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Electron Diffusion and Back Reaction in Dye-Sensitized Solar Cells: The Effect of Nonlinear Recombination Kinetics Julio Villanueva-Cab,† Hongxia Wang,‡ Gerko Oskam,† and Laurence M. Peter*,‡ †
Departamento de Física Aplicada, CINVESTAV-IPN, M erida, Yuc 97310, M exico, and Department of Chemistry, University of Bath, BA2 7AY, United Kingdom
‡
ABSTRACT The electron collection efficiency in dye-sensitized solar cells (DSCs) is usually related to the electron diffusion length, L = (Dτ)1/2, where D is the diffusion coefficient of mobile electrons and τ is their lifetime, which is determined by electron transfer to the redox electrolyte. Analysis of incident photon-to-current efficiency (IPCE) spectra for front and rear illumination consistently gives smaller values of L than those derived from small amplitude methods. We show that the IPCE analysis is incorrect if recombination is not first-order in free electron concentration, and we demonstrate that the intensity dependence of the apparent L derived by first-order analysis of IPCE measurements and the voltage dependence of L derived from perturbation experiments can be fitted using the same reaction order, γ ≈ 0.8. The new analysis presented in this letter resolves the controversy over why L values derived from small amplitude methods are larger than those obtained from IPCE data. SECTION Energy Conversion and Storage
method is invalid when γ ¼ 6 1. An indication of problems with the IPCE ratio method is the fact that the IPCE ratio (and hence the electron diffusion length) has been found experimentally to increase with light intensity.5 IPCE measurements in our laboratory confirm this. Solution of eq 1 with γ = 1 predicts that the diffusion length should be independent of intensity. However, the observed fall in IPCE ratio with decreasing intensity observed experimentally is consistent with γ < 1, as shown by Figure 1, which illustrates the intensity dependence of the IPCE ratio spectra predicted for γ = 0.8 (see Supporting Information for corresponding experimental IPCE spectra). If these spectral ratios are analyzed assuming γ = 1, an apparent diffusion length Lapp is obtained that increases with intensity. This increase in Lapp with intensity was seen experimentally in the present work, which confirmed the observations of Barnes et al.5 However, the analysis by Bisquert and Mora-Ser o makes it clear that the observed intensity dependence of Lapp contradicts the assumption made in the IPCE ratio analysis that the rate of recombination is first-order in electron concentration. Furthermore, the simple concept of diffusion length is no longer applicable in circumstances in which the recombination rate varies nonlinearly with the local electron concentration. It is worth remarking here that, if the IPCE is intensity dependent (i.e., γ ¼ 6 1), the usual linear correction for the
n a recent letter in this journal, Bisquert and Mora-Ser o1 showed that modeling of dye-sensitized cells (DSCs) should take into account the effect of nonlinear recombination kinetics involving transfer of electrons in the TiO2 to I3- ions in the electrolyte. The continuity equation, which has been widely used to treat both steady-state and time-dependent behavior of DSCs, needs to be written in the form Dnc D2 nc Df ð1Þ - kr ðnc - n0 Þγ ¼ GðxÞ þ D0 2 - Nt, 0 Dt Dt Dx
I
Here, nc is the concentration of free electrons, n0 is the equilibrium concentration of free electrons in the dark, G(x) is the position-dependent electron injection rate, D0 is the diffusion coefficient of free electrons, kr is the rate constant for the transfer of free electrons from the TiO2 to I3-, γ is a reaction order with respect to free electron concentration, and the third term on the right-hand side represents the trapping and detrapping of electrons from states in the band gap.2 Under steady-state conditions, the trapping/detrapping term is equal to zero, and, for γ 6¼ 1, eq 1 can be solved numerically for appropriate boundary conditions (see Supporting Information). Analytical solutions of eq 1 for short circuit conditions with γ = 1 were used by S€ odergren et al.3 to derive an expression for the ratio of incident photon-to-current efficiency (IPCE) values for illumination from the electrolyte (EE) and substrate (SE) sides. The EE/SE IPCE ratio method has been used by Halme et al.4 and by Barnes et al.5 to derive values of the electron diffusion length for DSCs. However Bisquert and Mora-Ser o have pointed out that the IPCE ratio
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Received Date: January 10, 2010 Accepted Date: January 26, 2010 Published on Web Date: January 29, 2010
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Figure 1. EE/SE IPCE ratio spectra calculated for γ = 0.8 and the intensities shown. See Supporting Information for parameter values and details of the calculation.
