyl alcohol mixture and dried under a N2 stream. To minimize rehydration of TiO2 from moisture in the ambient air, the electrodes were immersed in the dye solution while they were still warm (100-120 °C) from the annealing step. Pt counter electrodes with a mirror finish were prepared by electron beam deposition of a 60 nm layer of Pt on top of a 40 nm layer of Ti on a glass plate. The Pt electrode was placed over the dye-coated electrode and the edges of the cell were sealed with 0.5 mm wide strips of 25 µm thick Surlyn (Dupont, grade 1702). Sealing was accomplished by pressing the two electrodes together at a pressure of 900 psi and a temperature

J. Phys. Chem. B, Vol. 104, No. 9, 2000 2047 of about 100 °C. The redox electrolyte, consisting of 1.0 M LiI in acetonitrile with varying amounts of I2 (30-210 mM), was introduced into the cell through one of two small holes drilled in the counter electrode; the I2 concentration had no significant effect on experimental results in this study. The holes were then covered and sealed with small squares of microscope objective glass and Surlyn. The edges of the cell were further sealed with Torr Seal (Varian). The resulting cell had an active area of about 0.25 cm2. Apparatus. The setup for IMPS and IMVS measurements is described elsewhere.9,10 The samples were illuminated with 680 nm wavelength light from a laser-diode (SDL model 7421 H1). This wavelength of light is only weakly absorbed by the dye molecules, ensuring a homogeneous electron-injection rate throughout the cell. A laser diode driver (SDL model 800) was used to modulate the intensity of the output beam. A lockin amplifier (Stanford Research Systems model 830) was employed to control the light modulation and measure the current or photopotential signal. A second lock-in amplifier (SRS model 830), synchronized to the first, was used to measure the light modulation. The light modulation amplitude was lower than 0.01 of the bias light intensity. For IMPS measurements, a potentiostat (EG&G Princeton Applied Research model 283) was used to apply a bias voltage to the cell and to measure the current. The modulation frequency ω of the light intensity (radial frequency ω ) 2πf) covered the range 0.06-(6 × 105) s-1, which is well below the bandwidth of the potentiostat (6 × 106 s-1). EIS measurements were performed with the same basic setup used for IMPS. A potentiostat was used to apply a bias to the cell. A lock-in amplifier supplied the modulated voltage and monitored the current response. A second lock-in amplifier measured the voltage modulation, which was e20 mV in magnitude. The frequency range of EIS was the same as that used for IMPS and IMVS. Spectra Analysis. The IMPS and IMVS spectra were fitted to eq 7 with a complex nonlinear least-squares algorithm, where T is a transfer function, A is the magnitude of the IMPS or IMVS

1 T)A 1 + (iωτ)R

(7)

response at ω ) 0, τ is the time constant of a process, and R is related to the width of the time constant distribution function.33,34 All data points are equally weighted. The EIS spectra were fitted with the program Equivcrt,35 which allows for the frequencydependent weighting of data points. Results and Discussion Figure 4 shows a typical EIS spectrum of the cell. Two prominent features are evident. The spectrum shows a large (slightly flattened) semicircle at low frequencies and a small one at high frequencies. The presence of the small semicircle is more clearly seen in the inset in Figure 4, which displays the imaginary and real component of the impedance as a function of angular frequency. The frequency range and the size of the small semicircle (ca. 4 Ω wide) suggest36 that it is associated with kinetic processes at the Pt counter electrode of the solar cell. EIS measurements of a cell consisting of two platinum electrodes concurred with this interpretation. The semicircles in Figure 4 can be interpreted in terms of the equivalent circuit in Figure 5. Figure 5a displays the equivalent circuit for the photoelectrode based on the electrostatic arguments (Theory) and the EIS and IMVS measurements discussed above. The TCO/redox electrolyte interface is omitted

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Figure 4. Typical EIS spectrum and fit plotted in the complex plane. The inset shows same data plotted as imaginary (ZIm) and real (ZRe) parts of the impedance vs the modulation frequency ω; solid lines in the inset represent a fit to the equivalent circuit shown in Figure 5b. The cell was biased at 0.5 V and illuminated with 680 nm light (Jsc ) 12 mA cm-2). The redox electrolyte contained 1.0 M LiI and 120 mM I2 in acetonitrile.

