11194
J. Phys. Chem. B 2001, 105, 11194-11205
Nonthermalized Electron Transport in Dye-Sensitized Nanocrystalline TiO2 Films: Transient Photocurrent and Random-Walk Modeling Studies J. van de Lagemaat* and A. J. Frank* National Renewable Energy Laboratory, Golden, Colorado 80401 ReceiVed: May 14, 2001; In Final Form: August 14, 2001
A random-walk approach is developed to model the electron-transport dynamics in dye-sensitized TiO2 solar cells within a multiple-trapping framework, and the predicted results are compared with those measured by transient photocurrent. The illumination geometry and the wavelength of the probe light are used to create certain initial spatial distributions of photoinjected electrons in the TiO2 films. Both have a dramatic effect on the shape of the measured photocurrent transient. Cells are probed with light incident from either the collecting (substrate) electrode side or the counter-electrode side. Excellent correspondence between simulated and measured current transients is observed. When electrons are injected far from the collecting electrode, their diffusion is found to be classical, corresponding to thermalized (nondispersive) transport. Nonthermalized (dispersive) electron transport is shown to be important when electrons are injected near the collecting electrode, which corresponds to the illumination condition under which the cell normally operates. For strongly absorbed light incident from the collecting electrode side, it is estimated that about 80% of injected electrons are collected before they are within 95% of complete thermalization. Failure to account for the presence of nonthermalized electrons is shown to be a major limitation of previous theories of electron transport. The total density of trap states is estimated to be relatively small, on the order of 1 trap per particle. The average detrapping time is on the order of 10 ns. When electrons are generated far from the collecting electrode, they undergo an average of about 106 trapping events before being collected. Analytical expressions are derived that relate the experimentally measured collection time to other parameters affecting transport (e.g., trap density, light intensity, film thickness, and free-electron mobility). Experimental evidence is presented for ambipolar diffusion.
Introduction The electron-transport dynamics in dye-sensitized nanocrystalline solar cells is currently receiving much scrutiny1-18 from both a fundamental science and device application perspective. During the illumination of these cells, electrons are injected into the conduction band of TiO2 by photoexcited dye molecules adsorbed to the nanoparticle surface, which is in contact with a liquid electrolyte. Redox species in the liquid transport holes from oxidized dye molecules to the counter electrode. The transport of injected electrons to the collecting back contact (transparent conducting oxide: TCO) is believed to occur by diffusion because the macroscopic electric field across the film is negligible at normal solar light intensities due to screening by the electrolyte.7,13,19-21 Because the collection of injected electrons competes with recombination, slow transport can lead to a low charge-collection efficiency6,11,13,22 and consequently to a low overall conversion efficiency. Intensity-modulated photocurrent spectroscopy (IMPS),2-7,11,13,14 transient photocurrent measurements,1,2,7,12,17,18,23,24 currentvoltage (or conductivity) measurements,8,9,23,25 and electrical impedance spectroscopy23,26 have been used to probe the transport properties of nanocrystalline TiO2 films. At one-sun light intensity, the time constant for the collection of injected electrons in nanocrystalline films is in the millisecond domain,1,2,5-7,11,13,14 which is several orders of magnitude slower than is estimated from mobility data27-29 for single crystalline TiO2. The slow collection has been attributed to trap-limited * To whom correspondence should be addressed.
transport.1,3-10,14-16,18,23,24 In addition, the time constant for electron collection is observed to decrease with increasing light intensity,1,2,5-7,11,14 a phenomenon attributed to the trap states having an exponential distribution of energies.6,7,9,10,14 An exponential distribution of trap energies has also been used to explain10,13,30 the recombination kinetics of injected electrons with oxidized dye molecules and with redox species at the solid/ solution interface. In general, the traps are presumed to be on the surface because of the large surface-to-volume ratio of a nanoparticle. There is, however, no definite proof for this conjecture, and the identity and location of the traps are still speculative. In single-crystal anatase, exponential absorption tails (Urbach tails) extending from the band gap excitation threshold to lower energies are reported.31 The temperature dependence of the Urbach tails arises from strong electron-phonon interaction.27,31,32 Urbach tails are also observed in absorption and photocurrent action spectra of nanoparticle TiO2,9,18 where they have been attributed to band-state broadening33 induced by disorder, resulting from the presence of grain boundaries and interfaces.9 At thermal equilibrium, deep traps (those below the Fermi level) are predominantly filled and shallow traps (those above the Fermi level) are primarily empty. Because deep traps exhibit longer detrapping times than shallow traps, it is generally assumed that the rate of electron transport is mainly governed by the time electrons spend in traps located in the vicinity of the Fermi level.3-7,14,16 It has been estimated that, if the trap density depends weakly on energy, only traps within 4kT (0.1 eV at room temperature) of the quasi-Fermi level significantly
10.1021/jp0118468 CCC: $20.00 © 2001 American Chemical Society Published on Web 10/19/2001
