Evaluation of the Charge-Collection Efficiency of Dye-Sensitized

Evaluation of the Charge-Collection Efficiency of Dye-Sensitized Nanocrystalline TiO2 Solar. Cells. G. Schlichtho1rl, N. G. Park, and A. J. Frank*. Na...
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J. Phys. Chem. B 1999, 103, 782-791

Evaluation of the Charge-Collection Efficiency of Dye-Sensitized Nanocrystalline TiO2 Solar Cells G. Schlichtho1 rl, N. G. Park, and A. J. Frank* National Renewable Energy Laboratory, Golden, Colorado 80401 ReceiVed: July 22, 1998; In Final Form: October 30, 1998

Intensity modulated photovoltage spectroscopy (IMVS) and intensity modulated photocurrent spectroscopy (IMPS) are used to evaluate the charge-collection efficiency of dye-sensitized nanocrystalline TiO2 solar cells. The charge-collection efficiency of the photoinjected electrons from dye sensitization is estimated from the respective time constants for charge recombination at open circuit τoc and the combined processes of charge collection and charge recombination at short circuit τsc obtained by IMVS and IMPS measurements. Three models are developed for relating the charge-collection efficiency to τoc/τsc. The first model determines the charge-collection efficiency from τoc/τsc without considering the underlying physical processes measured by IMVS and IMPS. The second model obtains τoc/τsc by simulating the frequency response of IMVS and IMPS from the time-dependent continuity equation for simplified conditions. The third model determines the time constants for IMVS and IMPS from electron-concentration profiles calculated for constant light intensity and more realistic conditions. To obtain a realistic steady-state electron concentration profile, a nonlinear dependence of the rate of recombination on the electron concentration in the TiO2 film is considered. Furthermore, the continuity equation is modified to account for charge trapping and detrapping. For the first time, expressions are derived for calculating the time constants from the steady-state electron concentration profile. The validity of this method is demonstrated for the second model from which the exact IMPS and IMVS responses are calculated. The three models are compared with each other. A simple expression is derived for calculating the charge-collection efficiency from the measured values of τoc/τsc and the light intensity dependence of τoc.

Introduction Dye-sensitized nanocrystalline TiO2 solar cells with lightto-electrical power conversion efficiencies as high as 10% at AM 1.5 have been reported.1-3 Figure 1 shows the geometry and major transport processes of the most extensively studied cell. It consists of an 8-12 µm nanocrystalline TiO2 layer deposited onto a transparent conducting oxide glass substrate (TCO). The particles of the film are in contact with an electrolyte solution containing iodide and triiodide as a redox relay and are sandwiched by a second plate of glass covered with platinum. Sensitization is accomplished with a monolayer of a Ru-bipridyl-based charge-transfer dye adsorbed onto the TiO2 surface. Because of the large internal surface area of the TiO2 particle film and light scattering effects, the light-harvesting efficiency of the dye monolayer is high. Photoexcitation of the dye leads to electron injection into the conduction band of TiO2. The injected electrons diffuse through the interconnecting network of TiO2 particles and are collected at the transparent conducting glass substrate, where they pass through the external circuit and reenter the cell at the Pt counter electrode to reduce I3- to I- ions. The I- ions diffuse from the counter electrode into the pores of the TiO2 film, where they reduce the oxidized dye molecule, thus regenerating the original form of the dye and completing the oxidation-reduction cycle. The incident photon-to-current conversion efficiency (IPCE) in the spectral region, where the dye absorbs strongly, is reported to be high (ca. 85%).2,4,5 The IPCE can be expressed theoretically as the product of the light absorption efficiency of the dye, the quantum yield of electron injection, and the efficiency

Figure 1. The geometry and major transport processes of a dyesensitized nanocrystalline TiO2 solar cell.

of collecting the injected electrons at the conducting glass substrate η. Considering reflection losses at the glass surface, the reported high value of IPCE implies that the injection and collection efficiencies are close to 100%.2 High-efficiency dyesensitized nanocrystalline TiO2 solar cells show short-circuit photocurrent densities Jsc of about 18 mA/cm2 at 1 sun (AM 1.5).2,3 However, for many cells, Jsc is much lower, indicating a significant decrease of the IPCE. It is difficult to determine which of the three efficiencies (light absorption, charge injection, and charge collection) has decreased. Absorption of light by the dye in practical cells is difficult to measure. Even if the amount of light absorbed by the cell is measured precisely, it is difficult to determine quantitatively the fraction of incident light absorbed by the dye, the electrolyte, and the TiO2. Only

10.1021/jp9831177 CCC: $18.00 © 1999 American Chemical Society Published on Web 01/20/1999

