25504
J. Phys. Chem. B 2006, 110, 25504-25507
Analysis of Photovoltage Decay Transients in Dye-Sensitized Solar Cells† Alison B. Walker,*,‡ L. M. Peter,§ K. Lobato,§ and P. J. Cameron§,| Departments of Physics and Chemistry, UniVersity of Bath, Bath BA2 7AY, United Kingdom ReceiVed: July 29, 2006; In Final Form: September 19, 2006
It is shown that application of the so-called quasi-static approximation greatly simplifies the theoretical treatment of the open circuit photovoltage decay of dye-sensitized nanostructured solar cells (DSCs), since it removes the need to treat the kinetics of trapping and detrapping explicitly and leads to a straightforward analytical solution in the case of an exponential trap distribution. To identify the conditions under which the quasistatic approach is valid, transients calculated using the quasi-static approximation are compared with the results of numerical calculations that treat trapping and detrapping of electrons explicitly. The application of the quasi-static approach to derive the rate constant for the back-reaction of electrons from experimental photovoltage decay data is illustrated for an optimized DSC.
Introduction Dye-sensitized nanostructured solar cells (DSCs) have achieved energy conversion efficiencies exceeding 10%, making them competitive with conventional semiconductor photovoltaic devices.1-3 In the DSC, absorption of light by a monolayer of sensitizer dye anchored to the surface of a nanostructured porous film of TiO2 leads to rapid injection of electrons into the TiO2 film. The dye is regenerated from its oxidized state by a redox couple, usually I3-/I-, which also transports charge in the electrolyte phase to complete the regenerative cycle. The open circuit voltage generated under illumination is equivalent to the separation between the quasi-Fermi level of electrons in the TiO2 film and the redox Fermi level in the electrolyte.4-6 When the illumination is switched off, the conduction electron density in the TiO2 film decays at open circuit due to electron transfer to I3- ions in the electrolyte. Most of the photogenerated electrons are located in trap states7-15 in the nanostructured TiO2 film, leading to a trapped electron density that greatly exceeds the conduction electron density. These electrons must be released thermally to the conduction band before they can reach the interface and transfer to I3- ions. The rate at which the photovoltage decays therefore depends not only on the rate constant for electron transfer but also on the dynamic exchange of electrons between the conduction band and the trapping states. Our earlier work6,12-14 on DSCs incorporating a compact TiO2 blocking layer to prevent back-reaction via the conducting glass substrate showed that the intensity dependence of the photovoltage can approach the value 59 mV/decade over many orders of magnitude of light intensity associated with an ideal cell. (Typical values of the slope lie in the range of 60-80 mV/ decade, depending on the route used to prepare the nanocrystalline TiO2 film.) We infer from this that the loss of electrons from back-reaction in our cells occurs primarily via the conduction band, with surface states playing a less important †
Part of the special issue “Arthur J. Nozik Festschrift”. * Author to whom correspondence should be addressed. E-mail:
[email protected]. ‡ Department of Physics, University of Bath. § Department of Chemistry, University of Bath. | Present address: Max Planck Institute for Polymer Research, Postfach 3148, D-55021 Mainz, Germany.
