Influence of Grain Morphology on Electron Transport in Dye Sensitized

Finding an effective diffusion coefficient that gives good agreement with the ...... On global energy scenario, dye-sensitized solar cells and the pro...
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J. Phys. Chem. B 2003, 107, 113-119

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Influence of Grain Morphology on Electron Transport in Dye Sensitized Nanocrystalline Solar Cells M. J. Cass,† F. L. Qiu,† Alison B. Walker,*,† A. C. Fisher,‡ and L. M. Peter‡ Departments of Physics and of Chemistry, UniVersity of Bath, Bath, BA2 7AY, United Kingdom ReceiVed: August 20, 2002; In Final Form: October 31, 2002

We present a Monte Carlo model of electron transport in mesoporous films of TiO2 particles in which electrons execute a random walk through a chain of spherical grains with traps at the surface of the grains. This has been used to simulate transient photocurrents in dye sensitized nanocrystalline solar cells. By comparing our results with a model based on solutions of the continuity equation for the free and trapped electron densities in which the film is treated as a homogeneous medium, we find that necks between grains have a noticeable effect on reducing the photocurrent. Values of an effective electron diffusion coefficient have been deduced by comparing an analytical solution to the continuity equation for the free electrons, in which the traps and back-reaction are ignored, with the numerical results from the Monte Carlo and continuum models. To the authors' knowledge, this is the first time that the influence of the grain connectivity on electron transport has been modeled.

1. Introduction Dye sensitized nanocrystalline solar cells (Gra¨tzel cells) have shown great promise because they are composed of low cost materials and perform well under indirect radiation. An overall solar to electrical energy conversion efficiency of 10% has been obtained.1 They are based on a thin film of porous nanocrystalline TiO2 coated with a dye using a ruthenium complex that harvests the solar energy flux. Photoexcitation of the dye is followed by fast electron injection into the conduction band of the TiO2 nanoparticle electrode. The electrons then travel through the TiO2 to the collecting contact. The oxidized dye reacts with I- ions in the electrolyte, producing I3-, which diffuses to the Pt coated cathode where it is reduced to complete the cycle. The efficiency with which the incident photon flux is converted into current (the incident photon to current conversion efficiency IPCE) is determined in part by the competition between electron transport and back-reaction. Optimization of electron collection is therefore clearly important for any practical cell, regardless of whether it employs an electrolyte or another contacting phase such as a polymeric hole conductor. Electron transport in the TiO2 grains occurs by diffusion because the fields across the film are screened by the electrolyte and the small size of the particles does not support a built in field.2 An electron diffusion coefficient Dbare of 1 × 10-2 cm2 s-1 has been deduced from mobility data for single crystalline TiO2.3 The effective diffusion coefficient for electrons Deff for anatase TiO2 grains in Gra¨tzel cells has been deduced from IMPS and intensity modulated photovoltage spectra (IMVS).4-11 In ref 10 Deff is deduced for both anatase and rutile TiO2 grains from IMPS spectra. For example, Deff has been shown to vary with the background illumination intensity I0 as I00.68.9 Furthermore, the transit time for electrons across the films is also sensitive to I0, requiring several minutes at very low light and milliseconds at solar illumination intensities.4,12 The major * Corresponding author. E-mail: [email protected]. † Department of Physics. ‡ Department of Chemistry.

