Effective charge collection in dye-sensitized nanocrystalline TiO2

Masatoshi Yanagida; Youhei Numata; Keiichi Yoshimatsu; Shin Satoh; Liyuan Han

doi:10.1088/2043-6262/4/1/015006

1. Introduction

The charge transport in nanocrystalline semiconductor film electrodes has attracted considerable scientific interest [1–11]. The nanocrystalline TiO₂ film electrode consists of sintering 5–400 nm diameter particles on a transparent conductive oxide (TCO) glass, and has especially been utilized for dye-sensitized solar cells (DSCs). Normal DSCs (N-DSCs) generally consists of a dye-sensitized nanocrystalline TiO₂ film, an electrolyte solution containing an I₃^–/I^– redox couple, and a counter electrode (figure 1(a)). The thickness of the nanocrystalline TiO₂ film is about 10–30 μm. Because of its large surface area, the light harvesting efficiency of this film can be 100%, even when dyes adsorbs on TiO₂ surface at the monolayer [1–9].

**Figure 1.** Structure of (a) normal DSCs (N-DSCs), (b) back contact-type DSCs (BC-DSCs) and (c) sandwich contact-type DSCs (SW-DSCs).
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N-DSCs are illuminated with solar light from the TCO side, whereupon the excited dye injects an electron into the nanocrystalline TiO₂, and the resulting oxidized dye is reduced by an I^– ion in the electrolyte. The photo-generated electron diffuses in the TiO₂, eventually arrives at, and is collected by the TCO glass, which acts as a charge-collection electrode. The photoresponse of the short-circuit photocurrent density (J_SC) in N-DSCs is much slower (micro- to milliseconds) than that of classical inorganic semiconductor devices (nano- to picoseconds). The application of DSCs to commercial devices such as photovoltaics, photo-detector and sensor will be limited by the slow photoresponse. Two explanations for this slow photoresponse have been proposed. One is the slow electron diffusion in nanocrystalline TiO₂. The electron diffusion coefficient (<10^–3 cm² s^–1) is much smaller than that in single-crystalline TiO₂ (0.01 cm² s^–1) [10, 11]. Nanocrystalline TiO₂ has shallow energy trap sites (around 10¹⁸ cm^–3) or potential barrier (distance 6 nm) of grain boundary among the TiO₂ particles [12–22]. The electrons in the conduction band therefore move very slowly through nanocrystalline TiO₂. The second explanation for the slow photoresponse involves the position of the extraction electrode and the interfacial area between the extraction electrode and the TiO₂ film. The TCO glass serves not only as a light window but also as the front-collection electrode. Recently, the back contact-type DSC (BC-DSC), in which a charge collection electrode is placed on the TiO₂ film on the side opposite the optical window [23–37] has been proposed in order to remove highly costly TCO, and to improve the J_SC without reflection loss on the TCO layer (figure 1(b)).

In N-DSCs, the electrons are initially generated by irradiated light around TCO, and exponentially decrease from TCO to the counter electrode. In BC-DSC, the electrons are initially generated by irradiated light around the optical window (normal glass), and exponentially decrease from the optical window to the collection electrode. The distance between photo-generated part and the collection part in BC-DSCs is longer than that in N-DSCs. We expected that the photocurrent response of BC-DSCs is slower than that of N-DSCs. We recently showed the photocurrent response of sandwich contact-type DSCs (SW-DSCs) as shown in figure 1(c) [38]. The back collection electrode was placed on nanocrystalline TiO₂ film opposite the TCO glass. To investigate the effect of the position of the collection electrode on the photocurrent density, we evaluated photocurrent density of N-, BC- and SW-DSCs by measuring the photocurrent frequency response, as well as by calculating the incident monochromatic photon-to-current conversion efficiency (IPCE) response by means of an electron diffusion model.

