Electron Recombination Kinetics and the Analysis of Collection

Oct 15, 2010 - Piers R. F. Barnes* and Brian C. O'Regan. Department of Chemistry, Imperial College London, Exhibition Road, London SW7 2AZ. ReceiVed: ...
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J. Phys. Chem. C 2010, 114, 19134–19140

Electron Recombination Kinetics and the Analysis of Collection Efficiency and Diffusion Length Measurements in Dye Sensitized Solar Cells Piers R. F. Barnes* and Brian C. O’Regan Department of Chemistry, Imperial College London, Exhibition Road, London SW7 2AZ ReceiVed: July 8, 2010; ReVised Manuscript ReceiVed: September 2, 2010

A significant difference exists between two measured estimates of the electron diffusion length in dye sensitized solar cells. One method is based on spectral response at short circuit (L1) and the other on small perturbations (Ln). Following a recent theoretical treatment of the diffusion length for “non first order recombination”, one can propose an “effective diffusion length” at short circuit (λnj). Using simulation and measured values we show that L1 ≈ λnj, for a wide range of recombination orders, despite the first order assumption made when deriving L1. We also show Ln ≈ λnj, when Ln is derived from a time dependent full cell model incorporating non first order recombination. Thus the deviation of experimentally measured Ln from simulated Ln and L1 cannot be explained simply by the incorporation of nonlinear recombination. Our results give support to the spectral response method, but do not give a theoretical basis for rejecting the small perturbation method. Efficient collection of electrons is required for high performance dye sensitized solar cells (DSSCs). An electron diffusion length at least double the semiconductor film thickness is needed for efficient collection, since this allows electrons to reach the conducting substrate before recombination.1 Recent work has indicated that diffusion length estimates derived from the analysis of incident photon conversion efficiency (IPCE) measurements assuming first order conduction band electron recombination kinetics are smaller by a factor of ∼2-3 relative to those calculated by small perturbation measurements.2-4 A new letter by Bisquert and Mora-Sero´ suggested that sublinear recombination kinetics in dye sensitized solar cells (DSSCs) might be able to explain the discrepancy.5 Conduction band (free) electron recombination with an order less than 1 could also explain the variation in diffusion length with incident light intensity and electron Fermi level that has been observed.2,4 They showed that if electron recombination shows sublinear kinetics then diffusion length can only be rigorously defined (λn) for a specific electron concentration.5 This implied that the concept of diffusion length might not be meaningful for measurements at short circuit or the maximum power point where the concentration of free electrons (nc) varies significantly across the cell. Under these measurement conditions they suggested that the notion of diffusion length should be confirmed by full solution of the device model.5 In this article we present full numerical solutions of a device model incorporating sublinear recombination kinetics. These confirm that the diffusion length derived from analysis of IPCE (L1) is in good agreement with both the theoretical effective diffusion length (λnj) and simulated transient diffusion length (sLn) determined for the mean concentration of free electrons in the cell (njc). Furthermore we show that difference between IPCE diffusion lengths and those determined from small perturbation measurements (mLn) is not explained by the current model of sublinear recombination. Our observations suggest that the operational conditions under which a cell is measured influence the determination of diffusion length. The work herein is based on the approach and data detailed in a recent letter by * Corresponding author. E-mail: [email protected].

Villanueva-Cab et al.6 We note that there were numerical errors in their analysis of this data which led them to the opposite conclusions.7 We will discuss several theoretical, simulated and measured versions of the electron “diffusion length”, a measure of the average displacement of an electron before recombination. These terms are defined in Table 1. The relevant continuity equation describing the transport and recombination of conduction of electrons with recombination order b as a function of position (x) in a cell at steady state is

0 ) D0

∂2nc ∂x2

+ G(x) - krnbc

(1)

Here nc is the concentration of conducting electrons in excess of the background concentration n0, kr is the recombination rate constant, D0 is the conducting electron diffusion coefficient, n0 is the equilibrium concentration of conducting electrons in the dark, and G(x) is the position dependent electron generation rate in the model. The profile of G(x) is dependent on the absorption coefficient (R) of the dye on the TiO2, the absorption coefficient of the electrolyte in the pores (RI), the TiO2 film thickness (d), the incident photon flux including reflection losses (φ), the injection efficiency (ηinj, assumed to be constant9), the direction of illumination (SE or EE) and the transmission through the counter electrode and electrolyte to the edge of the EE interface. For substrate-electrode (SE) side illumination G(x) ) φRηinj exp[-(R + RI)x] and for electrolyte-electrode (EE) side illumination G(x) ) φRηinjTPt exp[-(R + RI)(d - x)]. When the cell is operating at short circuit the boundary conditions for eq 1 are n(x ) 0) ) n0 and dnc(x ) d)/dx ) 0. Solutions giving nc(x) satisfying eq 1 were calculated with the MATLAB numerical solver ode15s using the shooting method. The short circuit current density is given by the slope of the electron concentration at the substrate (x ) 0) allowing the IPCE for a