reason for such behavior is trapping and detrapping of the electrons. The detrapping time increases exponentially with the trap depth. As I0 increases, so does the generated electron density, and thus the traps fill up, so the electrons are only trapped in the shallower traps and they detrap much faster. Duffy et al.12,13 have observed that the intensity dependence of the electron lifetime measured from photovoltage transients appears to follow second order kinetics in which the rate of back-reaction varies as the square of the total (free + trapped) electron concentration. They argue that this result is either due to the mechanism of the two electron reduction of I3-, or to trapping, which will influence electron transport to surface reaction sites which could themselves be traps. Park et al.10 found that the crystalline structure of the TiO2 particles has a strong influence on the photoelectrochemical properties of Gra¨tzel cells through altering the morphology of the film; the rutile film is composed of rod shaped particles with an average size of 20 × 80 nm without a preferred orientation. It is possible that the rutile rods stack less efficiently than the anatase spheres, and so have a lower number of connections per volume and thus a smaller number of pathways for each electron from injection to the anode. By showing that differences in the number of surface states is not responsible for the variation in Deff between the films, the authors of ref 10 argued that the differences in connectivity of the rutile film and the anatase film acounted for the order of magnitude difference in Deff deduced from their measured intensity modulated photocurrent (IMPS) spectra. This result has implications for fabricating efficient solar cells to the extent that the morphology will vary. Recently, a new method has been developed for manufacturing a nanostructured porous layer of a semiconductor material on a conducting plastic substrate through compressing the particles that could reduce the costs of fabrication of Gra¨tzel cells and allow the possibility of large and flexible electrodes.14,15 The grain connectivity is likely to differ from the connectivity in cells in which the grains are sintered. Here, we show how the transient photocurrent response to a short illumination pulse can be used to identify how electron

10.1021/jp026798l CCC: $25.00 © 2003 American Chemical Society Published on Web 12/04/2002

114 J. Phys. Chem. B, Vol. 107, No. 1, 2003 transport is sensitive to the morphology in addition to trapping and back-reaction. Measurements of these transients have been interpreted by fitting Deff when comparing a measured photocurrent transient to a solution of the time dependent diffusion equation without an explicit mention of trapping/detrapping or back-reaction processes.16 In ref 16, it was shown that charge transport is sensitive to trapping/detrapping and to the electrolyte concentration. Cao et al.4 made measurements of photocurrent transients when turning on illumination through the substrate side and interpreted these data with a solution of the continuity equation assuming that Deff varies linearly with the electron density n(x), where x is the distance from the anode. They argued that there is a fast component due to injection of electrons close to the contact and a slow component related to the buildup of the concentration gradient to the steady state. The sensitivity of Deff deduced from measured photocurrent transients to the electrolyte composition and concentration has been explored by Nakade et al.17,18 A difficulty with large amplitude pulses is that as n changes, so does Deff, introducing nonlinearity in the continuity equation for free electrons and making interpretation of the transients more difficult. Duffy et al. addressed this problem by using small amplitude pulses superimposed on a steady background illumination that can be varied to obtain estimates of Deff(n).12 The back-reaction kinetics has also been looked at with transient absorption spectroscopy where the decay of the photoinduced cation following excitation by a laser is monitored.21 Kopidakis et al.22 use small amplitude pulses with an optical bias and interpret their photocurrent transients with an expression for Deff that takes into account diffusion of the electrolyte ions through an ambipolar model. They note that Deff measured in this way is about 10 times larger than Deff measured using large amplitude pulses. Multiple trapping in which transport occurs in the conduction band, punctuated by a series of trapping and detrapping events, and hopping models in which electrons hop between traps and any detrapping is ignored are commonly used to explain carrier transport in amorphous materials.23 Simulations on electron transport in Gra¨tzel cells with the multiple trapping are based on either solutions of the continuity equation5,9,24,25 or a Monte Carlo simulation, in which the electrons execute a random walk.21,26,27. Hopping models using Monte Carlo methods have been used to interpret transient data in Gra¨tzel cells in refs 11 and 28. For these calculations, we have adopted a multiple trapping approach. To understand the influence of the morphology on electron transport, we have compared predictions of the transient current from a Monte Carlo simulation, in which the electrons execute a random walk through a chain of spherical grains with traps at the surface of the grains, with results from a continuum model. In the latter model, the porous TiO2 grains are treated as a homogeneous material in which the 1-dimensional continuity equations for free electrons (with a diffusion coefficient equal to Dbare) and trapped electrons are solved. Only in a few cases have traps been considered explicitly in this way; see refs 24, 25, and 27 and other references by those authors. In ref 29, traps were allowed for by relating the trapped and free electron densities through Boltzmann statistics. In both our Monte Carlo and continuum models, the extent to which traps are occupied determines whether electrons are allowed to trap. Given the importance of understanding the loss of electrons through back-reaction with the I3- ions noted above, a loss term that is first order in the electron density has been included in both Monte Carlo and continuum models. Only