Nonthermalized Electron Transport in TiO2 Films affect the rate of electron transport.16 It has, therefore, been assumed that electron transport can be described in terms of a single trapping level.3-5,16 Recent studies suggest, however, that the density of traps depends strongly on energy.6,7,9,10,13,14,30 Data from current transients for gas-filled9 and electrolyte-filled7 TiO2 also indicate the presence of nonthermalized electrons. Numerical simulations,10 based on the continuous-time random-walk (CTRW) approach34 and the assumption of a complete exponential distribution of surface traps, have been shown to describe reasonably well the recombination kinetics of injected electrons with the oxidized dye and the photocurrent transient at a low light intensity (0.05 mW cm-2 514 nm). At a higher light intensity (4 mW cm-2), the simulation appears to deviate significantly from the experimental data. Nevertheless, this study establishes the capability of the CTRW approach to model both electron transport and recombination in TiO2 solar cells. It has been qualitatively acknowledged that electron diffusion in the TiO2 network is strongly coupled electrostatically to ionic motion in the electrolyte.12,35 Recently, the electron-diffusion coefficient was analyzed quantitatively in terms of an ambipolar diffusion model.7 It was pointed out that in an electrolyte-filled TiO2 cell electron and ion motion are coupled and that their diffusion coefficients cannot, therefore, be determined separately. However, because the density of ions in the electrolyte is much larger than the concentration of injected electrons in the TiO2 film, the effect of ionic mobility on electron diffusion has not been verified experimentally. Further evidence to support the concept of ambipolar transport is thus warranted. In this paper, we develop a random-walk approach to model the electron-transport dynamics of dye-sensitized TiO2 solar cells within a multiple-trapping framework, and the predicted results are verified by transient photocurrent measurements. It is found that nonthermalized (hot) electron transport is important under normal operating conditions of the cells. Failure to account for the presence of nonthermalized electrons is shown to be the main drawback of current theories of electron transport. A theoretical expression is developed for estimating the trap state density in films. Surprisingly, only about 1 trap per particle is found. Experimental evidence is also presented for the occurrence of ambipolar diffusion in dye-sensitized solar cells. Theory and Methodology In this section, electron transport is considered from the perspective of a random-walk model using computer simulations. The computer simulations consider both thermalized and nonthermalized modes of transport. In complete thermalization, the electron diffusion behaves classically (i.e., it obeys Fick’s laws). This means that the diffusion coefficient of electrons is independent of time and place in the film. When electrons are completely thermalized, it is possible to write analytical expressions, based on the random walk approach,36 that describe their behavior (Appendix A). In Appendix B, the random walk for incompletely thermalized electrons is considered. We neglect electron tunneling between trap states. From the analysis of plots of trapped electrons vs voltage,30 one can infer that trap sites of equal energy are too far apart for the occurrence of significant electron tunneling. The traps may be on the particle surface, intrinsic to TiO2, at grain boundaries between particles, or caused by the nanoporous structure of the film. Random-Walk Approach. The electron-transport dynamics are modeled computationally as a random walk using a Monte Carlo type procedure. The random-walk simulations describe numerically the trapping and detrapping of single excess charges (injected electrons) following their introduction onto a three-
J. Phys. Chem. B, Vol. 105, No. 45, 2001 11195
Figure 1. Three-dimensional grid that represents the porous TiO2 film. Each cell represents a single electron trap selected randomly from an exponential distribution. The depth of the traps varies with shades of gray. An electron performs a random walk by moving from trap-totrap over the grid until it is collected (arrives) at x ) 0. At x ) d, the electron is reflected back into the grid. The y and z directions are periodic.
dimensional grid (Figure 1) that represents the porous TiO2 film. The respective planes of the grid at the coordinates x ) 0 and d represent the collecting TCO substrate and the outermost TiO2 surface nearest the counter electrode. The electrons are initially placed on grid sites in a distribution that reflects the absorption profile of a small photoexcitation pulse (probe light). It is assumed that the number of electrons placed on the grid is sufficiently small to neglect electron-electron interaction and to leave the position of the Fermi level, established by the steady-state background (bias) light intensity, unchanged. Electrons are allowed to move randomly from grid site to grid site in any one of six directions determined by the boundary conditions.37 These conditions are, in part, that electrons reaching the TCO substrate (x ) 0) are collected and that no electron leaves the outermost TiO2 surface (x ) d). Because the lateral dimensions of a film are normally many orders of magnitude larger than the film thickness, periodic boundary conditions are applied in the y and z directions, such that electrons reaching a grid surface in the y or z directions reenter the grid at the opposing surface. The grid is divided into a finite number of cubic lattice cells. Each cell represents a trap state. Because the grid boundaries in the y and z directions are repetitive, the number of cells in these directions does not influence the simulations as long as it is sufficiently large that the total distribution of trap states is statistically represented. In the y and z directions, 100 cells is a sufficiently large number to satisfy this condition. The number of cells used in the x direction (Nx) is varied typically from 10 to 2000. The total density of traps in the simulated film is then given by Ntot ) (Nx/d)3. For example, for a film with a thickness of 10 µm, an Nx ranging from 10 to 2000 corresponds to a density of trap states between 1012 and 8 × 1018 cm-3. Drawing from results of intensity-modulated photovoltage spectroscopy (IMVS), IMPS, and photocurrent transient measurements, the energy of the intraband gap trap level ET associated with each cell is
