A Comparison of Different Methods To Determine the Electron

Sep 23, 2009 - (DSCs) is described and illustrated with data obtained using cells ... electron transfer) as at short circuit (where the majority of el...
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J. Phys. Chem. C 2009, 113, 18125–18133

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A Comparison of Different Methods To Determine the Electron Diffusion Length in Dye-Sensitized Solar Cells Hongxia Wang and Laurence M. Peter* Department of Chemistry, UniVersity of Bath, Bath BA2 7AY, United Kingdom ReceiVed: July 14, 2009; ReVised Manuscript ReceiVed: September 8, 2009

A new steady-state method for determination of the electron diffusion length in dye-sensitized solar cells (DSCs) is described and illustrated with data obtained using cells containing three different types of electrolyte. The method is based on using near-IR absorbance methods to establish pairs of illumination intensity for which the total number of trapped electrons is the same at open circuit (where all electrons are lost by interfacial electron transfer) as at short circuit (where the majority of electrons are collected at the contact). Electron diffusion length values obtained by this method are compared with values derived by intensity-modulated methods and by impedance measurements under illumination. The results indicate that the values of electron diffusion length derived from the steady-state measurements are consistently lower than the values obtained by the non-steady-state methods. For all three electrolytes used in the study, the electron diffusion length was sufficiently high to guarantee electron collection efficiencies greater than 90%. Measurement of the trap distributions by near-IR absorption confirmed earlier observations of much higher electron trap densities for electrolytes containing Li+ ions. It is suggested that the electron trap distributions may not be intrinsic properties of the TiO2 nanoparticles but may be associated with electron-ion interactions. Introduction 1-3

Dye-sensitized solar cells (DSC) have achieved AM 1.5 power efficiencies in excess of 11%, and efforts are now being made toward commercialization. The DSC differs from conventional photovoltaic devices because the light-absorbing component is a monolayer of molecular dye rather than a semiconducting solid. Efficient light harvesting is achieved by adsorbing the monolayer of dye on the high internal surface area of a sintered mesoporous layer of TiO2 (thickness typically 10-15 µm) permeated with an electrolyte (usually containing the I3-/I- redox couple). The photoexcited dye injects electrons into the conduction band of the oxide, and current collection involves diffusion of electrons through the network of interconnected particles to the anode contact. Following electron injection, dye molecules are regenerated rapidly in their original oxidation state by transfer of electrons from iodide ions in the electrolyte. The I3- ions formed in the regeneration process diffuse to the platinized cathode contact, where they are reduced to I- to complete the regenerative cycle. Despite the progress toward high efficiency cells, many aspects of the physics and chemistry of the DSC remain obscure or controversial. It is often necessary to make simplifying assumptions in modeling and data interpretation, and it is important to subject these assumptions to scrutiny. The present paper describes a number of different experimental approaches to the determination of the electron diffusion length in DSCs and examines the significance of the different results obtained. Theoretical Basis Electron Transport and Recombination in the DSC. Electrons will only be collected at the anode of the DSC if they are not lost by electron transfer to I3- ions (recombination) in the electrolyte. Electron transport through the electrolyte-filled * Corresponding author. E-mail: [email protected].

mesoporous TiO2 film in the DSC is still not fully understood: the slow response of the photocurrent to an illumination step4,5is attributed to trapping of electrons at states located in the forbidden gap of the oxide,6-9 and the illuminated film evidently contains a high density of electrons that are not free to move rapidly as they would in a bulk semiconductor. Screening of charges by the electrolyte appears to limit the build-up of macroscopic electrical fields in the mesoporous film,10-13and electron transport is usually modeled in terms of trap-limited diffusion, although the nature of the traps has not been established. Uncertainty also surrounds the kinetics of recombination in the DSC. Some models suppose that the rate at which electrons are transferred from the conduction band to the redox is first order in free electron concentration.3 However, it has also been suggested that the nonideal behavior of DSCs arises from the fact that the recombination process is mediated by electronic surface states below the conduction band of the oxide.14-16 It is important to distinguish here between trap states located in the ‘bulk’ of the TiO2 nanoparticles and surface states located at the surface of the oxide, which are able to exchange electrons with redox species as well as with the TiO2. Other authors have proposed that the recombination rate is limited by the slow transport of trapped electrons to the oxide/electrolyte interface.17 These uncertainties over the recombination mechanism mean that quantitative discussion is necessarily speculative to some extent as a consequence of the need to make simplifying assumptions. Definition of the Electron Diffusion Length. The electron diffusion length3,18-28 is a convenient measure of the distance an electron moves before it is lost by recombination. Efficient collection of electrons (collection efficiency,ηcoll, approaching 100%) requires the electron diffusion length to be greater than the film thickness. If recombination occurs predominantly via the conduction band and is first order in conduction band electron concentration, the electron diffusion length can be

10.1021/jp906629t CCC: $40.75  2009 American Chemical Society Published on Web 09/23/2009