2. Experiment

2.1. Materials

All chemicals were obtained from Wako Pure Chemical Industries (Osaka, Japan), Tokyo Chemical Industry Co. (Tokyo, Japan), or Sigma-Aldrich Japan Co. (Tokyo, Japan), and were used as supplied. N719 dye, TiO₂ pastes and TCO glass substrates were purchased from Dyesol (Queanbeyan, Australia).

2.2. DSC fabrication

Nanocrystalline TiO₂ photoelectrodes for the N-, BC- and SW-DSCs were prepared as follows [38]. TCO and normal glass substrates were washed with acetone, 37% hydrochloric acid and distilled water under ultrasonic irradiation for 30 min; and they were washed again with acetone and distilled water under ultrasonic irradiation for 30 min just prior to TiO₂ film fabrication. Nanocrystalline TiO₂ thin films were screen-printed onto the glass substrates. The thickness of the TiO₂ film was measured with a micro-figure measuring instrument (Surfcorder ET4000, Kosaka Laboratory, Tokyo, Japan). The thickness of the TiO₂ films used for this study was 9.7 μm (20 nm particle layer). For the BC- and SW-DSCs, we deposited Ti as the back electrode on the TiO₂ film by using a sputtering machine (Ulvac Technologies, Chigasaki, Japan) under a vacuum of 7.8 × 10^–2 Torr under a flow of Ar gas (20 sccm). The TiO₂-coated substrates were dipped into an acetonitrile/tert-butylalcohol (1:1 v/v) solution of N719 dye (3 mM) and heated for 70 h at 35 °C for adsorption of the dyes onto the TiO₂ surface. The coated substrates were clipped together with a platinized glass counter electrode with pin holes. The gap between the two electrodes was sealed with a thermal adhesive film (Surlyn-50 MS004620, Dyesol). Then an electrolyte consisting of 0.6 mM 1-methyl-3-propylimidazolium iodide, 0.1 mM LiI, 0.2 mM tert-butylpyridine and 0.05 mM I₂ in acetonitrile was injected into the gap.

2.3. Characterization of the DSCs

The J–V characteristics of the DSCs were measured with a WXS-90S-L2 super solar simulator (Wacom, Tokyo, Japan) under air mass 1.5 simulated solar illumination at 100 mW cm⁻². IPCE spectra were measured with a CEP-200BX spectrometer (Bunkokeiki, Tokyo, Japan). DSC devices with an active surface area of 0.25 cm² were created by means of a metal mask [39, 40].

Intensity modulated photocurrent spectroscopy was performed with a red diode laser (NEOARK, 650 nm, 35 mW) under short circuit condition. The power of modulated-light irradiation was reduced to <10% of the total photocurrent of the DSCs. When the frequency of the modulated-light irradiation was changed from 10 mHz to 100 kHz, the resulting changes in the amplitude and phase of the small sinusoidal voltage were detected with an impedance analyzer (Solartron 1255B) [17, 38].

3. Results and discussion

3.1. Light absorption profile

The calculated absorption profile in a nanocrystalline TiO₂ film sensitized with dyes is shown in figure 2. The light absorption is linearly related to the light intensity. The light absorption is largest at the optical window (x = 0 in figure 2). The light intensity exponentially decreases with an increase of thickness of TiO₂ The light absorption in TiO₂ film can be expressed as the following:

$\begin{equation} I(x) = I_{0}\,{\rm exp}( - \alpha x), \end{equation} \tag{ 1 }$

where x is the distance (cm) from the optical window in TiO₂ films, I(x) is light intensity at x, α is absorption coefficient (cm⁻¹) and I₀ is incident light intensity, respectively. Equation (1) is well known as Lambert–Beer law. Because the electron is generated by the absorbed photon, the electron generated by photon is distributed around optical windows. In case of large α, the generated electron distributes around x = 0. The probability of the collection by the collection electrode is large in N-DSC. The average distance (AD) between the position of a generated electron and a collection electrode in an N-DSC is very small because an electron is generated around the TCO glass. The probability of the collection in a BC-DSC is small because AD in BC-DSCs should be longer than that in N-DSCs. In the case of small α, the generated electrons widely distribute in the TiO₂ film. The AD in an N-DSC becomes larger. But AD in a BC-DSC becomes smaller. The electron collection in a BC-DSC should become favorable. On the other hand, the AD should be constant in an SW-DSC. The collection electrodes exist on both sides.