11196 J. Phys. Chem. B, Vol. 105, No. 45, 2001
van de Lagemaat and Frank
chosen from an exponential distribution of trap states N(E) extending from the conduction-band edge:6,7,9,10,13,14,30
N(E) ) Ntotmc-1e(E-EC)/mc
(1)
where EC is the energy of the conduction-band edge and the parameter mc, which is the average trap depth, describes the steepness of the trap-state distribution curve. Typically, values of mc range from 60 to 100 meV. To ensure that the final distribution of trap states is described by eq 1, a trap level ET for each cell is chosen randomly from an exponential distribution:
ET - EC ) mc ln(Rnd[0,1])
(2)
where Rnd[0,1] is a computer-generated random number between 0 and 1. At a given bias light intensity, the quasi-Fermi level for electrons EF(x) is determined by solving the continuity equation for the steady-state density of electrons in the conduction band at x.14 With knowledge of the energy of a trap associated with a particular cell at x, the Fermi-Dirac function (f(ET) ) 1/(1 + e(ET-EF(x))/kT)), which gives the probability that a trap of energy ET is filled, can be calculated. By comparing the value of f(ET) with a computer-generated random number, the occupancy of the trap level associated with the cell is determined:
f(ET) ) 1 (ET-EF(x))/kT
1+e
}
Rnd[0,1] < f(ET) f trap is filled (3) Rnd[0,1] > f(ET) f trap is empty
The amount of time an electron spends in a cell is determined by whether the trap is filled or empty and the trap depth ET. If the trap is empty, an electron entering the cell will become trapped. The residence time of a trapped electron τr is related to the thermally activated emission rate:38
τr ) (kTN(EC)Vcσc)-1e(ET-EC)/kT ) νth-1e(ET-EC)/kT
(4)
where N(EC) is the density of states at the conduction-band edge, Vc is the free-carrier velocity of electrons, and σc is the capture cross-section of the trap. Because of microscopic reversibility, the product kTN(EC)Vcσc equals the attempted-escape frequency νth (ca. 1012-1013 s-1).38 If the trap is filled, the amount of time an electron spends in the cell, before advancing to the next one, is negligible (∼10-12 s). Figure 2 shows a distribution of emission times as simulated using eqs 2-4. The combination of an exponential distribution of trap-state energies and an exponential dependence of the release (emission) time on energy is seen to lead to a powerlaw distribution of release times. The dashed lines, obtained from combining eqs 1, 3, and 4, shows the distribution of release times for traps above and below the Fermi level. As can be anticipated from eq 3, there is a drop off of emission times at τr(EF) for traps below the Fermi level. Even though there is a drop off, it can be deduced from Figure 2 that neglecting the release times for traps below the Fermi level10 can lead to errors in the random-walk simulations at long times. Although it is possible to assume initially a two-component power-law distribution of detrapping times for the simulations, such as the one shown in Figure 2, it is computationally intensive. A more facile computational approach is to use, as is done in the present study, an exponential energy distribution (eq 1) to create (via
Figure 2. Simulated distribution of emission times. The dashed lines show the expected distribution for times shorter (N ∝ τr-1-kT/mc) and longer (N ∝ τr-2-kT/mc) than the release time at the Fermi level. Values for the simulation parameters were mc ) 70 meV, EC - EF ) 0.25 eV, and T ) 300 K.
eqs 2-4) the appropriate time distribution. The two approaches are effectively equivalent, however, as can be inferred from Figure 2. The transport time τt of a single electron, starting from its point of origin in the three-dimensional lattice and ending at the coordinate x ) 0, where its collection is recorded, is given NT by the sum of the individual detrapping times: τt ) ∑i)1 τr,i; th the index i signifies the i trapping event and NT is the total number of trapping events during the transit of the electron. By simulating the random walk of many electrons (typically, ca. 105), a distribution of transit times is obtained. A histogram analysis of the number of electrons per time interval reaching the collecting substrate yields the predicted current transient. Transforming the current transient into frequency space (by a Fourier or a Laplace transform) gives a simulated IMPS spectrum. The average transit time for electron collection τc is determined by either fitting the generally observed exponential decay of the simulated photocurrent transient at long time to an exponential function (eq C.5) or fitting the low-frequency semicircle observed in the IMPS response to the corresponding IMPS function (eq C.5). The fraction of the transit time fτ that electrons spend in traps at a particular energy can be determined from the random-walk simulations using the following expression:
(
E ′)E
fτ(E) ) -d
/ )
∑ τr(E′) τt dE
E ′)EC
(5)
Equation 5 tracks the relative time that electrons spend in an energy range dE at E by summing up the trap release times for a single electron as a function of energy and subsequently averaging them for many electrons. Experimental Section Cell Fabrication. Anatase TiO2 slurries, consisting of 1520 nm sized particles, were prepared by hydrolyzing titanium tetraisopropoxide (Aldrich, 99.999%) in the presence of acetic acid, followed by autoclaving at 230 °C for 12 h.39 Conducting glass plates (TEC-8) were used as the substrate for depositing TiO2 films. To control the thickness of the deposited films and to protect the electrical contacts, transparent adhesive tape was used (nominal thickness 40 µm). The TiO2 slurry was then