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defined as L0 ) (D0τ0)1/2, where D0 is the diffusion coefficient of electrons in the conduction band of the TiO2, and τ0, the free electron lifetime, is determined by the rate constant for electron transfer from the conduction band and by the local I3concentration. The notation L0 is used here to distinguish the electron diffusion length defined in terms of reaction via the conduction band from the diffusion length Ln determined by small amplitude methods which effectively linearize the DSC response. Since the I3- concentration varies with distance under operating conditions where current flows in the DSC,29 the assumption of a constant value of τ0 is only valid if the concentration gradients are small (low light intensities). Bisquert et al. have recently discussed some of the problems in the definition of the electron lifetime that arise as a consequence of trapping, electron transfer via surface states, and the (unknown) reaction order of the recombination reaction with respect to electron concentration.16 The concept of electron lifetime may not be applicable for large amplitude perturbations, but in general it is possible to define a small amplitude lifetime16 that describes the (linearized) behavior of the DSC under conditions where the perturbation from the stationary state is sufficiently small, as is the case for impedance spectroscopy (IS) and intensity-modulated photovoltage/photocurrent spectroscopy (IMVS/IMVS). Determination of the Electron Diffusion Length by SteadyState Measurements. Halme and co-workers25,30 have shown that electron diffusion lengths derived from steady-state photocurrent IPCE (incident photon-to-electron conversion efficiency) spectra for illumination from the substrate and electrolyte sides are systematically lower than those derived by non-steady-state techniques, and Barnes et al.26,28 have also reached a similar conclusion. Both of these groups have concluded that the diffusion lengths derived from small amplitude methods are incorrect. Since in the case of the DSCs studied here, it was found that the diffusion lengths obtained by the IPCE method increase with bias illumination intensity to values where accurate measurement of the electron diffusion length is no longer possible, a different steady-state method was used in order to make comparisons with IS and IMVS/IMPS. The method is based on measuring the trapped electron concentration by near-IR absorbance31,32 under steady-state illumination at open circuit and at short circuit as a function of illumination intensity. The rationale behind the method and the assumptions made in the analysis are outlined below. In a DSC, the conversion of the incident photon flux I0 into a current density jphoto under short circuit conditions (illumination from the substrate side) depends on the product of four factors: Tsub, the transmission of the substrate, the light harvesting efficiency of the absorber layer ηlh, the electron injection efficiency ηinj, and the electron collection efficiency ηcoll.

jphoto ) Tsubηlhηinjηcoll qI0

(1)

The balance of volume rates for injection, recombination, and extraction of electrons can be expressed as

υext )

jphoto ) υinj - υrec qd

(2)

where d is the film thickness. Electrons are injected into the oxide at a rate given by

υinj )

TsubI0ηlhηinj d

(3)

Under short circuit conditions, the volume rate at which electrons are extracted (which corresponds to jsc/qd, where jsc is the short circuit current density) depends on the collection efficiency and is given by

υext )

jsc ) ηcollυinj qd

(4)

from which it follows that

ηcoll ) 1 -

υrec υinj

(5)

Now consider illumination at open circuit with a (lower) incident photon flux I0′ that is chosen so that the volume recombination rate υrec is the same as at short circuit at the higher photon flux I0. Since the extraction rate is zero at open circuit (provided that shunting via the substrate has been eliminated by a suitable blocking layer), the rates of injection and recombination must be equal (i.e., ηcoll ) 0).

′ ) υrec ) υinj

TsubI0′ηlhηinj d

(6)

Substitution of eqs 3 and 6 into eq 5 yields the result

ηcoll ) 1 -

I0′ I0

(7)

Determination of ηcoll therefore requires determination of pairs of intensities I0 (short circuit) and I0′ (open circuit) for which the volume recombination rate is the same. Here we introduce two assumptions. The first is that the rate of volume recombination is linearly proportional to the total number of free electrons in the film and independent of the distribution of electron concentration across the film. This will be true if the recombination rate is first order in electron density and changes in electrolyte concentrations between open circuit and short circuit conditions are small. The second assumption is that the total number of free electrons in the film at open and short circuit is approximately the same under conditions in which total concentration of trapped electrons is the same. The errors introduced by this approximation were checked for the first order case (and found to be small) by calculating the concentration profiles of free and trapped electrons for short circuit and open circuit conditions from the continuity equation, which in its timedependent form is given by33

∂nc ∂2nc ∂f ) ηinjRI0 exp(-Rx) - Nt,0 < > + D0 2 ∂t ∂t ∂x (nc - neq) τ0

(8)

It is important to note that eq 8 assumes that electron transfer to I3- is first order in electron concentration, so that the

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Figure 1. Electron collection efficiency ηcoll calculated from the continuity equation (eq 8) as a function of the electron diffusion length L0 ) (D0τ0)1/2. Substrate side illumination, R ) 500 cm-1, d ) 13 µm. The value of R was obtained from the molar absorption coefficient and loading of N719 dye.