10.1021/jp106329a  2010 American Chemical Society Published on Web 10/15/2010

Electron Recombination Kinetics in DSSCs

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TABLE 1: Description of Diffusion Length Terms Used in This Study symbol

description

Vr ) krnbc nc ) λn )

λn¯ ) L1 )

s

Ln )

m

1 d

rate of free electron recombination in terms of the conducting electron concentration (nc), rate constant (kr), and order (b)

∫0d nc(x) dx

mean free electron concentration in the cell with film thickness d

( ) ( ) D0n1-b c bkr

electron diffusion length based on the theory of small perturbations,5 hereafter referred to as the “small perturbation diffusion length”. D0 is the free electron diffusion coefficient.

1/2

D0nc1-b bkr

1/2

effective small perturbation electron diffusion length for a cell operating where nc(x) is not constant

( ) D0 kr

1/2

electron diffusion length derived from first order analysis of IPCE (collection efficiency) ratio (kr ) 1/τ0 where b ) 1, c.f. eq 3)

(

)

2d τn(nc) -1 π τj(nc)

(

1/2

1/2

)

Ln ) Dn(nc) τn(nc)

simulated small perturbation diffusion length derived from simulated values of the transient photovoltage lifetime (τn) and the transient photocurrent lifetime (τj) at matched njc. See Appendix A and the Supporting Information of ref 8. measured small perturbation diffusion length derived from small perturbation measurements of Dn and τn, at matched njc using a similar procedure as for sLn3,4

particular wavelength and illumination direction to be calculated according to

IPCE )

-D0 dnc(x ) 0) φ dx

(2)

Dividing the short circuit IPCE by the light harvesting efficiency and the assumed injection efficiency gives the electron collection efficiency at short circuit.1 If b ) 1 then eq 1 can be solved analytically such that the ratio of EE/SE IPCE is a function of only L1, d, R, and RI where L1 ) (D0/kr)1/2:1,2,10,11 IPCEEE IPCESE

)

TPt[(L1(R + RI) + 1)e2d/L1 - 2L1(R + RI)ed(R+RI+(1/L1)) + L1(R + RI) - 1] (L1(R + RI) - 1)e

d(R+RI+(2/L1))

+ (L1(R + RI) + 1)ed(R+RI) 2L1(R + RI)ed/L1

(3) If eq 1 is solved for both EE and SE side illumination, with any value of b, to give IPCE values using eq 2, then an estimation of the diffusion length can be found by solving eq 3 for L1 since all the other terms have been defined. Equation 3 assumes b ) 1 which is not theoretically correct when b < 1 in eq 1. The value of L1 determined in this way can be compared with the (theoretically derived) small perturbation electron diffusion length, λn, defined by Bisquert and Mora-Sero´ (see Table 1).5 This varies with electron concentration due to the sublinear recombination kinetics (b < 1). At short circuit nc varies significantly with x across the TiO2 so that λn(x) represents the average distance an electron starting at point x would travel before recombination if the regions neighboring x had the same concentration of conducting

electrons. Note that, for the commercial development of DSSCs, the parameters of most interest are the collection efficiency and diffusion length when the cell is operating close to maximum power point where there is also a large electron concentration gradient. Both IPCE measurements and some forms of small perturbation measurements, such as photocurrent transients or intensity modulated photocurrent spectroscopy (IMPS)12 which are used to derive the effective electron diffusion coefficient (Dn), are typically made at short circuit. To relate λn to cell measurements where nc varies we define λnj as the theoretical small perturbation diffusion length calculated using the mean concentration of conducting electrons in the cell, njc (see Table 1). As will be shown below, λnj gives an excellent approximation to sLn, the simulated small perturbation diffusion length derived from simulated open circuit photovoltage (τn) and short circuit photocurrent (τj) transients (Table 1). The transient decays were found by solving the time dependent form of eq 1 as described in the appendix A. The relationship sLn ≈ λnj holds even when b < 1 and when the influence of electron trapping and detrapping is considered by assuming the quasi-static approximation13,14 (Appendix A, Figure A3). The simulation of sLn represents the procedure used to determine the small perturbation diffusion lengths from measurements, mLn ) (Dnτn)1/2, where short circuit τj measurements are used to find Dn at a given njc. Studies deriving diffusion lengths using transient photocurrent and IMPS measurements (including ref 6) assume implicitly that the cell behavior at short circuit can be characterized by njc. However, for the data presented,6 we find that mLn * sLn. Figure 1a shows electron concentration profiles calculated by solving eq 1 at the same light intensity for SE side and EE side illumination. A recombination order significantly less than unity (b ) 0.6) and a low light intensity were chosen for this example to emphasize the effects of sublinear recombination. This leads to collection efficiencies of 0.63 and 0.21 for SE and EE side illumination respectively. Figure 1b shows λn(x) as a function of position corresponding to the nc profiles in

