Chinese Journal of Chemical Physics  2016, Vol. 29 Issue (6): 735-741

#### The article information

Yu-dan Wang, Zhe Sun, Ya-jun Ren, Yan Zhang, Mao Liang, Song Xue

Correlating Photovoltaic Performance of Dye-Sensitized Solar Cell to the Film Thickness of Titania via Numerical Drift-Diffusion Simulations

Chinese Journal of Chemical Physics, 2016, 29(6): 735-741

http://dx.doi.org/10.1063/1674-0068/29/cjcp1604090

### Article history

Accepted on: May 6, 2016
Correlating Photovoltaic Performance of Dye-Sensitized Solar Cell to the Film Thickness of Titania via Numerical Drift-Diffusion Simulations
Yu-dan Wanga, Zhe Suna, Ya-jun Renb, Yan Zhanga, Mao Lianga, Song Xuea
Dated: Received on April 28, 2016; Accepted on May 6, 2016
a. Tianjin Key Laboratory of Organic Solar Cells and Photochemical Conversion, School of Chemistry & Chemical Engineering, Tianjin University of Technology, Tianjin 300384, China;
b. School of Electric Engineering, Tianjin University of Technology, Tianjin 300384, China
*Author to whom correspondence should be addressed. Zhe Sun, E-mail:zhesun@tjut.edu.cn; Song Xue, E-mail:xuesong@ustc.edu.cn, Tel.:+86-22-60214250, FAX:+86-22-60214252
Abstract: The thickness of TiO2 film is vital to realize the optimization on photovoltaic performance of dye sensitized solar cells (DSSCs). Herein, the process of charge separation in DSSCs was simulated by using a drift-diffusion model. This model allows multiple-trapping diffusion of photo-generated electrons, as well as the back reaction with the electron acceptors in electrolyte, to be mimicked in both steady and non-steady states. Numerical results on current-voltage characteristics allow power conversion efficiency to be maximized by varying the thickness of TiO2 film. Charge collection efficiency is shown to decrease with film thickness, whereas the flux of electron injection benefits from the film thickening. The output of photocurrent is actually impacted by the two factors. Furthermore, recombination rate constant is found to affect the optimized film thickness remarkably. Thicker TiO2 film is suitable to the DSSCs in which back reaction is suppressed sufficiently. On the contrary, the DSSCs with the redox couple showing fast electron interception require thinner film to alleviate the charge loss via recombination. At open circuit, electron density is found to decrease with film thickness, which engenders not only the reduction of photovoltage but also the increase of electron lifetime.
Key words: Dye-sensitized solar cells    Titania    Film thickness    Simulation
Ⅰ. INTRODUCTION

Extensive explorations on green and sustainable energy sources have become urgent because of the swift surge of energy consumption in the past twenty years. Dye-sensitized solar cell (DSSC) has been recognized as a promising photovoltaic device for its capability of providing clean and renewable energy [1-4]. Compared with traditional silicon cell, DSSC is attractive to the researchers due to its simple architecture, easy fabrication and tunable optical responses [5]. For this reason, DSSC can be not only merely regarded as sunlight-to-electrical convertor, but also probably a low-cost solution to the global energy crisis [6].

The most successful DSSCs to date are fabricated as n-type, namely those devices are typically constituted by a dye-soaked TiO2 film as photoanode, a platinized FTO glass as counter electrode, and liquid electrolyte or hole-transport material as charge separation mediator [7, 8]. Very recently, the n-type DSSCs with cobalt complex based electrolyte have yielded the power conversion efficiency (PCE) exceeding 14 %, setting a new milestone in the records of device performance [9-13]. However, there is no denying the fact that the renewed cell efficiency is not quite larger than the record (~12 %) achieved by iodine electrolyte based DSSC [14-19]. As has been demonstrated, the advance of cell performance for cobalt complex based DSSCs is primarily impeded by the inefficient mass transport of the bulky redox shuttles in electrolyte solution [20-22]. The output of photocurrent in these DSSCs is constrained by diffusion-limited current density. Thereby, thinner TiO2 films (3-5 µm) in combination with high mole extinction coefficient organic sensitizers are extensively applied for facilitating charge transport in the bulk of electrolyte. It is known that those sensitizers are designed to show mole extinction coefficient higher than 104 (mol/L)-1cm-1 in wavelength range from 400 nm to 650 nm [19, 23]. Exceeding 650 nm, mole extinction coefficient has been found to decrease dramatically. From this perspective, thinner TiO2 films soaked by those sensitizers are less effective in harvesting the incident light of over 650 nm. In view of the large contribution of long-wavelength part in solar spectrum to photocurrent, this could engender considerable energy loss. To overcome such deficiency, one can reasonably utilize thicker TiO2 film. Meanwhile, fast transport redox shuttles, such as iodide/triiodide, disulfide/thiolate, and TEMPO/TEMPO+, need to be employed for preventing the occurrence of mass transport limitation inside thicker photoanode [6-8, 22].