Figure 5. (a) Equivalent circuit of the photoelectrode consisting of the TCO/TiO2 interface and the TiO2/redox electrolyte interface, where RTiO2,TCO is the resistance associated with flow of charge across the TiO2/TCO interface, Cacc is the capacitance of the accumulation layer of the TiO2 layer at the TCO/TiO2 interface, Rct is the resistance for charge transfer across the TiO2/redox electrolyte interface, Css is the surface-state capacitance of TiO2, and CH is the capacitance of the Helmholtz double layer. (b) Equivalent circuit of the cell consisting of the counter electrode (CE)/redox electrolyte interface and the TiO2/ redox electrolyte interface, where RCE and R are the respective resistances, Rs is the sheet resistance of TCO and Pt counter electrode and the resistance of the electrolyte, and CCE and C are the respective constant phase elements. It is assumed that the TiO2 nanoparticles are too small to support an electric field in the dark or when they are illuminated with light intensities up to 1 sun.

because, for the cells discussed in this paper, a thin TiO2 “passivating” layer was deposited to isolate the TCO surface from the redox electrolyte. In Figure 5a, the resistance RTiO2,TCO associated with flow of charge across the TiO2/TCO interface is expected to be relatively small because of the ohmicity of the contact (Theory section). The buildup of charge in the accumulation layer of the TiO2 layer at the TCO/TiO2 interface is represented by the capacitance Cacc. The charge transfer across the TiO2/redox electrolyte interface (i.e., recombination) is represented by a Randles equivalent circuit37 corresponding to a combination of a resistance (Rct) in parallel to the surfacestate capacitance of TiO2 (Css) and the capacitance of the

van de Lagemaat et al.

Figure 6. Applied bias dependence of the resistance (corrected for both ohmic losses at the conducting glass and overpotential at the counter electrode) of the low-frequency (large) semicircle in the EIS spectra. The lines show expected dVb/d ln(R) ) κkT/q for two possible situations: κ ) 1 for an injection barrier at the TCO/TiO2 interface and κ ) 2 for electron transfer across the TiO2/redox electrolyte interface (see text). The cell was illuminated with 680 nm light (Jsc ) 10 mA cm-2). The redox electrolyte contained 1.0 M LiI in acetonitrile with varying amounts of I2: 30 mM (b), 60 mM (0), 120 mM (O), 150 mM (2), and 210 mM (9).

Helmholtz double layer (CH). The time constant of such an RC circuit corresponds to the time constant for interfacial charge transfer.38 The surface-state capacitance is linearly proportional to the surface-state density at the Fermi level. The Helmholtz capacitance does not depend on bias. Furthermore, CH is larger than Css and can be ignored at the light intensities used in this study.9 Because the high-frequency (small) semicircle can be identified with the counter electrode/redox electrolyte interface (RC circuit on the left in Figure 5b), the low frequency (large) semicircle must be associated with the photoelectrode (Figure 5a) and correspond to charge transfer across either the TCO/ TiO2 interface or the TiO2/redox electrolyte interface. As a beginning point for identifying the cause of the large semicircle, one can estimate the frequency range and width of a semicircle associated with the TCO/TiO2 interface. Taking a reasonable value for the resistance of the ohmic contact of the TCO/TiO2 interface RTiO2,TCO (1-10 Ω) and the capacitance of the accumulation layer in TiO2, Cacc (1-10 µF), the frequency range for charge transfer across the TCO/TiO2 interface is estimated to lie at (0.1-1) × 106 s-1. This frequency range is quite different from that of the large semicircle (e0.01 × 106 s-1) in Figure 4. Furthermore, the estimated width of a semicircle for charge transfer across the TCO/TiO2 interface (1-10 Ω) is also quite different from that of the large semicircle (ca. 145 Ω). This simple analysis suggests that the large semicircle in Figure 4 is probably not associated with charge transfer across the TCO/ TiO2 interface represented by the RC circuit (RTiO2,TCO, Cacc) on the left in Figure 5a. More detailed information on the origin of the large semicircle can be obtained by analyzing the dependence of the resistance (width) of the large semicircle on applied bias. Figure 6 shows the applied bias dependence of the resistance of the low-frequency (large) semicircle for several cells. The resistance is seen to change over 4 orders of magnitude as the applied bias is varied between 0 and 0.55 V. The dependence