**Figure 2.** Absorption coefficient (α) dependent light absorption profile in nanocrystalline TiO₂ films sensitized with dye.
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3.2. Photovoltaic properties

We evaluated the photovoltaic properties of the SW- and N-DSCs. The short circuit current (J_SC), open circuit voltage (V_OC) and fill factor (FF) values of the N-DSC were 14.0 mA cm⁻², 0.751 V and 0.720, respectively; and the corresponding values of the BC-DSC were 10.9 mA cm⁻², 0.779 V and 0.767. In the SW-DSC, J_SC, V_OC and FF were 11.8 mA cm^–2, 0.761 V and 0.728, respectively. The overall conversion efficiencies of the N-, BC- and SW-DSCs were 7.6, 6.5 and 6.6%, respectively.

Incident monochromatic photon-to-current conversion efficiency (IPCE) spectra of an N-DSC, a BC-DSC and an SW-DSC are shown in figure 3. The maximum IPCE of the SW-DSC was comparable to that of the N-DSC. However, J_SC of the SW-DSC was slightly lower than that of the N-DSC, owing to light absorption by the back collection electrode at wavelengths >600 nm. In an N-DSC, light reflected and scattered by the counter electrode can impinge upon the TiO₂ film again. In contrast, in an SW-DSC and a BC-DSC, light transmitted through the TiO₂ film is immediately absorbed on the Ti back electrode. The loss of the reflected and scattered light reduced the J_SC value of the SW-DSC and the BC-DSC relative to that of the N-DSC.

**Figure 3.** Incident monochromatic photon to current conversion efficiency (IPCE) spectra of an N-DSC( $\square$ ), a BC-DSC(▵) and an SW-DSC( $\bullet$ ).
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In the BC-DSC, the maximum IPCE was lowest of the three different types of DSC. We now investigate the reason for the low J_SC. The total AD of BC-DSCs may be smallest. The optical properties of the optical window may also be changed when we change from a TCO glass to a normal glass.

3.3. Photocurrent frequency response

We normalized the IPCE of these DSCs at 650 nm to the value of the N-DSCs and investigated its dependence upon the frequency of applied modulating light (figure 4). The corrected IPCE response gradually decreased, starting at a frequency of around 20 Hz and decreasing to nearly 0 at frequencies >10 kHz. At a frequency of around 100 Hz, the IPCE response increased with increasing electron density (n) in TiO₂ (figure 4(a)). The electron density was measured by the charge extraction method at various light intensities [13, 14]. When electrons in TiO₂ diffuse in nanocrystalline TiO₂, the electrons are trapped and de-trapped thorough electron trap sites. The D value in TiO₂ films is smaller than that of bulk crystalline TiO₂. However, the number of electrons in TiO₂ increases with increasing light intensity, and these electrons fill up the traps. As a result, additional electrons can move smoothly through the conduction band without undergoing trapping and de-trapping. Therefore, the IPCE response increases with increasing n.