Nonthermalized Electron Transport in TiO2 Films spread on top of it. The thickness of the TiO2 film was 15 µm. The TiO2 covered glass was heated in air at 450 °C for 30 min and then allowed to cool. Subsequently, the TiO2 electrodes were immersed in acetonitrile/tert-butyl alcohol (50:50 v/v%) containing 3 × 10-4 M Ru[LL′(NCS)2] (L ) 2,2′-bypyridyl4,4′-dicarboxylic acid, L′ ) 2,2′-bipyridyl-4,4′-ditetrabutylammoniumcarboxylate) for 24 h at room temperature. The dyecovered electrodes were then rinsed with the acetonitrile/tertbutyl alcohol mixture and dried under a N2 stream. To minimize rehydration of TiO2 from moisture in the ambient air, the electrodes were immersed in the dye solution while they were still warm (100-120 °C) from the annealing step. Transparent counter electrodes were prepared by spreading a droplet of 5mM H2PtCl6 in 2-propanol onto conducting glass plates and subsequently annealing them at 380 °C for 20 min. The cells were sealed with 0.5 mm wide strips of 50 µm thick Surlyn (Dupont, grade 1702). Sealing was accomplished by pressing the two electrodes together at a pressure of 900 psi and a temperature of about 100 °C. The redox electrolyte, consisting of 0.6 M 1,3-methyl 2-hexyl imidazoliumiodide and 50 mM I2 in methoxyacetonitrile, was introduced into the cell through one of two small holes drilled in the counter electrode. The holes were then covered and sealed with small squares of microscope objective glass and Surlyn. The resulting cells had an active area of about 0.25 cm2, a film thickness of 15 µm, and an AM1.5 efficiency of about 7%. Transient Photocurrent Measurements. The cells were probed with a weak laser pulse superimposed on a relatively large background (bias) illumination at 680 nm that was incident from the substrate (collector) side. The bias light was supplied by a 0.5 W SDL model 7421 H1 laser diode. The intensity of the bias light was varied using neutral density filters. The probe laser pulses were incident from either the counter-electrode side or from the substrate side. The probe laser pulses were generated either by a 0.25 W SDL model 7421 H1 laser diode at 680 nm (pulse duration 1 µs) or a 30 mW frequency doubled Nd:YAG laser (Laser compact Co. LCS-DTL-112QT; λ ) 532 nm, pulse duration 10 ns). The current transients were measured using a Stanford Research Systems model SR570 low-noise current preamplifier, amplified by a Stanford Research Systems model SR560 low-noise preamplifier, and recorded on a LeCroy 9350A 500 MHz oscilloscope. The light intensity of the probe light was adjusted so that the collected charge due to the photocurrent transient was less than 1% of the steady-state charge as estimated from the product of the steady-state short-circuit photocurrent and the time constant for electron collection. Random-Walk Simulations. The random-walk simulations were performed using a C++ program compiled using gcc 2.95.240 on a 450 MHz PowerPC G4 running Mac OS X. The pseudo-random-number generator was a C++ adaptation of the Mersenne-Twister generator.41 This generator has an extremely large period of 219937-1 and is relatively fast with respect to other well-known pseudo-random-number generators. Before and several times during the simulations, a large matrix of detrapping times is determined. This matrix is represented in the computer memory by double-precision floating-point numbers. Because of the large size of the matrix (2000 × 100 × 100) and the necessity of keeping track of both the depth of the trap and its emission time, it requires about 170 MB of free available memory for the largest simulations. To avoid excessive loss of simulation speed owing to “swapping”, the simulation computers had a minimum of 256 MB random access memory. A typical random walk simulation takes approximately 4 h to complete.
J. Phys. Chem. B, Vol. 105, No. 45, 2001 11197
Figure 3. Photocurrent transients for cells illuminated with 532 nm probe-light pulses incident from either the collector side or the counterelectrode side. The bias light at 680 nm, which was incident from the collector side, generated a steady-state short-circuit current of 5 mA cm-2. The inset shows a linear plot of the photocurrent transient at short times for the probe light entering the cell at the counter electrode. The small current spike is caused by the absorption of reflected light at the collector side. The rise time of the spike yields the RC time constant of the cell. The film thickness was 15 µm.
Results and Discussion In the photocurrent transient experiments, the cells were probed with a low-intensity laser pulse superimposed on a large constant background (bias) illumination at 680 nm. The 680 nm bias light is only weakly absorbed by the dye (the effective absorption coefficient of the cell R ≈ 500 cm-1 for 680 nm light) and therefore provides a relatively constant electron injection current throughout the film. The background illumination was always incident from the substrate (collector) side, whereas the side of the cell exposed to the incident probe light varied. Figure 3 shows that the illumination geometry of the laser pulse has a dramatic effect on the shape of the photocurrent transients. When the cell is illuminated with a 532 nm laser pulse, incident from the counter-electrode side, the photocurrent transient displays a relatively fast rise time, followed by a peak maximum at about 2.7 ms and an exponential decay at long times. In marked contrast, when the cell is illuminated with a light pulse incident from the collector side, the photocurrent transient is present almost immediately after the pulse and then decays exponentially with time. In both cases, the total charge, obtained by integrating the current transients, was independent of the bias light intensity, which was varied over 3 orders of magnitude. The invariance of the total charge with bias light intensity is indicative of negligible recombination. The difference in the shape of the photocurrent transient curves in the first 5 ms can be understood in terms of the spatial and temporal relationship between the absorption profile of the light and the mechanism of detection of electrons in the external circuit. Because of the low intensity of the probe light and its strong absorption by the dye (the absorption coefficient of the cell R ≈ 5 × 103 cm-1 at 532 nm), the injected electrons are introduced into a narrow spatial region of the film, corresponding to where the probe light enters the film. The transport time of injected electrons will depend, therefore, on the distance that they traverse to reach the collecting substrate. Thus, when the 532 nm pulse enters the cell from the counter-electrode side, the bulk of injected electrons are produced near the outermost