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Figure 1. Schematic diagram showing how the grains are linked and the angle Rneck is defined. The shaded areas indicate the regions in which an electron changes from the middle grain to one of the side grains.

illumination through the anode rather than the cathode has been considered here, as this is the normal operating condition for the cell. Many previous approaches (e.g., see ref 4) have solved the continuity equation without traps and replaced Dbare by Deff. Deff has been obtained for our Monte Carlo and continuum models by fitting the transient currents at long times to a solution of the time dependent diffusion equation,12 which does not explicitly include trapping/detrapping or back-reaction processes. The computational resource requirements for our Monte Carlo model have been solved by using a novel approach with a combination of short time scale and long time scale simulations. This is described in detail below, as it is likely to be applicable to other systems. The layout of this paper is as follows: In section 2, the Monte Carlo and continuum models are specified. Section 3 gives comparisons between the transient currents predicted by these two methods and shows how they can be fitted to the solution of the continuum model in which traps are replaced by an effective diffusion coefficient Deff. Our conclusions are given in section 4. 2. Theory and Methodology 2.1. Details Common to Monte Carlo and Continuum Models. The coordination number of the grains in the TiO2 film is reported to be approximately 4,30 but to simplify our examination of the factors influencing transport time in our Monte Carlo simulation, we treat the film as chains of Ngrain uniform spherical grains of radius rgrain, which are parallel to each other, aligned normal to the anode and linked as shown in Figure 1. For the grain closest to the anode, the anode replaces the grain that would otherwise be to the left such that the grain and anode overlap by an amount shown by the shaded area in Figure 1. Only 100 grains per chain have been considered because of computer resource limitations. The chain length lchain has been set equal to the thickness of the TiO2 film in the continuum model to allow a meaningful comparison between the two models. The assumed chain spacing dchain leads to a contact area per chain Achain of 25 nm × 25 nm and, with a chain length of 2 µm, a grain concentration of 8 × 1016 cm-3 and a porosity of 0.34. Whereas most films are thicker, with thicknesses typically 10 µm, the smaller thickness is unlikely to materially affect our conclusions. A maximum theoretical short circuit current density of 23 mAcm-2 has been estimated for AM1.5 from the measured absorption spectrum of the wellknown N3 dye used in the Gra¨tzel cell. The calculation assumed that all the absorbed photons generate electrons that contribute to the short circuit current. This current density corresponds to an absorbed photon flux of 1.4 × 1017 cm-2 s-1 or 7.0 × 1020 cm-3 s-1 for a 2 µm thick film. With the measured second order recombination rate k2br ) 1.9 × 10-17 cm-3 s-1,20 application of the steady state condition at open circuit gives an initial

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J. Phys. Chem. B, Vol. 107, No. 1, 2003 115 and duration for the steps made by electrons in the random walk:

TABLE 1: Table Showing Values of Fixed Parameters Needed for the Monte Carlo and Continuum Models parameter

value

rgrain dchain Achain Ngrain lchain n0 EFn0 EFndark m* Dbare Nc ktrap kdetrap τr ET σ T tfstep Vth dfstep tcstep R kext k1br

10 nm 5 nm 6.25 × 10-12 cm2 100 2 µm 8 × 1016 cm-3 0.23 eV 1.0 eV 9 m0 1 × 10-2 cm2 s-1 6.775 × 1020 cm-3 4 × 107 s-1 3.2 × 106 s-1 3.2 × 10-7 s 0.3 eV 1 × 10-16 cm2 300 K 2 fs 5 × 106 cm s-1 0.1 nm 200 ps 2.3 × 103 cm-1 1 × 106 cm s-1 3 × 104 s-1

tfstep ) µbarem*/q

electron density n0 of 6 × 1018 cm-3. In the work reported here, an initial electron concentration corresponding to one electron per grain has been assumed, which is around 80 times less than the value deduced from the estimated short circuit current at AM1.5. Thus, our initial concentration corresponds to illumination levels of less than 0.1 sun. It is assumed to be uniform across the chain as lchain , 1/R. We have assumed that there are either one or two traps per grain, so that Nt0 is equal to or double the value for n0 quoted in Table 1. Though the authors of ref 11 have also assumed one trap per grain, this implies an Nt0 at least an order of magnitude less than commonly quoted values.2 For simplicity here, we have only one trap level energy ET. Although there is evidence that there is a distribution of trap levels,19,29 this allows us to understand the influence of the morphology in the first instance without the complexity introduced by such a distribution. Detrapping by thermal excitation is assumed, so the detrapping rate is

kdetrap )