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Figure 1. a. Example of electron concentration profiles for SE (blue) and EE (red) side illumination calculated for φ ) 1012 cm-2 s-1, d ) 13 µm, R ) 4000 cm-1, RI ) 0 cm-1, D0 ) 0.4 cm2 s-1, kr ) 4 × 109 cm-1.2 s-1, b ) 0.6, the total density of conducting states Nc ) 1020 cm-3, TPt ) 1, ηinj ) 1, T ) 298 K, and EC ) 1 eV vs electrolyte redox potential. The dotted lines show the calculated electron concentration profiles where b ) 1 and L1 ) 5.6 µm (with kr ) 1.3 × 106 s-1) for comparison. b. The small perturbation diffusion length λn as a function of position in the cell calculated for the b ) 0.6 electron concentration profiles. The inset table shows the magnitude of λnj calculated from the mean SE and EE electron concentrations and the value of L1 determined from the IPCE ratio data.

Figure 1a. Over most of the film thickness λn does not change rapidly. The inset table shows the magnitude of the effective small perturbation diffusion length for the cell, λnj, calculated using mean electron concentration in the cell (averaged over both illumination sides), njc. The value of λnj in Figure 1b is compared with L1 estimated using eq 3 with the ratio of the collection efficiencies determined from the solutions to eq 1. L1 is only 14% smaller than the characteristic small perturbation diffusion length based on the sublinear recombination kinetics. This shows that the first order IPCE analysis gives a good estimate of λnj despite significant difference in assumed recombination order and very significant collection losses. SE and EE side solutions to eq 1 were determined for a wide range of possible operation conditions and cell characteristics. For each set of conditions, taking the average of the resulting EE and SE concentration profiles (njc) enabled λnj to be calculated. This could be compared to the value of L1 estimated assuming first order recombination using eq 3. Figure 2 shows the relationship between λnj and L1 over the range of conditions examined. For more detail, Figure A4 in Appendix B shows corresponding examples of collection efficiency vs voltage curves calculated using the same parameters for SE and EE side illumination. Figure 2a shows calculations of λnj and the corresponding L1 for a case where b ) 0.6 and the light intensity is varied over 5 orders of magnitude up to an absorbed photon flux comparable to 1 sun intensity. Over this range there is not more than 16% difference between the corresponding lengths