$\blacksquare $ — **Figure 4.** Corrected photocurrent response as a function of frequency of applied modulating light at 650 nm. (a) Electron density (n) dependent photocurrent response of a SW-DSC. The response at n = 2.9 × 10¹⁷ ( $\blacksquare$ ), 5.0 × 10¹⁷ ( $\bullet$ ), 5.7 × 10¹⁷ ( $\hbox{\ding{169}}$ ), 6.6 × 10¹⁷ cm⁻³ ( $\blacktriangle$ ). The n at open circuit condition was measured by the charge extraction method under different irradiation powers. (b) Corrected photocurrent response of an N-DSC ( $\square$ ), a BC-DSC( $\vartriangle$ ) and an SW-DSC( $\bullet$ ) at n = 5.0 × 10¹⁷ cm⁻³.
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The corrected IPCE response becomes larger in the order of SW- > N- > BC-DSCs at frequencies from 10 to 500 Hz (figure 4(b)). Because the same TiO₂, N719 dye, electrolyte and counter electrode were used in both types of DSCs, the photovoltaic properties (e.g. D, τ and α) in the three types of DSCs can be expected to be identical. Therefore, the faster response of the IPCE in the SW-DSCs was likely to have been due to the addition of the back collection electrode. The difference in photocurrent response of these DSCs decreased with increasing light intensity. Because D increases with increasing light intensity, electrons in TiO₂ can be quickly collected by the front electrode at high light intensities, even when the back or the front electrode is not present.

At frequencies >500 Hz, the IPCE response of the SW-DSCs was smaller than that of the N-DSCs. The response at frequencies >500 Hz was not due to electron diffusion and can instead be explained in terms of the influence of the attenuation factor (AF(ω)), which is explained later [11].

The photocurrent density (J_SC) can be explained by using the electron diffusion model. We defined that the collection electrode existed at x = 0 in N-DSCs or BC-DSCs, at x = 0 and d in SW-DSCs when we consider the TiO₂ films with thickness of d. When DSCs are illuminated from the optical window, the density (n(x,t)) of electrons in TiO₂ is a function of the distance (x) from the interface between the optical window and the nanocrystalline TiO₂ and time (t) and is given by the following continuity equation [10–15]:

$\begin{equation} \frac{{\partial n\left( {x,t} \right)}}{{\partial t}} = D\frac{{\partial ^2 n\left( {x,t} \right)}}{{\partial x^2 }} - \frac{{n\left( {x,t} \right)}}{\tau } + \alpha I_0 {\rm e}^{ - \alpha y}, \end{equation} \tag{ 2 }$

where D is the electron diffusion coefficient, τ is the electron lifetime, y is d − x in BC-DSCs and x in N-DSCs or SW-DSCs, respectively. The J_SC can be described as the gradient of n value at the interface between a TiO₂ film and a collection electrode

$\begin{equation} J_{{\rm{SC}}} = qD\frac{{\partial n\left( {x_{\rm{C}} ,t} \right)}}{{\partial x}}\;\quad (x_{\rm{C}} = 0\;{\rm or}\;d), \end{equation} \tag{ 3 }$

where q is the quantity of charge on the electron.

Equation (2) can be solved under the boundary condition of each DSC. We can define the condition at the interface between a TiO₂ film and a collection electrode as the following equation:

$\begin{equation} k_{\rm{e}} n(x_{\rm{C}} ,t) = D\frac{{\partial n\left( {x_{\rm{C}} ,t} \right)}}{{\partial x}}\;\quad (x_{\rm{C}} = 0\;{\rm or}\;d), \end{equation} \tag{ 4 }$

where k_e is the rate constants of the electron collection from TiO₂ to the collection electrode. The distribution of electron at interface between a TiO₂ film and an insulated optical window or an electrolyte can be defined as the following equation:

$\begin{equation} \frac{{\partial n\left( {x_{\rm{O}} ,t} \right)}}{{\partial x}} = 0\,\quad (x_{\rm{O}} = 0\;{\rm or}\;d). \end{equation} \tag{ 5 }$

In the case of N-DSCs and BC-DSCs, we use x_O = d in equation (5) and x_C = 0 in equations (3) and (4). In the case of SW-DSCs, we use both x_C = 0 and d in equations (3) and (4), but need not use equation (5).

From equations (2), (4) and (5), the distribution of electron density (n) against x can be calculated, and then J_SC is calculated from each gradient of n(x) at the interface between a collection electrode and a TiO₂ film as shown in equation (3). Finally, we can calculate the photocurrent (IPCE) frequency response J(ω), which can be described as the following:

$\begin{equation} J(\omega) = qDu'\left( \omega \right), \end{equation} \tag{ 6 }$

where ω is frequency of modulated light. The equation of u'(ω) and the related parameters are summarized in table 1 [11, 38].