{

}

ktrapNc E C - ET exp Nt0 kBT

(1)

where the trapping rate for traps of cross-section σ and an electron thermal velocity Vth ≈ x3kBT/m* is given by

ktrap ) Nt0Vthσ

(2)

The trap residence time is

τr ) 1/kdetrap

(3)

2.2. Monte Carlo Simulations. In a Monte Carlo simulation, the trajectories of an ensemble of electrons are determined by a sequence of random numbers. The electrons move in straight lines until they are scattered. On scattering, a random direction is chosen for their subsequent motion. The average temporal and spatial separation of scattering events experienced by the electrons can be approximated from the electron mobility µbare and effective mass m* for crystalline TiO2, to give us a length

dfstep ) Vthtfstep

(4)

where q is the magnitude of the electron charge. From Table 1, tfstep) 2 fs and dfstep ) 0.1 nm. Hence dfstep is a small fraction of a grain diameter. As noted above, electrons can require milliseconds or longer to exit the TiO2 film so ∼1012 steps of 2 fs duration would need to be taken in each random walk. To reduce noise from random fluctuations, we must average over a large number of walks for each of hundreds of electrons moving in the chain, requiring unrealistically large computational resources. Thus, to simulate the passage of electrons through the chain of grains, we split the simulation into two stages as illustrated in Figure 2. First, a fine scale simulation, where an electron makes a random walk using steps of size tfstep, dfstep, is allowed to continue for a number of steps that is sufficiently large that the electron could move a few grains along the chain but sufficiently small that many walks are simulated in a reasonable time frame. A walk of 105 steps fulfills these requirements. By repeating such a walk many times, we have obtained probabilities for the number of grains moved and the likelihood of trapping. These can then be used in a coarse scale simulation, in which the position of each electron is recorded only in terms of which grain it is in and at each time interval the electron moves or is trapped according to the generated probabilities. The simulation code was written in C++. A full run with both coarse and fine scale simulation took approximately 24 h on a 350 MHz alpha Unix workstation with 640 Mb RAM. 2.2.1. Fine Scale Simulations. In the simulation used to determine the appropriate probabilities for the morphology being investigated, an electron starts at a random point within a grain. At each time step, its position changes by (∆x, ∆y, ∆z) given by eqs 5 using a coordinate system relative to the center of the grain that is currently occupied.

∆x ) dfstep sin(θfstep) cos(φfstep) ∆y ) dfstep sin(θfstep) sin(φfstep) ∆z ) dfstep cos(θfstep)

(5)

θfstep ) 2 sin-1(2ξ - 1)

(6)

φfstep ) 2πξ

(7)

θfstep and φfstep are angles used to calculate (∆x, ∆y, ∆z) and ξ is a random number with a uniform distribution between 0 and 1. We then test the new location; if it falls outside the grain the step is truncated so that the electron is at the grain surface. If the new location is within the region of the sphere that overlaps with its neighbor (the shaded area in Figure 1) then the electron is assumed to have changed grains. Traps are located at randomly selected points on the surface of the grains; if the electron comes within a set distance (determined by the trapping cross section σ) of these points, we record it as having interacted with the trap. Thus for each walk simulated, we can determine how many grains the electron has moved and how many traps it has encountered, allowing us to generate the probabilities required. Throughout most of the chain the probability of moving a given number of grains is the same but at the chain ends the probabilities may change. At the end of the chain from which the electrons exit the system there is no need to alter the probabilities because any electron moving past the last grain in