Barnes and O’Regan (λnj ≈ L1). Note that at high light intensities the EE/SE IPCE ratio approaches (but never reaches) 1. For example at φ ) 1017 cm-2 s-1 in Figure 2a the ratio IPCEEE/IPCESE ) 0.9925 so that L1 ()84 µm) . d ()13 µm). Experimentally this would be hard to measure reliably. The IPCE technique is only useful for determining L1 when there are experimentally measurable collection losses (i.e., when L1 < 2d). Figure 2b shows the variation of λnj and L1 with recombination order. Among the points plotted we see that when b ) 1 then λnj exactly equals L1 but when b < 1 then λnj is slightly greater than L1. This trend increases until the difference is 18% when b ) 0.2 (a reaction order much smaller than ever observed in DSSCs). Figure 2c shows the consequences of varying the light penetration depth (R-1) in the film. This varies the rate at which electrons are injected into the film as a function of x. When R is small there is little difference between the SE and EE profiles. It is clear from the figure that, for all R, λnj is slightly larger than L1 by an approximately constant amount (in this case ∼0.7 µm), an amount which is related to the value of b as seen in Figure 2b. Figure 2 also shows simulated values of sLn plotted against λnj. An approximately one to one relationship is also evident. Thus our calculations imply that the first order IPCE analysis (L1) gives a good approximation to λnj and also a good approximation to the diffusion length determined from simulated transients, sLn, even when the order of conducting electron recombination is less than 1 (λnj ≈ L1 ≈ sLn). We now show an application of the above analysis to the data presented previously by Villanueva-Cab et al.6 They measured the EE/SE IPCE ratio for a cell illuminated with monochromatic illumination (627 nm) at different intensities using either an LED or monochromator. The measurements are shown by the points in Figure 3a, the solid line is a fit to the data determined from solutions to eq 1 with kr ) 4 × 109 cm-1.2 s-1 and b ) 0.6. Note that we find the fitting parameters stated by Villanueva-Cab et al. (b ) 0.8 and kr ) 3 × 106 cm-0.6 s-1)6 cannot fit the data (see dotted and dashed red lines in Figure 3a), an error which led them to different conclusions.7 Using the same cell they also measured mLn using a combination of IMPS (short circuit) and IMVS (open circuit) measurements with 627 nm illumination at different intensities, shown by the open circles in Figure 3b, with matched internal voltage (equivalent to njc as described above). These measurements used an LED source at high light intensities. Also plotted in figure 3b are the L1 values calculated directly from the IPCE ratio data in figure 3a (points). These have been plotted on the internal voltage axis used by Villanueva-Cab et al. rather than against intensity (Figure 3 of ref 6) to allow direct comparison between the two measurement types. This direct comparison was not made in ref 6. The internal voltage for each point was calculated using qVinternal ) Ec + kBT ln[(njc + n0)/Nc] where njc was calculated from the model (eq 1) using the parameters of the line in Figure 3a. Compared on the same axis, values of mLn are 2-3 times larger than L1. Using the rate constant and order from Figure 3a we calculate λnj and plot it against the internal voltage of the cell. It is clear that λnj gives good agreement to the values of L1 derived from the IPCE data for Vinternal < 0.55 V. For higher internal voltage, here also corresponding to higher light intensity, the IPCE ratio approaches TPt which can limit the accuracy of the measured L1. In contrast the experimentally derived small perturbation data, mLn, are not well described by λnj derived from collection efficiency ratios measurements in Figure 3a. Note that the alignment of the L1 and λnj values on the Vinternal scale in Figure 3b is dependent on the choice of Nc. We chose Nc ) 1020 cm-3 for consistency with the plot shown

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Figure 2. First order diffusion length L1 calculated using IPCEEE/IPCESE for modeled cells where b < 1 (blue points) and the simulated transient diffusion length, sLn (red circles), plotted against the effective small perturbation diffusion length, λnj, determined from solutions to eq 1. The solid line indicates L1 and sLn ) λnj. Panel a shows the variation of L1 and sLn with λnj as the photon flux was varied between φ ) 1012 and 2 × 1017 cm-2 s-1 all other parameters are as described in the figure 1 caption. Panel b shows the variation of L1 and sLn with λnj as the electron recombination order was varied between b ) 0.2 and 1, the rate constant was adjusted so that kr ) (2.6 × 1015 cm-3 s-1)/(5.0 × 109 cm-3)b to approximately normalize the recombination rate. In these calculations φ ) 1013 cm-2 s-1 and all other parameters are as shown in the Figure 1 caption. c. The variation of L1 and sLn with λnj as dye absorption coefficient was varied between R ) 400 and 4000 cm-1, φ ) 1013 cm-2 s-1; all other parameters were as those in Figure 1.

Figure 3. a. EE/SE IPCE ratio data plotted against illumination intensity of wavelength 627 nm as measured by Villanueva-Cab et al. (points).6 The solid blue line shows the shows calculated relationship between IPCE ratio and intensity of 627 nm light where R ) 750 cm-1, TPt ) 0.93, and all other parameters are identical to those used in Figure 1 (kr ) 4 × 109 cm-1.2 s-1 and b ) 0.6). For comparison curves derived from the fitting parameters described in ref 6 are shown. The red dotted line shows b ) 0.8 and kr ) 3 × 106 cm-0.6 s-1, the red dashed line shows b ) 0.73 and kr ) 9 × 106 cm-0.81 s-1. b. The measured transient diffusion length, mLn (open circles) and L1 (points) calculated from the ratio data in panel a are plotted against the internal cell voltage. For the L1 values Vinternal was estimated from the parameters described in panel a, with Nc ) 1020 cm-3. The solid blue line shows λnj calculated as described in Table 1 with the parameters used in panel a (kr ) 4 × 109 cm-1.2 s-1 and b ) 0.6). The dotted red line shows the curve calculated using the parameters b ) 0.8 and kr ) 3 × 106 cm-0.6 s-1 if Nc ) 1020 cm-3 (not Nc ) 1021 cm-3 as stated in their caption which was a typographical error) as calculated by Villanueva-Cab et al.6

in reference 6, Figure 6. This gives an offset of around 150 mV between the open circuit and short circuit Vinternal for similar light intensities. We were unable to find a value of Nc that simultaneously gave a good fit of λnj to mLn with recombination parameters that also fit the IPCE EE/SE ratio data in Figure 3a. This is principally because a recombination order b ≈ 0.6 is required to fit L1 and b ≈ 0.8 is required to fit mLn.