The extension of light-harvesting to long-wave region by using thicker TiO2 film is inevitably accompanied by the acceleration of charge recombination. Even though the mass transport of redox shuttles is effective, the increase of film thickness can give rise to more active sites for interfacial recombination. This means that an optimized film thickness is of importance to realize more effective energy management. Otherwise, the improved photocurrent by thickening TiO2 film will be offset by the severe back reaction. Previous studies have demonstrated that the operating of DSSCs can be well described by using a drift-diffusion model [24-28]. Aside from predicting cell parameters, such as open circuit voltage ($V_{\textrm{oc}}$), short circuit current density ($j_{\textrm{sc}}$), and power conversion efficiency (PCE), this model is valid for mimicking the density profiles of free and trapped electrons in TiO2 film, as well as the distribution of redox couples in electrolyte solution. Also, this model allows the assessment on the impacts of film thickness on the recombination rate. Thereby, it is straightforward to establish the relationship between film thickness and electron lifetime. In this way, the variation of cell parameters with film thickness can be interpreted theoretically.

In this work, we evaluate the influence of film thickness on the photovoltaic performance of DSSCs on the basis of the drift-diffusion model. The electrons at the conduction band and the trapping states are considered so as to make a justified prediction on the electron lifetimes at open circuit condition. By solving the drift-diffusion model numerically, we aim to explain the variations of charge collection efficiency, electron lifetime, as well as film thickness, and uncover the origin of the loss of the separated charges. In addition, the film thickness is optimized under the conditions of strong and weak back reaction, by which we found that different film thickness should be employed according to kinetics of interfacial recombination.

Ⅱ. THEORETICAL MODEL

As illustrated schematically in Fig. 1, charge separation in DSSCs is simulated by using a drift-diffusion model. In this model, the operation of DSSCs is decomposed into four processes, i.e. (ⅰ) the injection of photo-generated electrons into TiO2 film, (ⅱ) the transport of the injected electrons to the layer of charge collector, (ⅲ) interfacial recombination between electrons and the acceptors in electrolyte, and (ⅳ) the recovery of the acceptors via diffusion. It is obvious that the first three processes are determinative to the accumulation and diffusion of the injected electrons inside TiO2 film. Resultantly, the kinetics of the conduction band electrons ($n_\textrm{c}$) at non-steady states is formulated as follows [28],

 $\begin{eqnarray} \frac{{\partial n_{\rm{c}} }}{{\partial t}}\left( {1 + \frac{{\partial n_{\rm{T}} }}{{\partial n_{\rm{c}} }}} \right) = D_{\rm{c}} \nabla ^2 n_{\rm{c}} - U_{{\rm{rec}}} + \Lambda _{\rm{e}} G_{{\rm{in}}} \end{eqnarray}$ (1)

where $D_\textrm{c}$ is the diffusion coefficient of conduction band electrons, $U_{\textrm{rec}}$ is the recombination rate, $G_{\textrm{in}}$ is the rate of electron injection, and $\Lambda _{\rm{e}}$ equals to unity, accounting for the stoichiometric coefficient. Moreover, we assume the diffusion of conduction band electrons obeys multi-trapping mechanism. Previous studies have addressed the ultra-fast exchange of the injected electrons between the conduction band and the trapped states. It follows that the relationship between $n_\textrm{c}$ and the density of the trapped electrons ($n_\textrm{T}$) satisfies the quasi-steady approximation. It is usual that $n_\textrm{c}$ is estimated according to the quasi-Fermi level of TiO2, that is