11198 J. Phys. Chem. B, Vol. 105, No. 45, 2001 surface of the TiO2 layer and must travel through almost the entire thickness of the TiO2 film, about 15 µm, before being collected at the TCO substrate. In contrast, electrons that are created near the TCO/TiO2 interface, as in the case where the 532 nm laser pulse enters the cell at the TCO substrate, have a smaller distance to travel before being collected. The presence of a peak maximum at about 2.7 ms after the laser pulse, incident from the counter-electrode side, suggests that a substantial amount of time passes before the bulk of injected electrons travels from their place of origin to the collector. This is strong evidence for the proposal that the current transient measured in the external circuit reflects the arrival of electrons at the collecting substrate7,19 and that electron transport in electrolytefilled nanoporous TiO2 can be described in terms of an ambipolar diffusion model.7 The evidence presented here for arrival-time detection challenges the notion3,4,16,42 that the current measured in the external circuit is in response to the motion of photoinjected electrons anywhere in the TiO2 film (a displacement current43). Arrival-time detection implies that the electric field associated with photoinjected electrons in the film is screened by the electrolyte7,19,35 and that only electrons reaching the thin electrical double layer at the TiO2/TCO interface influence the potential difference between the TCO and the counter electrode. During their transit through this double layer, the electrons will give rise to a current in the external circuit. The time required to cross this double layer is estimated to be much shorter (, 1 ns) than the RC time constant of the cell (≈ 30 µs; see below). Thus, the current detected in the external circuit correlates with the transmission of electrons across the TiO2/collector interface. The evidence for arrival-time detection supports the reasonableness of the ambipolar diffusion model.7 It implies that electron diffusion in the TiO2 network is coupled electrostatically to the motion of ions in the electrolyte. However, because the density of ions is normally much larger than the concentration of photoinjected electrons, the effect of ion mobility on electron diffusion can be neglected.7 The inset in Figure 3 shows the presence of a small current spike at short times ( kT, -(EF - EC) > mc, and that traps are distributed homogeneously throughout the film. We can write, with the aid of eqs A.1 and A.2, the time constant in a photocurrent transient experiment from the following expression:
τc ≈
6d 2ht ) 2.55d 2ht Ntot2/3 2 2.35λ
(A.3)
Equation A.3 shows that the collection time for thermalized electrons τc is proportional to the average detrapping time ht and scales with the total density of traps Ntot2/3. Equation A.3 also shows that any light intensity dependence of τc is caused by ht. The average detrapping or localization time ht is obtained by averaging the release time of all available traps between the valence band and the conduction band: ht ) ∫EEVC τr(E) ψτ(E) dE, where ψτ(E) is the normalized probability of an aVailable trap having an emission time τr(E).36 The term ψτ(E) is proportional to the density of states and the probability of a trap at E being empty (Fermi-Dirac function). The average detrapping time ht is a function of the quasi-Fermi level EF for electrons, which depends on the light intensity and its position x in the film. Because EF depends weakly on x,5,6 it can be approximated by an effective quasi-Fermi level EF,eff, which is independent of x. For the condition that the Fermi level is lower in energy (more negative) than the average trap energy (EF,eff e EC - mc), we can write, with the aid of eq 4, the following expression:
ht )
∫EE
1 (1 - e(EF,eff-EC)/mc)-1 mcνth
C
V
f ht dE
(A.4)
where f ht is the relative contribution of traps at E to the average detrapping time. Because the electron-collection time is proportional to the average detrapping time ht (eq A.3), the fraction
Nonthermalized Electron Transport in TiO2 Films
J. Phys. Chem. B, Vol. 105, No. 45, 2001 11203
Figure 11. Contribution of traps at energy E to the average detrapping time (f ht; eq A.5) as a function of trap depth, T ) 300 K, EC - EF,eff ) 0.4 eV, and mc ) 70 meV. The respective vertical lines show EF,eff (eq A.6) and E ht (energy where τr ) ht, eq A.8). The inset shows the same plot on a logarithmic scale.
of the transit time fτ that electrons spend in traps at a particular energy is proportional to f ht:
(
)
E - E C E - EC mc kT fτ ∝ f th ) E - EF,eff 1 + exp kT exp
(
)
(A.5)
The energy Epk, where fτ or f ht has a maximum, is given by Epk ) EF,eff + kT ln({kT}/{(mc - kT)}). Because the second term in this equation is generally on the order of -10 meV, the trap energy Epk that most strongly affects the average detrapping time ht occurs slightly below EF,eff. Changing the light intensity will cause EF,eff to shift and, therefore, alter the average detrapping time (eq A.4). For bias light incident from the collector side, the effective Fermi level is given by the equation14
EF,eff ) EC + kT ln
ncb ) NCB EC - kT ln NCB + kT ln
(
)
I0 fR (A.6) RbDn,free
where I0 is the incident intensity of the bias light and Rb is the absorption coefficient of the cell at the wavelength of the bias light. It is assumed that the electron-injection efficiency of the dye is unity and that recombination is negligible at short circuit.6,11,13,14 For the case where the bias light is incident from the collector side and the effective Fermi level equals the Fermi level at 0.5d, the factor fR is given by the expression14 fR ) 1 - 0.5Rbd exp(-Rbd) - exp(-0.5Rbd). For weakly absorbed light (1/Rb . d), fR ) 0.5Rb2 d2, and for strongly absorbed light (1/Rb , d), fR is unity. Equation A.5, which is plotted in Figure 11, shows the contribution of traps at energy E to the average detrapping time. The inset in Figure 11 shows the same data on a logarithmic scale. Every state in the exponential distribution is seen to contribute to the average detrapping time ht. This suggests that the trapping/detrapping kinetics cannot be expressed in terms a single trap level.3,4,6,16 Because eq A.4 has no simple algebraic solution, the average detrapping time must be determined nu-