Dye-Sensitized Nanocrystalline TiO2 Solar Cells

J. Phys. Chem. B, Vol. 103, No. 5, 1999 783

light absorbed by the dye is expected to contribute to the shortcircuit photocurrent. Light absorbed by TiO2 generates electrons and holes that can recombine. When light is absorbed by the dye, the excited electron can recombine to the ground state of the dye. This process can also be mediated by TiO2 when the electron is first injected into the semiconductor and then recaptured by the dye. Usually the recapture of an electron by the oxidized dye is prevented by an even faster neutralization of the oxidized dye by the redox electrolyte. When an electron is injected into TiO2 it can either react with the redox electrolyte or enter the external circuit via the charge-collecting conducting glass. Low photocurrent is therefore the result of inefficient light harvesting by the dye, inefficient charge injection into TiO2, or inefficient collection of injected electrons. There is no direct way to determine quantitatively the separate efficiencies of light absorption, charge injection, and charge collection. Intensity modulated photovoltage spectroscopy IMVS6 and intensity modulated photocurrent spectroscopy IMPS7-11 are used to measure the respective time constants for charge recombination at open circuit and the combined processes of charge collection and charge recombination at short circuit in nonsensitized and dye-sensitized nanocrystalline TiO2 cells. If charge collection is much faster than charge recombination, then the charge-collection efficiency will be high because the electrons are collected before they have time to recombine. As the time constants approach each other, the charge-collection efficiency declines. Unfortunately, no analytical expression can be derived that describes how the ratio of the time constants depends on the charge-collection efficiency. In the following, we consider three models for determining the relation between the charge-collection efficiency and the ratio of the time constants for charge recombination and the combined processes of charge collection and charge recombination. The first model does not consider the underlying physical processes measured by IMVS and IMPS. The second model simulates the frequency response of IMVS and IMPS from the time-dependent continuity equation for simplified conditions. The third model determines the time constants for IMVS and IMPS from electronconcentration profiles calculated for constant light intensity and more realistic conditions. For the first time, expressions are derived for approximating the time constants for IMPS and IMVS from the steady-state charge carrier-concentration profile. Theory Model 1 (Linear and Independent Processes). In model 1, it is assumed that the loss of injected electrons via both charge collection (photocurrent) and recombination are proportional to the electron concentration in TiO2 and that the rate constants for these processes are independent of each other. The time constant for charge recombination at open circuit τoc can be measured by IMVS.6 The time constant for charge collection τcc cannot be measured directly but is inferred from the relation between τoc and the time constant for the combined processes at short circuit τsc that is obtained from IMPS measurements.

1 1 1 ) + τsc τoc τcc

(1)

For this case, the charge-collection efficiency η is given by the expression

1 Jsc Jsc τsc τcc η) ) ) )11 Jinj Jsc+ Jr 1 τoc + τcc τoc

(2)

where Jinj is the electron-injection current density from the excited dye to TiO2 and Jr is the recombination-current density. Equation 2 is a rough estimation of η as function of τsc/τoc because recombination and charge collection are neither independent nor linear. Model 2 (Semilinear Model). In model 2, it is assumed that the loss of injected electrons via recombination is proportional to the electron concentration in TiO2 and that the electron diffusion current is proportional to the gradient of the electron concentration. Unlike model 1, the photocurrent of model 2 is not proportional to the electron concentration. An analytical solution of the dependence of η on τoc and τsc must consider charge transport and recombination in the TiO2 film. We assume that only free carriers contribute to electron transport. Considering that electron transport through the TiO2 film occurs by diffusion,7-9,12 that electric fields are negligible, and that free and trapped electrons are in thermal equilibrium,6,9 the transport equation can be expressed as

J(x) ) qDcb*(n)

dncb* dncb* dn dncb dn ) qDcb*(n) ) qDcb dx dn dx dn dx (3)

where n is the sum of the photoinduced electron concentration in the conduction band ncb* and in trap states, q is the unit charge, and Dcb*(n) is the diffusion coefficient of electrons in the conduction band; trap states include both bulk traps and surface states. The probability of an excess electron to move from one nanoparticle ( 0. This leads to an expression that approximates the time constant for IMPS.

τsc )

1 Jsc

n(x)

dJ dJ 1 x* dx ) dx ∫0x* G(x) dx ∫ τ(x) dx Jsc 0

η) R[2R exp(-Rd) - (R - L-1) exp(L-1d) - (R + L-1) exp(-L-1d)] [1 - exp(-Rd)][L-2 - R2][exp(L-1d) + exp(-L-1d)] (17)

For simulations, we can set mDcb ) 1 and d ) 1 because the units of time and length can be chosen independently. Also, inasmuch as both ∫n(x) dx and ∫G(x) dx are proportional to the light intensity, the time constants (eqs 11 and 16) for model 2 do not depend on φIo. Thus, it is noteworthy that only two adjustable parameters (kr and R) remain in model 2. The chargecollection efficiency (eq 17) can be expressed in terms of the rate constant for recombination kr (recall that L ) xmDcb/kr) for a given R. It can be shown that the ratio τoc/τsc approaches 2 when the charge-collection efficiency goes to 0 and that τoc/ τsc can be approximated by η/(1 - η) when η ≈ 1. Model 2 does not, however, explain the observation that the measured τoc and τsc depend strongly on the light intensity. This deficiency in model 2 is addressed in model 3 below. Model 3 (Nonlinear Model). Model 3 takes into account the observations6 that the rate of recombination does not depend linearly on the electron concentration and that ncb/n depends on n. When the cell has a low absorption coefficient for the incident light, the charge-generation term G is virtually independent of x, and at open circuit, diffusion processes can be neglected. Under these conditions, n is essentially independent of x, and one can determine the dependence of the rate constant for recombination on the electron concentration in TiO2 from IMVS measurements at different light intensities. The dependence of ncb on n can be deduced from the dependence of τsc on Io. IMVS and IMPS measurements show, respectively, that6-8,11