Figure 3. Points: Experimental EE/SE IPCE ratio measured at 627 nm for a DSC with a 13 μm thick TiO2 film. See Supporting Information for further details. Red line: Fit to nonlinear recombination kinetics with γ=0.73, kr =9 106 cm-0.81 s-1. Blue line: Fit to γ = 0.80, kr = 3 106 cm-0.6 s-1.
Figure 2. Normalized plots of the EE/SE IPCE ratio for λ=627 nm calculated as a function of intensity for different values of the reaction order γ as shown. Note that the IPCE ratio is independent of intensity only in the case where γ = 1.
Figure 4. Open and filled circles: Apparent intensity dependence of the electron diffusion length derived from the experimental data in Figure 3. Red line: Fit to nonlinear recombination kinetics obtained using γ = 0.73, kr = 9 106 cm-0.81 s-1. Blue line: Fit to γ = 0.80, kr = 3 106 cm-0.6 s-1.
incident light intensity as a function of wavelength is no longer possible. In practice, IPCE spectra need to be measured under conditions where the intensity of the bias light is higher than that of the variable wavelength light from the monochromator. Calculations based on eq 1 show that the IPCE ratio spectra tend toward a limiting form at high intensities (see the plot for 10 mW cm-2). Under these conditions, the collection efficiency tends toward 100%, and the IPCE ratio is determined only by light absorption losses associated with the platinized cathode and the I3- electrolyte. In the present study, the IPCE ratio was measured carefully as a function of intensity at a constant wavelength of 627 nm in order to avoid accuracy problems associated with optical corrections for light absorption by I3-. Figure 2 shows how the EE/SE IPCE ratio at 627 nm is predicted to increase with intensity for different γ values. For convenience, the theoretical plots have been normalized (see Supporting Information). It can be seen that the intensity dependence of the IPCE ratio becomes more pronounced as γ becomes smaller. The IPCE ratio of a DSC with a 13 μm TiO2 layer sensitized with N719 was measured as a function of intensity at 627 nm, and Figure 3 compares the experimental data as a function of intensity with plots calculated from eq 1 for γ = 0.73 and γ = 0.80, respectively. In both cases, kr was varied to give the best fit. The lower intensities were obtained using light output from the monochromator attenuated by neutral density filters. The higher intensities were obtained using a 627 nm light-emitting diode (LED) to provide additional illumination. The good fit to
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the data indicates that γ lies in the range 0.73-0.80. The small deviations at higher intensities could be due to errors in the transmission values used for the platinized cathode, reduction of the electron lifetime due to an increase in the concentration of tri-iodide, or a transition to recombination predominantly via the conduction band (with γ = 1) rather than via surface states. The experimental EE/SE IPCE ratio values shown in Figure 3 were also analyzed in the conventional way (i.e., with γ=1), and Figure 4 illustrates the intensity dependence of the apparent electron diffusion length values (Lapp) obtained. The figure also shows the fits to the Lapp data obtained for γ = 0.8 and γ =0.73, respectively. It is important to note at this point that the “diffusion lengths”, Lapp, obtained from IPCE data by assuming that γ = 1, have no simple physical interpretation. Full modeling requires incorporation of the nonlinear recombination term. However, the EE and SE IPCE spectra are still valuable individually, and a useful practical approach would be to measure the IPCE spectra for a bias illumination corresponding to 1 sun. In this case, the IPCE spectrum can be used to predict the short circuit current density for SE or EE illumination. However, if IPCE measurements made with lower bias illumination are used, the short circuit current is likely to be underestimated.
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Figure 5. Normalized plots for different γ values showing how the small amplitude electron diffusion length is predicted to increase with open circuit voltage. Note that the diffusion length is constant for the case γ = 1.
Figure 6. Points: Experimental small amplitude diffusion length as a function of open circuit voltage. Line calculated from eq 2 with γ=0.80, kr =3 106 cm-0.6 s-1, Nc =1021 cm-3, D0 =0.4 cm2 s-1, Ec - EF,redox = 1.0 eV, T = 293 K.