conclusion that the errors are small is only valid for this case. The first term on the right-hand side of the equation describes the local rate of injection of electrons into the oxide (R is the absorption coefficient and I0 is the incident photon flux corrected for reflection losses). The second term accounts for trapping and detrapping. Nt,0 is the total trap density and f is the probability of trap occupation The angular brackets in the trapping/detrapping term indicate an average over the trap-state energies (ET) weighted by the trap density, g(Et). The third term describes the time-dependent diffusion of electrons (D0 is the diffusion coefficient of conduction band electrons). The final term allows for the loss of electrons by back reaction with triiodide. The electron lifetime, τ0, corresponds to the inverse of the pseudo-first-order rate constant for the back reaction process, kbr[I3-]. Under conditions of steady illumination, ∂f/∂t and ∂nc/ ∂t are both zero, so that an analytical solution can be obtained. Since the IPCE ratio method34,35 is based on this analytical solution, it also assumes that recombination is first order in electron concentration, although this is rarely stated explicitly. In the present study, near-IR absorption measurements were used to identify intensities I0 and I0′ for which the concentration of trapped electrons was identical under short circuit and open circuit conditions, respectively. In a related study,36 we have determined the I0 and I0′ values using an interrupt technique developed by Boschloo and Hagfeldt37 to measure the mean quasi-Fermi level under illumination at short circuit. The nearIR measurement appears to be experimentally simpler and more reliable. The electron collection efficiency (eq 7) for substrate side illumination was calculated as a function of electron diffusion length by solving eq 8 with ∂nc/∂t and 〈∂f∂t〉 ) 0 and ηinj ) 1 to find jphoto, noting that the collection efficiency is then the ratio between the electron flux and the absorbed photon flux.

ηcoll )

jphoto qI0(1 - e-Rd)

(9)

Figure 1 illustrates the results of this calculation for d ) 13 µm and R ) 500 cm-1, a typical value determined for N719dyed TiO2 films at the red LED wavelength of 627 nm. The Electron Quasi-Fermi Level and Occupation of Electron Trap States. In the case of first order recombination, the spatially dependent concentration of free electrons and hence the corresponding quasi-Fermi level nEF(x) can be calculated

Figure 2. nT profiles calculated from the steady-state continuity equation for open and short circuit conditions. Nt,0 ) 1019 cm-3, T0 ) 1000 K, τ0 ) 10-5 s, R ) 500 cm-1, d ) 13 µm, L0 ) 20 µm.

by solving the steady-state continuity equation with appropriate boundary conditions. For a nondegenerate semiconductor, the quasi-Fermi level is related to the local conduction band electron density nc(x) by the Boltzmann expression

Ec - nEF(x) ) -kBT ln

nc(x) Nc

(10)

where Nc is the density of conduction band states. The concentration profile of trapped electrons nT(x) can then be obtained for any trap distribution by integration between EF,redox and nEF (x), and then integration of nT(x) over the film thickness gives the total number of trapped electrons that will be detected by near-IR absorbance. Figure 2 shows the results of a calculation of the trapped electron density profiles, nT(x), under open circuit and short circuit conditions for the case where the exponential trap distribution is described by eq 11 with the characteristic temperature T0 ) 1000 K.

g[Et] )

Nt,0 exp kBT0

[

Ec - Et kBT0

]

(11)

The preceding analysis is the basis for the steady-state method used to determine ηcoll and L0. The trapped electron density was measured by near-IR absorbance as a function of illumination intensity at open and short circuit. The pairs of intensities for which recombination rate is identical under open and short circuit conditions (cf. eq 7) were then identified by choosing intensities for which the trapped electron densities are the same at open and short circuit. Determination of the Electron Diffusion Length by Periodic Methods. The Quasistatic Approximation. As a result of trapping/detrapping, small amplitude transient and periodic methods generally measure effectiVe values of electron lifetime (τn) and diffusion coefficient (Dn) that vary with illumination intensity.4,11,18,19,27,38-41 Bisquert and co-workers21,42 have shown for the case of noninteracting trap states that it should be possible to derive the electron diffusion length from Dn and τn provided that the values are measured under conditions where the electron quasi-Fermi level nEF (and hence the trap occupancy) is the same.3,24 If this condition is satisfied, the approach of Bisquert et al. predicts that Ln ) (Dnτn)1/2 ) (D0τ0)1/2 ) L0 (here Ln is used to indicate electron diffusion length values obtained by non-steady-state techniques). However, it is important to realize that the equivalence Ln ) L0 is based on several assumptions