Table 1. List of equations for photocurrent response J(ω).

Type of cells	u' (ω)	A	B	C
N-DSC	−γA_N	A_N=C/γ	B_N=−C/γ
	+γB_N	[{(k_e+Dγ)α−(k_e−Dα)γe^−γd}	[{(k_e−Dγ)α+(k_e−Dα)γe^γd}
	+αC	/{k_e(e^γd+e^−γd)+Dγ(e^γd−e^−γd)}]	/{k_e(e^γd+e^−γd)+Dγ(e^γd−e^−γd)}]
BC-DSC	γA_BC	A_BC=−Ce^−αd/γ	B_BC=Ce^−αd/γ	αI₀/{D(γ²−α²)}
	−γB_BC	[{(k_e+Dγ)αe^αd+(k_e−Dα)γe^−γd}	[{(k_e−Dγ)αe^αd−(k_e−Dα)γe^γd}
	+αCe^−αd	/{k_e(e^γd+e^−γd)+Dγ(e^γd−e^−γd)}]	/{k_e(e^γd+e^−γd)+Dγ(e^γd−e^−γd)}]
SW-DSC	(1+e^γd)γA_sw	A_SW=C	B_SW=−C
	+(1+e^−γd)γB_SW	[{(k_e+Dα)(e^−αd−2e^γd+e^−γd)}	[{(k_e+Dα)(e^−αd−e^γd)}
	+(1+e^−αd)αC	/{(k_e−Dγ) (e^γd−e^−γd)}]	/{(k_e+Dγ) (e^γd−e^−γd)}]

In these equations, $\gamma = \{ (D\tau )^{ - 1} + (i\omega /D)\}^{1/2}$ , where i is (−1)^1/2.

The observed photocurrent response J_app(ω) in the intensity modulated photocurrent spectra is attenuated by the attenuation factor AF(ω) [11]:

$\begin{equation} J_{\rm app}( \omega) = {\rm AF}(\omega)\,\,J(\omega), \end{equation} \tag{ 7 }$

AF(ω) is given by

$\begin{equation} {\rm{AF}}\left( \omega \right) = \frac{1}{{1 + {\rm{i}}\omega RC}}\;, \end{equation} \tag{ 8 }$

where i is (−1)^1/2, R is the series resistance of the sheet resistance of the collection electrode and the electrolyte and C is capacitance, which depends on the surface area of the collection electrode and TiO₂ film in the DSC. The plot of J_app(ω) versus ω is shown in figure 5. When D value increases as shown in figure 5(a), the J_app(ω) becomes large. The collection electrode can collect electrons, when electrons fast diffuse in TiO₂. The relationship between calculated J_app(ω) and D is in good agreement with the results in figure 4(a). As shown in figure 5(b), the J_app(ω) becomes larger in the order of SW- > N- > BC-DSCs at frequencies from 10 to 500 Hz, which is in agreement with the experimental results (figure 4(b)). In good agreement between figures 4 and 5, the diffusion model can be applied for prediction of the properties of N-, BC- and SW-DSCs.

**Figure 5.** Calculated photocurrent response as a function of frequency of applied modulating light. (a) D dependence of calculated photocurrent response for SW-DSCs at D = 2.5 × 10^–4 ( $\blacksquare$ ), 5 × 10^–4 ( $\bullet$ ) and 2 × 10^–3 cm² s⁻¹ (▴), (b) Calculated photocurrent response of an N-DSC(□), a BC-DSC(▵) and an SW-DSC( $\bullet$ ) at D = 5 × 10^–4. The following parameter values were used for the calculations: d = 10 μm, R = 3 Ω, k_ext = 10 cm s⁻¹, α = 823 cm⁻¹, τ = 1 s.
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3.4. Photocurrent density at steady state