We now comment on the use of bias light or voltage with IPCE measurements. We will refer to this as differential IPCE measurement, where a cell’s spectral response is determined while operating with a constant background bias light or applied voltage.2,3,8,15 The differential IPCE is found by measuring the small additional photocurrent (∆j) generated by the superposition of a low intensity monochromatic light (∆φ) on a cell operating with constant background illumination or voltage (differential IPCE ) ∆j/q∆φ). This enables the IPCE and thus collection efficiency to be estimated at different light intensities and applied voltages. If b ) 1 (there is only linear recombination) the diffusion length determined from differential IPCE measurement is independent of the internal voltage in the cell such that differential IPCE ) -D0/φ dnc(x ) 0)/dx ) IPCE. In cases where b < 1 our calculations show that the differential IPCE is insensitive to the background distribution of electrons generated by the bias light or applied voltage and is primarily dependent on njc (or Vinternal) as expected. Analysis of simulated differential IPCE measurements using eq 3 approximately yields L1 determined from absolute IPCE measurements. In summary we have shown that the method used to derive diffusion lengths by analyzing IPCE data assuming b ) 1 yields physically meaningful values which are approximately equal to the small perturbation diffusion length predicted by theory for cells where b < 1. We find the measured small perturbation diffusion length does not appear to be consistent with the observed performance of the cell at short circuit, as has been shown previously.2-4 It is interesting to speculate on the cause of this discrepancy. Recently we have shown, using TiO2 water splitting electrodes, that IPCE measurements with low current and an applied potential yielded diffusion lengths consistent with transient diffusion lengths derived from open circuit photovoltage rise and decay measurements where nc(x) was approximately constant.8 We previously attributed this to the observation that b ≈ 1; however in light of the analysis herein it now possible that the agreement may be related to the relatively similar operational conditions for the two measurements. Recent work by Jennings and Wang also showed agreement between L1 determined from differential IPCE measurements made at a voltage equivalent to open circuit and the diffusion length determined from impedance measurements made entirely at open circuit both for DSSCs where b < 1 and where b ) 1.15 Thus we suggest that the observed inconsistencies between IPCE and transient diffusion length appear to be related to comparison of values measured at significantly different operational conditions

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(cell current, applied voltage and light intensity). This raises the possibility that differences reflect actual changes in electron diffusion length correlated with operational conditions. For example Nakade et al. have demonstrated the electron diffusion coefficient at short circuit is several times greater than at open circuit for the same electron concentration for large perturbation measurements at matched charge concentration.16 This may also be the case for small perturbation measurements. In a broader context, the mounting evidence that diffusion length in operating cells is shorter than previously thought and varies with electron concentration suggests that, when designing commercial cells, the TiO2 thickness should be tailored to the expected light conditions. Acknowledgment. We thank the authors of ref 6 for providing their original data for Figure 3 and their useful comments. We are also grateful to Dr. Matthias Schmid for his helpful discussion. This work was supported by the EPSRC (Grant No. EP/E035175/1) and the EU (Grant No. 212792). We are grateful Dr Wenhua Leng for his discussions on the interpretation of measurements.

Barnes and O’Regan

Figure A2. a. Open circuit photovoltage transient calculated using the parameters in Figure A1 but with φ ) 3.5 × 1011 cm-2 s-1, adjusted to give a mean electron concentration njc ) 1.2 × 108 cm-3 to match that in the calculations described in Figure A1 at short circuit. The photovoltage relaxation time is τn ) 7.1 × 10-7 s. b. The corresponding to the impedance spectrum (equivalent to an IMVS measurement, t ) 0 - 3 × 10-5 s), the time constant is τn ) 6.1 × 10-7 s.