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including the reaction order with respect to electrons and the absence of electron transfer via surface states. It is therefore useful to examine the extent to which the electron diffusion lengths derived from steady-state and non-steady-state methods agree. Solutions of eq 8 for non-steady-state conditions must be obtained numerically.17 Usually, however, the trapping term is omitted, and the resulting simplified form of the continuity equation is solved using intensity-dependent effectiVe values of the electron diffusion coefficient and lifetime, τn and Dn, respectively.19,38 Experimentally, it has been observed that Dn increases with intensity, whereas τn decreases with intensity.19 The intensity dependence of Dn and τn is attributed to relaxation of the density of trapped electrons that are exchanged with the mobile electrons in the conduction band. The time constants associated with this relaxation have been analyzed by Bisquert and Vikhrenko for the case where electrons are trapped at static sites and interactions between trap states are neglected.21 According to this analysis, Dn and τn are predicted to depend on the relative rates at which the densities of free and trapped electron density (nc and nt) change as the quasi-Fermi level, nEF, is perturbed by a small amount. Provided that band edge shifts due to charging can be neglected, the quasistatic approach yields the following relationships (see ref 42 for more general expressions that include surface charging effects).

(

∂nt Dn ) 1 + ∂nc

(

τn ) 1 +

)

Figure 3. Equivalent circuit used to fit the impedance.

Impedance Spectroscopy. Impedance spectroscopy has been widely used to characterize DSCs,45,51-56 and the theoretical background has been reviewed Bisquert and Fabregat-Santiago.44,45 The generalized equivalent circuit used in the present study to fit the impedance of DSCs under illumination at open circuit is shown in Figure 3. The transmission line element Ztl represents electron transport and back reaction in the TiO2 film. The Warburg impedance ZW, together with the charge transfer resistance Rcath and capacitance Ccath describe diffusion of ions and electron transfer/charging at the cathode. The series resistance Rser arises primarily from the FTO-coated glass. Ztl consists of a distributed series connection of resistance elements representing electron transport (rtrans) and parallel distributed impedance elements (rct in parallel with the chemical capacitance cµ) representing the back reaction. The following relationships were used to derive values of the electron lifetime, electron diffusion coefficient, and electron diffusion length.

τn ) RctCµ

-1

D0

)

∂nt τ ∂nc 0

(15)

(12) Dn )

(13)

d2 RtransCµ

( )

Ln Rct ) d Rtrans

where

(16)

1/2

(17)

Experimental Section

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

(14)

g(nEF), is the density of states function of the electron trap states. Nt,0, the total trap density (cf. eq 11) is typically 1019 1020 cm-3,9 and T0 is generally found to be considerably higher than ambient temperature (T0 ) 600-1500 K9,43). Since T0 > T, the trapped electron density increases more slowly than the conduction band electron density as the quasi-Fermi level is increased, giving rise to an increase in Dn with intensity and a corresponding decrease of τn. It follows from eqs 12 and 13 and the assumptions outlined previously that the product (Dnτn)1/2 ) Ln should be equal to the electron diffusion length (D0τ0)1/2 ) L0 provided that Dn and τn are determined under conditions of identical trap occupancy, (i.e., the same quasi-Fermi level). Ideally, therefore, Dn and τn should be determined simultaneously at the same trap occupancy: suitable methods include impedance measurements44-46 and photovoltage risetime measurements.27,47 Alternatively, the variations of Dn and τn with trap occupancy can be measured separately over a wide range of illumination intensities in order to allow the value of the diffusion length to be obtained after correction for the differing trap occupancies: this approach has been used for measurements employing intensity-modulated photovoltage and photocurrent spectroscopy (IMVS and IMPS).20,38,41,48-50

Cleaned fluorine doped tin oxide (FTO) conducting substrates (TEC 15, Libbey Owens Ford) were coated with a compact layer of TiO2 film by spray pyrolysis followed by a layer of colloidal TiO2 (Dyesol DSL-18-NR) by doctor-blading. The films were sintered at 500 °C for 30 min to form transparent mesoprous layers with a thickness of 13 µm. The TiO2 films were immersed in 40 mM aqueous solution of TiCl4 at 70 °C for 30 min, washed with Milli-Q water, dried with nitrogen, and then resintered at 450 °C for 30 min. The films were then immersed for 16 h in the N719 dye solution (2.5 × 10-4 M cis-dithiocyanato bis(2,2′bipyridine-4,4′-dicarboxylate) ruthenium(II) bis-tetrabutylammonium in 1:1, v/v acetonitrile/tert-butanol). The dye-coated TiO2 films were washed thoroughly with HPLC grade acetonitrile to remove excess dye before being assembled into cells. Thermally platinized cathodes were sealed onto the anodes using hot melt spacers (Surlyn, 25 µm, Solaronix). Electrolyte was introduced into the interelectrode gap through two holes drilled in the cathode electrode. The holes were then sealed with a microscope slip using Surlyn. The dye loading (determined by spectrophotometry following dye desorption) was used to calculate the absorption coefficient R at the wavelengths used in the characterization studies. Three different kinds of electrolyte were used in the cells. Cell A was filled with 0.06 M I2, 0.6 M 1-propyl-3methylimidazolium iodide (PMII), 0.5 M LiI, 0.5 M tertbutylpyridine (TBP), 0.1 M guanidinium thiocyanate (GuSCN) in methoxypropionitrile.