 $\begin{eqnarray} n_{\rm{c}} = N_\textrm{c} \exp \left( {\frac{{E_{{\rm{F, n}}} - E_{\rm{c}} }}{{k_{\rm{B}} T}}} \right) \end{eqnarray}$ (2)

where $E_\textrm{c}$ is the energy level of the band edge, $N_\textrm{c}$ is the density of the states at the conduction band, $k_\textrm{B}$ is the Boltzmann constant, and T is the absolute temperature. By assuming the exponential distribution of the trapped states, $n_\textrm{T}$ is expressed as the following,

 $\begin{eqnarray} n_{\rm{T}} = N_{\rm{T}} \exp \left( {\frac{{E_{{\rm{F, n}}} - E_{\rm{c}} }}{{k_{\rm{B}} T_0 }}} \right) \end{eqnarray}$ (3)

where $N_\textrm{T}$ is the density of the trapped states, T0 is the characteristic parameter determining the depth of exponential distribution. It is straightforward to express $n_\textrm{T}$ as a function of $n_\textrm{c}$ [24]:

 $\begin{eqnarray} n_{\rm{T}} = \frac{{N_{\rm{T}} }}{N_{\rm{c}}^{T/{T_0 }} n_{\rm{c}}^{{T/{T_0 }}}} \end{eqnarray}$ (4)
 FIG. 1 Schematic of the operation of the DSSC under irradiation. The thicknesses of TiO2 film and electrolyte phase are indicated as d and $d_{\textrm{el}}$, respectively.

The recombination reaction is assumed to be localized, which is written as

 $\begin{eqnarray} U_{{\rm{rec}}} = k_{\rm{r}} el_{{\rm{ox}}} n_{\rm{c}}^b \end{eqnarray}$ (5)

where $el_{\textrm{ox}}$ is the concentration of electron acceptor in electrolyte, $k_\textrm{r}$ is the rate constant of recombination reaction, and b is the apparent reaction order. The value of b is usually less than 1.0, implying the non-ideality of recombination reaction [29-31]. It has been suggested that this phenomenon is due to the involvement of the trapped electrons in recombination [32-34].

Here the DSSCs is assumed to be illuminated in x-direction (see Fig. 1). Considering the exponential decay of the absorption of dye-soaked TiO2 film, electron injection rate in Eq.(1) is given by:

 $\begin{eqnarray} G_{{\rm{in}}} = I_0 \eta _{{\rm{inj}}} \alpha \textrm{e}^ { - \alpha x} \end{eqnarray}$ (6)

where I0 is incident photon flux, $\eta _{{\rm{inj}}}$ is electron injection efficiency, and α is adsorption coefficient.

The kinetic equations for the diffusion-reaction of the redox shuttles in electrolyte are given as Eq.(7) and (8),

 $\begin{eqnarray} \frac{{\partial el_{{\rm{ox}}} }}{{\partial t}} \hspace{-0.15cm}&=&\hspace{-0.15cm}D_{{\rm{ox}}} \nabla ^2 el_{{\rm{ox}}} - U_{{\rm{re}}} + \Lambda _{{\rm{ox}}} G(x) \end{eqnarray}$ (7)
 $\begin{eqnarray} \frac{{\partial el_{{\rm{re}}} }}{{\partial t}}\hspace{-0.15cm}&=&\hspace{-0.15cm}D_{{\rm{re}}} \nabla ^2 el_{{\rm{re}}} - \Lambda _{{\rm{re}}} G(x) \end{eqnarray}$ (8)

where el represents the mole concentration, subscripts "ox" and "re" strand for the oxidized form (acceptor) and the reduced form (donor) in redox shuttles, respectively.

