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What Limits Photoconductance in Anatase TiO2 Nanostructures? A Real and Imaginary Microwave Conductance Study Maria C. Fravventura, Dimitrios Deligiannis,† Juleon M. Schins, Laurens D. A. Siebbeles, and Tom J. Savenije* Opto-electronic Materials Section, Department of Chemical Engineering, Delft University of Technology, Julianalaan 136, 2628 BL, Delft, The Netherlands S Supporting Information *

ABSTRACT: The macroscopic electron transport in porous films of sintered 9 nm sized nanocrystals (NC-TiO2) is known to be 4−6 orders of magnitude worse than in polycrystalline (Pol-TiO2) dense films. To obtain fundamental knowledge regarding this large difference, we investigated the effects of spatial confinement and electron trapping processes on the charge transport in these samples. We determined the timeresolved real and imaginary microwave photoconductance (TRMC) on pulsed excitation. Large amounts of the photoexcited electrons are readily immobilized in deep traps in NC-TiO2 as concluded by comparing the TRMC decay kinetics with previously published transient absorption measurements. Our results show that trapped electrons do not give rise to a microwave photoconductance response, nor do they affect the motion of conduction band electrons. Additionally, by comparing a bare NC-TiO2 film with a dye-sensitized NC-TiO2 sample, the influence of the photogenerated holes on the photoconductance as a function of their locus is investigated. The positive charges either inside or outside the TiO2 nanocrystals contribute insignificantly to the photoconductance. On a nanosecond time scale only a minor fraction (maximum 2%) of the photoexcited electrons resides in the conduction band of NC-TiO2. Importantly, in both NC-TiO2 and Pol-TiO2 these electrons have the same intraparticle microwave mobility of 1.7 cm2/(V s) due to frequent backscattering events at a mean time interval of 85 fs. This mobility value represents the upper limit for the trapfree dc electron mobility in anatase TiO2 irrespective of the crystallite size. Hence, the photoconductance across a NC-TiO2 layer can be strongly enhanced by reducing the electron trap density and by eliminating the relatively inefficient electron hopping steps between adjacent nanocrystals.

1. INTRODUCTION Titanium dioxide (TiO2) has received increasing interest in the past decades because of its applications in photocatalysis, water splitting, and solar energy conversion.1−4 Dye-sensitized solar cells (DSSCs) represent a cheap promising alternative to conventional crystalline silicon photovoltaic devices, reaching efficiencies over 12%.5 DSSCs are based on a nanoporous network of titanium dioxide nanocrystals (NC-TiO2) covered with a light absorbing dye. Optical excitation of the dye leads to formation of an excited electronic state, which decays over the sub-nanoseconds time scale by electron injection into the TiO2 nanoparticles. The power conversion efficiency of a DSSC is strongly affected by the electron transport properties through the TiO2 porous network.4,6 Electric studies on operating DSSCs showed that dc electron transport is heavily reduced in the porous network as compared to a dense TiO2 layer. The low reported dc mobilities, in the order of 10−4−10−6 cm2/(V s),7 are often explained in terms of the so-called multiple trapping model.8−17 According to that description, conduction band electrons get trapped in an exponential distribution of intra-bandgap electronic states and become mobile as they are released by thermal activation. © 2013 American Chemical Society

Hence, the increased trap density in nanocrystalline samples results in an intrinsic low mobility of conduction band electrons in NC-TiO2 with respect to dense bulk TiO2. This model, however, is not in agreement with temperature-dependent photoconductance measurements.18−21 Several studies using the time-resolved electrode-less real microwave18,22−25 and complex terahertz26−29 photoconductance technique on different TiO2 samples propose alternative descriptions for charge transport in TiO2. Both methods predict that the macroscopic electron mobility in NC-TiO2 is heavily reduced as transport proceeds via a temperature activated interparticle hopping mechanism over a distribution of energy barriers. However, no conclusive physical description of the intraparticle dynamics of the charge carriers in the TiO2 matrix has been developed. In various microwave photoconductance studies a difference of 2 orders of magnitude in the real photoconductance going from dense (poly)crystalline TiO2 to NC-TiO2 was observed. This large difference was attributed to Received: January 8, 2013 Revised: March 28, 2013 Published: March 29, 2013 8032

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als (DS-NC-TiO2) prepared with the N3 dye. Our intent is threefold: by comparing Pol-TiO2 and NC-TiO2 we aim (1) to study the effects of spatial confinement on electron transport and (2) to investigate the role of deeply trapped electrons on the complex photoconductance. (3) By comparing direct UV excitation of NC-TiO2 with visible excitation of DS-NC-TiO2, we attempt to clarify the contribution of photogenerated holes on photoconductance as a function of their location inside or outside the TiO2 nanocrystal. First, we present a model to deduce the time-dependent real and imaginary photoconductance (ΔG′ and ΔG″, respectively) from the experimental microwave data. Subsequently, upon application of the Drude−Smith model for photoconductance we describe the behavior of the mobile electrons in Pol-TiO2 and NC-TiO2, and we determine the yields of mobile charge generation, the effective electron masses, and the momentum scattering times.