Appendix A. Numerical Simulation of Photocurrent and Photovoltage Transients with and without Trapping. To examine the relationship between the small perturbation diffusion length, λnj, calculated using njc and simulated transient photocurrent and photovoltage perturbations we solve a time dependent continuity equation describing the generation, diffusion and recombination of electrons in a porous semiconductor film of the form

(

)

∂nc ∂nL ∂2nc ∂ 1+ ) D0 2 + G(x) - krnbc (nc + nL) ) ∂t ∂t ∂nc ∂x (A1) where nL is the concentration of localized trapped electrons so the total concentration is given by nc + nL. The two populations of electrons are assumed to be in instantaneous local equilibrium even during a perturbation (i.e., the quasi-static approximation).14 If the density of localized states in the cell has an exponential

Figure A1. a. Photocurrent transient calculated using the same parameters as shown in figure 1a for SE side illumination, with NL ) 0 (blue points). The initial illumination pulse added a perturbation concentration of ∆nc (t ) 0) ) 10 000 cm-3. The red line shows the single exponential fit to the tail of the decay, giving a time constant τj ) 4.4 × 10-7 s. b. The real (red) and imaginary (blue) parts of the Fourier transform of the decay in panel a. (t ) 0 - 2.5 × 10-5 s) in the frequency domain. This is approximately equivalent to the impedance spectrum that would be measured by IMPS, the time constant is τj ) 4.6 × 10-7 s.

Figure A3. a. Simulated and calculated small perturbation photocurrent (τj) and photovoltage (τn) transient lifetimes for a cell with the same parameters as in Figure 1 with NL ) 1019 cm-3 and β ) 0.33 for SE side illumination. The red squares show τn (simulated at open circuit for incident photon fluxes: φ ) 3.5 × 1011, 2.25 × 1012, 1.13 × 1013, 5.06 × 1013, 2.1 × 1014, and 8.5 × 1014 cm-2 s-1) and the red triangles show τj (simulated at short circuit for incident photon fluxes: φ ) 1012, 1013, 1014, 1015, 1016, and 1017 cm-2 s-1). Since these photon fluxes give matching njc the transient lifetimes are used to derive Dn using eq A6 (blue circles). The solid red line shows the τnj calculated using eq A9 and the solid blue line shows the Dn calculated using eq A10. b. Corresponding simulated and calculated diffusion lengths, with and without traps. Red circles shows sLn (eq A8) derived from the simulated transients above (eq A8). Black triangles shows sLn derived from simulated transients with no trapping (NL ) 0 cm-3). The solid black line shows the corresponding effective small perturbation diffusion length (λnj).

distribution below the conduction band energy (EC), then the concentration of localized electrons is given by

nL ) NLeβ(EF-Ec)/kBT

(A2)

where β is a temperature dependent parameter describing the slope of the distribution and NL is the total concentration of trapping

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Figure A4. Normalized current-voltage curves (collection efficiency-voltage) calculated for the parameters used in Figure 2 (d ) 13 µm, R ) 4000 cm-1, RI ) 0 cm-1, D0 ) 0.4 cm2 s-1, kr ) 4 × 109 cm-1.2 s-1, b ) 0.6, the total density of conducting states Nc ) 1020 cm-3, TPt ) 1, ηinj ) 1, T ) 298 K, and Ec ) 1 V vs electrolyte redox potential). a. Curves calculated for three different light intensities, φ ) 1013 (red), 1015 (green), and 1017 cm-2 s-1 (blue). The solid lines show calculations for illumination from the SE side, and the dotted lines show EE side illumination. The injected current for the different light intensities are qφηinjηLH ) 0.0016, 0.16, and 16 mA cm-2 respectively. b. Curves calculated for b ) 1 (blue), 0.6 (green) and 0.2 (red) for both SE (solid lines) and EE side (dotted lines) illumination with φ ) 1013 cm-2 s-1 and where kr ) (2.6 × 1015 cm-3 s-1)/(5.0 × 109 cm-3)b. The injected current is qφηinjηLH ) 0.0016 mA cm-2. c. Curves calculated with parameters as used in panel a with φ ) 1013 cm-2 s-1 and R ) 4000 (blue), 1000 (purple), and 400 cm-1 (red) for SE (solid lines) and EE (dotted lines) side illumination. The injected currents are qφηinjηLH ) 0.0016, 0.00116, and 0.00065 mA cm-2 respectively.

states (β ) T/T0 where T0 is a critical temperature). If the concentration of conduction band electrons is described by Boltzmann statistics and n0 , nc

nc ) Nce(EF-Ec)/kBT

(A3)

where Nc is the concentration of conduction states. Then with eqs A2 and A3, we can use the quasi-static approximation to relate the local concentrations of conducting and trapped electrons by

nL )

NL

nβ β c Nc

(A4) Dn )

Equation A4 can be differentiated with respect to nc so that the left-hand side of eq A1 can be written

(

)