Eq.(1), Eq.(7), and Eq.(8) in this work are employed for determining density profiles of the conduction band electrons and redox couples at both steady and transient (non-steady) conditions. These equations were solved numerically in one dimension by using Clark-Nicolsen method. In these numerical calculations, we applied reflective boundary conditions for $n_\textrm{c}$, $el_{\textrm{ox}}$, and $el_{\textrm{re}}$, i.e. ${\nabla n_{\rm{c}} }|_{x = d}$=0, ${\nabla el_{{\rm{ox}}} }|_{x = 0}$=0, ${\nabla el_{{\rm{ox}}} }|_{x = 0}$=0 [26]. At the contact between TiO2 film and TCO sheet (x=0), the interfacial density of the conduction band electron ($n_\textrm{c}^0$) is related to its Fermi-level ($E^0_{\textrm{F, n}}$). And the latter depends on bias voltage (V) and the energy level of counter electrode ($E_{\textrm{F, redox}}$) by the expression of $E_{{\rm{F, n}}}^{\rm{0}}$=qV+$E_{{\rm{F, redox}}}$, where q is the elementary charge. According to the definition, $E_{\textrm{F, redox}}$ is written as:

 $\begin{eqnarray} E_{{\rm{F, redox}}} = k_{\rm{B}} T\ln \left( {\frac{{el_{{\rm{ox}}}^{{\rm{eq}}} - el_{{\rm{ox}}}^{{\rm{CE}}} }}{{el_{{\rm{ox}}}^{{\rm{CE}}} }}} \right) \end{eqnarray}$ (9)

where $el_{{\rm{ox}}}^{\rm{eq}}$ and $el_{{\rm{ox}}}^{\rm{CE}}$ are the concentrations of the acceptors at counter electrode (x=d+$d_{\textrm{el}}$) under illumination and in the dark, respectively. Current density is estimated by the expression of ${\rm{ }}j{\rm{ = }}qD_{\rm{c}} \nabla n_{\rm{c}}^{\rm{0}}$ after calculating the density profiles.

In our simulations, the thickness of TiO2 film is varied for evaluating its impacts on cell performance, while the thickness of electrolyte layer is given as 10 µm. Iodide/triiodide is employed as redox couple for their fast mass transport in electrolyte solution. In addition, recombination rate constant is varied in the simulations so as to figure out the relation between recombination kinetics and film thickness. The employed parameters in the simulations are as follows: conduction band edge $E_\textrm{c}$=1.2 eV, reaction order of recombination b=0.8, density of the conduction band $N_\textrm{c}$=7×1020/cm3, density of the trapping states $N_\textrm{T}$=5×1019/cm3, diffusion coefficient for the conduction band electrons $D_\textrm{c}$=0.5 cm2/s, diffusion coefficient for eletron acceptors $D_{\textrm{ox}}$=3.6×10-5 cm2/s, diffusion coefficient for eletron donnors $D_{\textrm{re}}$=4.4×10-5 cm2/s, porosity p=0.6, thickness of electrolyte layer $d_{\textrm{el}}$=10 µm, temperature T=293.15 K, characteristic temperature T0=800 K, wavelength $\lambda$=520 nm, light intensity I0=1×1017/cm2, electron injection efficiency $\eta_{\textrm{inj}}$=0.98, absorption coefficient a=0.4 µm-1.

Ⅲ. RESULTS AND DISCUSSION

Current-voltage (j-V) characteristics of the DSCs with various film thicknesses were simulated by using the drift-diffusion model at "steady-state". Recombination rate constant in the simulation is kept as 1×10-12 cm3/s, indicating weak recombination in DSSCs. Figure 2(a) shows the j-V curves for the DSSCs under irradiation. Based on these curves, photovoltaic parameters are attained and plotted against film thickness in Fig. 2(b). It is shown that short circuit current density ($j_{\textrm{sc}}$) rises dramatically by thickening the TiO2 film from 1 µm to 8 µm. This is evident due to the increase of light harvesting efficiency ($\eta_{\textrm{lh}}$) with film thickness from 33 % to 96 %. Afterwards, the decrease of $j_{\textrm{sc}}$ is observed, which is attributed to the increased charge loss via interfacial recombination. Also open circuit voltage ($V_{\textrm{oc}}$) is found to decrease with film thickness. By varying the thickness of TiO2 film from 1 µm to 20 µm, the value of $V_{\textrm{oc}}$ drops about 60 mV. At open circuit, all the photo-generated electrons are constrained in TiO2 film. As a result, the electron density maintains at a high level. It is thus reasonable to assume the density profiles for the Fermi-level in TiO2 layer and the acceptors in electrolyte to be uniform. Under this assumption, $V_{\textrm{oc}}$ is given by