Dye-Sensitized Solar Cells Cell B was filled with 0.03 M I2, 0.6 M PMII, 0.5 M TBP, and 0.1 M GuSCN in acetonitrile/veleronitrile (85:15, v/v). Cell C contained a mixture of ionic liquids with the composition 0.05 M I2, 0.45 M N-methylbenzimidazole (NMBI), 0.1 M GuSCN in 1-propyl-3-methylimidazolium iodide (PMII) + 1-ethyl-3-methylimidazolium thiocyanate (EMISCN) (3:2, v/v). The transport and back reaction of electrons in the three cells were characterized by intensity-modulated photocurrent spectroscopy (IMPS) under a short-circuit condition and intensitymodulated photovoltage spectroscopy (IMVS) under an opencircuit condition, respectively. A red light emitting diode (LED, 627 nm) was used as the modulated light source to ensure a homogeneous distribution of the illumination in the TiO2 film (R ) 500 cm-1 at 627 nm). A Solartron 1260 frequency response analyzer was use in combination with a Solartron 1287 electrochemical interface for impedance characterization of electron transport and back reaction in the three DSCs at open circuit as a function of illumination intensity (627 nm LED). Impedance data were fitted using ZView 3.0a (Scribner Associates; calc-modulus weighting). Care was taken to ensure that the values obtained for the Warburg resistance and Warburg time constant of the electrolyte were consistent with the values calculated from diffusion coefficients of I3- in the three electrolytes and the spacer size. Similarly, the impedance of the cathode was checked to ensure that it was consistent with values obtained using a symmetrical configuration with two identical platinized electrodes. In cases where a free fit generated implausible values of either the Warburg or cathode impedance, the values were fixed to avoid errors. The concentrations of trapped electrons in the cells under short-circuit and open-circuit conditions were measured by nearIR transmittance spectroscopy. The cells were illuminated with a 627 nm LED, and changes in near-IR absorbance were measured using a Kodenshi GaAlAs IR LED (OPE5594, 940 nm, modulated at 10 kHz) and a silicon photodiode detector (Sharp, PD481PI) connected to a current amplifier. A GaAs filter was used to eliminate stray light from the 627 nm LED. The modulated signal from the photodiode amplifier was detected by a lock-in amplifier (SR850, Stanford Research System), and the output signal from the lock-in amplifier was recorded using an auxiliary input of an Autolab PG 12 potentiostat. The absorption cross-section of electrons at 940 nm was determined in previous work32,57 to be 5.4 × 10-18 cm2, which corresponds to a molar absorption coefficient of 3.25 × 103 dm3 mol-1 cm-1. Results and Discussion Derivation of ηcoll and L0 by Analysis of the Steady-State IR Data. Figure 4 illustrates the intensity dependence of the trapped electron density at open and short circuit determined by near-IR absorbance. The fact that the slopes of the two linear regression lines differ by less than 5% over three orders of illumination intensity indicates that the intensity dependence of the quasi-Fermi level is essentially the same at open circuit and at short circuit. This parallel behavior would not be expected if the recombination reaction was not first order in free electron density, since the local electron density should scale linearly with intensity at short circuit (provided that the majority of injected electrons are collected, which is the case here), whereas the electron density will scale nonlinearly with intensity if recombination is not first order in electron density, as is the case when recombination occurs via surface states.14,15,46,58 The same conclusion follows from the analysis based on the I0′/I0

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Figure 4. Trapped electron densities at open and short circuit as a function of incident photon flux for cell A. The lines are linear regressions.

TABLE 1: Electron Diffusion Length Values Derived by Different Methods for Cells A, B, and C cell