Conversion Efficiency of Nanocrystalline TiO2 Solar Cells of the resistance R on the applied bias Vb can be described by the following expression: R ∝ exp(qVb/κkT), where q is the electronic charge, k is the Boltzmann constant, T is the absolute temperature, and κ is an experimental parameter. From the inverse of the slope of the plot dVb/d ln(R), one can obtain information about the location of the current limiting process. If the resistance were associated with an injection barrier at the TCO/TiO2 interface, one would expect dVb/d ln(R) ) κkT/q ≈ 26 mV,39 corresponding to an ideality factor of κ ) 1. If the resistance were associated with electron transfer across the TiO2/ redox electrolyte interface, one would expect from the ButlerVolmer equation40 that dVb/d ln(R) ) κkT/q ≈ 52 mV, corresponding to κ ) 1/β ) 2,41 where β is the symmetry factor. The lines in Figure 6 depict these two possibilities. Analysis of the plot yields a dVb/d ln(R) of about 52 mV. Typically, for a diode, a dVb/d ln(R) of about 52 mV implies a κ value of 2, which is indicative of a space-charge limited recombination current.42 A κ value of 2 has been observed by others.16,21,43 A space-charge limited current is, however, without merit for dyesensitized nanocrystalline solar cells. Thus, the analysis of the ln(R) vs Vb data is consistent with a resistance associated with charge transfer across the TiO2/redox electrolyte interface. Specifically, one can identify the RC circuit (Rct, Css, and CH) on the right in Figure 5a with the large semicircle in Figure 4. Additional data to support this assignment are given below. Figure 5b displays the equivalent circuit of the cell, which includes elements associated with the counter electrode/redox electrolyte and TiO2/redox electrolyte interfaces. These are represented by resistances (RCE or R) in parallel with constant phase elements (CCE or C). The constant phase element33 accounts for the spatial distribution of the surface-state capacitance, arising from the gradient of the Fermi level at non-opencircuit conditions (Figure 3). The amplitude of the constant phase element C′ can be viewed as a capacitance because of the narrow spatial distribution of surface-state capacitance observed.44 The series resistance Rs represents the sheet resistance of the TCO (ca. 10 Ω) and the Pt counter electrode (ca. 4 Ω) and the resistance of the high ionic strength electrolyte (ca. 1 Ω). The phenomenon underlying the equivalent circuit for the counter electrode has been reported elsewhere.36 Of more relevance to this paper is to understand the kinetic processes and location of potential drops at the photoelectrode. In particular, we examine the bias dependence of the RC circuit on the right in Figure 5b associated with the TiO2/redox electrolyte interface. Figure 7 shows the bias dependence of the amplitude of the constant phase element C′ or effective capacitance.44 In terms of the geometric cell area, values of C′ range from about 50 µF cm-2 to 10 mF cm-2, as the applied bias increases from 0 to 0.55 V. At the highest light intensity (ca. 1 sun), C′ ) 3.4 mF cm-2 at open circuit. Although this value is reasonable for a capacitance distributed over the surface of the nanocrystalline film (roughness factor ≈ 800) used in this study, it is too high to be associated with a narrow accumulation layer (1-2 TiO2 particle layer thickness) at the TCO/TiO2 interface. At most, the magnitude of the capacitance of the accumulation layer is comparable to that of the Helmholtz layer (ca. 10 µF cm-2).30 It has been suggested18 that the accumulation of the photogenerated electrons in the TCO layer (with respect to the redox electrolyte) creates the photopotential of the cell. If this were the case, the net charge at the TCO/TiO2 interface would decrease with increasing photopotential. The decrease of the net charge at the interface stems from the accumulation of the photoinjected electrons that compensate the positive charge on