(

∂nc ∂nL ∂nc NL 1+ ) 1 + β β nβ-1 c ∂t ∂nc ∂t Nc

photocurrent transient measurement (and corresponding IMPS spectrum) calculated under conditions shown in Figure 1 of the main text for SE side illumination. Figure A2 shows a simulated photocurrent transient (and corresponding IMPS spectrum) for a cell operating at open circuit with the same parameters as those in Figure 1 for SE side illumination and the photon flux adjusted to φ ) 3.5 × 1011 cm-2 s-1 to give the same mean electron concentration: njc ) 1.2 × 108 cm-3. The time constants in Figures A1 and A2 can be used to calculate the diffusion coefficient Dn using (see the Supporting Information of ref 8 for the derivation)

)

(A5)

Using this expression, eq A1 can be solved numerically using the MATLAB pdepe solver function for open and short circuit boundary conditions in terms of nc. The concentration of trapped charge can then be found using eq A4 as required. To simulate an electron recombination small perturbation photovoltage transient measurement at open circuit and a small perturbation photocurrent transient measurement at short circuit, we must first solve for the electron concentration profile with steady state operating conditions. At short circuit the boundary conditions are as described in the main text. For open circuit conditions the boundary conditions are dnc(x ) 0)/dx ) 0 and dnc(x ) d)/dx ) 0. The steady state solutions of eq 1 provide the initial conditions for the transient simulation. The photocurrent and photovoltage transients are calculated by introducing a small uniform additional concentration of electrons to the initial concentration profile and monitoring the relaxation of the transient photovoltage (where ∆V ∝ ∆nc(x ) 0)) or transient photocurrent (where ∆jsc ∝ -D0 d∆nc/dx(x ) 0)). Initially we consider the case where no electrons are trapped by setting NL ) 0. Figure A1 shows an example of a simulated

(

4d2 1 1 2 τ τ π j n

)

(A6)

This gives Dn ) 0.52 cm-2 s-1 for the transient measurements and Dn ) 0.37 cm-2 s-1 for the impedance measurements, both close relatively to the input parameter of D0 ) 0.4 cm-2 s-1 despite the significant collection losses under the short circuit conditions (τn and τj are relatively similar). The simulated recombination time constant, τn, can be compared to the theoretical value expected given the mean electron concentration njc ) 1.2 × 108 cm-3 in the cell

τnj )

njc1-b bkr

(A7)

The simulated values are close to the value of τnj ) 7.1 × 10-7 s given by eq A7. The simulated time constants can also be used to calculate the small perturbation diffusion length by using eqs A6 and A7 with sLn ) (Dnτn)1/2

Ln )

s

( )

2d τn -1 π τj

1/2

(A8)

This gives Ln ) 4.1 µm for the transient measurements, close to the λnj ) 5.3 µm (for SE side illumination) determined using the expression on line 4 of Table 1 in the main text. We find the simulated transient diffusion length sLn ≈ λnj over a wide

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range of cell operating conditions, particularly when τn . τj, see red circles in Figure 2 (main text). Our calculations also indicated that the incorporation of traps (NL > 0) had no significant influence on this relationship other than slowing down both time constants in proportion. For example if NL ) 1019 cm-3 and β ) 0.33 (values typical of DSSCs), simulation of a cell with otherwise identical characteristics under the same intensities gave time constants of τj ) 1.8 s and τn ) 2.7 s yielding Ln ) 4.0 µm, similar to the value with no localized trapping states. The effect of the traps does not significantly influence the diffusion length because using eq A5 the effective diffusion coefficient becomes

(

Dn ) 1 +

∂nL ∂nc

)

(

-1

D0 ) 1 + β

)

-1

NL

njcβ-1 Nβc

D0

(A9)

and transient electron lifetime becomes

(

τnj ) 1 +

(

)

)

∂nL njc1-b NL njc1-b ) 1 + β β njcβ-1 ∂nc bkr bkr Nc

(A10)

Thus the terms in brackets cancel out when the small perturbation diffusion length is calculated by sLn ) (Dn τn)1/2. This can be seen in Figure A3. Figure A3a shows examples of the simulated small perturbation transients for a cell with the parameters shown in Figure 1 where the trapping is also included. These are compared with the calculated values determined using eqs A9 and A10, good agreement is observed. Figure A3b shows the corresponding values of sLn found from the simulated transients in Figure A3a. For comparison values of sLn determined from simulated transients where there is no trapping (NL ) 0) are also shown. There is good agreement between these points and also with λnj as seen in Figure 2. In summary, according to the theoretical calculations and simulations above for typical DSSC operating characteristics

( )

D0njc1-b λnj ≈ bkr

1/2

≈

(

)

2d τn(njc) -1 π τj(njc)

1/2

) sLn

(A11)