low yield of mobile charge carrier photogeneration due to the large density of deep electron traps18,22 and/or to a lower effective electron mobility because of the electron scattering at the boundaries of the nanocrystals.23,24 In all the aforementioned studies, the influence of trapped electrons on the photoconductance has been considered negligible, while in 2012, Dunn et al.25 attributed part of the microwave photoconductance to trapped charges. It appears clear that recording only the real component of the microwave photoconductance is not sufficient to fully understand the intraparticle electron kinetics in NC-TiO2 and the role of trapped charges. In the past only a few complex microwave photoconductance studies have been performed in order to study charge transport in nanoporous systems.30,31 Several other studies have been performed using timeresolved terahertz spectroscopy to determine the real and imaginary photoconductance in TiO2. However, conflicting models regarding the local mobility of electrons in TiO2 matrices were reported. Hendry et al. described the motion of conduction band electrons in single crystal rutile TiO226 and nanocrystalline TiO227 by the ordinary Drude model combined with the Maxwell−Garnett effective medium theory. In those works, the authors determined an electron mobility of 1 cm2/ (V s) in bulk rutile and an intrinsic upper limit for electron mobility of 0.01 cm2/(V s) in TiO2 nanocrystals of mixed 30% rutile and 70% anatase. In 2002, Turner et al.28 investigated dye-sensitized nanocrystalline TiO2 (mixed anatase and rutile phase) and applied an extended Drude−Smith model to explain the intraparticle motion of charge carriers. They determined an electron mobility of 15 cm2/(V s) at terahertz frequency. Additionally, in that work, the authors reported the evolution of the photoconductance in the dye-sensitized TiO2 sample from Drude-like to Drude−Smith-like over the picoseconds time scale. They attributed this change in charge motion to polaron formation (i.e., the perturbation of the positions of the nuclei due to the excess mobile electrons). In 2010, Richter et al.29 presented a time-resolved terahertz spectroscopy study on dyesensitized polycrystalline TiO2 nanotubes and determined equally low intrinsic electron mobilities in the nanotubes and in nanocrystalline TiO2. However, they attributed the low mobilities in the two materials to distinct phenomena: excitonlike trap states (exhibiting a sharp resonance at terahertz frequency) in the TiO2 nanotubes and backscattering-induced electron localization (in agreement with the Drude−Smith model) in the nanocrystalline TiO2 sample. Because of the disagreement in the literature, more information regarding the intrinsic charge transport properties and their relation to the morphology of TiO2 nanostructures is required, as this will eventually delineate the optimal dimensions and geometric configuration of the TiO2 film to be used in e.g. photovoltaic devices. For example, this will clarify whether the use of highly anisotropic TiO2 nanostructures (such as nanotubes29,32 or nanorods33,34) might be beneficial for charge transport as compared to a sintered nanoporous TiO2 network. In this work, we present a real and imaginary microwave photoconductance study using the time- and frequencyresolved microwave conductance (TRMC) technique. We perform complex photoconductance measurements on three types of samples: dense anatase TiO2 films composed of large (>100 nm) crystalline domains (Pol-TiO2) (see Supporting Information), porous networks of 9 nm sized anatase nanocrystals (NC-TiO2), and dye-sensitized anatase nanocryst-

2. EXPERIMENTAL SECTION Two types of anatase TiO2 are prepared: polycrystalline (PolTiO2) and nanocrystalline (NC-TiO2). Thin films of Pol-TiO2 (80 ± 10 nm) were purchased from Everest Coatings (Delft, The Netherlands) and were prepared by chemical vapor deposition on top of 1 mm thick quartz plates. Subsequent annealing at 450 °C for 2 h was performed to improve stoichiometry. To realize uniformly thick NC-TiO2 samples, we first spincoated a 0.01 M solution of Ti(i-OPr)4 in isopropanol on top of 1 mm thick quartz plates. After 20 min annealing at 450 °C, a solid TiO2 underlayer adhering to the quartz substrate is obtained. A colloidal suspension of anatase TiO2 (Solaronix SA Ti-Nanoxide HT, average particle size 9 nm) is spread with the doctor blading method over the Ti(i-OPr)4 layer. The obtained 2 μm thick films are annealed for 6 h at 450 °C. The sintered nanocrystals form an intricate network with average pore diameter of 9 nm. More detailed description of the sample preparation technique can be found elsewhere.35 For the DS-NC-TiO 2 , the N3 molecule cis-bis(isothiocyanato)bis(2,2′-bipyridyl-4,4′-dicarboxylato)ruthenium(II) was purchased from Solaronix (99% purity) and was used without further purification. To prepare the N3 solution, the dye was dissolved in a 1:1 (volume ratio) mixture of tert-butyl alcohol and acetonitrile with a concentration of 0.3 mM. The DS-NC-TiO2 sample was then prepared by dipping for 8 h the bare NC-TiO2 films in the N3 solution to ensure homogeneous dye adsorption onto the surfaces of the TiO2 nanoparticles. 3. METHOD AND MODEL The TRMC technique allows the determination of the complex photoconductance in weakly conducting materials using Xband microwaves (8−12 GHz) as probe.23,36,37 Upon excitation of the sample using a nanosecond laser pulse (3 ns full width at half-maximum, 10 Hz repetition rate), photophysical products like bound electron hole pairs and/or mobile charge carriers can induce two effects on the microwaves: (1) absorption of a fraction of the microwave power and (2) a phase shift of the microwave electric field due to a change of the dielectric constant of the excited sample. The measured normalized change in the microwave power ΔP(ω,t)/P as a function of time, t, and probe frequency, ω, is proportional to the complex 8033