(18a)

(18b)

where m1 and m2 are experimentally accessible constants. For most cells, Jsc is proportional to Io, and Io can be replaced by Jsc in eq 18. Figure 2 illustrates the dependence of Jsc on Jscτim oc and Jscτim sc . At open circuit, the rate of recombination ∫R(n) dx is proportional to the light intensity (∫R(n)dx ∼ Io). Recalling that n is independent of x and that Ioτoc ∼ n, one can conclude with the aid of eq 18a that

R(n) ) kronm1

(19)

where kro is a constant. At short circuit, to a first approximation, n ∼ Io1/m2 and ncb ∼ Io. From eq 18b, one can conclude that

(16)

where x* is the smallest positive value for which J(x) ) 0. At constant light intensity, the electron-concentration profile and Jsc are obtained with eqs 6, 13, and 14 by setting ω to 0 in both A and γ. Equation 16 provides the first description for approximating the time constants, obtained from the IMPS response, by using only the steady-state electron-concentration profile. At constant light intensity, the term 1/γ becomes the diffusion length of electrons L. To obtain η, Jscq-1 (eq 14) must be divided by the integral of the charge-generation term ∫G(x) ) ∫φIoR exp(-Rx) dx ) φIo[1 - exp(-Rd)] across the TiO2 film.

Io ∼ (Ioτoc)m1

Io ∼ (Ioτsc)m2

ncb ) nonm2

(20)

Substituting dncb/dn ) nom2 n(m2 - 1) and d(dncb/dn)/dx ) (nom2)(m2 - 1) (dn/dx) n(m2 - 2) from eq 20 and the recombination term R(n) (eq 19) into the modified continuity equation (eq 4) yields

[

]

dn d2n m2-1 dn dn n + (m2 - 1) nm2-2 + ) Dcbnom2 2 dt dx dx dx G(x) - kronm1 (21) For exponential light absorption across the cell (G(x) ) RφIoexp(-Rx)), eq 21 transforms to

dn ) n′′[Dcbnom2n(m2-1)] + n′n′[Dcbnom2(m2 - 1)n(m2-2)] + dt RφIoexp(-Rx) - kronm1 (22) where n′ ) dn/dx and n′′ ) d2n/dx2. Under stationary conditions, dn/dt ) 0, and one obtains

n′′ ) -C1(n′)2n-1 - C2 exp(-Rx)n(1-m2) + C3n(1+m1-m2) (23) where C1 ) (m2 -1), C2 ) RφIo/(Dcbnom2), and C3 ) kro/(Dcbnom2). Because eq 23 cannot be solved analytically, numerical simulations must be used to obtain the carrierconcentration profile n(x) under the boundary conditions for open circuit (eq 7) and short circuit (eq 12). From the carrierconcentration profiles, all other functions (J(x), dJ/dx) necessary for determining τoc (eq 11) and τsc (eq 16) can be calculated. In contrast to model 2, the diffusion length L of model 3 depends on x and can be expressed as

L(x) ) xDcbτcb )

x

n ncb Dcb ) kronm1 n

x

Dcbno (m2-m1) n (24) kro

where τcb is the lifetime of electrons in the conduction band before they recombine. The lifetime of electrons in the conduction band is expressed as the product of the lifetime of the electrons in the TiO2 particles (n/R(n)) and the probability of these electrons being in the conduction band (ncb/n). Thus the upper boundary of the integral in eq 11 must be modified to be independent of x. For model 3, we choose the upper limit of the integrals of eq 11 to be the largest x for which L*(x) ) d[1 - exp(-2.5L(x)/d)]/[1 + exp(-2.5L(x)/d)] > x. Equation 24 predicts that L depends on n and therefore also on Io. For cells fabricated in our laboratories, the exponent m2 - m1 is usually small. Considering that at short circuit n ∼ Io1/m2, the change of

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Figure 2. Relation of the short-circuit photocurrent density Jsc to the product of Jsc and the time constants obtained from the imaginary part im of the measured IMVS τim oc (eq 10) and IMPS τcs (eq 15) response for the same solar cell. The solid lines are fits according to the expressions im m2 m1 Jsc ) M1(Jscτim oc ) and Jsc ) M2(Jscτsc ) , where M1, M2, m1, and m2 are -4 fit parameter. M1 ) 6.37 × 10 , m1 ) 2.18, M2 ) 1.58 × 10-2, m2 ) 2.64. The solar cell consisted of [RuL2(NCS)2] (L ) 2,2′-bipyridyl4,4′-dicarboxylic acid)-coated 8-µm-thick nanocrystalline TiO2 electrode in CH3CN containing I2 (50 mM) and LiI (0.6 M).