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Figure 4. Probabilities of an electron not encountering further traps (clear bars) and encountering further traps (shaded bars). Panel A is for the case where if a trap is encountered, it is empty. Panel B is the case where one full trap has been encountered but, if a second trap is encountered, it is empty. Panel C is the case where two filled traps have been encountered but, if a third trap is encountered, it is empty. Figure 2. In the fine scale simulations (top panel), an electron moves a fixed distance from an initial position (x, y, z) shown with an open circle, to a position (x + ∆x, y + ∆y, z + ∆z) shown with a filled circle. Its motion in 3-dimensions is constrained only by the boundaries of the grains. In the coarse scale simulations (bottom panel), the position of an electron is only determined by the grain it resides on. Here the electron moves by taking steps of varying numbers of grains up and down the chain, as illustrated.

Figure 3. Probabilities of the electron moving the number of grains shown either without encountering a trap (clear bars) or encountering at least one trap (shaded bars). Panel A is for an electron in the middle of the chain. Panels B-F are respectively for an electron starting 4, 3, 2, 1, and 0 grains from the final grain.

the chain is considered to have exited. At the other end of the chain, however, the probabilities will alter due to proximity to the chain termination grain. Electrons cannot move beyond this grain and so have a greater probability of moving in the other direction, down the chain toward the exit. It is consequently necessary to generate separate probabilities for each case where an electron might reach the terminating grain, for example if the electron can move up to 4 grains along the chain in a single time step we must obtain a separate probability set for when the electron is 4 grains from the end of the chain, 3 grains from the end of the chain, etc. as shown in Figure 3. Separate probability sets are also required to deal with the situation where an electron encounters one or more filled traps. As we record that an electron has encountered a trap rather than halting its walk, we can establish the number of times the electron encounters traps during its walk. Consequently, we determine the probabilities for whether the electron encounters another trap by comparing the number of electrons that encounter only one trap with the number that encounters more than one trap. The same system can be applied to determine probabilities for electrons encountering two filled traps, etc.; see Figure 4. The asymmetry about zero increases as the number of trappings increases due simply to the reduction in sample

size from which the probabilities are calculated. As it is unlikely that the electron will encounter two traps, the sample size for calculating the probability of encountering a third trap is small. 2.2.2. Coarse Scale Simulations. Our starting point is a uniform electron distribution with concentration n0. The probability sets described in section 2.2.1 are used to simulate a random walk with a time step tcstep of 200 ps. After one of these time steps, an electron can (i) remain a free particle, (ii) move grains, (iii) trap, or (iv) back-react. Each of these four possibilities is chosen with a weighting determined by the appropriate probabilities. For each electron in turn we randomly select the number of grains through which the electron moves at a given time step, and use a further random number to determine whether the electron encounters a trap with the appropriate probability. If the electron encounters a trap, we must establish whether the electron becomes trapped. If the trap already contains an electron it cannot trap additional electrons; consequently, we must use the probability set shown in Figure 3 to dictate the electron behavior. If the trap is empty, the electron becomes trapped and will remain at its current location for a time τr determined by the depth of the trap. Trapped electrons may back-react with the electrolyte with a probability pbr, removing them from the system. We record the number of electrons N(t) that exit the chain in the time interval tcstep and use these data to deduce the anode photocurrent density.

j(t) )

qN(t) Achaintcstep

(8)

The Monte Carlo results presented in section 3 have been smoothed by averaging j(t) over a time 2 × 10-5 s. 2.3. Continuum Model. The time dependent current can also be calculated from the gradient of the electron density profile n(x,t) where x is the distance from the anode and is determined from the continuity equation for conduction electrons. Its rate of change with time is determined by

∂2n ∂n ) Dbare 2 + RI0 exp(-Rx) + kdetrap Nt0 f - ktrap n(1 - f ) ∂t ∂x (9) Here the first term on the right-hand side is the electron flux gradient assuming a diffusion coefficient without traps Dbare, the second term the generation rate from light of intensity I0 and absorption coefficient R, and the third term the net trapping rate whose terms are defined below. Here, we have neglected the possibility of back-reaction by free electrons, as this has been argued in ref 1 to be unlikely. Equation 9 is coupled with the continuity equation for the probability f that a trap level is