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Figure 1. Calculated response of a microwave cavity in terms of ΔP/P as a function of frequency and time in the case of (a) purely real, (b) purely imaginary, and (c) complex photoconductivity, imposing for all of them a monoexponential decay time of 1 μs. The three lower panels show the frequency dependency of ΔP/P at 40 ns, corresponding to the dashed lines in the upper panels.

photoconductance ΔG(ω,t) induced in the sample, according to ΔP(ω , t ) = −K (ω)ΔG(ω , t ) P

To analyze the time- and frequency-dependent experimental transients, a mathematical description of the microwave response due to a time-dependent change in complex photoconductance is required. To this end, Schins et al.39 proposed a model based on the parallelism between an electromagnetic resonator (TRMC cavity) and an electrical resonator (RLC circuit). The frequency- and time-dependent microwave power reflected by the microwave resonant cavity is mathematically described as the power reflected by such circuit, in which the inductor L and the resistor R are time-dependent components undergoing a perturbation due to optical excitation and an eventual decay. More specifically, the model generates the frequency- and time-dependent normalized changes in reflected microwave power (ΔP(ω,t)/P). It requires as input parameters the maximum changes ΔR and ΔL of the circuit components, which are proportional to the changes in real and imaginary photoconductance (ΔGmax′) and ΔGmax″), together with a time profile describing the rise and relaxation of ΔR and ΔL. Figures 1a−c display the simulated ΔP(ω,t)/P obtained by application of the RLC model considering instantaneous perturbation of ΔR and ΔL and subsequent monoexponential decay with time constant of 1 μs. Figure 1a shows how a purely real photoconductance induces a symmetric shape of ΔP(ω)/P around the resonance frequency. In the case of a purely imaginary photoconductance an antisymmetric shape with a positive and a negative lobe around the resonance frequency is observed (see Figure 1b). Finally, Figure 1c displays the response upon a change in conductance containing both real and imaginary components. Time-dependent changes in the

(1)

In eq 1, K(ω) is the frequency-dependent sensitivity factor of the system. The time-dependent real component of the photoconductance ΔG′ induced in the sample can be determined by mounting the sample inside a nonresonant microwave cell (denoted as open cell) and measuring the quantity ΔP(t)/P at a single probe frequency. In this experimental condition, the sensitivity of the system is determined to be Koc = 103 S−1.22,38 Using this approach, the time resolution is 3 ns, limited by the electronics and the duration of the laser pulse. To measure not only the real but also the imaginary photoconductance, the samples are placed inside a microwave resonant cavity. The normalized microwave power reflected from such cavity follows an inverted Lorentzian profile as a function of the microwave frequency. At minimum reflected power (i.e., at the resonance frequency) corresponding with maximum electric field strength within the cavity, most sensitive photoconductance measurements can be performed. At off-resonance frequencies, information regarding changes in the dielectric constant on photoexcitation can be obtained.36 Note that, depending on the frequency of the microwave probe, not only the sensitivity but also the time resolution of the system changes, ranging from 18 ns at resonance to 3 ns far from resonance. 8034

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ΔG″/ΔG′ ratio are indicative of formation of different photophysical products with distinct decay kinetics. For the calculations shown in Figure 1, the ratio ΔG″/ΔG′ was kept constant as a function of time, corresponding to the formation of a single photophysical product. In practice, the following approach was used to obtain the ΔG′ and ΔG″ induced in various TiO2 samples. First open-cell microwave transients were recorded to obtain the decay profiles of the real part of the photoconductance with temporal resolution of 3 ns. Subsequently, we performed frequency- and time-dependent measurements using a microwave cavity, yielding the ΔP(ω,t)/P data set for each sample. The experimental open-cell decay profiles were used as starting inputs to fit the time- and frequency-dependent ΔP(ω,t)/P generated by the RLC model to the experimental ΔP(ω,t)/P. When needed, additional decay components were included to take into account the contribution of different species such as Coulombically bound electron−hole pairs. Finally, by adjusting the fitting parameters ΔR and ΔL corresponding to ΔGmax′ and ΔGmax″ for each species we could accurately reproduce the experimental ΔP(ω,t)/P.