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Figure 7. Bias dependence of the amplitude of the constant phase element C′ or effective surface-state capacitance at two light intensities represented by short-circuit photocurrents Jsc of 3 mA cm-2 (O) and 10 mA cm-2 (b). The applied bias is corrected for potential drops over Rs and RCE (Figure 5b), and C′ is given in terms of the geometric area of the TiO2 electrode. The redox electrolyte contained 1.0 M LiI and 210 mM I2 in acetonitrile.

the TCO surface, causing a decrease of the electrical potential drop (or band bending) at the TCO/TiO2 interface. Thus, if this model were valid, one would predict that the capacitance of the cell would decrease with increasing photovoltage or increasing light intensity. Figure 7 shows, however, just the opposite. The capacitance at open circuit increases with light intensity. The same trend is observed in IMVS measurements of charge (Q) vs open-circuit photovoltage (Voc).9 The Q vs Voc plot9 shows that the charge increases exponentially with increasing Voc. It follows that the photocapacitance (dQ/dVoc) increases with light intensity. This suggests that the accumulation of the photogenerated electrons and therefore the increase of Voc is associated with the TiO2 film rather than with the TCO layer. Thus, the photopotential of the cell is due to the change of the electrochemical (or chemical) potential of TiO2 rather than due to the electrical potential drop across the TCO/TiO2 interface. In other words, the photopotential of the cell is due to Fermi level movement in the TiO2 film (with respect to that of the redox electrolyte), arising from the buildup of photoinjected electrons in the film. Furthermore, because there is no significant change in the electrical potential drop across the TiO2/electrolyte interface at light intensities up to 1 sun,9 the upward movement of the Fermi level in TiO2 causes a decrease of the electrical potential drop across the TCO/TiO2 interface. Thus, both the photopotential of the cell and the decrease of the electrical potential drop across the TCO/TiO2 interface are caused by the buildup of photoinjected electrons in the TiO2 film. Figure 7 shows that as the applied bias increases, a steep rise of the C′ vs Vb plot is observed. Near open circuit (ca. 0.4 V), dV/d lnC′ ≈ 110 mV, which is the same value observed for dV/d lnQ at open circuit as determined by IMVS.9 The latter investigation shows that Q corresponds to the amount of charge in surface states at open circuit. It has been pointed out that in principle the measured capacitance is the sum of the capacitance due to charge buildup in the conduction band and in trap states.45 In practice, however, the amount of charge in the conduction band is negligible compared with that in surface states.9 Because the capacitance C ) dQ/dV, it can also be shown that the conduction band capacitance is negligible compared with the

2050 J. Phys. Chem. B, Vol. 104, No. 9, 2000

Figure 8. Comparison of the imaginary component of the IMVS and EIS response at open circuit for the same cell and light intensity (680 nm; Jsc ) 1 mA cm-2). The inset shows a plot of the time constant of EIS vs the time constant of IMVS at open circuit over a range of light intensities (Jsc varies from 0.1 to 10 mA cm-2). The redox electrolyte contained 1.0 M LiI and 120 mM I2 in acetonitrile.