At steady state under illumination, the diffusion equation (2) can be described as the following [10]:

$\begin{equation} 0 = D\frac{{\partial ^2 n\left( {x,t = \infty } \right)}}{{\partial x^2 }} - \frac{{n\left( {x,t = \infty } \right)}}{\tau } + \alpha I_0 {\rm e}^{ - \alpha y} . \end{equation} \tag{ 9 }$

We defined n₀ as the electron density in TiO₂ under dark conditions. When we solve equation (9) by using Laplace transform and the boundary condition of equations (4) and (5), we can get the equation of the electron density (n(x, t = ∞)) in steady state, and J_SC for N-, BC- and SW-DSCs [10, 38]. In SW-DSCs, electrons generated in the TiO₂ diffuse to both the front and the back collection electrodes. The electrons in TiO₂ are separately collected by the front and back electrodes. But the collected electrons at both sides finally flow through the same outer circuit. Therefore, we assumed that the total current density can be expressed as the sum of the current densities at the front electrode and the back electrode for a given TiO₂ thickness d. These equations for n and J_SC are summarized in table 2. When we input the parameters such as α, d, D and τ into these equations in table 2, we can simulate n(x, t = ∞) in TiO₂ at short circuit condition, and J_SC.

Table 2. List of the equations for the electron density n(x, t = ∞) in steady state and short circuit photocurrent (J_SC).

Type of cells	n(x, t=∞)	J_SC
N-DSC	n₀{1−cosh(x/L)}	qDPZ'(x=d)/cosh(d/L)
	+P[Z(x)−LZ'(x=d)sinh(x/L)/cosh(d/L)]
BC-DSC	n₀	−qDPY'(x=d)/cosh(d/L)
	+P[Y(x)−LY'(x=d) sinh(x/L)/cosh(d/L)]
SW-DSC	n₀	qDP[Z'(x=d)−Z(x=d){cosh(d/L)−1}/
	+P[Z(x)−LZ(x=d) sinh(x/L)/sinh(d/L)]	{Lsinh(d/L)}]

In this table 2, $L = \sqrt {D\tau }$ , $P = \frac{{I_0 \alpha L}}{{2D}}$ , $Z(x) = \frac{{{\rm{e}}^{ - \alpha x} - {\rm{e}}^{ - {x/L}} }}{{1/L - \alpha }} + \frac{{{\rm{e}}^{ - \alpha x} - {\rm{e}}^{{x/L}} }}{{1/L + \alpha }}$ , $Y(x) = {\rm{e}}^{ - \alpha d} \left\{ {\frac{{{\rm{e}}^{\alpha x} - {\rm{e}}^{ - {x/L}} }}{{1/L + \alpha }} + \frac{{{\rm{e}}^{\alpha x} - {\rm{e}}^{{x/L}} }}{{1/L - \alpha }}} \right\}$ , $Z'(x) = {{{\rm{d}}Z(x)}/{{\rm{d}}x}}$ and $Y'(x) = {{{\rm{d}}Y(x)}/{{\rm{d}}x}}$ , respectively.

The calculation of n(x, t = ∞) in TiO₂ are shown in figure 6. The α, d, n₀, D and τ values used for the calculations were 823 cm^–1, 10 μm, 10¹³ cm⁻³, 5 × 10^–4 cm² s⁻¹, and 1 s, respectively. In figure 6, light is irradiated from x = 0. The collection electrode in N-DSCs, and BC-DSCs exists at x = 0 and d, respectively. According to the distribution of electrons in the TiO₂ of N-DSCs, the n value at x = 0 is smallest. The electron collection rate around the collection electrode was faster than the electron diffusion rate through the TiO₂ film. Therefore, the electron density decreased at the interface between the TiO₂ film and the electrode and an electron density gradient was generated around the collection electrode. The J_SC is determined by the diffusion rate, and not by the electron transfer rate (electron collection) at the interface between a TiO₂ film and a collection electrode in the diffusion model.