laser pulse excites the samples in order to avoid charge accumulation. As discussed in previous works,23 the electron mobility in anatase TiO2 is at least 10 times higher than the hole mobility, and thus electrons dominate the TRMC signal. The slow exponential decay observed in Pol-TiO2 has been previously attributed to electron−hole recombination.22 The decay kinetics of the NC-TiO2 sample is 2 orders of magnitude faster and follows a multiexponential profile. In 2004, Yoshihara et al. monitored the electron−hole recombination kinetics by transient absorption (TA) spectroscopy upon UV excitation of similar bare NC-TiO2.40 In that work they observed the hole decaying with recombination time of 1 μs (estimated from the period required for the signal to decay by 50%), which is more than 1 order of magnitude slower than the TRMC decay in Figure 2a. From this comparison we reason that electron−hole recombination is not the only decay mechanism of the TRMC trace and that an additional contribution has to be considered: interparticle hopping toward unsaturated traps in adjacent nanocrystals reducing the overall population of conduction band electrons.18,22,23 Once localized in deep traps, electrons are no longer detectable by the TRMC setup. Apart from the decay profiles of the TRMC traces, the amplitude of the signal provides additional information. Although in Figure 2a the excitation fluences differ by 2 orders of magnitude, the maxima of the (ΔP/P) signal for the two samples are comparable. This aspect will be discussed in the last part of the Results and Discussion section. Figure 2b shows the fluence dependency of (ΔP/P)max/I0 for the two samples on excitation at 300 nm. This quantity is directly proportional to the product of electron mobility and photogeneration yield. As explained previously,22 for Pol-TiO2, the (ΔP/P)max/I0 value remains constant for fluences lower than 1012 photons/cm2 since only first-order decay processes occur below this threshold. Above this threshold, the (ΔP/P)max/I0 values become gradually smaller due to sub-nanosecond high-order recombination processes, resulting in a faster decay kinetics as already shown by Savenije et al.22 For NC-TiO2, the value of (ΔP/P)max/I0 first increases, it reaches a maximum at incident fluence of 1013 photons/cm2, and then decreases again. The small signal detected in the low fluence range has been attributed to deep trap filling reducing the population of mobile detectable electrons. At low excitation density a large fraction of photogenerated electrons are trapped and thus immobile. Note that no accumulation of charges was induced on repetitive laser pulses. Upon increasing excitation density high-order recombination processes on the sub-nanosecond time scale result in a reduced signal as discussed in full detail by Kroeze et al.22,23 All the TRMC experimental data reported in the remaining part of the paper have been recorded using fluences of 2.3 × 1011 and 1.2 × 10 13 photons/cm 2 for Pol-TiO2 and NC-TiO2 , respectively. The frequency- and time-dependent experimental data recorded using the microwave cavity upon photoexcitation at 300 nm are displayed in Figures 3a and 3b for Pol-TiO2 and NC-TiO2, respectively. The experimental data show that the microwave photoconductance in the two samples exhibits both real and imaginary contributions, since the 3D contour plots in Figures 3a and 3b are neither fully symmetric nor antisymmetric around the resonance frequency. The two contour plots are comparable in terms of absolute signal size, position of the maximum of ΔP/P, and extent of the asymmetry around the resonance frequency. The absolute position of the resonance frequency varies for the two samples

4. RESULTS AND DISCUSSION Complex Photoconductance of Pol-TiO2 and NC-TiO2. Figure 2a shows the transient ΔP/P(t) for 300 nm excitation of Pol-TiO2 and NC-TiO2 recorded in the open-cell at incident photon fluence I0 = 2.3 × 1011 photons/cm2 and I0 = 1.2 × 1013 photons/cm2, respectively. Both curves exhibit a characteristic 3 ns rise time, corresponding to the duration of the laser pulse, followed by a subsequent decay. It was experimentally verified that the TRMC signal decayed back to zero before the next

Figure 2. (a) Open-cell ΔP(t)/P transients for 300 nm excitation of Pol-TiO2 and NC-TiO2 upon 2.3 × 1011 photons/cm2 and 1.2 × 1013 photons/cm2 illumination, respectively. (b) Fluence dependency of (ΔP/P)max/I0 upon 300 nm excitation of the two TiO2 samples. 8035

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Figure 3. Top: experimental contour plots of the frequency- and time-dependent behavior of ΔP/P for 300 nm excitation of (a) Pol-TiO2 and (b) NC-TiO2 using incident photon fluence of 2.3 × 1011 and 1.2 × 1013 photons/cm2, respectively. All the experimental traces are recorded in the resonant cavity microwave system. The solid black horizontal line indicates the resonance frequency in each graph. Bottom: fits of the RLC model to ′ and ΔGmax ″ reported in the insets and the experimental contour plots of (c) Pol-TiO2 and (d) NC-TiO2. This fits are obtained by applying the ΔGmax the multiexponential decay profiles obtained from the open-cell transients in Figure 2a.