surface-state capacitance.46 Thus, the capacitance in this region of the C′ vs Vb plot is attributed to the buildup of charge in surface states. Furthermore, this analysis indicates that EIS measurements at open circuit, as a function of light intensity, can be used to obtain the surface-state energy distribution function, which has been shown9 to be exponential over the energy range investigated. Away from open circuit, the measured capacitance cannot be easily interpreted in terms of the density of surface states at the Fermi level because the Fermi level is not constant across the film (Figure 3). Figure 8 compares the imaginary component of the IMVS and EIS response at open circuit for the same cell and light intensity. The EIS is biased to produce no current flow in the external circuit. It can be seen that the EIS and IMVS spectra coincide, displaying a minimum at the same frequency and thus showing the same time constant. Because IMVS yields the time constant for recombination τr at open circuit, the time constant from EIS τEIS can be attributed to recombination, i.e., τEIS ) τr. This is not surprising since the low-frequency (large) semicircle (Figure 4) corresponds to the RC circuit for the TiO2/ electrolyte interface (Figure 5b), and thus its time constant corresponds to that of interfacial charge transfer.38 The inset in Figure 8 shows a plot of the time constant of EIS47 vs the time constant of IMVS at open circuit over a range of light intensities. A one-to-one relationship between the time constants is observed. This agreement can be understood by considering the factors that determine the EIS and IMVS response. The IMVS response is given by the ratio of the voltage response (V ˜ ) to the modulated light intensity (I˜0) (IMVS ) V ˜ /qI˜0). The EIS response can be expressed quite similarly. It is given by the ratio of the complex voltage modulation (V ˜) to the modulated current density response (J˜) (Z ) V ˜ /J˜). The difference between the two methods is the location of the current source. In EIS, the current is supplied by an external voltage source, whereas in IMVS, the current source is the adsorbed dye on theTiO2 surface (i.e., the current is generated in situ). In IMVS, factors (e.g., sheet resistance) associated with the conducting glass and counter electrode do not contribute to the signal. However, EIS can be done at any biassit is not limited to open circuit. Thus, with EIS, one can examine the recombination kinetics over a wide range of operating conditions of the cell. Figure 9 shows the dependence of J and IPCE on applied bias for a cell illuminated with 680 nm light. Near short circuit conditions, the current density and IPCE curves are at a

van de Lagemaat et al.

Figure 9. Dependence of current density (O) and IPCE (b) on applied bias for a cell illuminated with 680 nm light (Jsc ) 11.5 mA cm-2). The IPCE is normalized to its short-circuit value as determined from the low-frequency limit of IMPS. The bias voltage is not corrected for ohmic losses. The redox electrolyte contained 1.0 M LiI and 120 mM I2 in acetonitrile.

maximum. With increasing bias, both J and IPCE decrease. A decrease of the photocurrent with bias has been observed by others.16,45 This observation is consistent with a decrease of the IPCE. At open circuit, the IPCE is only about 45% of its shortcircuit value, significantly less than unity, implying that the cell behaves as a nonideal photodiode. In the vicinity of the power point of the cell at about 0.3 V, the drop off of J and IPCE values with increasing bias coincide, suggesting a cause-andeffect relation. These results indicate that the drop off of current is due solely to the decrease of IPCE and, in particular, to one of the three determinants of the IPCE (light absorption, charge injection, and charge collection). Normally, one would not expect the light-harvesting efficiency ηlh to exhibit a significant dependence on the applied bias for the Ru-bipyridyl-based dye used in this study. When the applied bias is cycled between short circuit and open circuit, no hysteresis is observed, indicating that the drop off of IPCE is not due to dye degradation. Furthermore, a decrease of the charge-injection efficiency ηinj with applied bias can only occur if the band edges shift with respect to the dye position, such that electron injection from the adsorbed dye to the conduction band of TiO2 is disfavored energetically. However, we have shown9 that no significant band edge movement occurs in these cells at 1 sun illumination, ruling out a potential dependent charge-injection efficiency.48 Thus, the drop off of IPCE can be attributed to the decrease of the charge-collection efficiency with applied bias. Thus, at open circuit, this implies that ηcc ≈ 0.45. A decrease of ηcc would be manifested by a bias dependence of the time constants for charge transport and/or recombination (eq 6). Figure 10 shows the time constants for EIS and IMPS as a function of applied bias. At a bias close to short circuit (Vb ≈ 0 V), the time constant for recombination (τEIS) is more than 2 orders of magnitude larger than that for charge transport (τIMPS), indicating that the charge-collection efficiency must be close to unity. With increasing bias, it can be seen that τEIS decreases, indicating that recombination becomes faster. At the same time, τIMPS increases, implying that the collection of photoinjected electrons becomes slower. At open circuit, only a factor of 4 separates the time constants, suggesting that the recombination kinetics is competitive with charge collection. One can understand the dependence of τEIS on the applied bias from the strong