 $\begin{eqnarray} V_{{\rm{oc}}} {\rm{ = }}E_{\rm{c}} - E_{{\rm{F, redox}}} - \frac{{k_{\rm{B}}T}}{{bq}}\ln d + A_0 \end{eqnarray}$ (10)

where A0 is a factor relative to light intensity and recombination rate constant. Eq.(10) indicates that the increase of thickness unavoidably engenders the loss of photovoltage. As indicated in Fig. 2(a), the DSSC with thicker TiO2 film shows stronger dark current. It implies faster recombination reaction in such device. That could be the major reason for the decrease of $V_{\textrm{oc}}$. Moreover, the variation of power conversion efficiency (PCE) with film thickness is indicated in Fig. 2(b). It is shown that the maximum PCE of 19.8 % is attained at the film thickness of 8 µm. When film thickness is higher than this value, considerable drop of PCE is observed. It is interesting that the film thickness for the maximum PCE is roughly consistent with that for the maximum $j_{\textrm{sc}}$. This is because PCE is just the product of $V_{\textrm{oc}}$, $j_{\textrm{sc}}$, and fill factor (ff). And it is seen that the variations of $V_{\textrm{oc}}$ and ff with film thickness are not as large as that of $j_{\textrm{sc}}$.

 FIG. 2 (a) J-V characteristics of the DSCs with various film thickness under illumination and in the dark. (b) Plots of photovoltaic parameters as a function of film thickness. I: $J_{\textrm{sc}}$ is short circuit current density, II: PCE is power conversion efficiency, Ⅲ: $V_{\textrm{oc}}$ is open circuit voltage, and IV: ff is fill factor.

Density profiles of the conduction band electrons at short circuit are plotted in Fig. 3(a) so as to figure out how the injected electrons flow in TiO2 film. It is shown that the density of the conduction band electrons rises with film thickness from 1 µm to 12 µm. The maximum value of the electron densities is observed at the interface between TiO2 film and electrolyte layer. It indicates that all the injected electrons have the possibility to be extracted to the external circuit. In contrast, density profiles for the film thickness over than 12 µm show the peak inside TiO2 film. For example, the peak of electron density is roughly at 8 µm for a 16 µm thick TiO2 film, which implies that the electrons injected at the position over 8 µm are entirely consumed by recombination reaction, namely these electrons have no contribution to photocurrent output.

 FIG. 3 (a) Density profiles of the conduction band electrons at short circuit. (b) Dependence of the fluxes of electron injection and interfacial recombination on film thickness at short circuit. (c) Plots of average electron density versus bias voltage.

Herein the fluxes of electron injection and interfacial recombination at short circuit are plotted against film thickness in Fig. 3(b). We can see that the injection flux increases rapidly. And it is quite close to its theoretical limit (9.8×1016 cm-2s-1) when film thickness is over 6 µm. Also, a persisting increase of recombination flux is observed. Since $j_{\textrm{sc}}$ is essentially determined by the flux difference between injection and recombination, actual output of photocurrent is reduced in thicker TiO2 film. This indicates that the enlargement of film thickness has negative effect on the photocurrent improvement. Furthermore, it is noted that recombination flux is the integral of the recombination rate given in Eq.(5). The enlargement of recombination flux indicated in Fig. 3(b) does not always lead to the increase of the density of the conduction band electrons. On the contrary, a slight reduction of the averaged electron density, shown in the inset of Fig. 3(c), is observed when film thickness exceeds 12 µm. From this point, we conclude that the major loss of photocurrent after thickening TiO2 film arises from severe recombination reaction. As a consequence, a part of the separated charges near the contact of TiO2/electrolyte are not extracted but recombined inside the bulk of TiO2. Moreover, the average of the conduction band electrons ($\bar n_\textrm{c}$) is also plotted as a function of bias voltage (V) in Fig. 3(c). Not surprisingly, dramatic increase of $\bar n_\textrm{c}$ is observed at high bias voltage since the energy barrier at TCO/TiO2 is quite large for hindering charge extraction and hence resulting in electron accumulation.