because of the difference in film thickness and density. The contour plots in Figures 3c and 3d are the modeled traces obtained with the RLC model using as input the open-cell temporal profiles shown in Figure 2a and adjusting ΔG′max and ΔG″max until a good match to the experimental traces is obtained. The best fits were obtained with the ΔG′max and ″ values reported in the insets and considering for both ΔGmax samples constant ΔG″/ΔG′ ratios as a function of time (see Supporting Information for discussion on fits quality). No additional decay components related to species with a different ΔG″/ΔG′ ratio were required, indicating that in both samples the microwave photoconductance is due to a single photophysical species (i.e., conduction band electrons). Most interestingly, for both samples negative values of ΔG″ in accordance with previous studies28,31 and almost identical ΔG″/ΔG′ ratios are needed to model the two experimental contour plots of Figures 3a and 3b. The fact that the same ΔG″/ΔG′ ratio reproduces the experimental data for the two samples indicates that the mobile electrons in Pol-TiO2 and inside the nanocrystals in NC-TiO2 behave in similar fashion. Apparently, the motion of electrons is not significantly affected by the spatial restrictions of the nanocrystals in the NC-TiO2 sample. This aspect will be discussed in more detail in the last part of the Results and Discussion section. Complex Photoconductance of NC-TiO2 and DS-NCTiO2. In this section we will examine to what extent the lightinduced positive charges contribute to the TRMC signal. To this aim we compare the complex photoconductance of a NCTiO2 sample excited at 300 nm with that of an N3 DS-NCTiO2 sample upon photoexcitation at 550 nm. For bare NCTiO2 the holes are located in the TiO2 nanoparticles, while for DS-NC-TiO2 the positive charges remain immobile on the dye

molecules at the surface of the nanoparticle. The comparison between these two materials should reveal whether the close proximity of the photogenerated holes to the conduction band electrons in the TiO2 matrix of the nanocrystalline sample results in a contribution to the complex photoconductance. As control experiment we recorded open-cell TRMC transients upon 300 nm excitation of the DS-NC-TiO2 at various incident photon fluences (see Supporting Information Figure 3S). No difference in signal size and decay kinetics was observed in comparison to bare NC-TiO2 under the same excitation conditions, ensuring that direct formation and decay of conduction band electrons is independent of the presence of the dye. Open-cell ΔP/P(t) transients have been recorded for 300 nm excitation of NC-TiO2 and 550 nm excitation of DS-NC-TiO2 at incident photon fluence I0 = 1.2 × 1013 and 1.3 × 1014 photons/cm2, respectively (data not shown). The signals decay in the two samples with identical profiles. Haque et al.41 studied the decay of the dye cation in comparable dye-sensitized NCTiO2 samples using TA spectroscopy. This TA decay is 4 orders of magnitude slower than our TRMC signal, and it has been entirely attributed to recombination of the injected electrons with the dye cation. The large difference in decay times supports our previous proposition that apart from charge carrier recombination, also trapping shortens the TRMC decay profile. Figures 4a and 4b show the experimental frequency- and time-dependent ΔP/P upon UV and visible excitation of the NC-TiO2 and DS-NC-TiO2 samples, respectively. Despite differences in signal size, which are due to different incident photon fluences (see Supporting Information), the overall shapes of the two contour plots in Figure 4 are almost 8036

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details of the complex photoconductance in the two bare TiO2 samples only. Analysis of the Complex Photoconductance of PolTiO2 and NC-TiO2. In this section, different photoconductance models will be discussed to understand the motion of electrons in Pol-TiO2 and NC-TiO2. Conductance in metals and inorganic semiconductors is usually well described by the Drude model, which represents electrons as particles experiencing scattering events at mean time interval τ, leading to randomization of the momentum. According to this model the imaginary photoconductance as a function of frequency is always positive. Since we found negative values of ΔGmax ″ for all samples, the Drude model is not appropriate to describe the properties of excess electrons in Pol-TiO2 and NC-TiO2. In 2001, Smith proposed a modification of the Drude model to take into account the persistence of the velocity after a collision experienced by a charge.44 The simplified Drude−Smith formula for complex photoconductivity is ⎤ ε0ωp2τ ⎡ c σ(ω) = ⎢1 + n⎥ 1 − iωτ ⎣ (1 − iωτ ) ⎦

(2)

in which ε0 is the permittivity in vacuum, τ is the momentum scattering time, ω = 2νπ is the radial frequency, and ωp2 = Ne2/ (ε0m*) is the plasma frequency, N is the mobile charge carrier density, and m* is the effective mass of the carriers. Negative values of c describe backscattering events implying the reduction of the real photoconductivity Δσ′ and a shift of the maximum of Δσ′ from dc to higher frequency. For values of c < −0.5, the imaginary photoconductivity Δσ″ is negative for ω ≪ 1/τ. In the extreme case of c = −1 the dc conductivity vanishes, and the maximum of Δσ′ appears at τω = 1 . In the Supporting Information the influence of c on the real and imaginary conductivity as a function of frequency is shown. With the modification to the Drude formula proposed by Smith, the negative imaginary photoconductivity observed by Turner et al. in 25 nm sized NC-TiO2 at terahertz frequencies could be described.28 From eq 2 the change in photoconductance (ΔG) is derived according to ΔG = β∫ d0Δσ(x) dx, β = 2.08 being a known parameter related to the inner dimensions of the microwave waveguide and d the thickness of the material:

Figure 4. Experimental contour plots of the frequency- and timedependent behavior of ΔP/P in the case of (a) bare NC-TiO2 at 300 nm excitation and (b) DS-NC-TiO2 at 550 nm excitation. The incident photon fluences have been taken such as the photoconductance signal is maximized and the high-order recombination processes are minimized. These conditions correspond to 1.2 × 1013 and 1.3 × 1014 photons/cm2 for (a) and (b), respectively (see Supporting Information for more details).