Bisquert and Mora-Ser o have also shown that a physically meaningful small amplitude electron diffusion length λn can be defined under conditions in which linear expansion of the recombination term is permissible and the 'background' density of electrons is effectively constant. This applies to a range of small amplitude methods involving perturbation of voltage or light intensity at open circuit. The general expression for λn is !1=2 D0 n1b - γ λn ¼ ð2Þ γkr
band energy of the TiO2, or a transition to recombination predominantly via the conduction band (i.e., γ = 1). An important conclusion at this point is that the plots of apparent diffusion length Lapp (obtained from the EE/SE ratio) and the plots of λn (obtained by IMVS/IMPS) can be fitted with the same values of reaction order and rate constant. It can be seen that the λn values lie in the range 40-110 μm, whereas over the same intensity range, Lapp varies from 7 to 27 μm. It is therefore clear that the apparent discrepancy between diffusion length values obtained by the IPCE and small amplitude methods can be explained by taking into account recombination with a reaction order smaller than 1, as suggested by Bisquert and Mora-Ser o. The preceding discussion highlights the importance of considering the reaction order when analyzing data obtained by both steady state and transient/periodic methods. However, the origin of the nonlinear recombination kinetics remains to be established. Possibilities include electron transfer via surface states7,8 and the effects of the transfer coefficient9 for the electron transfer process. In the case of small amplitude measurements, the results may also be affected by shifts in the conduction band energy under illumination.10 In practical terms, the reaction order will influence the fill factor and thus the DSC performance, so that it would be useful to discover the origin of the nonlinear reaction order in order to improve device performance.
Here, nb is the “background” free electron density, which is much larger than the perturbation of electron concentration caused by the external stimulus. Note that the units of kr depend on the value of γ. For the special case that γ=1, eq 2 reduces to the familiar expression for the electron diffusion length: 1=2 D0 ¼ ðD0 τ0 Þ1=2 ð3Þ λn ¼ L ¼ kr Equation 2 shows that λn will increase with nb (and hence with intensity or open circuit voltage, Uphoto) for γ < 1. Since nb is related to Uphoto by the Boltzmann expression qUphoto nb ¼ neq exp ð4Þ kB T eq 2 can be used to construct plots of λn versus voltage for comparison with experimental data. The normalized plots in Figure 5 show the dependence of λn on Uphoto for a range of γ values less than 1. Figure 6 shows that the small amplitude diffusion length obtained by the analysis of intensity modulated photocurrent and photovoltage spectroscopy (IMPS and IMVS) data following the method described elsewhere6 (see also Supporting Information) can be fitted with γ = 0.8 and kr = 3 106 cm-0.6 s-1 (i.e., the same values that were used to fit Lapp in Figure 3) for photovoltages up to 0.7 V. Similar values of λn (although over a smaller range of voltages) were obtained from impedance measurements under illumination at open circuit. It is evident from Figure 6 that the diffusion length falls slightly at higher open circuit voltages. This may be due to accumulation of I3- at high light intensities (which will increase the rate of recombination), shifts in the conduction
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SUPPORTING INFORMATION AVAILABLE Details of the numerical solution of eq 1. Experimental EE/SE IPCE spectra as a function of intensity. Details of theoretical calculation of EE/SE IPCE spectra. Details of normalization of calculated EE/SE ratios in Figure 2. Details of cell fabrication and IPCE measurements. Details of determination of λn by IMVS/IMPS. This material is available free of charge via the Internet at http://pubs.acs.org.
AUTHOR INFORMATION Corresponding Author: *To whom correspondence should be addressed. E-mail: l.m.peter@ bath.ac.uk.
ACKNOWLEDGMENT This work was supported by EPSRC (Grant No. EP/E035469/1), the Consejo Nacional de Ciencia y Tecnología
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(CONACYT, Mexico) under Grant No. 80002 and a Mexico Research Links Project funded by the British Council. J.V.C. thanks CONACYT and the Fondo Yucat an (Cinvestav-M erida) for a travel grant. The authors wish to thank Juan Bisquert for helpful discussions.
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