Figure 4 illustrates the relationship between power conversion efficiency and film thickness. Simulations were carried out at various recombination rate constants for mimicking different kinetics d of interfacial recombination. It is seen that the plots of PCE vs. d show the peaks under the condition that the recombination rate constant ranges from 1×10-10 cm3/s to 1×10-13 cm3/s. And there is a movement of the peak toward larger d when reducing recombination rate constant. To be specific, the DSSC with only 2 µm thick TiO2 film yields the maximum PCE in case of fast recombination ($k_\textrm{r}$=1×10-10 cm3/s). The device with sluggish back reaction ($k_\textrm{r}$=1×10-13 cm3/s) requires the film thickness of 8 µm to attain an optimized efficiency. It indicates that redox shuttles, such as iodide/triiodide, disulfide/thiolate, showing sluggish recombination, allow thicker TiO2 film to be employed. By contrast, the redox shuttles with fast electron interception, e.g. cobalt complexes, ferrocene/ferrocenium, and TEMPO/TEMPO+, need to be combined with thinner film. The study of Bach et al. dealing with ferrocene/ferrocenium ($F_\textrm{c}$/$F_\textrm{c} ^+$) has shown that the optimized film thickness is only 2.2 µm since their self-exchange rate constant is~107 (mol/L)-1s-1, higher than iodide/triiodide by two orders of magnitude [35, 36]. In all events, large recombination rate constant tends to deteriorate cell performance, as indicated in Fig. 4. Thereby, it is always effective to improve PEC by retarding recombination reaction. But film thickness should be adjusted accordingly. This is particular useful for the redox shuttles suffering from fast recombination.

 FIG. 4 Plot of PCE versus film thickness.

Figure 5 illuminates the impact of film thickness on charge collection efficiency ($\eta_{\textrm{col}}$). In this figure, considerable reduction of charge collection efficiency is observed. It reflects the negative role of thickening the TiO2 film, namely the enhancement of recombination rate. At short circuit, the reduction of charge collection efficiency, denoted as $\eta^0_{\textrm{col}}$, in the case of slow charge recombination is less significant than that for strong recombination. It is seen that $\eta^0_{\textrm{col}}$ is nearly linear to film thickness. When strong recombination occurs in DSSCs, a sharp decrease of $\eta^0_{\textrm{col}}$ with film thickness is detected. It implies that cell performance is sensitive to film thickness when the redox shuttles with fast electron interception are employed, or alternatively interfacial recombination is not well retarded via the modification on TiO2 surfaces. Indeed the definition of $\eta_{\textrm{col}}$ allows this parameter to be expressed as the flux ratio of charge extraction to electron injection, which is given by:

 $\begin{eqnarray} \eta _{{\rm{col}}} = \frac{j}{q[I_0 \eta _{\rm{inj}} (1-\textrm{e}^{-\alpha d} )]} \end{eqnarray}$ (11)
 FIG. 5 (a) Plots of charge collection efficiency at short circuit versus film thickness. (b) Dependence of charge collection efficiency on bias voltage.

It is obvious that $\eta_{\textrm{col}}$ is dependent on bias voltage. The plots of $\eta_{\textrm{col}}$-V indicated in Fig. 5(b) are in agreement with previous studies. We can see that the reduction of $\eta_{\textrm{col}}$ with film thickness is valid in the whole scope of bias voltage. It seems that surface modification to suppressing recombination could be particularly effective when thicker TiO2 film is used as photoanode.

The effects of recombination characteristics on the average density of electrons ($\bar n$) at short circuit are illustrated in Fig. 6. In our simulations, both the conduction band electrons and the trapped electrons were considered. If the profiles of $\bar n$ in Fig. 6 are compared with the plot of $\bar{n}_\textrm{c}$ in the inset of Fig. 3(c), we find that the contribution of the conduction band electrons to electron density is negligible, namely $\bar n$ can be approximated by the density of trapped electrons. This result is frequently observed in experimental studies. For the DSSCs suffering severe recombination (large value of $k_\textrm{r}$), Fig. 6 shows the reduction of $\bar n$ when film thickness exceeds 5 µm. As mentioned above, there is little increase to the flux of electron injection by thickening TiO2 film over 8 µm. It follows that TiO2 film thicker than such value only leads to net loss of the separated charges. In view of the rapid exchange of the electrons between the conduction band and the trapping states, it is reasonable to estimate fewer electrons are localized in the trapping states when film thickness exceeds 8 µm. On the other hand, weak recombination in DSSCs ($k_\textrm{r}$=1×10-13 cm3/s) results in a persisting increase of $\bar n$ with film thickness. In fact, the $j_{\textrm{sc}}$ is found to decrease when film thickness is over 9 µm, which implies the increased charges in TiO2 film cannot be extracted, instead those are merely consumed by the acceptors in electrolyte.