In BC-DSCs, electrons also diffuse to the collection electrode. The maximum n value in BC-DSCs is the largest in the three types of DSCs. In particular, the maximum n value in BC-DSCs is larger than that in N-DSCs. Electrons in TiO₂ are generated around the optical window (x = 0) of BC-DSCs. Electrons in TiO₂ have to transport from x = 0 to the back collection electrode (x = d). The AD of BC-DSCs is longer than that of N-DSCs. In addition, the D value in TiO₂ films is much smaller than that of bulk crystalline TiO₂. Therefore, electrons are accumulated around x = 0 in BC-DSCs when the τ is very large.

The maximum n value in SW-DSCs is smallest of the three types of DSC. In the TiO₂ film of an SW-DSC, the n at x = 0 and d were smallest because electrons are collected by the front and the back electrodes. The AD of SW-DSCs is shortest in these three structures. The electrons can quickly be collected as shown in figures 4 and 5. Therefore the maxmun n value in SW-DSCs is smallest at short circuit. The D and τ in DSCs depend on the electron density in TiO₂. In particular, the τ decreases in increase with n. The large n induces the charge recombination. Therefore, we assumed that the charge recombination can be minimized because of the small n in SW-DSCs.

In SW-DSCs, we assumed that the total current density can be expressed as the sum of the current densities at the front electrode and the back electrode for a given TiO₂ thickness d. In the diffusion model, we can estimate the contribution of the front and back electrodes to the collection of electrons in a TiO₂ film. We can separately calculate the photocurrent density as the following from equation (3):

$\begin{equation} J_{{\rm{SC}}}^{\rm{F}} = \frac{{qI_{0} \alpha }}{{2\sinh \left( {{d/L}} \right)}}Z\left( {x = d} \right), \end{equation} \tag{ 10a }$

$\begin{equation} J_{{\rm{SC}}}^{\rm{B}} = \frac{{qI_{0}\alpha L}}{2}\left( {Z'\left( {x = d} \right) - Z\left( {x = d} \right)\frac{{\cosh \left( {{d/L}} \right)}}{{L\sinh \left( {{d/L}} \right)}}} \right), \end{equation} \tag{ 10b }$

where $J_{{\rm{SC}}}^{\rm{F}}$ $J_{{\rm{SC}}}^{\rm{F}}$ , and $J_{{\rm{SC}}}^{\rm{B}}$ $J_{{\rm{SC}}}^{\rm{B}}$ are the photocurrent density of the front and back electrodes, respectively. The value of Z(x = d) and Z'(x = d) are defined in table 2. The contribution of the $J_{{\rm{SC}}}^{\rm F}$ $J_{{\rm{SC}}}^{\rm F}$ and $J_{{\rm{SC}}}^{\rm{B}}$ $J_{{\rm{SC}}}^{\rm{B}}$ to the total photocurrent density can be estimated by changing the light absorption property as α (figure 7). The $J_{{\rm{SC}}}^{\rm{F}}$ $J_{{\rm{SC}}}^{\rm{F}}$ equals to $J_{{\rm{SC}}}^{\rm{B}}$ $J_{{\rm{SC}}}^{\rm{B}}$ below α = 100. Many electrons can be generated at x = d when the light is illuminated from x = 0 as shown in figure 2. When the α increases over 100 cm⁻¹, electrons generated around x = 0 can be preferably collected by the front electrode. Therefore, the contribution of $J_{{\rm{SC}}}^{\rm{B}}$ $J_{{\rm{SC}}}^{\rm{B}}$ gradually decreases. Light harvesting efficiency of around 100% can be obtained at x = 4/α from equation (1). But $J_{{\rm{SC}}}^{\rm{B}}$ $J_{{\rm{SC}}}^{\rm{B}}$ is kept at around 10–20% of the total photocurrent density even when the value of 4/α approaches the thickness of TiO₂ (10 μm). The contribution of $J_{{\rm{SC}}}^{\rm{B}}$ $J_{{\rm{SC}}}^{\rm{B}}$ is kept at around 10% even at α = 10⁴ cm⁻¹. The calculation means that electrons also diffuse from the front electrode to the back electrode even when α is large. The back electrode can effectively collect electrons in SW-DSCs.