the diffusion length with light intensity is typically about L ∼ Io0.1 The small change of L is not expected to have a significant influence on the collection efficiency. Mirror Effect on G(x). It is noteworthy that for efficient solar cells, a reflective back surface (mirror) is used. The chargegeneration profile in the presence of a mirror G*(x) will be the superpositon of the absorption of the incoming and outgoing light and can be estimated by G*(x) ) φIo*R* [exp(-R*x) + exp(R*x - 2R*d)], where Io* is the incident photon flux density and R* is the light absorption coefficient. In our calculations, G*(x) is approximated by G(x) ) φIoR exp(-Rx). A reasonable approximation of G*(x) can be obtained when the condition G*(0)/G*(d) ) G(0)/G(d) is fulfilled. From this condition, one obtains the dependence of R on R*.

(

)

1 1 + exp(-2R*d) R ) ln d 2 exp(-R*d)

(25)

The fraction of absorbed light can be calculated from the expressions

Iabs ) 1 - exp(-2R*d) Ioo* Iabs ) 2[1 - exp(2Rd) + exp(Rd)xexp(2Rd) - 1] Ioo

(26a)

(26b)

where Ioo is the photon flux entering the TiO2 film. Although eq 25 does not represent a least-mean-square fit of G*(x) with an exponential function, the condition G*(0)/G*(d) ) G(0)/ G(d) yields an analytical expression for R. Figure 3 shows the dependence of the absorption coefficient of the cell on the fraction of light absorbed without a mirror [Iabs/Ioo ) 1 exp(-Rd)] and with a mirror (eq 26). In the presence of a mirror, for Rd ) 1, more than 96% of the light incident on the dyecovered TiO2 film is absorbed. The Rd value of high efficiency dye-sensitized TiO2 solar cells (with a mirror) will not be much larger than 1 because of several limitations. For Rd ) 1, a further increase of cell thickness will lead to only a moderate increase of absorbed light. However, a thicker cell will increase the distance of electron and ion diffusion across the cell, leading to a lower fill factor. A thicker cell will also increase the surface

Figure 3. Relation of the absorption coefficient of the solar cell to the fraction of light absorbed in the absence and in the presence of a mirror at the back. The cases with a mirror are calculated with eq 26.

area of the TiO2 film and therefore will increase the number of recombination centers, resulting in a lower Voc. Experimental Section Numerical Simulations. For model 2, the frequency response of the cell was calculated for ω ranging from 10-5 to 105 in 100-equidistant steps on the logarithmic scale. The cell thickness d and the expression mDcb were set equal to unity. To calculate the respective IMVS (eq 9) and IMPS (eq 14) response, φIoR and qφIoR were also set equal to unity. Simulations were made for values of R ) 0.1, 0.5, 1.0, 2.0, and 3.0. The value of kr was varied to yield a charge-collection efficiency of 10%, 20%, ..., 90%, and 95%. For R ) 0.5, a 5% increment of the chargecollection efficiency was used. The frequency response was fitted according to eqs 10 and 15. For model 3, a Runge-Kutta-Nystro¨m algorithm15 was used for calculating n(x) and n′(x) (eq 23). Double precision variables were used for all simulations. The cell thickness d and the term Dcbnom2 were set equal to unity. The term φIo was set equal to 1/[1 - exp(-R)] in order that ∫10, G(x)) dx ) φIo[1 exp(-R)] across the TiO2 film equals one. At short circuit, one boundary condition was n′(d) ) 0. The second boundary condition (n(0) ) 0) was approximated by adjusting n(d) until n(0) was smaller than 10-3 n(d). At short circuit, a step width of 1/3000 was employed. For calculating the electron-concentration profiles at open circuit, n′(0) was set equal to 0, and n(0) was adjusted until J(d) was smaller than 10-4 Jinj. At open circuit, the step width was increased to 1/1000. For both open circuit and short circuit, decreasing the step width to 1/10000 did not yield significant differences in the resulting profiles, charge-collection efficiencies, and time constants. For each parameter set, different charge-collection efficiencies were obtained by using kro ) 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1.0, 2.0, 5.0, 10.0, 20.0, and in several cases, 50.0. In the case of dyesensitized TiO2 solar cells fabricated in our laboratory,6,11 the values of m1 typically range between 1.5 and 3.0 with an average of about 2.2; the values of m2 usually lie between 2.0 and 3.5 with an average of about 2.7. The standard set of parameters employed was R ) 1, m1 ) 2.2, and m2 ) 2.7. Only one of these parameters was changed at a time. Other values used were R ) 0.3 and 3, m1 ) 1.5 and 3.0, and m2 ) 2.0 and 3.5. Cell Fabrication. Details of dye-sensitized nanocrystalline TiO2 solar cell fabrication are given elsewhere.16 The solar cell consisted of [RuL2(NCS)2] (L ) 2,2′-bipyridyl-4,4′-dicarboxylic acid)-coated 8-µm-thick nanocrystalline TiO2 electrode in CH3CN containing I2 (50 mM) and LiI (0.6 M).