Conversion Efficiency of Nanocrystalline TiO2 Solar Cells

Figure 10. Dependence of the time constants for EIS and IMPS on applied bias for a cell illuminated with 680 nm light (Jsc ) 11.5 mA cm-2). The bias voltage is corrected for ohmic losses due to series resistance. The redox electrolyte contained 1.0 M LiI and 120 mM I2 in acetonitrile.

dependence of the rate of recombination on the electron concentration in the film. Because the concentration of photoinjected electrons in the film is much higher at open circuit than at short circuit, the rate of recombination is expected to increase with applied bias. The basis for the dependence of τIMPS on applied bias is more difficult to understand. It has been proposed that the bias dependence of τIMPS is due to RC limitations16 rather than due to slow electron collection. If this were the case, τIMPS would increase with capacitance as the applied bias increases. However, Figure 10 shows that the IMPS time constant passes through a maximum around open circuit, whereas the capacitance continues to increase with bias (Figure 7). Furthermore, Figure 10 shows that in the bias range (0.2-0.4 V) in which τIMPS increases, dV/d ln τIMPS is about 200 mV. In this same bias range, however, Figure 7 shows that dV/d ln C′ is about 60 mV. These observations imply that τIMPS is not RC limited. Thus, the increase of τIMPS reflects slower electron collection. The basis for this slow electron collection is currently under study. Figure 11 displays the dependence of τEIS/τIMPS on the chargerecombination efficiency (1 - ηcc) and the charge-collection efficiency ηcc. The values for the τEIS/τIMPS vs (1 - ηcc) plot were obtained for three cells in which the applied bias was varied from short-circuit to open-circuit conditions. Over a wide range of charge-collection efficiencies (ηcc > 20%), the dependence of τEIS/τIMPS on ηcc can be described by an inverse linear relation (eq 6), leading to a straight line in the doublelogarithmic plot of τEIS/τIMPS vs (1 - ηcc). The good agreement between the prediction of eq 6 and the experimental data implies that the statistical model describes reasonably well the principal underlying phenomena for charge transport and recombination in dye-sensitized nanocrystalline solar cells. Conclusions The role of the electrical potential, charge transport, and recombination on the photopotential and photocurrent conversion efficiency IPCE of dye-sensitized nanocrystalline solar cells was studied. Up to 1 sun illumination, no significant electrical potential drop is shown to exist within the porous TiO2 film when it contacts the electrolyte. Electrical impedance

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Figure 11. Dependence of τEIS/τIMPS on the charge-recombination efficiency (1 - ηcc) and the charge-collection efficiency ηcc for three cells illuminated with 680 nm light (Jsc ) 11.5 mA cm-2). The applied bias was varied from short-circuit to open-circuit conditions. The redox electrolyte contained 1.0 M LiI in acetonitrile with varying amounts of I2: 30 mM (b), 120 mM (O), and 210 mM (9).

spectroscopy EIS is shown to provide information on the electrical and electrochemical potential distribution and the energy distribution of surface states. Analyses of EIS and other data indicate that both the photopotential of the cell and the decrease of the electrical potential drop across the TCO/TiO2 interface are caused by the buildup of photoinjected electrons in the TiO2 film. EIS and intensity-modulated photocurrent spectroscopy (IMPS) are used to measure the time constants for the recombination and collection of the photoinjected electrons. As the applied bias is varied from short-circuit to open-circuit conditions at 1 sun illumination, recombination becomes faster, the collection of electrons becomes slower, and the IPCE decreases. The drop off of IPCE correlates with the decline of the charge-collection efficiency ηcc, which is determined from the time constants for the recombination and collection of the photoinjected electrons. Over a wide range of charge-collection efficiencies (ηcc > 20%), the dependence of the ratio of the respective time constants on ηcc can be described by an inverse relation (eq 6). Significantly, at open circuit, ηcc is only 45% of its short-circuit value, indicating that the dye-sensitized nanocrystalline TiO2 solar cell behaves as a nonideal photodiode. Acknowledgment. This work was supported by the Office of Science, Division of Chemical Sciences (J.v.d.L., and A.J.F.), and the Office of Utility Technologies, Division of Photovoltaics (N.-G. P), U.S. Department of Energy, under contract DE-AC3699GO10337. References and Notes (1) O’Regan, B.; Gra¨tzel, M. Nature 1991, 353, 737. (2) Barbe´, C. J.; Arendse, F.; Conte, P.; Jirousek, M.; Lenzmann, F.; Shklover, V.; Gra¨tzel, M. J. Am. Ceram. Soc. 1997, 80, 3157. (3) Nazeeruddin, M. K.; Kay, A.; Rodicio, I.; Humphry-Backer, R.; Mu¨ller, E.; Liska, P.; Valchopoulos, N.; Gra¨tzel, M. J. Am. Chem. Soc. 1993, 115, 6382. (4) Fillinger, A.; Parkinson, B. A. Unpublished data. (5) Park, N.-G.; Schlichtho¨rl, G.; van de Lagemaat, J.; Cheong, H. M.; Mascarenhas, A.; Frank, A. J. J. Phys. Chem. B 1999, 103, 3308. (6) Ferber, J.; Luther, J. Sol. Energy Mater. Sol. Cells 1998, 54, 265.