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occupied,

∂f Nt0 ) ktrap n(1 - f ) - kdetrapNt0 f - k1br(ntrap - ntrapdark) (10) ∂t where k1br is the first order back-reaction rate for trapped electrons with the I3- ions. A first order back-reaction rate has been assumed here to allow comparison with the Monte Carlo simulations. The trapped electron density in the dark is determined by the occupation fraction in the dark, fdark via ntrapdark ) Nt0 fdark. To find fdark, Fermi statistics have been used:

1 fdark ) 1 + exp{(ET - EFndark)/(kBT)}

1 1 + exp{(ET - EFn)/(kBT)}

∂n(x,t) |x)0 ∂x

(13)

j(t) ) qkext(n(0,t) - ndark)

(14)

where ndark ) Nc exp[EFndark/(kBT)], kext ) xkBT/(2m/nπ) for IMPS spectra (short circuit) and kext ) 0 for IMVS spectra (open circuit). This is the Schottky boundary condition31 with a recombination velocity Vr ) kext. At the cathode side of the TiO2 network (x ) lchain), the concentration gradient is set equal to zero. It is important to note here that the trap density was chosen to correspond to the value used in the Monte Carlo simulation as shown in Table 1. Similarly, the trap depth was chosen to give the same thermal release rate as used in the Monte Carlo simulations. This is essential for a meaningful comparison of the results obtained by the two methods. If the effect on electron transport from traps were solely to change the diffusion coefficient, eq 9 can be solved analytically by treating the cell as consisting of one electrode and a parallel insulating plate, giving12 (a factor Deff is missing from eq 4 of ref 12)

j(t) )

lchain

∞

(

∑ exp

k)0

)

-Deff (2k + 1)2π2t 4lchain2

(15)

An estimate of the back-reaction rate k1br can be made from the probability pbr that an electron back-reacts when it traps from the following argument. In a single time step tcstep in the coarse time scale simulation, the number of electrons in the chain back-reacting Nbr must be equal to pbrNtrapping, where Ntrapping is the number of electrons that were free and then become trapped in time tcstep. Nbr is also equal to k1brNtrappedtcstep, where Ntrapped is the number of electrons already trapped. Hence,

k1br )

pbr〈Ntrapping〉 tcstep〈Ntrapped〉

1000

Ntrapped )

(0.8)n0.2Ntot ≈ Ntot ∑ n)0

(18)

where the sum is over the number of steps in τr, Nstep ) 1000. The approximation made is to sum to n ) ∞. Hence, the average number of electrons trapped over a time interval τr,

Ntot Nstep

(19)

Ntot/1000 ) 1 × 10-3 Ntot

(20)

〈Ntrapping〉 ) Hence

The anode boundary condition requires

2qDeff n0

(17)

Thus, the total number of electrons trapped in time τr is

(12)

The photocurrent density j is determined from the concentration gradient at the anode:

j(t) ) qDbare

Ntrapping ) (0.8)n0.2Ntot

(11)

For a constant light intensity resulting in a steady state electron density nss and zero back-reaction, this result can be generalized to

fss )

To find Ntrapping, we consider a time period equal to the residence time in the trap τr. At the start of this period, the total number of free electrons is Ntot. After n coarse time steps, if the probability of being trapped ≈0.2 (as can be seen from Figure 3),

(16)