indistinguishable in terms of the position of the maximum, the extent of the asymmetry around the resonance frequency, and the decay profile. Using the RLC model discussed above, a good fit to the contour plot of the DS-NC-TiO2 is obtained on application of the same ΔG″/ΔG′ ratio and the same decay profile used to model the ΔP(ω,t)/P of the bare NC-TiO2. This implies that in the two samples the microwaves probe the same species (i.e., mobile electrons) showing the same dynamic behavior indicated by the open-cell decay profile. We can thus conclude that in both samples the microwave complex photoconductance is not influenced by the photogenerated holes, which do not form any bound pair with electrons nor do they give any direct appreciable contribution to the signal. As already mentioned, the quantity (ΔP/P)/I0 is proportional to the product of electron mobility and quantum yield η of mobile charge photogeneration. In the case of visible excitation of the N3 dye, η can be expressed as the product of two terms: the efficiency of electron injection from the dye to the TiO2 nanocrystals, ϕI, and the fraction of mobile electrons over the number of injected electrons, ϕM. The efficiency of charge injection from the N3 dye has been previously found to be close to unity;42,43 hence, upon visible excitation of the dye, η ≅ ϕM. Since (ΔP/P)I0 is almost identical in NC-TiO2 and DS-NC-TiO2 we conclude that in the two samples the yield of mobile electron photogeneration η is the same upon UV and upon visible excitation, respectively. The electron mobility is expected to be comparable in the two samples because of the absence of any observable contribution of the positive holes to the photoconductance. Given the similarities between NCTiO2 and DS-NC-TiO2, in the following we will discuss the

ΔG′ =

ΔG″ =

βI0ηe 2τ m*(1 + ω 2τ 2)2

[1 + c + ω 2τ 2(1 − c)]

βI0ηe 2τ 2ω [1 + 2c + ω 2τ 2] m*(1 + ω 2τ 2)2

(3a)

(3b)

In eqs 3a and 3b, I0 is the photon fluence entering the sample and η is the number of mobile charge carriers generated per incident photon. To fit the Drude−Smith model to the values of ΔG′max and ΔG″max determined for Pol-TiO2, we used as input parameter τ = 84 fs as reported previously.28 In ref 28, this value of τ has been determined from terahertz photoconductance on 25 nm nanoparticles of mixed anatase and rutile phase. The lower-lying conduction band edge of anatase with respect to rutile TiO2 favors the accumulation of the mobile electrons in the anatase rather than in the rutile phase.45 This allows a direct comparison between the sample studied in ref 28 and the pure anatase samples investigated in the present work. For the given value of τ, the other fitting parameters η (0 < η ≤ 1), m*, and c (− 1 < c < −0.5) in eqs 3a and 3b could be 8037

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immobile and give no significant contribution to the photoconductance, as already suggested by other authors.18,22 In contrast with the observations from Dunn et al. in which part of the imaginary microwave photoconductance was attributed to trapped charges,25 our measurements demonstrate that large amounts of trapped electrons do not affect the motion of conduction band electrons nor do they contribute to an additional response in our microwave measurements. The experimental conditions used in that work,25 however, were quite different from those in the present study, especially for the presence of an electrolyte allowing control of the Fermi level and the different microwave frequency utilized as probe, making a direct comparison difficult. In the present study the Fermi level is not fixed and the trapped electron density is not known. By application of the Drude−Smith model using the parameters listed in Table 1, the ac mobility of conduction band electrons at 9 GHz is 1.7 cm2/(V s) in both Pol-TiO2 and NC-TiO2. By extrapolating the ac mobility to 1 THz, we determine a value of 18 cm2/(V s), in good agreement with the results of Turner et al. on 25 nm NC-TiO2.28 Because of the weak frequency dependence of mobility between 0 and 9 GHz, the upper limit for trap-free dc mobility remains constant at 1.7 cm2/(V s). This value compares with what has been previously reported for room-temperature Hall mobility in single-phase anatase thin films,46 while it is significantly higher than the mobilities of 10−4−10−6 cm2/(V s),7 determined from time-offlight measurements in NC-TiO2 films. Our study shows that this large difference in macroscopic long-range transport is not determined by intraparticle low mobility. As limiting factors for macroscopic electron transport we propose the large trap density reducing the population of conduction band electrons and the thermally activated interparticle hopping mechanism between adjacent nanocrystals.

obtained independently from each other and are listed in Table 1. Table 1. Input Parameters Used To Reproduce the Real and Imaginary Photoconductance ΔG′max and ΔG″max Using the Drude−Smith Formulas in Eqs 3a and 3b, for I0 = 2.3 × 1011 and 1.2 × 1013 photons/cm2 for Pol-TiO2 and NC-TiO2, Respectively momentum scattering time τ (s) quantum yield η electron effective mass m* (kg) coefficient c