Dye-Sensitized Nanocrystalline TiO2 Solar Cells

Figure 4. Comparison of the calculated IMVS and IMPS response with fits using a single time constant for model 2. (a) The calculated real and imaginary parts of the IMVS frequency response according to eq 9 with parameters R ) 0.5, φIo ) 2, d ) 1, kr ) 0.37, and mDcb ) 1 (Note: γ ) (kr + iω)1/2). The theoretical charge-collection efficiency for these parameters is 90%. The solid lines show the fits of the frequency response according to eq 10. These fits yield parameters X1 im ) 1.0712, τre oc ) 2.6720, X2 ) 1.0653, and τoc ) 2.6926. (b) The calculated real and imaginary parts of the IMPS frequency response according to eq 14 with parameters R ) 0.5, qφIo ) 2, d ) 1, kr ) 0.37, and mDcb ) 1 (Note: γ ) (kr + iω)1/2). The theoretical chargecollection efficiency for these parameters is 90%. The solid lines show the fits of the response according to eq 15. These fits yield the im parameters X3 ) 0.7019, τre sc ) 0.2080, X4 ) 0.5847, and τsc ) 0.2781.

Results and Discussion Neither the IMPS8,9c nor IMVS response of dye-sensitized TiO2 solar cells is expected to show an ideal semicircle in the complex plane, even for the simplified conditions (ncb/n ) constant and R ∼ n) in model 2 (eqs 9 and 14). However, the real or imaginary part or both parts of the frequency response can be fitted by using an equation for a single time constant for the response.17 In this case, the fitting procedure is described by eqs 10 and 15. In Figure 4a, the IMVS response for the semilinear model (model 2) shows virtually no deviation between the fit and the calculated frequency response. In contrast, Figure 4b shows that the fit of the IMPS response is less ideal. Furthermore, τre sc and τim sc do not have the same value. Because the time constants for IMPS and IMVS have never been determined from the steady-state electron-concentration profile, the validity of eqs 11 and 16 was examined for the semilinear model (model 2) for which the frequency response can be calculated analytically (eqs 9 and 14). In Figure 5, the time constant τoc (eq 11), obtained from the steady-state electronconcentration profile, is compared with the time constants im obtained from the real part τre oc and the imaginary part τoc of the frequency response (eq 10). These time constants can be approximated by the reciprocal of the rate constant for recombination 1/kr, and for the reason of scaling, the product τkr is shown in Figure 5. Only for high absorption coefficients R and moderate charge-collection efficiencies η, major deviations between time constants and 1/kr are observed. However, it can im be seen that for a given η and absorption coefficient, τre oc, τoc , and τoc are about the same. For all investigated absorption coefficients, the deviation between τoc and the time constants im τre oc and τoc is below 6%. These results indicate that eq 11 can indeed be used to determine the time constant from the steadystate electron-concentration profile.

J. Phys. Chem. B, Vol. 103, No. 5, 1999 787

Figure 5. The products of the recombination rate constant kr and time im constants τre oc and τoc vs the charge-collection efficiency for model 2. re im τoc and τoc are obtained from respective fits of re(∆Voc) and im(∆Voc) according to eq 10. The curves re(∆Voc) and im(∆Voc) are calculated from eq 9 for d ) mDcb ) 1 and combinations of R and kr. Values of R are given in the figure. The charge-collection efficiency is calculated from eq 17 for each set of parameters (d, mDcb, R, and kr). The solid lines show the product τoc kr, where τoc was calculated from the steadystate electron-concentration profile (constant light intensity, ω ) 0) according to eq 11 with the same parameters used to calculate re(∆Voc) and im(∆Voc).

im Figure 6. Dependence of time constants τre sc and τsc on the chargere im collection efficiency for model 2. τsc and τsc are obtained from the respective fits of re(∆Jsc) and im(∆Jsc) according to eq 15. The re(∆Jsc) and im(∆Jsc) curves are calculated from eq 14 for d ) mDcb ) 1 and combinations of R and kr. Values of R are given in the figure. The charge-collection efficiency is calculated from eq 17 for each set of parameters (d, mDcb, R, and kr). The solid lines show τsc calculated from the steady-state electron and photocurrent profiles (constant light intensity, ω ) 0) according to eq 16 with the same parameters used to calculate re(∆Jsc) and im(∆Jsc).

im Figure 6 shows τre sc, τsc , and τsc for various charge-collection efficiencies for model 2. Typically, τre sc is about 30% less than τim sc . For high light-absorption coefficients and high chargecollection efficiencies, the deviation increases up to about 50%. However, τsc is always close to τim sc , and the deviation is typically not more than 5% and never exceeds 11% in our simulations. These results indicate that eq 16 can be used to approximate τim sc . Figure 7 shows the dependence of τoc/τsc on the chargecollection efficiency for the semilinear model (model 2) for different absorption coefficients. It can be shown analytically that, as η becomes very small, τoc/τsc approaches 2. For low charge-collection efficiencies (η e 50%), the variation of τoc/τsc with η is small. For high charge-collection efficiencies (η > 50%), τoc/τsc follows, to a good approximation, the relation

τoc/τsc ≈ b1(1 - η)b2

(27)