role. Back-reaction via surface states16 would give a nonideal slope since the surface electron concentration shows a different variation with the quasi-Fermi level from the conduction electron density. Another characteristic of these optimized cells is that a plot of the photovoltage decay versus log(time) is almost linear at long times.12 Here we show that this behavior is consistent with the presence of a distribution of trap states g(ET) that varies exponentially with trap energy ET. Although recently the multiple trapping model has been questioned,17,18 no other model has been framed that predicts this decay behavior. In the DSC, trapping and detrapping of electrons reduce the rate at which electrons reach the anode and also the rate at which electrons can react with I3- ions in solution. As a consequence, the apparent diffusion coefficient, Dn, of electrons is many orders of magnitude smaller than that in bulk TiO2. Similarly, the apparent electron lifetime, τn, is much longer than that of free conduction band electrons. The retardation of electron transport and the back-reaction depend on the extent to which the trap states are filled by photogenerated electrons, so that Dn increases with light intensity whereas τn decreases. It has been found that the apparent values of the electron diffusion coefficient and electron lifetime for electrolyte-based15 and solid-state19 DSCs vary in opposite senses with illumination intensity according a power law. The product Dnτn remains almost constant, so that the electron diffusion length Ln ) (Dnτn)1/2 is independent of light intensity. This means that electron trapping does not degrade the performance of cells. Bisquert and Vikhrenko20 have provided an elegant explanation of the observation that the electron diffusion length is independent of light intensity. They have used a quasi-static condition to relate the time dependence of the conduction band electron density ∂nc/∂t in the nanocrystalline oxide to the corresponding rate of change of the density of trapped electrons ∂nt/∂t
∂nc ∂nt ∂nc ) ∂t ∂nc ∂t
(1)
where ∂nt/∂nc is independent of time (quasi-static). The relaxation time constant for electrons in the conduction band is often referred to as the electron lifetime or apparent
10.1021/jp064860z CCC: $33.50 © 2006 American Chemical Society Published on Web 10/21/2006
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J. Phys. Chem. B, Vol. 110, No. 50, 2006 25505
electron lifetime, but it is important to recognize that it is determined not only by the rate of loss of electrons in the backreaction but also by changes in electron trap occupancy. For ∂nt/∂nc . 1, the relaxation time constant can be defined as20
∂nt 1 ∂nt τn ) ) τ ∂nc kcb ∂nc 0
(2)
Here we show that the quasi-static approximation can be used to derive a convenient expression for the transient photovoltage decay Uphoto(t) that relates the slope of a plot of Uphoto(t) versus log(t) to the characteristic temperature, Tc, of the electron trap distribution. It is therefore of interest to explore the limits of validity of the quasi-static approximation by numerical solution of the continuity equations for free and trapped electron densities.
Uphnum(t) )
( )
kBT nc(0,t) ln q nc,eq
(6)
Calculation of the free and trapped charge per unit area by integration of eqs 3 and 4 across the device and over time showed that the numerical accuracy was ∼0.1%. The photovoltage decay also can be obtained using the quasistatic approximation. In this case, the influence of traps is dealt with by removing the term ∂nt/∂t in eq 4 and replacing D0 and kcb by their effective values Dn and kcb,eff
Dn ) D0
∂nc ∂nt
kcb,eff ) kcb
(7)
∂nc ∂nt
(8)
Theory The DSC is treated as a quasi-homogeneous medium, in which fields are screened out by the electrolyte and in which, as noted in the Introduction, there is no back-reaction from traps. The continuity equations for the density of conduction electrons nc(x,t) and trapped electrons nt(x,t) are21,22
∂nt ∂2nc ∂nc ) D0 2 + RI0 exp(-Rx) - kcb(nc - nc,eq) (3) ∂t ∂t ∂x
〈〉
∂nt ∂f ) 〈ktnc(1 - f ) - kdNt,0f〉 ) Nt,0 ∂t ∂t
(4)
Here, illumination is from the anode side and x is the distance from the anode (0 e x e d) where d is the thickness of the TiO2 layer, D0 is the diffusion coefficient of electrons in the conduction band of TiO2, R is the absorption coefficient, nc,eq ) NC exp[-(EC - EF,redox)/(kBT)] is the equilibrium electron density, T is the absolute temperature, NC is the conduction band density of states, f(ET) the trap occupation fraction, and kt and kd are the rate constants for trapping and detrapping, respectively. Angular brackets indicate an average over ET for the quantity in brackets weighted by the normalized trap distribution g(ET)