〈Ntrapping〉 〈Ntrapped〉

≈

If we assume pbr ≈ 0.001 and tcstep ≈ 100ps, eq 16 gives k1br ≈ 107 × 10-3 ) 104. This estimate for k1br gives the same order of magnitude as kdetrap and is considerably greater than k1br deduced from k2br in ref 20 assuming small amplitude perturbations of 2n0k2br ) 3 s-1 9 (eq 6a in ref 9 has a factor of 2 missing). We have chosen a much larger back-reaction rate to demonstrate the effects that back-reactions will have in our system. 2.4. Parameter Values. EFndark is at the same energy relative to Ec as the I3-/I- redox Fermi level which is 1.0 eV below Ec. The value of m* is in units of the bare electron mass m0 and is taken from ref 3. Dbare comes from the value of µbare in ref 3 via the Einstein relation Dbare ) µbarekBT/q. 3. Results and Discussion In the figures presented below, we have shown the transient anode photocurrent density j in units of A cm-2 using log-log plots, which make differences between the predictions much easier to see than linear plots. Furthermore, these plots emphasize the long time behavior where the continuum and analytic models show behavior closer to the Monte Carlo simulations, as discussed below. The analytic predictions for j come from varying Deff in eq 15 to give a good fit by eye at long times to the numerical results for j. It is possible from the change in slopes in Figure 5 at different times to identify the two regimes noted in ref 4: short times when the electrons close to the contact exit and longer times when the electron concentration throughout the film relaxes from its initial value of n0. From Figure 5, the continuum model has an initially much higher current density than the Monte Carlo simulations and the electrons exit faster, showing that the effect of the necks is to slow the electrons down considerably. This effect is quantified by noting that Deff for the continuum model is a factor of 15 larger than for the Monte Carlo data. Deff is deduced by matching the analytic solution for j in eq 15 to j from either the Monte Carlo or continuum results. A good match between j from the analytic solution and j from either of the numerical models is found except at short times, where the analytic solution always underestimates j. The

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Figure 5. Simulated photcurrent transients, comparing Monte Carlo, continuum, and analytic models: (a) dotted line, continuum model with mono-energetic traps (see Table 1 for values used in the calculation); (b) chained line (one dot), analytic solution for Deff ) 9.0 × 10-4 cm2 s-1 corresponding to the best fit to the continuum model; (c) chained line (two dots), Monte Carlo results with Rneck ) 0.1 and 1 trap per grain; (d) solid line, analytic solution for Deff ) 6.0 × 10-5 cm2 s-1, which gives the best fit to the Monte Carlo results.

discrepancy between the analytic solution and Monte Carlo model at short times is due to the small number of trapping events that take place before the electrons leave the anode. In such a case, the analytic model, which can only work when many trapping events are averaged over, is a poor description of the physics. A good fit between analytic and continuum models can also only be obtained at long times, when Deff is less. This reduction in Deff at long times occurs because the electron density throughout the grains has decreased considerably from its initial value n0, because, as time goes on, charge is drained out of the cell. The electron density determines EFn and hence the extent to which the traps are filled (in equilibrium, this is given by eq 12) A lower electron density gives rise to a lower EFn so the electrons are more likely to trap, leading a lower value of j. Our values for Deff bracket the value in Figure 1 of ref 22 of 2 × 10-4 cm2 s-1 at the density n0 used here. This value of Deff comes from fitting a continuity equation with an ambipolar diffusion coefficient to their measured photocurrent transients. The difference between the ambipolar and electron diffusion coefficients is estimated in ref 22 as 15% and so can be neglected here. Kopidakis et al. have shown that their values for Deff(n0) agree well with IMPS data from.7 Franco et al.6 quote a Deff of about 3 × 10-5 cm2 s-1 at a short circuit current density of 1.8 mA cm-2 deduced from their IMPS data. Using the same arguments as in section 2.1, this is equivalent to an intensity I0 of 1 × 1016 cm-2 s-1 and n0 ) 2 × 1017 cm-3, a factor of about 3 greater than the intensity considered here. One would therefore expect their value of Deff at the density n0 used here to be even less. There are many possible reasons for Deff to vary, including the grain connectivity and the trap density and depth as we show below. The electrolyte concentration and composition have also been shown to influence Deff.17,18 We vary the connectivity of the grains as measured by Rneck. Figure 6 shows that by changing Rneck we alter the rate of extraction of carriers. Reducing the size of the neck between grains increases the amount of time required for carriers to exit the chain and here reduces Deff by a factor of about 2.

Cass et al.

Figure 6. Simulated photcurrent transients, varying Rneck: (a) chained line (two dots), Monte Carlo results with Rneck ) 0.1 and 1 trap per grain; (b) solid line, best fit analytic solution for Deff ) 6.0 × 10-5 cm2 s-1; (c) dashed line, Monte Carlo results with Rneck ) 0.01 and 1 trap per grain; (d) dotted line, best fit analytic solution to Monte Carlo results, from which Deff ) 2.7 × 10-5 cm2 s-1.