ηcoll

A B C

0.990 0.948 0.897

L0 from I0′/ I0 ratio/µm Ln IMPS/IMVS /µm Ln impedance /µm 73 30 20

142-148 53-55 27-32

160-250 65-102 30-34

ratio (cf. eq 7): this too is constant within experimental error over the whole range of intensity. This result is significant because it suggests that the nonideal behavior of DSCs that is seen in the intensity dependence of the photovoltage and the voltage dependence of the recombination resistance in impedance measurements16 may not be due to a higher reaction order of the recombination reaction with respect to electron concentration. Instead the nonideality may reflect nonideal behavior of the free electron gas in a system with strong electron-ion interactions on the nanoscale. The values of I0 and I0′ required to obtain ηcoll using eq 7 were derived from the plots in Figure 4. ηcoll was found to be 0.990 for this cell. The corresponding value of L0 was then obtained using the relationship between ηcoll and L0 shown in Figure 1. Similar parallel plots were obtained for the other cells (difference in regression slopes less than 4%), and the collection efficiencies and corresponding electron diffusion lengths obtained are summarized in Table 1. The diffusion length values were: cell A, 73 µm; cell B, 30 µm; cell C, 20 µm. Measurement of Ln by Periodic Methods. IMPS/IMVS Measurements. IMPS and IMVS are well-established methods for the determination of Dn and τn, respectively. However, derivation of Ln requires values of Dn and τn to be taken under conditions of identical trap occupancy. The potential problem here is that IMPS measurements are performed under short circuit conditions, where nT varies with distance. Consequently, the value of nT derived from the near-IR measurements at short circuit corresponds to an aVerage over the film thickness. However, numerical solutions of the continuity equation including trapping/detrapping show that the Dn value derived from the IMPS response should be close to the value derived by using the average quasi-Fermi level.59 Figure 5 shows values of Dn and τn plotted as a function of trapped electron density for cell B. The data in the region of overlap between the two plots were used to obtain the value of Ln, which is around 50 µm. It can be seen that Ln is only weakly dependent on trap occupation (and hence on the position of the quasi-Fermi level). Similar plots were obtained for the other two cells, and the values of Ln for all three cells determined using IMPS and IMVS in the

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Figure 5. Dn (closed circles) and τn (open circles) plotted as a function of the trapped electron density for cell B. The calculated electron diffusion length, Ln, is shown as a continuous line in the region of overlap between the Dn and τn plots.

Wang and Peter

Figure 7. Plots showing variation of lifetime τn as a function of trapped electron density. Note that at constant nT, τn decreases in the order cell A > cell B > cell C. However, the comparison is not absolute since the electron trap distributions for the three cells are different (cf. Figure 6).

Figure 8. Complex plane impedance plot and fit (line) for cell C under illumination at open circuit (Uphoto ) Voc ) 0.59 V).

Figure 6. Plots showing variation of Dn as a function of trapped electron density. Note that the plots for cells B and C almost coincide, indicating that the trap distributions are similar, whereas the plot for cell A (lithium electrolyte) is displaced toward higher nT values, indicating a different trap distribution.

overlap region of the plots were as follows: cell A, 142-148 µm; cell B, 53-55 µm; cell C, 27-32 µm. The electron diffusion coefficients and electron lifetimes for all three cells are contrasted by plotting them as a function of trapped electron density as shown in Figures 6 and 7. This approach is commonly used to deconvolute real changes in recombination kinetics from effects due to shifts in the conduction band energy relative to EF,redox brought about, for example, by additives60,61 or surface treatments such as TiCl4.62 While this method may be appropriate if the electrolyte composition is varied only to a minor extent, the present results show that major changes in electrolyte composition alter the density and distribution of electron trap states. This is evident from Figure 6, which shows that the plot of Dn vs nT for the lithium-based electrolyte cell A is quite different from the plots for cells B and C. This is consistent with the observation that the density of trapped electrons in cell A is nearly 1 order of magnitude higher than in cells B and C at any given open circuit voltage (see Figure S1 in the Supporting Information). Kopidakis et al.17 have shown that lithium intercalates irreversibly into the mesoporous TiO2 film under solar illumination at open circuit. In the present work, lower intensities were used, and since no hysteresis was observed in the near-IR absorbance during voltage scans,32 it seems unlikely that the higher trap density observed in our measurements is due to irreversible intercalation of lithium. An alternative explanation is that the electron traps may not be due to defect states in the TiO2 but arise instead

from electrostatic interactions with the cations in the electrolyte. If this is the case, the results could indicate that the small lithium cation is particularly effective at causing trapping. Open Circuit Impedance Measurements. In the present work, electron diffusion lengths were determined by impedance measurements carried out at open circuit under illumination at different intensities. Impedance measurements made under illumination at open circuit have several advantages over measurements in the dark under forward bias.44,55 The first is that the quasi-Fermi level is almost flat. The second is that transport and back reaction are characterized for the same quasiFermi level. The third is that the Warburg and cathode impedances should be constant. The disadvantage of the impedance method is that it becomes difficult to measure the transport resistance at high light intensities, particularly in the case of cells with long diffusion lengths, where Rtrans becomes much smaller than the series and cathode impedances at open circuit voltages and fitting becomes unreliable. Figure 8 illustrates a typical impedance response for low light intensities, where the characteristic transmission line behavior can be seen at high frequencies (region 45° slope). The corresponding Bode plot is shown in Figure 9. In both cases, the fit to the equivalent circuit shown in Figure 3 is excellent. The fitting of the impedance response of cell C at higher light intensity is illustrated in the Supporting Information, which also presents the impedance plots for the other cells. Analysis of the impedance responses of cell C obtained using different illumination intensities gave the Rtrans and Rct data shown in Figure 10. It can be seen that Rct and Rtrans decrease in parallel as the open circuit voltage increases. The value of Ln derived from eq 17 is only weakly intensity-dependent, ranging from 30-34 µm. Analysis of the impedance response of cells A and B gave the following values of Ln: cell A, 160-250 µm; cell B, 65-103 µm. In both cases, Ln appears to increase with increasing open circuit voltage as shown in Figure

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Figure 12. Comparison of values of Dn (filled symbols) and τn (open symbols) derived from IMPS/IMVS and impedance for cell C. The Dn values obtained by IMPS have been plotted taking into account the fact that the quasi-Fermi level is 67 meV lower on average at short circuit compared to open circuit. Figure 9. Bode plot and fit (lines) for cell C under illumination at open circuit (Uphoto ) 0.59 V).