g(ET) )
Nt,0/kBTc 1 - exp[-(EC - EF,redox)/(kBTc)]
×
[
exp -
]
(EC - ET) (5) kBTc
Nt,0 is the total trap density, kB is the Boltzmann constant, EC is the conduction band edge, EF,redox is the redox Fermi energy, and Tc is a characteristic temperature that determines the width of the exponential distribution. Since we are considering open circuit and zero back-reaction at the substrate, the gradient of nc at x ) 0 and x ) d is set to zero. We assume that an initial steady-state condition has been established by an illumination level I0 that ceases at t ) 0. The coupled eqs 3 and 4 were solved numerically with a fully implicit approach employing a relaxation method using the differential equation solver solvde.23 To obtain a convenient spread of data on a log(t) time scale, at the ith time step, we set t ) 10-6+δi, where δ is chosen to give adequate time resolution. The photovoltage, Uphnum, is calculated from nc(x ) 0) and the electron charge magnitude q by
In the steady state, the solution to eq 4 is
f(ET) )
1 ≈ 1 + exp{(ET - EF)/(kBT)} θ(EF - ET) if kBT , |ET - EF| (9)
where θ(x) is the step function. The detrapping rate kd(ET) is assumed equal to its thermal equilibrium value, kd(ET) ) ktNCeet/ Nt,0 from eqs 4 and 9, where eet ) exp[-(EC - ET)/(kBT)]. Using the approximate expression in eq 9 and the trap distribution in eq 5
nt ≈ Nt,0
{( ) ( ) } nc NC
T/Tc
nc,eq NC
-
T/Tc
(10)
Hence
()
∂nt Nt,0T nc ) ∂nc NCTc NC
T/Tc-1
(11)
At open circuit, nc does not vary strongly with x, and even at t ) 0 where nc takes its maximum value, nc/NC ≈ 10-5, and so at all times ∂nt/∂nc . 1. An analytic solution to eq 3 within the quasi-static approximation can be found by the following arguments: (i) The first term on the right-hand side can be neglected at open circuit since for our parameter values the gradient in nc is small everywhere and is zero at the extracting electrode; (ii) once the illumination is switched off the second term is zero; (iii) for much of the subsequent decay time, nc,eq , nc, so the third term can be approximated by -kcbnc. In the discussion below eq 11, we showed that ∂nt/∂nc . ∂nc/∂t. Thus in eq 3, the term on the left-hand side can be neglected and so
∂nt ) -kcbnc ∂t
(12)
Hence from eqs 6 and 11, the quasi-static approximation to Uph(t) is
Uph,qs(t) ) where
kBT/q ln[c1 + c2t] 1 - T/Tc
(13)
25506 J. Phys. Chem. B, Vol. 110, No. 50, 2006
c1 ) exp
[
]
(T/Tc - 1)qUph(0) kBT
Walker et al.
and
[ ] ( )
c2 ) kcb 1 -
T NCTc nc,eq Tc Nt,0T NC
(1-T/Tc)
(14)
The quasi-static approximation breaks down when c1 + c2t > 1 since here Uph,qs(t) < 0. The term c1 can be neglected for times t when c2t . c1, which for the parameters considered here is most times of interest. For such times, the following simplification holds
Uph,qs(t) ≈ -
kBT/q (ln[c2] + ln[t]) 1 - T/Tc
(15)
It follows from this that at long times a plot of Uph,qs versus ln(t) should have a slope
kBT/q dUph,qs(t) ≈d ln t 1 - T/Tc
(16)
This offers a simple way of determining Tc from photovoltage decay transients in the case where a simple exponential trap distribution is present.
Figure 2. Photovoltage decay plots calculated numerically (full lines) and via the quasi-static approximation (broken lines) for three different values of Nt,0 as shown. kcb ) 104 s-1. For the numerical calculations, R ) 500 cm-1, and D0 ) 0.4 cm2 s-1 from ref 26. Other parameter values are as in Figure 1.
Experimental Section The procedures for fabrication of DSCs with a compact blocking layer and measurement of photovoltage decay transients have been described previously.12 Photovoltage decay measurements were made using a high impedance operational amplifier voltage followed with very low offset current to prevent errors at long times. Results Figure 1 illustrates the effect on the photovoltage decay of changing the rate constant for the transfer of conduction band electrons to I3-. The plots were calculated using the quasi-static approximation (eqs 13 and 14). It can be seen that lowering kcb
Figure 3. Photovoltage decay plots calculated numerically (full lines) and via the quasi-static approximation (broken lines) for two different values of Nt,0 as shown. kcb ) 104 s-1. Other parameter values are as in Figure 1.