Values of the parameters b1 and b2 for different light-absorption coefficients are given in Figure 7. For high charge-collection efficiencies (η g 95%), τoc/τsc approaches 1/(1-η) At open circuit, all carrier-concentration profiles (n(x)) are more or less flat, except at low charge-collection efficiencies and high light-absorption coefficients. Figure 8a shows how the nonlinearity of model 3 affects the carrier-concentration profile

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Figure 7. Dependence of τoc/τsc on the charge-collection efficiency for various absorption coefficients R for model 2. The solid lines represent a fit for η g 55% according to τoc/τsc ) b1(1 - η)b2. The fit parameter b1 was 1.16, 1.18, 1.22, 1.61, and 2.14, and the parameter b2 was -0.953, -0.936, -0.934, - 0.842, and -0.764 for the respective R ) 0.1, 0.5, 1.0, 2.0, and 3.0.

Figure 9. Dependence of calculated time constants τoc on kro for model 3. The procedure and conditions for simulations are described in the Experimental Section. Standard parameters (φIo ) [1 - exp(-R)] -1, d ) 1, R ) 1.0, m1 ) 2.2, and m2 ) 2.7) were used unless indicated otherwise. The hollow circles were obtained from the same calculations with the standard parameters. The solid lines correspond to kro-1/m1.

vicinity of the charge-collecting electrode.

τoc ≈ Figure 8. (a) Electron-concentration profiles and (b) photocurrent density profiles calculated for model 2 and model 3. For both models and profiles, the conditions used were R ) 0.5, d ) 1 and the rate constants for recombination were adjusted to produce a chargecollection efficiency of 44.7%. The electron concentration at x ) d was normalized to unity. For model 3, m1 ) 2 and m2 ) 3.5 were used.

at short circuit. For the conditions described in Figure 8, the carrier-concentration profile of the nonlinear model (model 3) is more square shaped than that of the semilinear model (model 2). For a constant photocurrent density across the cell, the slope of the carrier-concentration profile is constant for model 2. In contrast, for model 3, when the term m2 (eq 20) is larger than 1, the slope decreases with increasing electron concentration. A comparison of the photocurrent profiles (Figure 8b) calculated for model 2 and model 3, with the same absorption coefficient and charge-collection efficiency, shows they are almost the same. Because model 2 is a special case of model 3 with m1 ) 1 and m2 ) 1 (cf. eqs 5 and 21), one can conclude that the photocurrent profile is not significantly influenced by m1 and m2 in model 3 for the same charge-collection efficiency. For cells with a high charge-collection efficiency, the time constant at open circuit in model 3 depends on the average rate of electron injection throughout the cell (∫G(x)/d ) φIod-1 (1 - exp(-Rd)) and on the dependence of the recombination rate on the electron concentration (m1 and kro, eq 19).

τoc ≈

[] [ 1 kro

1/m1

]

φIo (1 - exp(-Rd)) d

(1/m1-1)

(28a)

For low charge-collection efficiencies, ∫G(x)/d must be replaced by G(0) ) φRIo, the electron-injection rate in the

[] 1 kro

1/m1

[φRIo](1/m1-1)

(28b)

In the simulations, the term [φIod-1 (1 - exp(-Rd)] was set equal to unity in order to keep the rate of charge generation independent of R. When Rd , 1, τoc calculated from either eq 28a or eq 28b yields approximately the same value, and the proportionality factor between τoc and kro-1/m1 is about the same for all kro. Figure 9a shows this proportionality for R ) 0.3. As R increases, the τoc versus kro plot deviates negatively from a straight line at high kro values. With increasing kro, the chargecollection efficiency declines, and over some range of kro, the transition from eq 28a to eq 28b can be observed. For kro g 10, the ratio of τoc to kro-1/m1 is close to the theoretical value of (φRIo)(1/m1-1) (eq 28b). As expected, m1 (Figure 9b) strongly influences the dependence of τoc on kro, whereas m2, which influences the electron-diffusion process, has no significant effect on it (Figure 9c). Figure 10 shows the dependence of τoc on the chargecollection efficiency of the cell for model 3. The time constant τoc was calculated from the electron-concentration profile n(x) at open circuit, and η was calculated from n(x) at short circuit for the same cell parameters (kro, m1, and m2) and the same electron generation profile G(x). For kro values yielding high charge-collection efficiencies, the relation of τoc on η is independent of R and m2 but varies strongly with m1. For kro values giving low charge-collection efficiencies, the relation of τoc to η depends on R and m2 but is independent of m1. Figure 11 illustrates the dependence of τsc on the chargecollection efficiency of the cell for model 3. Figure 11a shows that τsc decreases with increasing R for a given charge-collection efficiency. The decline of τsc with increasing R is consistent with a decrease of the average distance that an injected electron must diffuse to the charge collecting electrode. It is seen in Figure 11b that m1 has no effect on the relation of τsc to the

Dye-Sensitized Nanocrystalline TiO2 Solar Cells

Figure 10. Dependence of calculated time constants τoc on the chargecollection efficiency for model 3. The procedure and conditions for simulations are described in the Experimental Section. Standard parameters (φIo ) [1 - exp(-R)] -1, d ) 1, R ) 1.0, m1 ) 2.2, and m2 ) 2.7) were used unless indicated otherwise. The hollow circles were obtained from the same calculations with the standard parameters.