merically to obtain a value for the collection time τc (eq A.3) or the effective diffusion coefficient Dn that can be compared to experimental results. Insight into the factors that control τc can be obtained by studying the behavior of f ht as a function of energy. Figure 11 shows that a broad range of trap energies (>0.1 eV half width of f ht) affect the average detrapping time and therefore influence the collection time. The trap energy Epk that most strongly influences the average detrapping time ht is found to be slightly below EF,eff. Figure 11 also shows that the energy E ht where the emission time of a trapped electron (eq 4) equals the average detrapping time (τr ) ht) occurs significantly above EF,eff. Thus, although the majority of traps that influence ht lie close to EF,eff, the average detrapping time is not the detrapping time at the effective Fermi level. From the numerical variation of the three parameters (mc, T, and EF,eff with respect to EC) in eq A.4, one finds that ht can be approximated by a product of the emission time of traps τr(EF,eff)and the relative density of trap states N(EF,eff)/Ntot at the effective Fermi level:
ht ≈ Cmc
N(EF,eff) τr (EF,eff) Ntot
(A.7)
where the proportionality constant C ) 1.22 ( 0.16. Because traps below a critical energy Ecrit (see below) do not appreciably affect ht, a more realistic value of C is 1.10 (Results and Discussion section), which is within error of the theoretical value. Substituting ht (eq A.7) for the emission time of traps τr(EF,eff) and the energy for the average detrapping time E ht for ET in eq 4 yields the relation
E th ≈ EF,eff - kT ln(C) -
kT (E - EC) mc F,eff
(A.8)
Equation A.8 shows that the energy for the average detrapping time E ht is roughly halfway between the effective Fermi level EF,eff and the conduction band EC because mc is typically between 2kT and 4kT. Combining eqs 4, A.6, and A.7 yields the relation c νth-1 ht ≈ CN1-kT/m CB
(
I 0 fR RbDn,free
)
kT/mc-1
(A.9)
Substituting eq A.9 and eq A.2 into eq A.3 gives expressions for the collection time: c Ntot2/3 νth-1 τc ≈ 2.55Cd 2 N1-kT/m CB
(
I 0 fR RbDn,free
)
kT/mc-1
(A.10a)
c-1 τc ∝ I kT/m 0
(A.10b)
c τc ∝ µ1-kT/m n
(A.10c)
τc ∝ d 2kT/mc for 1/Rb . d for 1/Rb , d τc ∝ d 2
(A.10d)
Equation A.10a shows that the collection time for electrons τc depends on the light intensity I0 (eq A.10b), the free-electron mobility (eq A.10c), and the film thickness (eq A.10d), among other factors. Recently, we obtained the same relation between τc and I0 (eq A.10b) by considering the light-intensity dependence of τc on the trapped-electron population.14 This relation predicts that as the light intensity increases the collection time for electrons becomes shorter. From eq A.10a and the relation Dn,free ) (kT/e)µn, τc can be related to the free-carrier mobility µn (eq A.10c). When mc > kT, which always appears to be the
11204 J. Phys. Chem. B, Vol. 105, No. 45, 2001
van de Lagemaat and Frank
case, eq A.10c predicts that as the mobility of free electrons increases the rate of electron transport becomes slower. This prediction seems counterintuitive, but it can be understood in terms of the relation between EF,eff and µn (eq A.6 and ref 14): increasing the free-carrier mobility causes the Fermi level to move downward. The resulting availability of deeper traps slows electron transport. Equation A.10d links the collection time to the film thickness for both weakly and strongly absorbed bias light. Using a typical value of mc (70 meV), eq A.10d predicts the relation τc ∝ d0.74 for weakly absorbed light. For strongly absorbed bias light, eq A.10d predicts that τc ∝ d 2. For moderately absorbed light, the power-law dependence of τc on d is expected to be intermediate with an exponent between 2kT/ mc and 2. Combining eqs A.1, A.2, and A.9 yields the effective diffusion coefficient for completely thermalized electrons:
Dn ≈
(
I 0 fR 1 kT/mc-1 -2/3 Ntot νth N CB 6C RbDn,free
)
1-kT/mc
(A.11)
The effective diffusion coefficient in eq A.11 corresponds to Dn in eqs C.3 and C.4, which describe the IMPS response of the cell for completely thermalized electrons. Rearrangement of eq A.10a (or eq A.11) yields Ntot, which can be estimated experimentally from the dependence of τc (or Dn) on I0:
Ntot ≈ 0.25
(
c-1 τcνthN kT/m CB
Cd
2
14.7
(
[
] ) [ ] )
I 0 fR RbDn,free
1-kT/mc 3/2
c-1 νthN kT/m I 0 fR CB CDn RbDn,free
1-kT/mc 3/2
(A.12)
In order for electrons in the conduction band to be in thermal equilibrium with all available traps, the electrons must undergo a sufficient number of trapping and detrapping events to interact with all available traps. The average number of trapping events N h T, which is determined by the trap density and average path length of electrons, may, however, be insufficient to achieve complete thermalization. Because both the trap density and the probability of traps being empty drop off strongly with trap depth, it is expected that there is a certain critical trap energy Ecrit below which the number of trapping events is insufficient for electrons to become trapped during their transit. The critical energy Ecrit is the energy at which only one trapping event is expected. Below this energy, a strong deviation of the contribution of traps to the average detrapping time from eq A.5 is expected. The expected number of trapping events 〈T〉 that electrons undergo at some energy E ( kT is given by the expression:
[ (
)][
E - EC kT exp mc mc
(
1 + exp -
)]
E - EF,eff kT
Ecrit ≈
( )