2052 J. Phys. Chem. B, Vol. 104, No. 9, 2000 (7) Rothenberger, G.; Comte, P.; Gra¨tzel, M. Sol. Energy Mater. Sol. Cells 1999, 58, 321. (8) Grunwald, R.; Tributsch, H. J. Phys. Chem. B 1997, 101, 2564. (9) Schlichtho¨rl, G.; Huang, S. Y.; Sprague, J.; Frank, A. J. J. Phys. Chem. B 1997, 101, 8141. (10) Schlichtho¨rl, G.; Park, N.-G.; Frank, A. J. J. Phys. Chem. B 1999, 103, 782. (11) de Jongh, P. E.; Vanmaekelbergh, D. Phys. ReV. Lett. 1996, 77, 3427. (12) Vanmaekelbergh, D.; Iranzo-Marı´n, F.; van de Lagemaat, J. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 616. (13) de Jongh, P. E.; Vanmaekelbergh, D. J. Phys. Chem. B 1997, 101, 2716. (14) Vanmaekelbergh, D.; de Jongh, P. E. J. Phys. Chem. B 1999, 103, 747. (15) Cao, F.; Oscam, G.; Meyer, G. J.; Searson, P. C. J. Phys. Chem. 1996, 100, 17021. (16) Dloczik, L.; Ileperuma, O.; Lauermann, I.; Perer, L. M.; Ponomarev, E. A.; Redmond, G.; Shaw, N. J.; Uhlendorf, I. J. Phys. Chem. B 1997, 101, 10281. (17) Zaban, A.; Ferrere, S.; Sprague, J.; Gregg, B. A. J. Phys. Chem. B 1997, 101, 55. (18) Schwarzburg, K.; Willig, F. J. Phys. Chem. B 1999, 103, 5743. (19) Bisquert, J.; Garcia-Belmonte, G.; Fabregat-Santiago, F. J. Solid State Electrochem. 1999, 3, 337. (20) Solbrand, A.; Lindstro¨m, H.; Rensmo, H.; Hagfeldt, A.; Lindquist, S.-E.; So¨dergren, S. J. Phys. Chem. B 1997, 101, 2514. (21) Huang, S. Y.; Schlichtho¨rl, G.; Nozik, A. J.; Gra¨tzel, M.; Frank, A. J. J. Phys. Chem. B 1997, 101, 2576. (22) Levy, B.; Liu, W.; Gilbert, S. E.. J. Phys. Chem. B 1997, 101, 1810. (23) Boschloo, G.; Goossens, A.; Schoonman, J. J. Electroanal. Chem. 1997, 428, 25. (24) Zaban, A.; Meier, A.; Gregg, B. A. J. Phys. Chem. B 1997, 101, 7985. (25) Eichberg, R.; Willig, F. Chem, Phys. 1990, 141, 159. (26) Moser, J.-E.; Gra¨tzel, M.; Durrant, J. R.; Klug, D. R. Femtochemistry: Ultrafast Chemical and Physical Processes in Molecular Systems; World Scientific: River Edge, NJ, 1996. (27) Ellingson, R. J.; Asbury, J. B.; Ferrere, S.; Ghosh, H. N.; Sprague, J. R.; Lian, T. Q.; Nozik, A. J. Phys. Chem. B 1998, 102, 6455. (28) Nelson, J. Phys. ReV. B 1999, 59, 15374. (29) Hagfeldt, A.; Gra¨tzel, M. Chem. ReV. 1995, 95, 45. (30) Morrison, S. R. Electrochemistry at Semiconductor and Oxidized Metal Electrodes; Plenum: New York, 1980; p 60. (31) Fonash, S. J. Solar Cell DeVice Physics; Academic Press: San Diego, 1981; p 122. (32) Smestad, G. Sol. Energy Mater. Sol. Cells 1994, 32, 273. (33) Macdonald, J. R. Impedance Spectroscopy; John Wiley & Sons: New York, 1987; p 39.