Figure 1. Linear-log plots of the photovoltage decay calculated from the quasi-static approach for different values of kcb, the rate constant for the back-reaction of electrons with I3- (cf. eq 13). NC ) 1021 cm-3, from a parabolic band model24 with an effective mass of approximately 9 times the bare electron mass.25 Ec - EF,redox ) 1.0 eV, T ) 298 K,12 Tc ) 800 K, and Uph(0) ) 0.712 V. The total trap density, Nt,0, was estimated using the charge extraction method to be 1018 cm-3.
shifts the plots to the right on the logarithmic time scale. The slope of the semilogarithmic decay plot is not affected. This is consistent with eq 16, which shows that this slope depends only on the ratio T/Tc. This suggests that a useful diagnostic approach is to use the slope to estimate T/Tc and hence to compare the values obtained by the charge extraction method12 or intensitymodulated photovoltage spectroscopy (IMVS).6 The utility of the quasi-static approximation is demonstrated by Figure 2, which shows that at trap densities typical for DSCs the numerical solution predicts a decay plot that is very similar to the one derived from eqs 13 and 14. If Nt,0 is reduced to 1017 cm-3, then agreement is worse. Such a low value of Nt,0 has not been observed for any of the DSCs fabricated in our laboratories, but it is conceivable that other preparation routes could lead to lower trap densities. The effect of changing Tc is illustrated in Figure 3 for both numerical and quasi-static calculations. It can be seen that the slope of the linear decay region in the quasi-static plots increases
Photovoltage Decay Transients in DSCs
J. Phys. Chem. B, Vol. 110, No. 50, 2006 25507 circuit, where the quasi-Fermi level is lowered due to extraction of electrons than that at open circuit where the quasi-Fermi level is much higher. Conclusions The quasi-static approximation leads to a simple expression for the photovoltage decay that can be used to analyze experimental decay transients to derive the rate constant for the electron transfer. This approach should be useful for studies in which attempts are made to reduce the rate of electron transfer by surface modification by blocking layers or molecular adsorbates. Numerical calculations have established that the quasi-static approximation is a useful and simple approach to data analysis, but deviations at short times are significant, particularly if the density of trapping states is small. The extension of the numerical and quasi-static approaches to describe the IMVS response of DSCs will be presented elsewhere.28
Figure 4. (Open circles) Experimental photovoltage decay (data sampled in 10 steps per decade of time). (Line) Nonlinear regression fit of the data to eq 13. See text for the fit values.
as Tc decreases (eq 16). At the same time, the difference between the numerical and the quasi-static plots becomes more marked as Tc is reduced. The numerical solution for Tc ) 600 K also exhibits the asymptotic behavior expected for the last part of the decay. The absence of this asymptotic behavior in the quasistatic plot arises from the assumption that nc exceeds nc,eq. Clearly this assumption no longer holds as Uph(t) approaches zero. Figure 4 shows an experimental photovoltage decay transient for a cell incorporating a thin TiO2 blocking layer to prevent back-reaction of electrons via the conducting glass substrate. The decay is in excellent agreement with the behavior predicted by the quasi-static approximation (eqs 13 and 14), and a nonlinear regression fit of the experimental data (shown as a line) gave Uph(0) ) 0.74 V, Tc ) 849 K (thus β ) 0.35), c1 ) 7.7 × 10-9, and c2 ) 4.95 × 10-7 s-1. Through the use of a value of the conduction band density of states, NC, equal to 1021 cm-3, eq 14 gives kcb ) 25 s-1, corresponding to a conduction