mc mc mckT kT + + ln E E (B.2) mc + kT N mc + kT F,eff mc + kT C h TkT Evaluation of eq B.2 indicates that for typical values of N h T, which is about 106 (see below), Ecrit is several kT below EF,eff. It is estimated that the error that is made in calculating the average detrapping time from eqs A.4, A.7, and A.9 is, therefore, very small. Consequently, even if all electrons do not thermalize completely, eqs A.4, A.7, and A.9 are still applicable. Because of the existence of a critical energy, a measurement of the diffusion coefficient of electrons or the time constant for electron collection is insensitive to the density of states below this energy. It is, therefore, expected that eqs A.10b-d apply, even though there may be a large number of deep trap states below the critical energy that do not fit the exponential tail (eq 1). Appendix C. Classical Electron-Diffusion Model It is assumed that photoinjected electrons move by classical diffusion (nondispersive transport) and that recombination is negligible at short circuit.7 The continuity equation for electrons in the conduction band is given by14
∂2ncb ∂ncb ) G(x,t) + Dn, free 2 - T(x,t) + E(x,t) (C.1) ∂t ∂x
)
Appendix B. Random Walk for Incompletely Thermalized Electrons
〈T〉 ) N hT
(EF,eff - Ecrit . kT, which is generally true because N h T . 1), Ecrit can be estimated from the relation
-1
(B.1)
where the second term on the right is the probability that the energy of a trap is E and the third term is the probability that a trap at E is empty. For strongly absorbed light incident at the counter electrode, the average total number of trapping events can be estimated from N h T ≈ Nx2 where Nx is the number of cells (traps) in the random-walk simulations along the x axis. At the critical energy (E ) Ecrit), 〈T〉 is unity. When the critical energy Ecrit is much lower than the effective Fermi level EF,eff
where, at spatial and temporal coordinates x and t, ncb is the density of electrons in the conduction band, G(x,t) is the rate of electron injection into TiO2 by the excited dye, and Dn, free is the diffusion coefficient for free (nontrapped) electrons. The diffusion coefficient is connected with the free-carrier mobility through Einstein’s relation: Dn, free ) (kT/e)µn, where k is the Boltzmann constant, T is the temperature, and µn is the mobility of electrons in the conduction band; µn is limited by electron scattering at the surface, grain boundaries, impurity sites, and possibly polaronic transport.50 The terms T(x,t) and E(x,t) are the respective rates at which electrons are captured by traps and thermally emitted back to the conduction band. Recombination of electrons from trap states to the redox electrolyte, which is the dominant electron-loss mechanism in dye-sensitized solar cells at open circuit,30 is implicitly contained in eq C.1 as the difference between the electron-trapping and electron-emission terms ((T(x,t) - E(x,t)). It is assumed by the use of eq C.1 that the electrolyte-filled mesoporous TiO2 film can be treated as a continuous phase and that the film porosity does not influence electron transport.37 If electron trapping occurs over a wide energy distribution of traps, it is not possible to solve eq C.1 analytically. The effect of trapping T(x,t) and detrapping E(x,t) is therefore taken into account by using an effective electron-diffusion constant Dn.5 The diffusion coefficient and the effective quasi-Fermi level EF,eff are assumed5 to be constant throughout the film. With these assumptions and in the absence of recombination at short circuit,7 eq C.1 becomes
∂2 n ∂n ) G(x,t) + Dn 2 ∂t ∂x
(C.2)
where n is the total electron density of the film and is given as the sum of the density of trapped and free electrons. For a
Nonthermalized Electron Transport in TiO2 Films
J. Phys. Chem. B, Vol. 105, No. 45, 2001 11205
small modulated light intensity φ˜ 0 of frequency ω superimposed on a constant (bias) light intensity, G(x,t) can be expressed as G(x,t) ) G(x) + G ˜ (x,ω). Recognizing that, as a result of the ambipolar condition, the external current density J (and therefore its modulated component J˜) is determined by the flux of electrons at the collector, the IMPS function J˜/eφ˜ 0 can be calculated from eq C.2. The solution to this problem and the necessary boundary conditions are described elsewhere.5 Here we rewrite the IMPS function for the case where the modulated light is incident from the counter-electrode side and recombination is negligible as follows:
1 J˜ R ) × 1 + iωτ R + γ eφ˜ 0 RC
[
e-Rd tanh[γd] +
]
R e(γ-R)d - 1 (C.3) γ - R cosh[γd]
where γ ) xiω/Dn and φ˜ 0 is the modulated component of the incident light (the injection efficiency is assumed to be unity). The factor (1 + iωτRC)-1 in eq C.3 accounts for RC attenuation of the photocurrent.5 An analogous expression (eq C.4) is obtained for the situation where the modulated light is incident from the collector side:5
R 1 J˜ × ) eφ˜ d 1 + iωτRC R + γ
[
tanh[γd] +
]
R e-Rd - e-γd (C.4) γ - R cosh[γd]
At low modulation frequencies, the IMPS functions for the modulated light incident from the counter electrode (eq C.3) and the collector (eq C.4) can be expressed in terms of a single equation:
J˜ 1 w J(t) ) A′e-t/τc )A 1 + iωτ eφ˜ c
(C.5)
where τc is the time constant for electron collection and is approximately τc ≈ d 2/2.35Dn; τc is independent of R and the illumination direction of the probe light. A and A′ are constants. Equation C.5 shows the relation between J˜ measured in frequency space at low modulation frequencies and the inverse Fourier transform of J˜ corresponding to J(t) measured in the time domain at long times. Because of the difficulty of transforming eqs C.3 and C.4 from the frequency domain to the time domain, it is convenient to present and analyze the transient photocurrent measurements as an IMPS response in frequency space. References and Notes (1) Schwarzburg, K.; Willig, F. Appl. Phys. Lett. 1991, 58, 2520. (2) Cao, F.; Oskam, G.; Meyer, G. J.; Searson, P. C. J. Phys. Chem. 1996, 100, 17021. (3) de Jongh, P. E.; Vanmaekelbergh, D. Phys. ReV. Lett. 1996, 77, 3427. (4) de Jongh, P. E.; Vanmaekelbergh, D. J. Phys. Chem. B 1997, 101, 2716. (5) Dloczik, L.; Iluperama, O.; Lauermann, I.; Peter, L. M.; Ponomarev, E. A.; Redmond, G.; Shaw, N. J.; Uhlendorf, I. J. Phys. Chem. B 1997, 101, 10281. (6) Fisher, A. C.; Peter, L. M.; Ponomarev, E. A.; Walker, A. B.; Wijayantha, K. G. U. J. Phys. Chem. B 2000, 104, 949.