van de Lagemaat et al. (34) Equation 7 is the equivalent of a constant phase element in electrical impedance spectroscopy. This equation models a distribution of time constants. For further information see: Macdonald, J. R. Impedance Spectroscopy; John Wiley & Sons: New York, 1987; pp 13 (eq 1) and 34. (35) Boukamp, B. A. Faculty of Chemical Technology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands. (36) Papageorgiou, N.; Maier, W. F.; Gra¨tzel, M. J. Electrochem. Soc. 1997, 144, 877. (37) Macdonald, J. R. Impedance Spectroscopy; John Wiley & Sons: New York, 1987; p 102. (38) Because the capacitance C ) dQ/dV and the resistance R ) dV/dJ ) dV/dJr (where Jr is the recombination current as given by Jr ) J - Jsc), the time constant of the RC circuit is given by τEIS ) RC ) dQ/dJr. Furthermore, from ref 9, Q ) Jrτr, implying that dQ/dJr ) τr (where τr is the time constant for recombination). Thus, τEIS ) τr. (39) Sze, S. M. Semiconductor DeVices. Physics and Technology; John Wiley and Sons: New York, 1985; p 166. (40) Bard, A. J.; Faulkner, L. R. Electrochemical Methods, Fundamentals and Applications; John Wiley & Sons: New York, 1980; p 101. (41) The symmetry factor is determined from the Tafel slope in: Stanley, A.; Matthews, D. Aust. J. Chem. 1995, 48, 1294. (42) Sze, S. M. Semiconductor DeVices. Physics and Technology; John Wiley and Sons: New York, 1985; p 128. (43) So¨dergren, S.; Hagfeldt, A.; Olsson, J.; Lindquist, S.-E. J. Phys. Chem. 1994, 95, 5522. (44) The impedance Z of a constant phase element33 is given by the relation Z ) 1/(C′(iω)R), where C′ is the amplitude of the constant phase element and R (0 < R < 1) is related to the width of the spatial distribution of the surface-state capacitance. For a value of R close to unity, Z corresponds to a single capacitor, implying that the width of the spatial distribution function goes to zero. For the cells reported in this study, R is practically ideal, with values ranging from 0.93 to 0.99. Thus the general units of C′ (FR ΩR-1 cm-2) can be approximated by F cm-2. (45) Franco, G.; Gehring, J. Peter, L. M.; Ponomarev, E. A.; Uhlendorf, I. J. Phys. Chem. B 1999, 103, 692. (46) From eq A.34 in ref 9, it can be shown that the amount of charge in surface states Qss is much larger than that in the conduction band Qcb because (Qcb + Qss)/Qcb ≈ 1000 for an open-circuit photovoltage of Voc ) -700 mV and a conduction band edge potential Vcb ) -900 mV at 1 sun. This implies that the ratio of the measured (or total) capacitance to the conduction band capacitance is Ctotal/Ccb ≈ 260, indicating that Ccb can be ignored and that Ctotal can be identified with the surface-state capacitance Css. (47) The time constant for EIS is obtained from the relation τEIS ) (RC′)1/R, where R is defined in ref 36. (48) The charge-injection efficiency ηinj is defined as the fraction of excited dye molecules that inject an electron into TiO2. This definition differs from that used in ref 45.