Figure 13. Rct, Rtrans, and Ln derived from impedance measurements at open circuit under illumination for cell A. Figure 10. Rct, Rtrans, and Ln derived from impedance measurements at open circuit under illumination for cell C.

Figure 11. Values of Ln derived from impedance measurements on all three cells. The limited data range for cell A arises from the fact that Rct becomes very small above 0.55 V (see Figure 13).

11, which compares the Ln values as a function of open circuit voltage for all three cells. The values of diffusion length can be compared with those reported recently by Wang et al. for different types of DSC.56 The high efficiency cell (volatile electrolyte, Z907 dye), η ) 10.1%) studied by Wang et al. gave Ln values of around 100 µm, which is similar to the values for our cells A and B. Their least efficient liquid cell (ionic liquid, η ) 7.1%) gave Ln values around 20-30 µm. However, in contrast to the results shown in Figure 11, Wang et al. observed that Ln decreased with voltage by about a factor of 2 over 300 mV.

In order to compare the Dn and τn values derived by impedance spectroscopy and by IMPS/IMVS, it is necessary to take into account the shift in quasi-Fermi level between open and short circuit in the case of the latter method. The plots of nT vs light photon flux (cf. Figure 4) were converted to plots of nT vs open circuit voltage using the measured intensity dependence of the open circuit (plots of open circuit voltage vs log10 I0 for all three cells are given in the Supporting Information). The displacement of the short circuit nT plot along the voltage axis relative to the open circuit nT plot corresponds to the difference in (mean) quasi-Fermi level between open circuit and short circuit. In the case of cell C, the shift in quasi-Fermi level was found to be 67 meV. The values of Dn and τn have been plotted in Figure 12 as a function of the difference between the quasi-Fermi level and the redox Fermi level, EF,redox. The plots show that shifting the IMPS data by 67 meV relative to the open circuit voltage brings the plots of Dn into acceptable coincidence. The good agreement between the sets of data for cell C appears to validate the methodology adopted to treat the IMPS/ IMVS data. However, the agreement is less satisfactory for cells A and B, and this is evident from the higher values obtained for Ln by impedance (see Table 1 below for a summary of data). The problems encountered with cells that have high Ln values are illustrated in Figure 13 for cell A. It can be seen that Rtrans becomes very small (less than 1 Ω cm2) for open circuit voltages above 0.55 V, and the apparent flattening off of the plot probably reflects the fact that the fitting has become unreliable because Rtrans is comparable with Rcath and 1 order of magnitude smaller than Rser (cf. Figure 3). Ln was also found to increase with open

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circuit voltage in the case of cell B (see Figure S9 in the Supporting Information). Comparison of Different Methods for Determining the Electron Diffusion Length. Table 1 compares the values of the electron diffusion length obtained by the different methods. It can be seen that the agreement is best for cell C, which has the shortest electron diffusion length. In the case of cell A, by contrast, the periodic methods give electron diffusion length values that are considerably greater than the value estimated from the steady-state measurement. The values obtained for cell A from the impedance measurements appear to show a substantial increase with open circuit voltage, but since only the three lowest voltage points are fitted, the errors are large. The steady-state and periodic methods give values of the electron diffusion coefficient that show the same trends for the three cells. However, as the diffusion length becomes longer, the difference between the results obtained by the different methods become more marked. The Ln values obtained by impedance measurements at low light intensities agree reasonably well with the values obtained by the IMPS/IMVS analysis, but in all cases the steady-state method gives values that are consistently lower. A similar, although larger, discrepancy between values of electron diffusion length derived from the ratio of IPCE values for illumination from the electrolyte and substrate sides has been noted by other workers.25,26,28,30 It is worth noting that if the L0 values derived in the present work from the steady-state analysis are correct, the IPCE ratio method would not be reliable since the errors become large when L0 exceeds twice the film thickness. This highlights the fact that the cells studied are not ideal for comparison of steady-state and periodic methods. For this reason, we plan experiments using outer sphere redox systems to replace the I3-/I- couple in order to reduce the electron lifetime so that the condition L < d is met. The origins of the differences between the L0 and Ln values obtained by steady-state and periodic methods, respectively, probably lie in the assumptions made in the analysis. The steadystate method used here (and the IPCE ratio method) relies on the assumption that recombination is first order in electron density so that the continuity equation can be used with a constant τ0 term. This assumption will not be valid if recombination occurs via surface states, giving rise to a superlinear reaction order and hence a voltage-dependent τ0 value as discussed recently by Bisquert et al.16 The experimental evidence presented here suggests that at least in the case of the three cells studied, the reaction order with respect to electron density is close to unity. It is also assumed that the conduction band position is fixed, i.e. that surface charging is negligible. Band edge unpinning might be expected at high light intensities, but it seems unlikely that it can influence the results over a wide intensity range. The periodic methods give a small amplitude lifetime, so a nonlinear dependence of the recombination rate on free electron density will appear as an intensity dependence of Ln. In a recent paper, Bisquert et al. have taken the approach of assuming that D0 is constant, so that the measured diffusion lengths are used to derive voltage-dependent τ0 values.16 If this approach is used in the case of our data, the increase in Ln with voltage could be interpreted as evidence for electron transfer via surface states, in which case the first-order approximations made in the case of steady-state methods no longer hold. However, the variation in diffusion length over the entire voltage (and hence electron density) range is small (less than a factor 2), so the use of the first-order approximation is probably still justified. Finally the equivalence between Ln and L0 obtained from the quasi-static approximation relies on a number of