Pol-TiO2

NC-TiO2

8.4 × 10−14 1 2.25me −0.974

8.5 × 10−14 0.02 2.25me −0.970

The electron effective mass we determine in Pol-TiO2 is m* = 2.25me, me being the electron rest mass. This value is in agreement with previous studies.46 As already suggested by the similarity of the experimental contour plots in Figures 3a and 3b, the values of ΔG′max and ΔG″max for the NC-TiO2 sample can be reproduced by applying the same τ, m*, and c fitting parameters used for Pol-TiO2. Only the yield η is reduced by 2 orders of magnitude, implying that most photoexcitations in NC-TiO2 result in rapidly trapped electrons having such low mobility to give a negligible contribution to the photoconductance signal. As mentioned above, the incident photon fluence was chosen in order to maximize the TRMC signal for the NC-TiO2 sample. From the fact that η is as low as 0.02 it follows that no full saturation of traps is accomplished at the used fluence. Interestingly, although additional scattering of the conduction band electrons is expected at the surface of the nanoparticles in the NC-TiO2 sample, no adjustment of τ, m*, and c is required. Hence, no evidence of charge confinement in the volume of the 9 nm nanocrystals is found. This is in agreement with previous studies determining an exciton Bohr radius shorter than 1 nm in anatase TiO2.47 The mean free path of the mobile electrons can be determined as l = vthτ, with the thermal velocity vth = (3kBT/m*)1/2, kB being the Boltzmann constant and T the temperature. Using the values of τ and m* from Table 1, we obtain a mean free path for electrons in both Pol-TiO2 and NC-TiO2 of 6.5 nm. Turner et al.28 reported an electron mean free path at 77 K of 1.6 nm in 25 nm sized NCTiO2 (anatase and rutile mixture). However, in that work, the authors used an electron effective mass of 10me (determined from rutile single crystals) even though the anatase/rutile mixture should result in a lower effective mass, as already discussed previously.46 Since both Pol-TiO2 and NC-TiO2 studied in this work are constituted of pure anatase, a common effective mass of m* = 2.25me is used in our analysis. Our findings indicate that irrespective of the size of the TiO2 crystallites, the electrons experience frequent backscattering events over distances of 6.5 nm, and no additional scattering from the surface of the nanoparticles is observed at 9 GHz frequency. This might be due to sintered TiO2 nanoparticles forming crystal domains for electron transport larger than 9 nm. The total amount of mobile conduction band electrons is given by the product ηI0 and is comparable for the Pol-TiO2 and the NC-TiO2 samples. The 2 orders of magnitude difference in the yield of mobile charge carrier generation η can be explained by considering that most conduction band electrons formed on excitation of the NC-TiO2 sample decay by deep trapping and/or bimolecular recombination on a subnanosecond time scale. The deeply trapped electrons are

5. CONCLUSIONS By applying frequency- and time-dependent microwave conductance measurements, we determined the real and imaginary components of the conductance on pulsed excitation of anatase TiO2 samples with different crystallite size. We can conclude that conduction band electrons in the TiO2 samples exhibit the same type of motion: no reduction in mobility due to spatial confinement is found. The electron mobility is bulk limited by the short momentum scattering time of 85 fs and by the electron effective mass of 2.25me. The yield of mobile charge carrier generation in NC-TiO2 is found to be 2 orders of magnitude lower than in Pol-TiO2 due to the large number of deep traps in NC-TiO2. The trapped charges in NC-TiO2 do not contribute to the complex photoconductance signal at least at the used frequency of 9 GHz. It is shown that the photogenerated positive charges do not give rise to an appreciable photoconductance signal. With this study we demonstrate that the limiting factor for dc electron transport in sintered networks of TiO2 nanocrystals utilized in DSSCs is not low intraparticle mobility of conduction band electrons. We rather propose that the large trap density reduces the population of conduction band electrons and that the thermally activated interparticle hopping between adjacent nanocrystals lowers long-range electron mobility. According to our findings, the local electronic transport along highly anisotropic nanostructures (e.g., nanotubes or nanorods) will not be improved by the nanomorphology. In contrast, the macroscopic dc mobility might be 8038