band electron lifetime of 40 ms. That there is a low value for kcb may be seen from Figure 4 where noticeable decay only takes place after approximately 0.01 s. Our deduced value is much smaller than the values generally assumed for DSCs (∼104 s-1), but it may be noted that a low value for kcb for DSCs with blocking layers was also found in ref 14. From eq 14, kcb ∝ Nc-T/Tc, and thus kcb would increase to 50 s-1, with a lifetime of 20 ms, if Nc were reduced by a factor of 10. The value of Nc is uncertain, coming from an effective mass m* obtained from spectroscopic measurements and the parabolic band approximation. We have observed elsewhere27 the difficulty of ascertaining Nc and that its value is important in the interpretation of quasi-Fermi level measurements. With the value of D0 of 0.4 cm2 s-1 used for our numerical model, the electron diffusion length Ln is 900 µm, much larger than previous estimates based on effectiVe lifetime and diffusion coefficient values determined respectively by IMVS and intensity-modulated photocurrent spectroscopy (IMPS) data.15 The IMVS/IMPS approach significantly understimates Ln since the effective diffusion coefficient is lower at short
Acknowledgment. This work was supported by the U. K. Engineering and Physical Sciences Research Council. A.B.W. thanks the Royal Society for an Industry Fellowship. References and Notes (1) Gra¨tzel, M. C. R. Chim. 2006, 9, 578. (2) Gra¨tzel, M. J. Photochem. Photobiol. 2005, 44, 6841. (3) Gra¨tzel, M. J. Photochem. Photobiol. 2004, 164, 3. (4) Bisquert, J.; Greenshtein M; Mora-Sero, I. J. Am. Chem Soc. 2004, 126, 13550. (5) Duffy, N. W.; Peter,L. M., Rajapakse, R. M. G.; Wijayantha, K. G. U. J. Phys. Chem. B 2000, 104, 8916. (6) Cameron, P. J.; Peter, L. M. J. Phys. Chem. B 2005, 109, 7392. (7) Bisquert, J.; Zaban, A. Appl. Phys. A: Mater. Sci. Process. 2003, 77, 507. (8) Nelson, J.; Chandler, R. E. Coord. Chem. ReV. 2004, 248, 1181. (9) Kambili, A.; Walker, A. B.; Qiu, F. L.; Fisher, A. C.; Savin, A. D.; Peter, L. M. Physica E 2002, 14, 203. (10) Peter, L. M.; Duffy, N. W.; Wang, R. L.; Wijayantha, K. G. U. J. Electroanal. Chem. 2002, 524, 127. (11) van de Lagemaat, J.; Kopidakis, N.; Neale, N. R.; Frank, A. J. Phys. ReV. B 2005, 71, 035304. (12) Bailes, M.; Cameron, P. J.; Lobato, K.; Peter, L. M. J. Phys. Chem. B 2005, 109, 15429. (13) Cameron, P. J.; Peter, L. M. J. Phys. Chem. B 2003, 107, 14394. (14) Cameron, P. J.; Peter, L. M.; Hore, S. M. J. Phys. Chem. B 2005, 109, 930. (15) Fisher, A. C.; Peter, L. M.; Ponomarev, E. A.; Walker, A. B.; Wijayantha, K. G. U. J. Phys. Chem. B 2000, 104, 949. (16) Bisquert, J.; Cahen, D.; Hodes, G.; Ru¨hle, S.; Zaban, A. J. Phys. Chem. B 2004, 108, 8106. (17) Kopidakis, N.; Benkstein, K. D.; van de Lagemaat, J., Frank, A. J.; Yuan, Q.; Schiff, E. A. Phys ReV B 2006, 73, 043526. (18) Nissfolk, J.; Hagfeldt, A.; Boschloo G. J. Phys. Chem. B 2006, 110, 17715. (19) Kruger, J.; Plass, R.; Gra¨tzel, M.; Cameron, P. J.; Peter, L. M. J. Phys. Chem. B 2003, 107, 7536. (20) Bisquert J; Vikhrenko, V. S. J. Phys. Chem. B 2004, 108, 2313. (21) Vanmaekelbergh, D.; de Jongh, P. E. J. Phys. Chem. B 1999, 103, 747. (22) Cass, M. J.; Qiu, F. L.; Walker, A. B.; Fisher, A. C.; Peter, L. M. J. Phys. Chem. B 2003, 106, 113. (23) Press, W. H.; Teukolsky, S. A.; Vettering, W. T.; Flannery, B. P. Numerical Recipes in Fortran, 2nd ed.; Cambridge University Press: Cambridge, U. K., 1992. (24) Singh, J. Semiconductor DeVices; McGraw-Hill: New York, 1994. (25) Enright, B.; Fitzmaurice, D. J. Phys. Chem. 1996, 100, 1027. (26) Peter, L. M.; Walker, A. B.; Boschloo, G.; Hagfeldt, A. J. Phys. Chem. B 2006, 110, 13694. (27) Lobato, K.; Peter, L. M.; Wu¨rfel, U. J. Phys. Chem. B 2006, 110, 16201. (28) Walker, A. B.; Peter, L. M.; Lobato, K., to be submitted for publication.