Wang and Peter assumptions that need to be examined. At this stage it seems difficult to reach a definitive conclusion about which of the electron diffusion length values are correct. Each experimental method and corresponding data analysis has its problems, and more work will be needed before the discrepancies revealed by this and other studies can be resolved satisfactorily. Conclusions The near-infrared absorption method is a convenient way of determining the trapped electron density at open circuit and short circuit. Plots of trapped electron density vs photon flux are parallel for open circuit and short circuit conditions over a wide intensity range for all the cells studied. This is interpreted as evidence that the recombination process is close to first order with respect to electron concentration. The near-IR absorbance measurements form the basis of a new steady-state method of determining the collection efficiency and hence the electron diffusion length. The study has revealed significant and reproducible differences between the electron diffusion length values obtained by steady-state and periodic perturbation techniques. The differences appear to become more acute as the electron diffusion length increases. The results obtained by IMPS/IMVS agree reasonably well with those obtained from impedance measurements at open circuit, so the discrepancy is evidently not connected with the fact that the quasi-Fermi level varies with distance under the short circuit conditions used for IMPS measurements. In fact, the Dn values obtained by IMPS and impedance are similar. The origin of the different electron diffusion length values is not clear at present, but it is likely to be related to the assumptions made in analyzing the data obtained by the different methods. Acknowledgment. This work was supported by the UK Engineering and Science Research Council (EPSRC). The authors acknowledge useful discussions with Alison Walker, Eric Maluta, Julio Villanueva, Juan Bisquert, Francisco Fabregat, and Janne Halme. Supporting Information Available: nT vs Uphoto plots for all three cells, discussion of the nonideality factor m, plots of nT vs I0 for cells B and C, plots showing derivation of Ln from IMPS/IMVS data for cells A and C, and fitting of impedance of cell C at high light intensity. This information 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) Kroon, J. M.; Bakker, N. J.; Smit, H. J. P.; Liska, P.; Thampi, K. R.; Wang, P.; Zakeeruddin, S. M.; Gra¨tzel, M.; Hinsch, A.; Hore, S.; Wu¨rfel, U.; Sastrawan, R.; Durrant, J. R.; Palomares, E.; Pettersson, H.; Gruszecki, T.; Walter, J.; Skupien, K.; Tulloch, G. E. Prog. PhotoVoltaics 2007, 15, 1. (3) Peter, L. M. J. Phys. Chem. C 2007, 111, 6601. (4) Cao, F.; Oskam, G.; Meyer, G. J.; Searson, P. C. J. Phys. Chem. 1996, 100, 17021. (5) Goossens, A.; Boschloo, G. K.; Schoonman, J. Mater. Res. Soc. Symp. Proc. 1997, 452, 607. (6) Willig, F.; Schwarzburg, K. J. Phys. Chem. B 1999, 103, 28. (7) Boschloo, G. K.; Goossens, A. J. Phys. Chem. 1996, 100, 19489. (8) Nelson, J. Phys. ReV. B 1999, 59, 15374. (9) Bailes, M.; Cameron, P. J.; Lobato, K.; Peter, L. M. J. Phys. Chem. B 2005, 109, 15429. (10) Zaban, A.; Meier, A.; Gregg, B. A. J. Phys. Chem. B 1997, 101, 7985. (11) Kopidakis, N.; Schiff, E. A.; Park, N. G.; van de Lagemaat, J.; Frank, A. J. Phys. Chem. 2000, 104, 3930. (12) Bisquert, J.; Zaban, A. Appl. Phys. A: Mater. Sci. Process. 2003, 77, 507. (13) Nelson, J.; Chandler, R. E. Coord. Chem. ReV. 2004, 248, 1181.

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