Figure 11. Dependence of calculated time constants τsc on the chargecollection efficiency for model 3. The procedure and conditions for simulations are described in the Experimental Section. Standard parameters (φIo ) [1 - exp(-R)] -1, d ) 1, R ) 1.0, m1 ) 2.2, and m2 ) 2.7) were used unless indicated otherwise. The hollow circles were obtained from the same calculations with the standard parameters.

charge-collection efficiency. Because the diffusion conditions (including m2) in Figure 11b are fixed, the photocurrent profile determines the electron-concentration profile. For a given η, the photocurrent profile is relatively independent of m1 as shown in Figure 8b. It follows therefore that the electron-concentration profile is also independent of m1 for a given η. For the same photocurrent and electron concentration profile, eq 16 predicts the same value for τsc. Figure 11c shows that, as m2 increases,

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Figure 12. Dependence of τoc/τsc on the charge-collection efficiency η for model 3. The procedure and conditions for simulations are described in the Experimental Section. Standard parameters (φIo ) [1 - exp(-R)] -1, d ) 1, R ) 1.0, m1 ) 2.2, and m2 ) 2.7) were used unless indicated otherwise. The hollow circles were obtained from the same calculations with the standard parameters.

τsc becomes larger for a given charge-collection efficiency. The increase of τsc is related to the stronger dependence of the diffusion term on n with increasing m2 in the modified transport equation (eq 21). As discussed in connection with Figure 8b, for the same charge-collection efficiency, the photocurrent profile is not significantly affected by m1 and m2. With increasing m2, the electron-concentration profile becomes more square shaped for the same photocurrent profile, resulting in a larger value of n(x)/G(x) (eq 16) and a higher value of τsc. Figure 12 shows that τoc/τsc varies only moderately with the charge-collection efficiency for η e 50% for model 3. Consequently, the estimation of the charge-collection efficiency from τoc/τsc will have a large margin of error for values of η less than 50%. It can be seen in Figure 12 that τoc/τsc exhibits a small increase with increasing R and decreasing m1 and m2. These variations of τoc/τsc are small compared to the individual changes of τoc and τsc with R and m2 observed in Figures 10 and 11. At low charge-collection efficiencies, τoc/τsc approaches values between 1.3 and 1.5 which contrasts with the situation for model 2 in which τoc/τsc approaches 2 (Figure 7). Figure 13 displays the dependence of τoc/τsc on the charge recombination efficiency (1 - η) and the charge-collection efficiency for η g 50% for model 3. Over this range of chargecollection efficiencies, the dependence of τoc/τsc on η can be described by a power law (eq 27), leading to a straight line in the double-logarithmic plot of τoc/τsc vs 1 - η. The parameters b1 and b2 are virtually independent of m2 (Figure 13c) and, to some extent, independent of R (Figure 13a). However, when Rd ) 3, the fit differs from those for R ) 0.3 and R ) 1.0. Only for charge-collection efficiencies above 90% is the dependence of τoc/τsc on (1 - η) essentially the same for values of R e 3. The parameter b2 depends strongly on m1, which describes how fast the rate of recombination increases with the electron concentration. Values of b1 and b2 are given in Table 1. The relation τoc/τsc ≈ 1.2(1 - η)-0.9/m1 yields, to a good approximation, the dependence of τoc/τsc on η. In our experience, the most precise method to determine m1 is from the relation

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Schlichtho¨rl et al. tion (eq 19) in model 3. At low charge-collection efficiencies, the nonlinear dependence of the diffusion term (eqs 3 and 21) also contributes to the decrease of τoc/τsc values in model 3. Conclusions

Figure 13. Dependence of τoc/τsc on the recombination efficiency (1 - η) and the charge-collection efficiency η for model 3. The procedure and conditions for simulations are described in the Experimental Section. Standard parameters (φIo ) [1 - exp(-R)] -1, d ) 1, R ) 1.0, m1 ) 2.2, and m2 ) 2.7) were used unless indicated otherwise. The hollow circles were obtained from the same calculations with the standard parameters. The solid lines represent a fit according to τoc/τsc ) [b1(1 - η)-b2]. The fit parameters are given in Table 1.

TABLE 1: Fit Parameters Describing the τoc/τsc Dependence on η (eq 27) for Model 3a,b R ) 0.3 R ) 1.0 R ) 3.0 m1 ) 1.5 m1 ) 3.0 m2 ) 2.0 m2 ) 3.5 b1 1.176 1.222 1.676 1.274 1.222 1.198 1.210 b2 -0.4058 -0.3996 -0.3452 -0.5910 -0.2860 -0.4155 -0.3962 b

a Unless indicated otherwise, R ) 1.0, m ) 2.2, and m ) 2.7. 1 2 Figure 13 compares the fit and calculated values of τoc/τsc.