Increasing the number of traps per grain also acts to retard the passage of electrons through the chain. The effect of this change will depend on the number of electrons in the system as the extent to which electrons are trapped is determined by the ratio of the number of electrons to the number of traps in the grain. If there are more traps than electrons in each grain, then increasing the number of traps will have a less marked effect than it would in the case where there are initially more electrons than traps. This is because if there are initially more electrons than traps, increasing the number of traps means that electrons that would previously have been trapped little, if at all, are trapped much more often. If, however, there are more traps than electrons, all the electrons would spend most of their time trapped and, once excited out of the trap, are more likely to be trapped again. Figure 7 shows the effect of doubling the number of traps, so that there are twice as many traps as electrons, does act to slow the passage of electrons particularly at short times. This is to be expected because an increase in the rate of trapping has the greatest effect on those electrons that are least often trapped. Deff decreases with increasing the number of traps per grain, indicating slower transport, as expected. Finding an effective diffusion coefficient that gives good agreement with the simulated j becomes difficult when we include the back-reaction of trapped electrons, because those electrons that manage to traverse the chain are likely to be the ones that have had least interaction with traps and therefore for which the assumption of an effective diffusion coefficient is least valid. Figure 8 shows the effect of increasing the probability pbr of a trapped electron to back-react. We see that with the back-reaction rate k1br in Table 1 and the chosen values of pbr, the back-reaction significantly changes the shape of the photocurrent curve. With a 1% back-reaction probability, it is unlikely that electrons any distance into the chain will reach the contact, and only those that hardly interact with the traps will exit the chain. The differences between the continuum and Monte Carlo results seen in Figure 5 are obvious here, thus the morphology is important in slowing the electrons down even with a back-reaction.

Electron Transport in Nanocrystalline Solar Cells

J. Phys. Chem. B, Vol. 107, No. 1, 2003 119 influence of a first order back-reaction of the trapped electrons with the electrolyte. By choosing parameter values that match as far as possible between the models, we have shown that the morphology and back-reaction can have a significant effect on the transport time, although hitherto largely ignored in models of electron transport in dye sensitized cells. The effects of the morphology and back-reaction have been quantified by fitting the results to an analytical expression that comes from solving the continuum model with the neglect of traps and back-reaction. Acknowledgment. This work was supported by the U.K. Engineering and Physical Sciences Research Council. References and Notes

Figure 7. Simulated photcurrent transients, varying the number of traps per grain. Here Rneck ) 0.1 rad. Key: (a) dashed line, Monte Carlo results with 2 traps per grain; (b) dotted line, fitted analytic solution for Deff ) 3.0 × 10-5 cm2 s-1; (c) chained line (two dots), Monte Carlo results with 1 trap per grain; (d) solid line, fitted analytic solution for Deff ) 6.0 × 10-5 cm2 s-1.

Figure 8. Simulated photcurrent transients, varying the back-reaction probability and comparing with the continuum model. Here Rneck ) 0.1 rad and there is one trap per grain. Key: (a) long dashed line, Monte Carlo results with pbr ) 0.001; (b) dotted line, continuum model with k1br ) 3 × 104 s-1; (c) chained line (one dot), Monte Carlo results with pbr ) 0.0001; (d) chained line (two dots), Monte Carlo results with pbr ) 0; (e) solid line, analytic solution fitted to Monte Carlo results with pbr ) 0 with Deff ) 6.0 × 10-5 cm2 s-1.

4. Conclusions A novel random walk simulation has been developed for electrons diffusing through a porous film of nanoparticles, in which the grain morphology is explicit, which covers transport over a wide range of time scales by combining a fine time scale with a coarse time scale simulation. The simulation has been used to calculate the transient photocurrent in a dye sensitized cell arising after illumination is removed with that predicted by a continuum model in which the porous film is treated as a homogeneous medium. We have also solved the time dependent continuity equations for free and trapped electrons. In both random walk and continuum models, we have studied the

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