 FIG. 6 Dependence of averaged electron density at short circuit on film thickness.

Transient photovoltage was simulated by using the drift-diffusion equations under non-steady conditions. In the simulations, photovoltage is monitored by setting the DSSCs at open circuit. A 40 ns monochrome pulse (520 nm) is irradiated across the counter electrode for generating small photovoltage variation. After the flash, transient photovoltage is recorded until the system is back to its equilibrium state. To ensure an exponential decay, the photovoltage transient is constrained in 5 mV. Figure 7 illustrates the decay of photovoltage transient. And the inset indicates a good linearity between the logarithm of photovoltage increment (ln$\Delta V_{\textrm{oc}}$) and decay time (t). Electron lifetime ($\tau_{\textrm{n}}$) equals reversely to the slope of the linear part of the ln$\Delta V_{\textrm{oc}}$-t plot. The variations of $\tau_\textrm{n}$ with film thickness are depicted in Fig. 7(b). It is seen that the DSSC showing weak recombination has longer electron lifetime, indicating the justification of lifetime in evaluating the kinetics of back reaction. Moreover, it is found that the DSSC with higher film thickness also engenders longer lifetime. As indicated in Fig. 7(b), $\tau_{\textrm{n}}$ prolongs about two times when film thickness extends from 1 µm to 20 µm. Certainly, the variation of film thickness does not alter the recombination kinetics in essence. Instead, electron density is reduced by increasing film thickness. Figure 7(c) shows that the extension of film thickness from 1 µm to 20 µm results in the decrease of electron density by ca.16 %. According to the drift-diffusion model in our work, $\tau_\textrm{n}$ can be related to the averaged electron density approximately as follows,

 $\begin{eqnarray} \tau _{\rm{n}} = \frac{{TN_{\rm{T}}^{bT_0 /T} }}{{bT_0 k_{\rm{r}} N_{\rm{c}}^b }}\bar n^{1 - bT_0 /T} \end{eqnarray}$ (12)
 FIG. 7 (a) Simulation on photovoltage transient at open circuit. Inset shows the logarithmic photovoltage for fitting electron lifetime. (b) Plots of electron lifetime versus film thickness. (c) Plot of electron lifetime versus the average of electron density.

It is noted that Eq.(12) is based on the assumption that electron density is invariable to the position in TiO2 film. This is acceptable because the gradient of electron density at open circuit can be ignored. By inserting the data of T=293.15 K, T0=800 K, and b=0.8 into Eq.(12), we found that $\tau_\textrm{n}$ is proportional to $\bar n^{ - 1.18}$. Thereby it is reasonable to observe the decrease of $\tau_\textrm{n}$ with the averaged electron density.

Ⅳ. CONCLUSION

In this work, we employed a drift-diffusion model for illuminating the impacts of the thickness of TiO2 film on photovoltaic performance of DSSCs. Simulation results indicate that the optimization of film thickness is of importance in maximizing device performance. Current-voltage characteristics, as well as the calculations on the flux of back reaction, show the accelerated recombination by thickening TiO2 film, which unavoidably reduces charge collection efficiency. Actual output of photocurrent relies on the rate difference between electron injection and recombination reaction. Thereby the optimized film thickness is attained, over which excessive loss in photocurrent engenders the deterioration of cell performance. Moreover the optimized film thickness is shown to depend on the kinetics of recombination. Weak recombination in DSSCs allows thicker TiO2 film to ensure sufficient light-harvesting. By contrast, thinner film is critical to impede the back reaction for the devices suffering from strong recombination. In addition, electron lifetime is shown to increase with film thickness, which is due to the reduction of the averaged electron density.

Ⅴ. ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China (No.21103123).

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a. 天津理工大学化学化工学院, 天津市有机太阳能电池与光化学转换重点实验室, 天津 300384;
b. 天津理工大学自动化学院, 天津 300384