The charge-collection efficiency of nanocrystalline dyesensitized TiO2 solar cells can be estimated for reasonable charge-collection efficiencies η (g50%) from combined IMVS and IMPS measurements. The IMVS and IMPS are used to measure the respective time constants for charge recombination at open circuit τoc and charge collection at short circuit τsc. For low charge-collection efficiencies, the variation of τoc/τsc with the charge-collection efficiency is too small to obtain reliable values of η. For estimating the charge-collection efficiency, the parameters τoc, τsc, and m1 must be determined. The time constant τoc can be obtained by fitting either the real or imaginary part of the IMVS response (eq 10). To obtain τsc, as defined in eq 16, the imaginary part of the IMPS response must be fitted according to eq 15b. For estimating m1, which describes the nonlinearity of the recombination rate on the electron concentration in the TiO2 film (eq 19), τoc must be determined from the IMVS response at different bias-light intensities. In our measurements, a very good linear relation between log(Jsc) and log(Jscτoc) is always found (e.g., Figure 2), and the parameter m1 can be calculated as the slope d log(Jsc) /d log(Jscτoc). With τoc/τsc and m1, the charge-collection efficiency can be calculated using eq 27. The simulations to obtain τoc show that for a light absorption coefficient with Rd e 3 (d equals the TiO2 film thickness) and charge-collection efficiencies above 50%, the time constant is independent of R (Figure 10). Under these conditions, the electron-concentration profile at open circuit is essentially constant, and it is sufficient to consider the recombination current (Jr ) Jinj) and the accumulated charge (Jinjτoc) to determine the dependence of the rate of recombination on the electron concentration. Only for a high absorption coefficient and low η must one evaluate the electron-concentration profile and the distribution function of the recombination current across the TiO2 film. Acknowledgment. We are grateful to Dr. Barton Smith and Dr. Ehud Poles at the National Renewable Energy Laboratory for their valuable discussions. This work was supported by the Office of Basic Energy Sciences, Division of Chemical Sciences, U.S. Department of Energy, under Contract DE-AC3683CH10093. References and Notes

Figure 14. Dependence of τoc/τsc on the charge-collection efficiency for models 1 to 3. For model 2, R ) 1.0 and d ) 1 were used. For model 3, standard parameters (φIo ) [1 - exp(-R)] -1, d ) 1, m1 ) 2.2, and m2 ) 2.7) were used.The following fit functions and validity ranges are used: model 1, τoc/τsc ) 1.0 (1 - η)-1.0(η g 0%); model 2, τoc/τsc ) 1.2 (1 - η)-0.93(η g 50%); model 3, τoc/τsc ) 1.2 (1 η)-0.9/m1(η g 50%).

between the measured time constant at open circuit and the light intensity: Jsc ∼ (Jscτoc)m1 (Figure 2). Figure 14 compares the dependence of τoc/τsc on η for the three models. Both model 1 and model 2 deviate significantly from model 3, which uses more realistic conditions. Thus neither model 1 nor model 2 describes correctly the τoc/τsc dependence on η. The τoc/τsc values obtained with model 3 are lower than those calculated with model 2. At high charge-collection efficiencies, this is caused predominantly by the nonlinear dependence of the recombination rate on the electron concentra-

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Dye-Sensitized Nanocrystalline TiO2 Solar Cells (10) Goossens, A.; Boschloo, G. K.; Schooman, L. Mater. Res. Soc. Symp. Proc. 1997, 452, p 607(Advances in Microcrystalline and Nanocrystalline Semiconductors, 1996). (11) Schlichtho¨rl, G.; Frank, A. J. Unpublished results. (12) (a) Solbrand, A.; Lindstro¨m, H.; Rensmo, H.; Hagfeldt, A.; Lindquist, S.-E.; So¨dergren, S. J. Phys. Chem. B 1997, 101, 2514. (b) So¨dergren, S.; Hagfeldt, A.; Olsson, J.; Lindquist, S.-E. J. Phys. Chem. 1994, 98, 5552. (13) Sze, S. M. Physics of Semiconductor DeVices, 2nd ed.; WileyInterscience: New York, 1981; p 51. (14) The time constant τoc is not very sensitive to the exact shape of the function L*(L,d), where L is the diffusion length of electrons and d is

J. Phys. Chem. B, Vol. 103, No. 5, 1999 791 the thickness of the TiO2 film. The requirements for L*(L,d) are that L* increases monotonically with L, L* is about equal to L when L is much smaller than d, and L* approaches d when L is much larger than d. Thus L*(L,d) ) min(L,d) would exhibit similar behavior for τoc. (15) Kreyszig, E. AdVanced Engineering Mathematics, 6th ed.; WileyInterscience: New York, 1988; p 1078 (16) Huang, S. Y.; Schlichtho¨rl, G.; Nozik, A. J.; Gra¨tzel, M.; Frank, A. J. J. Phys. Chem. B 1997, 101, 2576. (17) There are several methods to extract the time constants from the IMPS and IMVS response. The IMPS and IMVS response can be fitted by using the same scale factor and the same time constant for the real and imaginary part of the response or by using the phase shift of the response.