**Figure 7.** Absorption coefficient (α) dependent photocurrent density of the front collection electrode (- - -), the back collection electrode (······) and total photocurrent density (—) calculated from equation (10). The following parameter values were used for the calculations: d = 10 μm, D = 9 × 10^–5 cm² s⁻¹ and τ = 1 s.
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3.5. The effect of the diffusion length

The diffusion length is the specific distance of electron transport in the medium, and can be expressed by L = (Dτ)^1/2. The long L is needed to improve the energy conversion efficiency of DCSs. The large D and long τ result in the long L. In this section, we investigated the effect of L on the properties of N-DSCs, BC-DSCs, and SW-DSCs.

Plots of the D dependence of the calculated IPCEs of the N-, BC- and SW-DSCs were comparable at d = 10 μm (figure 8(a)). The IPCE values increased with increasing D. The IPCE of the N- and BC-DSCs becomes smaller than that of the SW-DSC at D values <10^–5 cm² s⁻¹. The electrons in the TiO₂ film could not be efficiently collected by the front electrode in the N-DSC or the back electrode in the BC-DSC at D < 10^–5 cm² s^–1 but could be collected by both the front and the back electrodes in the SW-DSC. Plots of the τ dependence of the IPCEs of the N-, BC- and SW-DSCs were comparable at d = 10 μm (figure 8(b)). The IPCE values increased with increasing τ. The IPCE of the SW-DSC was the largest of the three structures at 10^–6 < τ < 10^–2 s. The electrons in the TiO₂ film could not be efficiently collected by the front electrode in the N-DSC or the back electrode in the BC-DSC at 10^–4 s, but they could be collected by both the front and the back electrodes in the SW-DSC. The dependence of the calculated IPCEs on D and τ show that a high J_SC could be maintained when the electron diffusion length in the TiO₂ film was shortened.

4. Conclusion

We report the comparison for photocurrent density (J_SC) for N-, BC- and SW-DSCs to understand the role of structure of the electron collection electrode. Photocurrent response in SW-DSCs was the fastest of the three structures at frequencies between 10 and 500 Hz. We explained this difference in terms of the distance traveled by electrons in the conduction band of TiO₂: electrons far from the collection electrode had to diffuse for a long distance in N- and BC-DSCs, whereas in SW-DSCs, the electrons were collected by the front and back electrodes. In the SW-DSC, photogenerated electrons in the TiO₂ film were collected by both the front electrode and the back electrode. These results agreed with results calculated using the diffusion model.

In SW-DSCs, the average distance between the position of a photogenerated electron and an extraction electrode was half that in N- and BC-DSCs. The electron diffusion model shows electrons are effectively collected by the front and the back electrodes in SW-DSCs. The back electrode can effectively collect electrons in SW-DSCs even when α is large. Calculations based on the model suggested that a high short-circuit photocurrent could be maintained in SW-DSCs even when the electron diffusion length in the TiO₂ film was shortened.

Acknowledgments

This work was partially supported by Precursory Research for Embryonic Science and Technology (PRESTO) and Core Research for Evolutional Science and Technology (CREST) of Japan Science and Technology agency. This preparation of the back electrode was supported by Nano-Integration Foundry (NIMS) in 'Nanotechnology Platform Project' operated by the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. The authors gratefully acknowledge the kind help for the preparation of devices by Mr Takeshi Sugita, Mr Masayuki Chikatsu, Ms Naomi Itoh and Ms Toshiko Harimoto.

Effective charge collection in dye-sensitized nanocrystalline TiO₂

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Abstract

1. Introduction