B. Simulation Current-Voltage Curves and Collection Efficiency. The current-voltage characteristics of simulated cells were calculated by solving eq 1 with a series of different boundary conditions to give different cell potentials. The boundary conditions were n(x ) 0) ) nboundary and dnc(x ) d)/dx ) 0. Here the concentration of electrons at the x ) 0 boundary is related to the cell potential V by

nboundary ) Nce(qV-Ec)/kBT

(A12)

where q is the electronic charge. For different substrate electron concentrations, nboundary, the cell currents (j) corresponding to different cell voltages (V) were found (j ) -qDo dnc/dx(x)0)). The j values could be normalized by dividing by the light harvesting efficiency (ηLH ) 1 - e-Rd, in the case where RI ) 0), the injection efficiency (ηinj), the charge on the electron, and the incident photon flux φ to give the collection efficiency

ncol )

j qφηinjηLH

(A13)

Examples of the collection efficiency-voltage curves calculated for the parameters used in Figure 2 for SE and EE side illumination at different light intensities, different recombination orders and different absorption coefficients are plotted in Figure A4. Figure A4a demonstrates the difference in cell performance between SE and EE side illumination increases as the light intensity is reduced. In the case where φ ) 1017 cm-2 s-1, at short circuit the collection efficiencies are ηcol ) 0.9841 and 0.9916 for EE and SE side illumination respectively. This corresponds to an IPCE EE/SE ratio of 0.9925, corresponding to L1 ) 84 µm. This ratio would be very hard to measure experimentally, the use of IPCE to determine L1 is generally only reliable when L1 is less than around 2d. Figure A4b indicates the fill factor at low light intensity drops as the electron recombination order decreases as has been discussed by Bisquert and Mora-Sero´.5 Note that in this figure the recombination rate constant kr is changed for each value of b to approximately normalize the total amount of recombination. Figure A4c shows that the collection efficiency for the two illumination sides becomes more similar as the absorption coefficient in the cell decreases, corresponding to a more uniform generation across the cell thickness. References and Notes (1) Halme, J.; Boschloo, G.; Hagfeldt, A.; Lund, P. J. Phys. Chem. C 2008, 112, 5623. (2) Barnes, P. R. F.; Anderson, A. Y.; Koops, S. E.; Durrant, J. R.; O’Regan, B. C. J. Phys. Chem. C 2009, 113, 1126. (3) Barnes, P. R. F.; Liu, L.; Li, X.; Anderson, A. Y.; Kisserwan, H.; Ghaddar, T. H.; Durrant, J. R.; O’Regan, B. C. Nano Lett. 2009, 9, 3532. (4) Wang, H.; Peter, L. M. J. Phys. Chem. C 2009, 113, 18125. (5) Bisquert, J.; Mora-Sero´, I. J. Phys. Chem. Lett. 2010, 1, 450. (6) Villanueva-Cab, J.; Wang, H.; Oskam, G.; Peter, L. M. J. Phys. Chem. Lett. 2010, 1, 748. (7) Villanueva-Cab, J. ; Wang, H. ; Oskam, G. ; Peter, L. M. correction to ref 6. (8) Leng, W. H.; Barnes, P. R. F.; Juozapavicius, M.; O’Regan, B. C.; Durrant, J. R. J. Phys. Chem. Lett 2010, 1, 967. (9) Koops, S. E.; O’Regan, B. C.; Barnes, P. R. F.; Durrant, J. R. J. Am. Chem. Soc. 2009, 131, 4808. (10) Sodergren, S.; Hagfeldt, A.; Olsson, J.; Lindquist, S. E. J. Phys. Chem. 1994, 98, 5552. (11) Barnes, P. R. F.; Anderson, A. Y.; Koops, S. E.; Durrant, J. R.; O’Regan, B. C. J. Phys. Chem. C 2009, 113, 12615. (12) Dloczik, L.; Ileperuma, O.; Lauermann, I.; Peter, L. M.; Ponomarev, E. A.; Redmond, G.; Shaw, N. J.; Uhlendorf, I. J. Phys. Chem. B 1997, 101, 10281. (13) van de Lagemaat, J.; Frank, A. J. J. Phys. Chem. B 2000, 104, 4292. (14) Bisquert, J.; Vikhrenko, V. S. J. Phys. Chem. B 2004, 108, 2313. (15) Jennings, J. R.; Li, F.; Wang, Q. J. Phys. Chem. C 2010, 114, 14665. (16) Nakade, S.; Saito, Y.; Kubo, W.; Kanzaki, T.; Kitamura, T.; Wada, Y.; Yanagida, S. J. Phys. Chem. B 2004, 108, 1628.

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