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(10) Agrell, H. G.; Boschloo, G.; Hagfeldt, A. Conductivity Studies of Nanostructured TiO2 Films Permeated with Electrolyte. J. Phys. Chem. B 2004, 108, 12388−12396. (11) Dloczik, L.; Ileperuma, O.; Lauermann, I.; Peter, L. M.; Ponomarev, E. A.; Redmond, G.; Shaw, N. J.; Uhlendorf, I. Dynamic Response of Dye-Sensitized Nanocrystalline Solar Cells: Characterization by Intensity-Modulated Photocurrent Spectroscopy. J. Phys. Chem. B 1997, 101, 10281−10289. (12) van de Lagemaat, J.; Frank, A. J. Effect of the Surface-State Distribution on Electron Transport in Dye-Sensitized TiO2 Solar Cells: Nonlinear Electron-Transport Kinetics. J. Phys. Chem. B 2000, 104, 4292−4294. (13) Bisquert, J. Chemical Diffusion Coefficient of Electrons in Nanostructured Semiconductor Electrodes and Dye-Sensitized Solar Cells. J. Phys. Chem. B 2004, 108, 2323−2332. (14) Bisquert, J.; Vikhrenko, V. S. Interpretation of the Time Constants Measured by Kinetic Techniques in Nanostructured Semiconductor Electrodes and Dye-Sensitized Solar Cells. J. Phys. Chem. B 2004, 108, 2313−2322. (15) Fisher, A. C.; Peter, L. M.; Ponomarev, E. A.; Walker, A. B.; Wijayantha, K. G. U. Intensity Dependence of the Back Reaction and Transport of Electrons in Dye-Sensitized Nanacrystalline TiO2 Solar Cells. J. Phys. Chem. B 2000, 104, 949−958. (16) Wang, Q.; Ito, S.; Gratzel, M.; Fabregat-Santiago, F.; Mora-Sero, I.; Bisquert, J.; Bessho, T.; Imai, H. Characteristics of High Efficiency Dye-Sensitized Solar Cells. J. Phys. Chem. B 2006, 110, 25210−25221. (17) Tiwana, P.; Parkinson, P.; Johnston, M. B.; Snaith, H. J.; Herz, L. M. Ultrafast Terahertz Conductivity Dynamics in Mesoporous TiO2: Influence of Dye Sensitization and Surface Treatment in SolidState Dye-Sensitized Solar Cells. J. Phys. Chem. C 2010, 114, 1365− 1371. (18) Savenije, T. J.; Huijser, A.; Vermeulen, M. J. W.; Katoh, R. Charge Carrier Dynamics in TiO2 Nanoparticles at Various Temperatures. Chem. Phys. Lett. 2008, 461, 93−96. (19) Boschloo, G.; Hagfeldt, A. Activation Energy of Electron Transport in Dye-Sensitized TiO2 Solar Cells. J. Phys. Chem. B 2005, 109, 12093−12098. (20) Peter, L. M.; Walker, A. B.; Boschloo, G.; Hagfeldt, A. Interpretation of Apparent Activation Energies for Electron Transport in Dye-Sensitized Nanocrystalline Solar Cells. J. Phys. Chem. B 2006, 110, 13694−13699. (21) Kopidakis, N.; Benkstein, K. D.; van de Lagemaat, J.; Frank, A. J.; Yuan, Q.; Schiff, E. A. Temperature Dependence of the Electron Diffusion Coefficient in Electrolyte-Filled TiO2 Nanoparticle Films: Evidence Against MultipleTrapping in Exponential Conduction-Band Tails. Phys. Rev. B 2006, 73, 045326. (22) Savenije, T. J.; de Haas, M. P.; Warman, J. M. The Yield and Mobility of Charge Carriers in Smooth and Nanoporous TiO2 Films. Z. Phys. Chem. 1999, 212, 201−206. (23) Kroeze, J. E.; Savenije, T. J.; Warman, J. M. Electrodeless Eetermination of the Trap Density, Decay Kinetics, and Charge Separation Efficiency of Dye-Sensitized Nanocrystalline TiO2. J. Am. Chem. Soc. 2004, 126, 7608−7618. (24) Katoh, R.; Huijser, A.; Hara, K.; Savenije, T. J.; Siebbeles, L. D. A. Effect of the Particle Size on the Electron Injection Efficiency in Dye-Sensitized Nanocrystalline TiO2 Films Studied by Time-Resolved Microwave Conductivity (TRMC) Measurements. J. Phys. Chem. C 2007, 111, 10741−10746. (25) Dunn, H. K.; Peter, L. M.; Bingham, S. J.; Maluta, E.; Walker, A. B. In Situ Detection of Free and Trapped Electrons in Dye-Sensitized Solar Cells by Photo-Induced Microwave Reflectance Measurements. J. Phys. Chem. C 2012, 116, 22063−22072. (26) Hendry, E.; Wang, F.; Shan, J.; Heinz, T. F.; Bonn, M. Electron Transport in TiO2 Probed by THz Time-Domain Spectroscopy. Phys. Rev. B 2004, 69, 081101. (27) Hendry, E.; Koeberg, M.; O’Regan, B.; Bonn, M. Local Field Effects on Electron Transport in Nanostructured TiO2 Revealed by Terahertz Spectroscopy. Nano Lett. 2006, 6, 755−759.

positively influenced by the formation of large crystalline domains, reducing the hopping-like character of long-range transport. Moreover, the smaller surface to volume ratio of nanotubes and nanorods might lower the number of deep electron traps, resulting in a higher yield of conduction band electrons with respect to NC-TiO2.

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ASSOCIATED CONTENT

S Supporting Information *

(1) AFM picture and XRD diagram of Pol-TiO2; (2) time- and frequency-dependent contour plots of experimental data, modeled data, and residual for Pol-TiO2 upon 300 nm excitation; (3) intensity dependency of (ΔP/P)max/I0 upon 300 nm excitation of the three samples and 550 nm excitation of the DS-NC-TiO2 sample; (4) frequency dependency of the Drude−Smith real and imaginary photoconductance for different values of the coefficient c. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Present Address †

D.D.: Photovoltaic Materials and Device Section, Department of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, Mekelweg 4, 2628 CD, Delft, The Netherlands. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Work of M.C.F. is part of the research program of the “Stichting voor Fundamenteel Onderzoek der Materie (FOM)”, which is financially supported by the “Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO).

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