(7) Kopidakis, N.; Schiff, E. A.; Park, N.-G.; van de Lagemaat, J.; Frank, A. J. J. Phys. Chem. B 2000, 104, 3930. (8) Ko¨nenkamp, R.; Wahi, A.; Hoyer, P. J. Phys. Chem. 1993, 97, 7328. (9) Ko¨nenkamp, R. Phys. ReV. B 2000, 61, 11057. (10) Nelson, J. Phys. ReV. B. 1999, 59, 15374. (11) Schlichtho¨rl, G.; Park, N.-G.; Frank, A. J. J. Phys. Chem. B 1999, 103, 782. (12) Solbrand, A.; Lindstro¨m, H.; Rensmo, H.; Hagfeldt, A.; Lindquist, S.-E.; So¨dergren, S. J. J. Phys. Chem. B 1997, 101, 2514. (13) van de Lagemaat, J.; Park, N.-G.; Frank, A. J. J. Phys. Chem. B 2000, 104, 2044. (14) van de Lagemaat, J.; Frank, A. J. J. Phys. Chem. 2000, 104, 4292. (15) Vanmaekelbergh, D.; de Jongh, P. E. J. Phys. Chem. B 1999, 103, 747. (16) Vanmaekelbergh, D.; de Jongh, P. E. Phys. ReV. B 2000, 61, 4699. (17) Solbrand, A.; Henningsson, A.; So¨dergren, S.; Lindstro¨m, H.; Hagfeldt, A.; Lindquist, S.-E. J. Phys. Chem. B 1999, 103, 1078. (18) Dittrich, T. Phys. Stat. Sol. A 2000, 182, 447. (19) Schwarzburg, K.; Willig, F. J. Phys. Chem. B 1999, 103, 5743. (20) Bisquert, J.; Garcia-Belmonte, G.; Fabregat-Santiago, F. J. Solid State Electrochem. 1999, 3, 337. (21) Papageorgiou, N.; Gra¨tzel, M.; Infelta, P. P. Sol. Energy Mater. Solar Cells 1996, 44, 405. (22) Lindstro¨m, H.; Rensmo, H.; So¨dergren, S.; Solbrand, A.; Lindquist, S.-E. J. Phys. Chem. 1996, 100, 3084. (23) Dittrich, T.; Weidmann, J.; Timoshenko, V. Y.; Petrov, A. A.; Koch, F.; Lisachenko, M. G.; Lebedev, E. Mater. Sci. Eng. B 2000, 69-70, 489. (24) Dittrich, T.; Lebedev, E. A.; Weidmann, J. Phys. Stat. Sol. A 1998, 165, R5. (25) Dittrich, T.; Weidmann, J.; Koch, F.; Uhlendorf, I.; Lauermann, I. Appl. Phys. Lett. 1999, 75, 3980. (26) Bisquert, J.; Garcia-Belmonte, G.; Fabregat-Santiago, F.; Ferriols, N. S.; Bogdanoff, P.; Pereira, E. C. J. Phys. Chem. B 2000, 104, 2287. (27) Forro, L.; Chauvet, O.; Emin, A.; Zuppiroli, L.; Berger, H.; Le´vy, F. J. Appl. Phys. 1994, 75, 633. (28) Enright, B.; Fitzmaurice, D. J. Phys. Chem. 1996, 100, 1027. (29) Gra¨tzel, M.; Frank, A. J. J. Phys. Chem. 1982, 86, 2964. (30) Schlichtho¨rl, G.; Huang, S. Y.; Sprague, J.; Frank, A. J. J. Phys. Chem. B 1997, 101, 8141. (31) Tang, H.; Le´vy, F.; Berger, H.; Schmid, P. E. Phys. ReV. B 1995, 52, 7771. (32) Tang, H.; Berger, H.; Schmid, P. E.; Le´vy, F.; Burri, G. Solid State Comm. 1993, 87, 847. (33) Street, R. A. Hydrogenated amorphous silicon; Cambridge University Press: Cambridge, U.K., 1991; p 90. (34) Scher, H.; Montroll, E. W. Phys. ReV. B 1975, 12, 2455. (35) Zaban, A.; Meier, A.; Gregg, B. A. J. Phys. Chem. B 1997, 101, 7985. (36) Scher, H.; Lax, M. Phys. ReV. B 1973, 7, 4491. (37) The random-walk approach does not explicitly take into account the porosity of the medium. Electrons “hop” from grid site to grid site in any one of six directions of the 3-D lattice with equal probability. This approach is likely valid because trapping and detrapping appear to limit electron transport (Introduction), and thus, morphological factors do not play a significant role. Further support for this conjecture will be provided in an upcoming paper. (38) Street, R. A. Hydrogenated amorphous silicon; Cambridge University Press: Cambridge, U.K., 1991; p 118. (39) Zaban, A.; Ferrere, S.; Sprague, J.; Gregg, B. A. J. Phys. Chem. B 1997, 101, 55. (40) http://www.gnu.org/software/gcc/gcc.html. (41) Matsumoto, M.; Nishimura, T. ACM Trans. Model. Comput. Sim. 1998, 8, 3. (42) Vanmaekelbergh, D.; Iranzo-Marı´n, F.; van de Lagemaat, J. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 616. (43) Scher, H. In PhotoconductiVity and related phenomena; Mort, J., Pai, D. M., Eds.; Elsevier: Amsterdam, 1976; p 92. (44) MacDonald, J. R. Impedance Spectroscopy; emphasizing solid materials and systems; John Wiley & Sons: New York, 1987; p 40. (45) Pandya, R.; Schiff, E. A. Philos. Mag. B 1985, 52, 1075. (46) Park, N.-G.; van de Lagemaat, J.; Frank, A. J. J. Phys. Chem. B 2000, 104, 8989. (47) Hagfeldt, A. Personal communication. (48) The values used for µn and NCB are not known to a high degree of accuracy, and the value for νth has not been measured for nanocrystalline anatase TiO2. Also, the present technique is not sensitive to the presence of traps below the critical energy. Such states would, however, not contribute measurably to the transport kinetics. (49) Duffy, N. W.; Peter, L. M.; Rajapakse, R. M. G.; Wijayantha, K. G. U. J. Phys. Chem. B 2000, 104, 8916. (50) Schelling, P. K.; Halley, J. W. Phys. ReV. B 2000, 62, 3241.
